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\begin{document}

\frontmatter
\title{Econ 871: LECTURE NOTES}
\author{Lukasz Drozd}
\date{Fall 2008}
\maketitle

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\mainmatter

\chapter{International Trade}

\section{Introduction}

International trade theory addresses the question why in the long-run
countries engage in the exchange of goods and services. Depending on the
answer, models of trade can be classified into three major categories: (i)\
Ricardian models, (ii)\ Hecksher-Ohlin models, and (iii) Monopolistic
competition models.

The first two categories are referred to as the \textit{traditional trade
theory}. In these models, countries trade because there are intrinsically
different, and by the logic of Ricardian comparative advantage, trade allows
them to take advantage of these differences. In particular, in the Ricardian
models the technologies to produce each good differ, and in the
Hecksher-Ohlin models, the factor endowments differ. Both features result in
a situation of comparative advantage, and lead to a partial or a complete
specialization.

The monopolistic competition models, referred to as the \textit{new trade
theory,} depart from this traditional approach in at least two important
respects. First, in the new trade theory countries are no longer
intrinsically different (ex-ante), but still trade and specialize (ex-post).
The key idea is that trade and specialization allows them to access a larger
variety of goods, while at the same time exploit economies of scale present
in producing them.

The second important difference is the modeling methodology. In contrast to
the traditional trade theory, the new trade theory is a strictly positive
theory\footnote{%
Positive theory directly characterizes what is. Normative theory focuses on
what ought to be.}. Namely, it attempts to directly describe and mirror the
exact market and institutional structure that we see in the data, and is
silent about the deep-rooted frictions that could give rise to this
structure endogenously. The weakness of such approach is the need for a more
extensive data-based justification of its more complex structure, but its
strength is a more natural mapping between the theory and the data. In trade
theory, it has paid off by allowing researchers to extend trade facts to
producer-level facts, and as Samuel Kortum puts it, \textquotedblleft 
\textit{by building on the firm-level stylized facts, the resulting
aggregate theory is likely to be more credible both as a description of
reality and as a tool for policy analysis.}\textquotedblright

We should also mention that initially the new trade theory was a theoretical
response to the empirical observation that most trade takes place between
very similar countries (industrial countries), and most importantly, the
observation that these countries tend to trade very similar categories of
manufactured goods (called intra-industry trade). Popular at the time
Hecksher-Ohlin models could not sensibly deal with observation. In this
respect, monopolistic competition models still have an edge over other
theories. Even though Ricardian models can sensibly deal with intra-industry
trade, they are silent about the source of the underlying technology
differences that lead to this phenomenon. In the future, the ongoing
integration of the Ricardian theory with the theory of innovation and growth
is likely to fill this gap.

\section{Patterns of Trade in the Aggregate Data}

[to be completed]

\section{Armington Model}

Armington model is a Ricardian model.\footnote{%
Since this model is the baseline framework adopted in the open economy
macro, it is particularly important to know how it fits into the broadly
defined trade theory.} Each country in the Armington model is assumed to be
efficient in producing just one good, and infinitely inefficient in
producing all the other. This assumption makes the comparative advantage
structure somewhat trivial, but the model becomes very tractable.

In this section, we will study the predictions of a basic multicountry
Armington model, and apply them to understand trade flows between countries.

\subsection*{Model Economy}

There are $N$ countries (or regions)\ and $N$ goods in the world. Each
country has the technology to produce only one good from the set $1,..,N,$
and can not produce all the other goods by assumption. Production factors
are assumed immobile across countries, and all markets are competitive. In
terms of notation, we assume that country $n$ produces good $n.$

Geography is modeled here by an iceberg transportation cost that is intended
to capture the notion of trade barriers between countries (regions). Iceberg
transportation cost $d_{ni}$ between country $n$ and $i$ implies that $%
d_{ni} $ units of good must be shipped from country $n$ in order for one
unit to arrive in country $i$. In what follows, the following properties of
the iceberg cost will be assumed: (i) symmetry $d_{ni}=d_{in}\geq 1$, (ii)\
no cost within the country $d_{ii}=1$, (iii)\ triangle inequality 
\begin{equation*}
d_{ni}\leq d_{nj}+d_{ji}\ \text{\ all \ }i,j,n=1..N.
\end{equation*}

\subsubsection*{Households}

In each country $n=1,...,N$, there is a stand-in household that has
preferences described by a CES aggregator given by%
\begin{equation}
U_{n}=(\sum_{i=1..N}\alpha _{i}^{\frac{1}{\sigma }}c_{ni}^{\frac{\sigma -1}{%
\sigma }})^{\frac{\sigma }{\sigma -1}},  \label{Eq1U}
\end{equation}%
where $\sigma $ is the elasticity of substitution between the goods ($\sigma
>1),$ and $\alpha _{i}$ is the weight of each good ($\sum_{i}\alpha _{i}=1$%
). Each household is assumed to inelastically supply its endowment of $L_{n}$
units of labor.

Given market wage $w_{i},$ and a schedule of prices $p_{ni}$ for each good,
the problem of the household (in country $n)$ is to maximize (\ref{Eq1U})
subject to the budget constraint given by%
\begin{equation}
\sum_{i=1...N}p_{ni}c_{ni}=w_{n}L_{n}+\Pi _{n},  \label{Eq1a}
\end{equation}%
where $\Pi _{n}$ are the profits paid out by the local firms (in equilibrium 
$\Pi _{n}$ will be zero).

\subsubsection*{Firms}

In each country, there is a stand-in competitive firm that takes all prices
as given. The firm employs labor supplied by the home households, produces
goods, and sells these goods both at home and abroad. Production technology
is assumed to be subject to constant returns to scale.

The profit function of a stand-in firm from country $i$ is given by 
\begin{equation}
\Pi _{i}=\sum_{i=1..N}p_{ni}y_{ni}-w_{i}l_{i},  \label{Eq2}
\end{equation}%
where $y_{ni}$ is the amount of good $i$ sold in each country $n$ (sold
there at price $p_{ni})$, and $l_{i}$ is labor input. The firm's objective
is to maximize (\ref{Eq2}) subject to production constraint%
\begin{equation}
\sum_{i=1..N}d_{ni}y_{ni}\leq l_{i},  \label{Eq3}
\end{equation}%
where the left-hand-side denotes the total quantity produced, and the right
hand side is the production function.

\subsubsection*{Market Clearing and Feasibility}

Market clearing requires that the supply of each good equals the demand for
each good,%
\begin{equation}
c_{ni}=y_{ni},\text{ all }n,i.  \label{Eq4}
\end{equation}%
and the supply of labor equals to the demand for labor,%
\begin{equation}
L_{n}=l_{n},\text{ all }n.  \label{Eq5}
\end{equation}

\subsection*{Equilibrium}

The definition of equilibrium is as follows.

\begin{definition}
\textbf{Competitive equilibrium} in this economy is:

\begin{itemize}
\item prices $w_{n},p_{ni}$,

\item and allocation $c_{ni},y_{ni},l_{n},$
\end{itemize}

such that

\begin{itemize}
\item given prices, $c_{ni}$ solves the household's problem,

\item given prices, $y_{ni},l_{n},$ solve the firm's problem,

\item and all markets clear.
\end{itemize}
\end{definition}

\begin{proposition}
The competitive equilibrium allocation exists and is unique.
\end{proposition}

\begin{proof}
Competitive equilibrium allocation, if exists, must be Pareto efficient by
1st Welfare Theorem, and so the allocation must solve the planning problem
given by maximization of (\ref{Eq1U}) subject to (\ref{Eq3}), (\ref{Eq4})
and (\ref{Eq5}). Since this planning problem involves a maximization of a
continuous and concave objective function subject to a convex and compact
constraint set, the solution to the planning problem is unique. Thus, by 2nd
Welfare Theorem, the competitive equilibrium exists and is unique.
\end{proof}

\begin{exercise}
Show that in equilibrium the following version of the law of one price must
hold:%
\begin{equation*}
p_{ni}=d_{ni}p_{ii},\text{ all }n,i=1,\text{...},N.
\end{equation*}
\end{exercise}

\subsection*{\label{GravityArmington}Predictions for Trade}

In its general formulation, the Armington model can not be solved
analytically, and so we have to resort to a partial characterization of the
equilibrium. The proposition below derives the model's key predictions the
patterns of trade and how they depend on geography. We will refer to this
prediction as the \textit{gravity equation}.\footnote{%
In applied and atheoretical contexts, a similar equation has been
extensively used link trade to income, distance and other characteristics of
countries. It proved to be successful in capturing the actual patterns of
trade. Here, we will look at these results in light of the predictions of
the model. The simplest empirical gravity equation regresses the volume of
trade between bilateral pairs of countries (regions) on their bilateral
distance, income, and various\ dummy variable (common border, language,
etc...). It works really well in terms of fitting the data. However, since
such simple gravity equation is different from the one derived from our
model --- and in principle we would like to use it to perform counterfactual
experiments, the first order task is to understand the theory behind it
first.} In general, the gravity equation is an equation characterizing how
trade shares (expenditure shares of one country on some other country's
goods) are related to income levels, and various measures capturing trade
costs.

\begin{proposition}
In the Armington model, the share of expenditures of country $n$ on goods
imported from country $i$ in total expenditures of country $n$ is given by
the following equation: 
\begin{equation}
\frac{X_{ni}}{X_{n}}=\frac{X_{i}}{\sum_{n}X_{n}}(\frac{d_{ni}}{P_{i}P_{n}}%
)^{1-\sigma },  \label{Eq0}
\end{equation}%
where $X_{ni}=p_{ni}c_{ni}$ are expenditures of country $n$ on goods from
country $i,$ $X_{n}=\sum_{i}X_{ni}$ are total expenditures of country $i$ on
all goods (equal to country $n^{\prime }s$ income $w_{n}L_{n}$), and $%
P_{i}=(\sum_{i}p_{ni}^{1-\sigma })^{\frac{1}{1-\sigma }}$ is the ideal CPI
price index (price level weighted by the actual consumption share of each
good).
\end{proposition}

\begin{proof}
Note that the household's problem can be summarized by the following
Lagrangian:%
\begin{equation}
\mathcal{L}_{n}=(\sum_{i=1..N}\alpha _{i}^{\frac{1}{\sigma }}c_{ni}^{\frac{%
\sigma -1}{\sigma }})^{\frac{\sigma }{\sigma -1}}-\frac{1}{P_{n}}\left(
\sum_{i=1..N}p_{ni}c_{ni}-w_{n}\right) ,  \label{Eq6}
\end{equation}%
where by definition of the Lagrange multiplier, $\frac{1}{P_{n}}$ is the
shadow value of one unit of income in terms of the composite consumption $%
U_{n}$, and $P_{n}\ $is the shadow price of a unit of composite consumption $%
U_{n}$. Using the order conditions to this problem, 
\begin{equation}
\frac{\partial \mathcal{L}_{n}}{\partial c_{ni}}=U_{n}^{\frac{1}{\sigma }%
}\alpha _{i}^{\frac{1}{\sigma }}c_{ni}^{\frac{-1}{\sigma }}-\frac{p_{ni}}{%
P_{n}}=0,\text{ all }i  \label{Eq7}
\end{equation}%
It is easy to link this multiplier to prices:\ 
\begin{eqnarray*}
P_{n}^{\sigma -1}\alpha _{i}p_{ni}^{1-\sigma } &=&U_{n}^{\frac{1-\sigma }{%
\sigma }}\alpha _{i}^{\frac{1}{\sigma }}c_{ni}^{\frac{\sigma -1}{\sigma }} \\
P_{n}^{\sigma -1}\sum_{i}\alpha _{i}p_{ni}^{1-\sigma } &=&U_{n}^{\frac{%
1-\sigma }{\sigma }}\sum_{i}\alpha _{i}^{\frac{1}{\sigma }}c_{ni}^{\frac{%
\sigma -1}{\sigma }} \\
P_{n}^{\sigma -1}\sum_{i}\alpha _{i}p_{ni}^{1-\sigma } &=&(\sum_{i}\alpha
_{i}^{\frac{1}{\sigma }}c_{ni}^{\frac{\sigma -1}{\sigma }})^{-1}\sum_{i}%
\alpha _{i}^{\frac{1}{\sigma }}c_{ni}^{\frac{\sigma -1}{\sigma }},
\end{eqnarray*}%
\begin{equation}
P_{n}=(\sum_{i}\alpha _{i}p_{ni}^{1-\sigma })^{\frac{1}{1-\sigma }}.
\label{Eq7a}
\end{equation}%
We refer to it as an ideal price index.

Next, from the first order conditions%
\begin{eqnarray*}
\frac{p_{ni}}{P_{n}} &=&U_{n}^{\frac{1}{\sigma }}\alpha _{i}^{\frac{1}{%
\sigma }}c_{ni}^{\frac{-1}{\sigma }}, \\
\frac{p_{nj}}{P_{n}} &=&U_{n}^{\frac{1}{\sigma }}\alpha _{j}^{\frac{1}{%
\sigma }}c_{nj}^{\frac{-1}{\sigma }},
\end{eqnarray*}%
and the definition of expenditures $X_{ni}\equiv p_{ni}c_{ni}$, we derive%
\begin{equation*}
\frac{X_{nj}}{X_{ni}}=\frac{\alpha _{j}}{\alpha _{i}}(\frac{p_{nj}}{p_{ni}}%
)^{1-\sigma }.
\end{equation*}%
Summing up the above expression wrt $j$, we obtain:\footnote{%
The demand for each good $i$ in country $n$ is given by 
\begin{equation*}
c_{ni}=\alpha _{i}\left( \frac{p_{ni}}{P_{n}}\right) ^{-\sigma }\left( \frac{%
X_{n}}{P_{n}}\right) .
\end{equation*}%
}%
\begin{equation}
\frac{X_{ni}}{X_{n}}=\alpha _{i}(\frac{p_{ni}}{P_{n}})^{1-\sigma }.
\label{Eq8}
\end{equation}%
Multiplying both sides of equation (\ref{Eq8}) by $X_{n},$ and summing up
wrt $n,$ we use the law of one price 
\begin{equation}
p_{ni}=d_{ni}p_{ii},  \label{Eq9}
\end{equation}%
and the balanced trade condition (implied by (\ref{Eq1a}))%
\begin{equation}
X_{i}=Y_{i}=\sum_{n}X_{ni},  \label{Eq10}
\end{equation}%
to obtain%
\begin{equation}
\alpha _{i}p_{ii}^{1-\sigma }=\frac{X_{i}}{\sum_{n}(\frac{d_{ni}}{P_{n}}%
)^{1-\sigma }X_{n}}.  \label{Eq11}
\end{equation}

The above equation links prices (\ref{Eq8}) to aggregate variables. We use
it to substitute for prices in (\ref{Eq8}) (after using (\ref{Eq9})), and
derive:%
\begin{equation}
X_{ni}=\alpha _{i}p_{ii}^{1-\sigma }(\frac{d_{ni}}{P_{n}})^{1-\sigma }X_{n}=%
\frac{X_{n}X_{i}}{\sum_{n}(\frac{d_{ni}}{P_{n}})^{1-\sigma }X_{n}}\left( 
\frac{d_{ni}}{P_{n}}\right) ^{1-\sigma }.  \label{Eq12}
\end{equation}

In addition, in the special case when the iceberg transportation cost ($%
d_{ni}=d_{in})$ is symmetric, we can show that%
\begin{equation}
\sum_{n}(\frac{d_{ni}}{P_{n}})^{1-\sigma }X_{n}=P_{n}^{1-\sigma
}\sum_{i}X_{i},  \label{Pn0}
\end{equation}%
and instead of (\ref{Eq12}) obtain an even simpler expression (cumbersome to
derive):%
\begin{equation*}
X_{ni}=\frac{X_{n}X_{i}}{\sum_{n}X_{n}}\left( \frac{d_{ni}}{P_{i}P_{n}}%
\right) ^{1-\sigma }.
\end{equation*}
\end{proof}

\begin{exercise}
\label{Exercise2}Consider the following expenditure minimization problem:%
\begin{equation*}
E(U)=\min_{(c_{i})_{1..N}\geq 0}\sum_{i=1}^{N}p_{i}c_{i}
\end{equation*}%
subject to%
\begin{eqnarray*}
\sum_{i=1}^{N}(\alpha _{i}^{\frac{1}{\sigma }}c_{i}^{\frac{\sigma -1}{\sigma 
}})^{\frac{\sigma }{\sigma -1}} &=&U, \\
c_{i} &\geq &0,\text{ all }i=1..N,
\end{eqnarray*}%
where $p_{i}^{\prime }s\ $denote prices, $E(U)$ are total expenditures
(given $U)$, $\alpha _{i}^{\prime }s\ $are the preference weights, $\sigma $
is the elasticity of substitution, and $U\ $is the `composite good'
consumption level (or simply utility). Assume that $p_{i}^{\prime }s,\alpha
_{i}^{\prime }s,\sigma $ and $U$ are all strictly positive.

\noindent a. Show that $E(U)$ is homogenous of degree 1 ($E(\mu U)=\mu E(U),$
all $\mu >0),$ and thus takes the form $P\times U$ where $P=E(1)$.

\noindent b. Prove the Envelope Theorem in the context of the problem stated
in (a), i.e. show that $E^{\prime }(U)=\lambda ,$ where $\lambda $ is the
Lagrange multiplier on the constraint in (a). Then, use the conclusion from
point (a) to say $E^{\prime }(U)=P,$ and thus by Envelope Theorem to say $%
\lambda =P.$ Using it, solve for $E(1)$, which together with (a) shows 
\begin{equation*}
E(U)=(\sum_{i=1}^{N}\alpha _{i}p_{i}^{1-\sigma })^{\frac{1}{1-\sigma }}U.
\end{equation*}%
(This is an alternative way of deriving the price index to the one we did in
the proof of the proposition above.)

\noindent c. Show that the expenditures minimization problem with $E(U)=$%
Income, is equivalent the underlying utility maximization problem given by: 
\begin{equation*}
U=\max_{c_{i}\geq 0}(\sum_{i=1..N}\alpha _{i}^{\frac{1}{\sigma }}c_{i}^{%
\frac{\sigma -1}{\sigma }})^{\frac{\sigma }{\sigma -1}},
\end{equation*}%
subject to%
\begin{equation*}
\sum_{i=1...N}p_{i}c_{i}=\text{Income}.
\end{equation*}
\end{exercise}

\begin{exercise}
\label{Exercise1}Suppose that the preferences of the household are instead
described by: 
\begin{equation*}
U_{n}=(C^{NT})^{\gamma }(\sum_{i=1..N}\alpha _{i}^{\frac{1}{\sigma }}c_{ni}^{%
\frac{\sigma -1}{\sigma }})^{\frac{(1-\gamma )\sigma }{\sigma -1}},
\end{equation*}%
where $C^{NT}$ is the consumption of the local non-tradable good (services).
Assume that production technology of the non-tradable good is linear and
assume that labor is perfectly mobile across the two sectors. In the
extended model, derive the gravity equation by modifying each step in the
above proof accordingly. HINT:\ Use the fact that this is a Cobb-Douglas
aggregator, and so it implies that non-tradable goods have a constant share
in the overall consumer expenditures. You should get exactly the same
gravity equation with total expenditures replaced by total expenditures on
all tradable goods.
\end{exercise}

The existence of the first few terms in equation (\ref{Eq0}) should be
intuitive. In fact, we should expect the share of expenditures on good $i$
in total expenditures of country $n$ are positively related to the size of
country $i$ (measured by income or labor endowment), and negatively related
to the bilateral trade barrier $d_{ni}$ between them---with the strength of
the latter effect depending on the elasticity of substitution $\sigma $.
However, there are more terms in the gravity equation. Trade flows turns out
to additionally depend on the endogenous product of price indices of the two
countries $P_{n}P_{i}$ --- a term referred to by Anderson and Wincoop as
`gravitas'. Our next task is to link this term to the primitives in the
model.

\subsection*{Gravity with Gravitas}

Let's first take a look at the formula for the price level in country $n$, 
\begin{equation*}
P_{n}=(\sum_{i=1..N}\alpha _{i}(d_{ni}p_{ii})^{1-\sigma })^{\frac{1}{%
1-\sigma }}.
\end{equation*}%
and think what makes a country price level high. Since all countries face
the same $p_{ii}$'$s,$ we observe that high P can arise as consequence of:
(i)\ high overall level of $d_{ni}$'s, and/or (ii) high positive correlation
of $d_{ni}$'s with $p_{ii}$'s$.$ Thus, if we think of the iceberg cost $%
d_{ni}$'s in terms of distance between countries in some space, (i) means
that a country is distant from all other countries, and (ii)\ means that a
country is distant from the countries that are least distant from the rest
of the world\footnote{%
Because the price $p_{ii}$ is high, the good produced by the country must be
in high demand. This happens when the country is close to all the other
countries (rest of the world).}. Clearly, both (i) and (ii) are an
indication of isolation.

Figure \ref{Fig1} illustrates an example of such situation, which will
naturally arise when we are dealing with regions of a large country and
regions of a small country. An obvious example would the case of the states
of the US (large country), and the provinces of Canada (small country).



\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=15,scale=.44]{Figures.pdf} 
 \caption{An example of isolated small
country, and implications for aggregate price level.}%
\label{Fig1}
\end{figure}


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%country, and implications for aggregate price level. }}{\Qlb{Fig1}}{Figure}{%
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%TRUE;display "USEDEF";valid_file "T";width 145.8pt;height 145.8pt;depth
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%'KD2WCH00.wmf';tempfile-properties "XPR";}}

Upon closer inspection of equation (\ref{Eq0}), we note the following:

\begin{itemize}
\item Observation 1: The multilateral resistance term ($\frac{1}{P_{i}P_{n}}$%
) in the gravity equation makes the small country (two isolated dots in
Figure 1) to trade relatively more with each other.

\item Observation 2:\ The multilateral resistance term makes the large
country (dots that are not isolated in Figure 1) to trade relatively more
with the small country.
\end{itemize}

Formally, we can derive the above two observations as follows. For the sake
of argument, let's simply denote the two isolated regions provinces of
Canada, and rest of the regions the states of the US ($d$ stands for the
cost of crossing the national border), and set the following notation for
their underlying price levels: $P_{CA}=high,$ $P_{US}=low.$ Simplifying also
the notation for iceberg transportations,\ $d_{CA,US}=d>0,$ $%
d_{CA,CA}=d_{US,US}=1,$ we obtain from gravity equation:%
\begin{eqnarray*}
\frac{X_{CA,US}}{X_{CA,CA}} &=&\frac{\frac{X_{CA,US}}{X_{CA}}}{\frac{%
X_{CA,CA}}{X_{CA}}}=\frac{X_{US}\left( \frac{d}{high\times low}\right)
^{1-\sigma }}{X_{CA}\left( \frac{1}{high\times high}\right) ^{1-\sigma }}=%
\frac{X_{US}}{X_{CA}}d^{1-\sigma }(\frac{low}{high})^{\sigma -1}, \\
\frac{X_{US,CA}}{X_{US,US}} &=&\frac{\frac{X_{US,CA}}{X_{US}}}{\frac{%
X_{US,US}}{X_{US}}}=\frac{X_{CA}\left( \frac{d}{high\times low}\right)
^{1-\sigma }}{X_{US}\left( \frac{1}{low\times low}\right) ^{1-\sigma }}=%
\frac{X_{CA}}{X_{US}}d^{1-\sigma }(\frac{high}{low})^{\sigma -1}.
\end{eqnarray*}

As we can see, the additional endogenous term $(\frac{low}{high})^{\sigma
-1} $ does make $\frac{X_{CA,US}}{X_{CA,CA}}$ higher and it does make $\frac{%
X_{US,CA}}{X_{US,US}}$ lower as claimed above.

\paragraph{The Economics Behind Gravitas}

As we explain below, in the context of the example illustrated in Figure \ref%
{Fig1}, the key feature that the asymmetric size between the small country
and the large country is that the demand\ for imported goods is more elastic
in the large country than in the small country, and that the supply of
foreign goods is more elastic in the small country than in the large country.

To see this conclusion, simply note that the households from the larger
country can more effectively shift their expenditures from the foreign goods
towards the domestic goods. For instance, in the context of the example
considered above (Figure 1), when the US consumers cut spending on each
Canadian good by \$1, they must shift only \$2 of spending on 12 home goods.
However, when the Canadian consumers cut spending by \$1 on each US\ good,
they must shift as much as \$12 on only 2 Canadian goods. Now, because the
marginal utility from consumption of each good declines, the immediate
consequence of this property is a more elastic demand for Canadian goods in
the US than the demand for American goods in Canada.\footnote{%
In simple words, when all Canadian goods become more expensive in the US,
the US\ households can shift to a wide variety of home goods, but if all
American goods become more expensive in Canada, Canadians have to take the
hit.} (The opposite conclusion applies to the supply side because the US
producers face a much smaller decline in the price as move their sales from
Canada (i.e. one unit from each province) to US than the Canadian producers
face as they move their sales from the states (i.e. one unit from each
state) to Canada.)

\begin{exercise}
\label{Exercise3}To formalize the above argument, solve for the demand from
the following problem:%
\begin{equation*}
\max (q_{A}^{\frac{\sigma -1}{\sigma }}+Nq_{B}^{\frac{\sigma -1}{\sigma }})^{%
\frac{\sigma }{\sigma -1}}
\end{equation*}%
subject to 
\begin{equation*}
p_{A}q_{A}+Np_{B}q_{B}=I.
\end{equation*}%
Specifically, derive the demand for good B, and show that for large $N$ the
price index will be affected by the price $p_{B}$ -- implying a lower
measured price elasticity of demand. HINT: Derive an equation analogous to (%
\ref{Eq8}). Calculate the price index when $N$ is infinite.
\end{exercise}


\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=1,scale=.44]{Figures.pdf} 
 \caption{Canadian Producers and
Consumers Pay the Iceberg Cost.}%
\label{Fig2}
\end{figure}

%
%\FRAME{ftbpFU}{364.0667pt}{197.0667pt}{0pt}{\Qcb{Canadian Producers and
%Consumers Pay the Iceberg Cost.}}{\Qlb{Fig2}}{Figure}{\special{language
%"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
%"USEDEF";valid_file "T";width 364.0667pt;height 197.0667pt;depth
%0pt;original-width 434.2179pt;original-height 234.1789pt;cropleft
%"0";croptop "1";cropright "1";cropbottom "0";tempfilename '../Documents and
%Settings/Lukasz Drozd/Desktop/graphics/Figure2.wmf';tempfile-properties
%"XNPR";}}

Given the described above implication of relative size on elasticities, it
should not surprise that an increase in the iceberg transportation cost $d$
might have a very different effect on the two countries. These elasticities
determine who bears the burden of this cost. As illustrated in Figure \ref%
{Fig2}, in this case these are the consumers and producers of the small
isolated country who will pay for it. Thus, isolation implies that the terms
of trade (price of imports in terms of exports) of the small country worsens
relative to the large country, and the worsened terms of trade makes the
small country shift relatively more spending towards the home goods than the
large country. This is the economic intuition why the two endogenous terms
that appear in the theoretical gravity equation convey, and it is a
beautiful example how general equilibrium considerations sometimes matter.

To formalized the above idea, let's push the previously used argument to the
limit and make US arbitrarily large relative to Canada (the inelastic demand
and supply lines become vertical). In such case, assuming each region of the
same size normalized to 1 ($L_{i}=L_{j}=1),$ as number of regions in the
large country goes to infinity, we have%
\begin{equation*}
\frac{p_{CA,CA}}{p_{US,US}}=\frac{P_{CA}}{P_{US}}=d,
\end{equation*}%
and thus 
\begin{eqnarray*}
\frac{X_{CA,US}}{X_{CA,CA}} &=&\frac{X_{US}}{X_{CA}}d^{1-\sigma }(\frac{%
P_{US}}{P_{CA}})^{\sigma -1}=\frac{p_{US,US}}{p_{CA,CA}}d^{1-\sigma }(\frac{%
P_{US}}{P_{CA}})^{\sigma -1}=d^{-1}d^{1-\sigma }d^{1-\sigma }=d^{1-2\sigma },
\\
\frac{X_{US,CA}}{X_{US,US}} &=&\frac{\frac{X_{US,CA}}{X_{US}}}{\frac{%
X_{US,US}}{X_{US}}}=\frac{X_{CA}\left( \frac{d}{high\times low}\right)
^{1-\sigma }}{X_{US}\left( \frac{1}{low\times low}\right) ^{1-\sigma }}%
=dd^{1-\sigma }d^{\sigma -1}=d.
\end{eqnarray*}%
If, however, we did not have the `gravitas', we would have obtained instead:%
\begin{eqnarray*}
\frac{X_{CA,US}}{X_{CA,CA}} &=&\frac{X_{US}}{X_{CA}}d^{1-\sigma }=\frac{%
p_{US,US}}{p_{CA,CA}}d^{1-\sigma }=d^{-1}d^{1-\sigma }=d^{-\sigma }, \\
\frac{X_{US,CA}}{X_{US,US}} &=&\frac{\frac{X_{US,CA}}{X_{US}}}{\frac{%
X_{US,US}}{X_{US}}}=\frac{X_{CA}\left( \frac{d}{high\times low}\right)
^{1-\sigma }}{X_{US}\left( \frac{1}{low\times low}\right) ^{1-\sigma }}%
=dd^{1-\sigma }=d^{2-\sigma }.
\end{eqnarray*}

\subsection*{Does the Model Fit the Data?}

The simplest test of the model is to look at the predicted asymmetric effect
of trade costs on trade between a small country like Canada and a large
country like US (that are somewhat isolated from the rest of the world). In
such case, the Armington model predicts that if crossing the national border
involves a cost (tariff and non-tariff barriers), then the impact of this
cost should be asymmetric. In particular, such cost should drastically
reduce trade of each Canadian province with the US, but should not reduce as
much the trade of each US state with the Canadian provinces. By running 2
simple regressions, we can check if this is the case.

The empirical specification that we are going to adopt will simply assume
that the iceberg cost of transportation is a function of distance and the
national border. We will do the same for gravitas. Formally, we are going to
have: $d_{ni}=\exp (\tau b_{ni})\delta _{ni}^{\rho },$ $P_{n}P_{i}=\exp
(vb_{ni}),$ where $b_{ni}$ a border dummy (1 if there is national borders
between regions $n$ and $i$, 0 otherwise), and $\delta _{ni}$ is the
distance between $n$ and $i,$ $\tau $ is border cost affecting bilateral
trade barrier, and $v$ is the border effect operating through the
multilateral resistance term.

Given the specification of the iceberg cost and gravitas, plugging into (\ref%
{Eq0}), we thus need to estimate the following equation:\ 
\begin{equation*}
\log X_{ni}=\kappa +A\log X_{n}+B\log X_{i}+C\log \delta _{ni}+Db_{ni},
\end{equation*}%
where $C=\rho (1-\sigma )$ and $D=(1-\sigma )(\tau +v).$

The equation takes into account the multilateral resistance term $P_{i}P_{n}$
in the form of a border dummy. Based on our previous analysis, we should
expect this term to capture well the notion of isolation when regressed from
the US side and the Canadian side separately. In particular, based on our
discussion, we should expect to find that D is much higher when we run the
regression from the Canadian side (CA-CA and CA-US observations), than when
we run it from the US\ side (US-US and US-CA observations). The results are
as follows (Replicated Table 1 from Anderson and Wincoop, 2003):

%TCIMACRO{\TeXButton{B}{\begin{table}[t] \centering}}%
%BeginExpansion
\begin{table}[t] \centering%
%EndExpansion
\caption{Comparing CA-US and US-CA gravity equations.}%
\begin{equation*}
\begin{tabular}{lll}
\hline
Parameter & Regression from CA side & Regression from US side \\ \hline
$\kappa $ & 2.80 (.12) & .41 (.05) \\ 
$A$ & 1.22 (.04) & 1.13 (.03) \\ 
$B$ & .98 (.03) & .98 (.02) \\ 
$C$ & -1.38 (.07) & -1.08 (.04) \\ 
$D$ & -16.4 (2.0) & -1.5 (.08) \\ \hline
R$^{2}$ & .76 & .85 \\ \hline
\end{tabular}%
\end{equation*}

\label{ArmingtonT1}%
%TCIMACRO{\TeXButton{E}{\end{table}}}%
%BeginExpansion
\end{table}%
%EndExpansion

\begin{exercise}
Go to Anderson's website. Download the zip file with the dataset supporting
the paper. Replicate the above regressions using this dataset.
\end{exercise}

As we can see, the estimated values do exhibit strong asymmetry. In fact, $D$
is by far more negative in the regression from the Canadian side than in the
regression from the US side. Each Canadian province, controlling for income
and distance, trades $1600\%$ more with another Canadian province than with
a US state. Given such huge asymmetry in the regression, our next question
should be whether the model is quantitatively capable of generating it.

\begin{exercise}
\label{Exercise4}(Numerical experiment with the model)\ Consider the
Armington model with the following parameter setting: $N=100,$ $\alpha =1/N,$
$\sigma =11,$ $L_{i}=L_{j}=1$, all $\ i,j=1..N.$ Assume that the first 90 of
the $N$ regions are in a large country (US), and the last 10 are in a small
country (Canada), which roughly corresponds to the ratio of Canadian GDP\ to
the US GDP. Furthermore, assume that the transportation cost between the
regions within the same country is zero, i.e. $d_{ni}=1$ whenever $i,n\in
US, $ or $i,n\in CA,$ and assume that the iceberg transportation cost
between the regions within two different countries is 20\%, i.e. $d_{ni}=1.2$
whenever $i\in US$, $n\in CA$ or $i\in CA$, $n\in US.$

\noindent a. Use the following equilibrium relation from the model%
\begin{equation*}
p_{ii}L_{i}=\sum_{n}X_{ni}=\sum_{n}\alpha _{i}(\frac{X_{ni}}{X_{n}}%
)X_{n}=\sum_{n}\alpha _{i}(\frac{d_{ni}p_{ii}}{P_{n}})^{1-\sigma }p_{nn}L_{n}
\end{equation*}%
to construct an iterative algorithm that solves the model in MATLAB.%
\footnote{%
If it is a contraction, then it will converge to the fixed point.} Using the
algorithm, compute the overall price level of a representative US\ region
and Canadian region, and the prices of the corresponding goods. (HINT: The
algorithm may be unstable unless you slow down the updates a bit. To be on
the safe side, I suggest to divide both sides by $L_{i}p_{ii}^{1-\sigma },$
compute $p_{ii},$ and use the updating rule that puts .5 weight on the old
value and only .5 weight on the newly solved value:\footnote{%
This way you enlarge the domain on which out mapping is contraction. You
then do not need a very precise guess for convergence to the fixed point to
occur.} $p_{i+1}=.1p^{\prime }+.9p_{i},$ where $i$ is the iteration number,
and $p_{i}$ is used to solve for the vector $p^{\prime }$ in iteration $i$.
Don't forget to evaluate the convergence, and the residuals of equilibrium
conditions at the end. Remember that $p_{NN}$ is the numeraire. (Print out
the code and hand in with the HW.)

\noindent b. What is the home-bias from the US\ side (defined as $%
X_{US,US}/X_{US,CAN})$ and from the Canadian side (defined as $%
X_{CA,CA}/X_{CA,US}).$

\noindent c. Using data generated by the model, suppose you run the
following regression of trade flows on the border dummy (referred to as the
`McCallum regression'):%
\begin{equation*}
\ln \frac{X_{ni}}{X_{i}X_{n}}=\kappa +A\times \text{border\_dummy}%
+\varepsilon ,
\end{equation*}%
where $n\in CA,$ $i\in CA$ or $US$. What is the value of the regression
coefficient on the border dummy?

\noindent d. Suppose you run the same regression as in point \textit{c} but
from the US side, i.e. $n\in US,$ $i\in CA$ or $US$. What is the value of
the regression coefficient on the border dummy?

\noindent e. How do your answers to \textit{c} and \textit{d} compare to the
coefficients that Anderson and Wincoop found in the data by running
McCallum's regression separately from the US\ side and the Canadian side?
Explain briefly the implications of your findings.

\noindent f. Redo points \textit{c} and \textit{d} with $\sigma =8$.

\noindent g. Comparing the answers in \textit{e} and \textit{f,} what
fraction of the border effect is accounted for by the endogenous
multilateral resistance term?

\noindent h. What is the average share of trade with the US for a
representative Canadian province in the model (measure it by ($90X_{CA,US})$/%
$(10X_{CA,CA}+90X_{CA,US})$)? Consider two levels of trade cost: $%
d_{US,CA}=1.2$ (same as before), and $d_{US,CA}=1.175$. Given that the
median and average value of this object in the data is about\footnote{%
Pulled out form the data available from Anderson's website.} .45, which
level of the border cost accounts better for this number?

\noindent i. Would the answers to \textit{b-g} change if instead you had 100
Canadian regions and 900 US\ regions$?$ Explain your answer analytically.
\end{exercise}

\subsubsection*{Structural Estimation}

Anderson and Wincoop (2003) structurally estimate the model using a set of
10 provinces, 30 states of US and 20 OECD\ countries. Their exercise is
meant to address the question whether the model can quantitatively fit the
data for plausible parameter values. An alternative approach to theirs would
be a detailed calibration of the model in the spirit of the numerical
example you solved above.

\paragraph*{Specification}

To structurally estimate the model, Anderson and Wincoop use the following
specification for the iceberg transportation cost: 
\begin{equation}
d_{ni}=\exp (\beta b_{ni})\times \delta _{ni}^{\rho },  \label{modeldni}
\end{equation}%
where $b_{ni}$ is the national border dummy ($1$ if there is a national
border between region $n$ and region $i\,,$ 0 otherwise), $\delta $ is the
distance between regions $n$ and $i$ in miles, $\rho $ is the impact
parameter of distance between on the implied iceberg transportation cost,
and $\beta $ is the impact parameter of the national border on the implied
iceberg transportation cost.

Substituting out $d_{ni}$ in the theoretical gravity equation, they obtain
the following empirical specification of the model: 
\begin{eqnarray}
\ln \frac{X_{ni}}{Y_{i}Y_{n}} &=&k+a_{1}\ln (\delta _{ni})+a_{2}b_{ni}-
\label{reg} \\
&&-\ln P_{i}^{1-\sigma }-\ln P_{n}^{1-\sigma }+\varepsilon _{ni},  \notag
\end{eqnarray}%
where $a_{1}=(1-\sigma )\rho ,$ $a_{2}=(1-\sigma )\beta $, and the vector of
aggregate prices $(P_{n})_{n}$ solves to the fixed point problem given by (%
\ref{Pn0}):\footnote{%
Income data is assumed to be nomralized so that $\sum_{n}X_{n}=1.$}\ \ 
\begin{eqnarray}
P_{n}^{1-\sigma } &=&\sum_{i=1,..,N}\frac{\exp (a_{1}\ln (\delta
_{ni})+a_{2}b_{ni})}{P_{i}^{\sigma -1}}X_{i},\text{ }n=1\text{,..,}N-1,
\label{FPP} \\
P_{N} &=&1.  \notag
\end{eqnarray}

Note that the observable data includes distance matrix $(\delta _{ni})_{ni}$%
, border dummy matrix $(b_{ni})_{ni}$, multilateral expenditure shares $(%
\frac{X_{ni}}{Y_{i}Y_{n}})_{ni},$ and income vector $(Y_{i})_{i}.$ The price
vector $(P_{n})_{n}$ is unobservable, and so we must use theory to solve for
it.

\paragraph*{Numerical Algorithm to Estimate the Model (\protect\ref{reg})}

\begin{itemize}
\item Set $\sigma =6$ in consistency with the estimates of the long-run
impact of a change in tariff rates on trade from the literature.\footnote{%
Note that $\sigma $ can not be identified separately from the other
parameters.}

\item Set the values of $a_{1},a_{2}$, and solve for the vector of prices $%
(P_{n})_{n}\ $from the fixed point problem given by (\ref{FPP}).

\item Plug in the price vector $(P_{n})_{n}$ into the regression equation (%
\ref{reg}), and find the constant $\kappa $ that minimizes the squared sum
of regression residuals. Given residual minimizing value of $\kappa $,
evaluate the squared sum of residuals $r=\sum_{ni}\varepsilon _{ni}^{2}.$

\item Repeat steps 2-4 above by choosing $a_{1},a_{2}$ to minimize the
residual $r$ calculated in step 3.
\end{itemize}

\paragraph*{Results}

The results of estimating the structural model are presented in the table
reproduced below (Table 2 in the paper). As we can see, the model does an
magnificent job in account for the asymmetry and the border puzzle. In the
two country case (second column of Table 2), it underpredicts trade between
Canadian province on average by only 17\%, and overpredicts trade of US
states with other US states by 6\%. Given that Canadian provinces trade
1600\% more with each other than with US\ states, this is a huge success. In
addition, Anderson and Wincoop show that when the model is extended to
include other countries, it does an even better job (see last column in
Table 2 in the paper).
 

\subsection*{A Note on the Literature}

This part was based on two influential papers: McCallum (1995) and Anderson
and Wincoop (2003). McCallum's paper shows that an ad hoc gravity equation
on trade between Canada and US\ (from Canadian side) yields puzzling
results. Namely, after controlling for distance and income, Canadian
provinces trade 2200\% more with another Canadian province than with US\
state. The original paper interprets this finding as possibly suggesting an
enormous cost of crossing the border, and is referred to as `the border
puzzle'. Anderson and Wincoop (2003) is a response to this finding. Anderson
and Wincoop show that according to the theory, the specification of the ad
hoc gravity model in most applications s incomplete, and so the results may
be biased

\section{Dornbusch, Fisher and Samuelson Model}

Dornbusch, Fisher and Samuelson model is the most general version of the
Ricardian model (DFS model hereafter) for the case of two countries. The key
idea is to span goods on a unit interval, and thus summarize the endogenous
equilibrium specialization pattern by two cutoff values (pivotal goods)
defining \textit{the set of} \textit{goods that are produced only by country
1 and the set of goods that are produced only by country 2}. \ 

The DFS model nests a two country Armington model, it also nests the two
country Eaton and Kortum model discussed in the next section. Interestingly,
the DFS model is particularly difficult to extend to a multicountry
framework in full generality, and it wasn't until Eaton and Kortum
parameterization that this framework took off as a basis for any
quantitative analysis.

The exercise below will walk you through a simple symmetric version of the
DFS model. In particular, you will establish here its relation to the
Armington model, and solve for the cutoff values. Later, we will find all
these results useful to understand the intuition behind the Eaton and Kortum
(2002) model.

\begin{exercise}
\label{Exercise5}(Dornbusch, Fisher and Samuelson (1977)) Consider a world
with two\textbf{\ symmetric} countries and a continuum of goods indexed on a
unit interval. Preferences in each country are identical and given by\textbf{%
\ }%
\begin{equation*}
U_{i}=(\int_{0}^{1}\ln c_{i}(\omega )d\omega ,\text{ }i=1,2,
\end{equation*}%
and all markets are perfectly competitive. Assume each country has access to
a linear technology to produce each good using labor,%
\begin{equation*}
y_{i}(\omega )=z_{i}(\omega )l_{i}(\omega ),
\end{equation*}%
where $z_{i}(\omega )$ is the efficiency level in producing good $\omega $
in country $i$, and $l_{i}(\omega )$ is the labor input. Assume that the
labor endowment of the stand-in household in each country is one$,$ and the
production efficiency schedules are given by the following functions: 
\begin{eqnarray}
z_{1}(\omega ) &=&e^{1-\omega },  \label{EqES} \\
z_{2}(\omega ) &=&e^{\omega }.  \notag
\end{eqnarray}%
In addition, assume there is a positive tariff rate $T$ between the two
countries that amounts to 10\% of the value of the transported goods across
the border. The revenue from the tariff is lump-sum rebated to the
households.

\noindent a. Define competitive equilibrium for this economy.

\noindent b. \textbf{Refers to point a above. }Compute the competitive
equilibrium you have defined in a. HINT:\ Find 2 cutoffs that divide the
space of goods into 3 categories: (i)\ traded and produced in country 1,
(ii)\ traded and produced in country 2, and (iii)\ not traded (both
countries produce them for home market only). Exploit symmetry to say that
wages must be 1 in both countries. Use the fact that in the case of log
utility the share of expenditures on each good is always a constant fraction
of total expenditures on all goods.

\noindent c. Apply NIPA\ rules to compute\ the GDP of each economy. What
happens to the GDP in equilibrium when the tariffs are increased? HINT:\
Read handbook of NIPA\ accounting available from BEA website.\footnote{%
See http://www.bea.gov/national/pdf/NIPAhandbookch1-4.pdf.}

\noindent d. How would you have to modify the assumed efficiency schedules
stated in (\ref{EqES}) to effectively obtain a symmetric two-country
Armington model. Based on your answer, what is the key qualitative
difference between the Armington model and the DFS model.
\end{exercise}

\section{Hecksher-Ohlin Model}

The following exercise will walk you through the setup of the 2x2
Hecksher-Ohlin Model. In this version of the Hecksher-Ohlin model countries
have access to the same technologies to produce 2 goods, but differ in
factor endowment of capital and labor. Because technologies to produce each
good use these two factor at different intensities, in equilibrium countries
partially specialize in the production of the good more intensive in the
abundant factor. The specialization leads to a very peculiar result: Despite
the fact that factors are immobile across countries, trade in goods leads to 
\textit{factor price equalization} across countries (wages and interest
rates are the same).\footnote{%
These two results are referred to as the Hecksher-Ohlin Theorem and the
Factor Price Equalization Theorem.}

\begin{exercise}
\label{Exercise6}(Hecksher-Ohlin model) Consider the world with 2 countries
and 2 tradable goods. Preferences of the stand-in household in each country
are%
\begin{equation*}
U_{i}=\sum_{j=1,2}\log C_{i}^{j},\text{ }i=1,2
\end{equation*}%
where $C_{i}^{j}$ denotes consumption in country $i$ of good $j.$ The
stand-in household in country 1 has 2 units of labor ($L$) and 3 units of
capital ($K$), and the stand-in household in country 2 has 3 units of labor
and 2 units of capital. Firms in each country have access to the same CRS
technology to produce both goods. The technology to produce good 1 is $Y=K^{%
\frac{1}{3}}L^{\frac{2}{3}}$ (sector 1)$,$ and good 2 is $Y=K^{\frac{2}{3}%
}L^{\frac{1}{3}}$ (sector 2)$.$ For simplicity assume there is no
transportation cost.

\noindent a. Assume factors are perfectly mobile across countries. Define
the competitive equilibrium.

\noindent b. \textbf{Refers to equilibrium defined in a. }It can be shown,
using the First and the Second Welfare Theorems, that the competitive
equilibrium is unique up to the \textbf{undetermined allocation of capital
and labor across countries within sectors}, and it solves the following
planning problem for $\mu =\frac{1}{2}$:%
\begin{equation*}
\max_{(C_{i}^{j},K_{i}^{j},L_{i}^{j})_{i,j=1,2}}\mu \sum_{j=1,2}\log
C_{1}^{j}+(1-\mu )\sum_{j=1,2}\log C_{2}^{j}
\end{equation*}%
subject to 
\begin{eqnarray*}
\sum_{i,j=1,2}K_{i}^{j} &=&5,\text{ }\sum_{i,j=1,2}L_{i}^{j}=5, \\
\sum_{i=1,2}C_{i}^{1} &=&\sum_{i}\left( K_{i}^{1}\right) ^{\frac{1}{3}%
}\left( L_{i}^{1}\right) ^{\frac{2}{3}}, \\
\sum_{i=1,2}C_{i}^{2} &=&\sum_{i}\left( K_{i}^{2}\right) ^{\frac{2}{3}%
}\left( L_{i}^{2}\right) ^{\frac{1}{3}}.
\end{eqnarray*}%
\noindent Compute the competitive equilibrium you defined in \textit{a}.
(HINT:\ Remember that allocation is undetermined wrt to allocation of
production across countries (who produces what). Exploit symmetry to argue
that the relative price between goods and factors must be 1. Then, introduce
an aggregate firm that produces the entire world output (max output), and
compute K,L from its problem.)

\noindent c. Assume factors are immobile across countries. Define the
competitive equilibrium.

\noindent d. \textbf{Refers to equilibrium defined in c. }It can be shown
that there is a unique competitive equilibrium, and it solves the following
planning problem for $\mu =\frac{1}{2}$:%
\begin{equation*}
\max_{(C_{i}^{j},K_{i}^{j},L_{i}^{j})_{i,j=1,2}}\mu \sum_{j=1,2}\log
C_{1}^{j}+(1-\mu )\sum_{j=1,2}\log C_{2}^{j}
\end{equation*}%
subject to 
\begin{eqnarray*}
(\ast )\text{ }\sum_{j=1,2}K_{1}^{j} &=&3,\text{ }\sum_{i,j=1,2}L_{1}^{j}=2,%
\text{ }\sum_{j=1,2}K_{2}^{j}=2,\text{ }\sum_{j=1,2}L_{2}^{j}=3, \\
\sum_{i,j=1,2}K_{i}^{j} &=&5,\text{ }\sum_{i,j=1,2}L_{i}^{j}=5, \\
\sum_{i=1,2}C_{i}^{1} &=&\sum_{i}\left( K_{i}^{1}\right) ^{\frac{1}{3}%
}\left( L_{i}^{1}\right) ^{\frac{2}{3}}, \\
\sum_{i=1,2}C_{i}^{2} &=&\sum_{i}\left( K_{i}^{2}\right) ^{\frac{2}{3}%
}\left( L_{i}^{2}\right) ^{\frac{1}{3}}.
\end{eqnarray*}%
Compute the competitive equilibrium you defined in \textit{point c}. (%
\textbf{HINT}: Guess that the solution to the planning problem above solves
a relaxed problem with (*) constraints omitted (like in the planning problem
in point b). Verify the guess by showing that factor markets clear -- use
(*) in combination with the factor demand functions you derived in problem 2
to find market clearing production pattern (write it in matrix form, will be
easier...).

\noindent e. What is the pattern of trade in the competitive equilibrium you
found in \textit{d}? More precisely, in which good the labor abundant
country is a net exporter?

\noindent f. Note that `the trick' you used in \textit{d} to solve for the
equilibrium would not work in general, i.e. for an arbitrary distribution of
factor endowment levels\footnote{%
The range of endowment vectors for which `the trick' works is referred to as
the cone of diversification.}. Show which step of your solution in \textit{e}
would break down if this was not true, and explain why. HINT: Recall that
there are non-negativity constraints on all the variables.
\end{exercise}

\section{Eaton and Kortum Model}

Essentially, Eaton and Kortum (2002) model is a versatile and tractable
probabilistic parameterization of the Ricardian model with a continuum of
goods due to Dornbusch, Fisher and Samelson (1977). EK model extends the DFS
framework to a multicountry context, and allows for an explicit derivation
of the gravity equation.

\subsection*{Model Economy}

Goods are indexed on a unit interval $\omega \in \lbrack 0,1]$, and the
world is comprised of $N$ countries (regions). Every country can produce
every good from the continuum, but the labor requirement to produce each
good differs. Unlike in the DFS model, the productivity schedules are
described probabilistically. Namely, it is assumed that the efficiency of
producing a good in country $n$ is a realization of an i.i.d. Frechet
distributed random variable $\mathcal{Z}_{n}$:\ 
\begin{equation}
F_{n}(\mathcal{Z}_{n}\leq z)=\exp (-T_{n}z^{-\theta }),  \label{1Eq0}
\end{equation}%
where $T_{n}$ and $\theta $ are parameters governing the mean and the
dispersion\footnote{%
We will use the convention of denoting a random variable by a caligraphic
capital letter. By the law of large numbers, note that $F\left( z\right) $
is also the fraction of goods produced at efficiency $z$ or lower.}, and $n$
is the country index.

As before, geography is modeled by an iceberg transportation cost obeying
three standard properties: (i) symmetry $d_{ni}=d_{in}\geq 1$, (ii)\ no cost
within the country $d_{ii}=1$, and (iii)\ the triangle inequality 
\begin{equation}
d_{ni}\leq d_{nj}+d_{ji}\ \text{\ all \ }i,j,n=1..N.
\end{equation}

\subsubsection*{Probabilistic Notion of Comparative Advantage}

Figure \ref{FigFrechet} illustrates the plots of the Frechet density
function. The moments of this distribution are given by: (i)\ mean=$T^{\frac{%
1}{\theta }}\Gamma (1-\frac{1}{\theta }),$ (ii) coefficient of variation
(standard deviation/mean) = $exp(\frac{\pi }{\theta \sqrt{6}})$. Because $%
\theta $ unambiguously determines the coefficient of variation, the two
parameters have a natural interpretation: (i) $T_{i}$ characterizes the
overall level of technology of a country (absolute advantage), and (ii) $%
\theta $, a parameter common to all countries, characterizes the dispersion
of efficiency across goods (comparative advantage).


\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=7,scale=.44]{Figures.pdf} 
 \caption{Frechet density function.}%
\label{FigFrechet}
\end{figure}

%\FRAME{ftbpFU}{263.3333pt}{180.4pt}{0pt}{\Qcb{Frechet density function.}}{%
%\Qlb{FigFrechet}}{Figure}{\special{language "Scientific Word";type
%"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width
%263.3333pt;height 180.4pt;depth 0pt;original-width
%731.3333pt;original-height 500.0667pt;cropleft "0";croptop "1";cropright
%"1";cropbottom "0";tempfilename 'KD2WCH01.wmf';tempfile-properties "XPR";}}

Frechet distribution, or in general any exponential distribution, has the
following \textit{four} properties that will greatly simplify our analysis
of the model:

\begin{itemize}
\item \textit{Property 1 (Frechet distributed extreme values): Let }$(%
\mathcal{Z}_{1})_{i=1..N}\ $be a vector of Frechet distributed random
variables with parameters $(T_{i},\theta _{i})_{i}.$ Then, 
\begin{equation}
\mathcal{Z}=\max_{i}\{\mathcal{Z}_{i}\},
\end{equation}%
is Frechet distributed with parameter $T=\sum_{i}T_{i},$ and $\theta .$

\item \textit{Property 2 (Mean determined winning probability): Let }$(%
\mathcal{Z}_{i})_{i=1..N}\ $be a vector of Frechet distributed random
variables with parameters $(T_{i})_{i},\theta .$ Then, 
\begin{equation}
\Pr (\mathcal{Z}_{s}\geq \max_{i\neq s}\{\mathcal{Z}_{i}\})=\frac{T_{s}}{%
\sum_{i}T_{i}}.
\end{equation}

\item \textit{Property 3 (Memorylessness): Let }$\mathcal{Z}\ $be a Frechet
distributed random variables with parameters $(T,\theta ).$ Then, the
conditional distribution is equal to the unconditional distribution 
\begin{equation*}
\Pr (\mathcal{Z}_{i}\leq z_{2}|\mathcal{Z}\leq z_{1})=e^{-Tz_{2}^{-\theta }}.
\end{equation*}

\item \textit{Property 4 (Scale invariant dispersion): Let }$\mathcal{Z}\ $%
be a Frechet distributed random variables with parameters $(T,\theta ).$
Then, the distribution of a random variable $a\mathcal{Z}$ (a$\in R_{+})$ is
Frechet with parameters $(a^{-\theta }T,\theta ).$
\end{itemize}

\begin{proof}
Property 1: 
\begin{equation*}
\Pr (\max_{i}\{\mathcal{Z}_{i}\}\leq z)=\Pi _{i}\Pr (\mathcal{Z}_{i}\leq
z)=e^{-\sum_{i}T_{i}z^{-\theta _{i}}}
\end{equation*}

Property 2: 
\begin{eqnarray*}
\Pr (\mathcal{Z}_{s} &\geq &\max_{i\neq s}\{\mathcal{Z}_{i}\})= \\
\int_{0}^{\infty }\Pi _{i}\Pr (\mathcal{Z}_{i} &\leq
&z_{s})dF(z_{s})=\int_{0}^{\infty }\Pi _{i}\Pr (\mathcal{Z}_{i}\leq
z_{s})F^{\prime }(z_{s})dz_{s}= \\
&=&\int_{0}^{\infty }\theta z_{s}^{-\theta -1}T_{i}e^{-\sum_{i\neq
s}T_{i}z_{s}^{-\theta }}e^{-T_{i}z_{s}^{-\theta }}dz_{s}= \\
&=&\int_{0}^{\infty }\theta z_{s}^{-\theta
-1}T_{i}e^{-\sum_{i}T_{i}z_{s}^{-\theta }}dz_{s}= \\
&=&\frac{T_{i}}{\sum_{i}T_{i}}\int_{0}^{\infty
}(\sum_{i=1..N}T_{i})z_{s}^{-\theta -1}e^{-\sum_{i}T_{i}z_{s}^{-\theta
}}dz_{s} \\
&=&\frac{T_{i}}{\sum_{i}T_{i}}[e^{-z_{s}^{-\theta
}\sum_{i}T_{i}}]_{0}^{\infty }=\frac{T_{i}}{\sum_{i}T_{i}}
\end{eqnarray*}

Property 3 follows directly from the Bayes rule. 

Property 4 follows by rearranging the formula for the Frechet distribution.
\end{proof}

\subsubsection*{Households}

To state the household's problem formally, we need to transform this problem
to guarantee integrability of the utility function and the budget
constraint.\ To this end, given the equilibrium distribution of prices $%
G_{n}(p)$ in country $n,$ we exploit the fact that the distribution of
prices tells us the measure of goods that are available at price $p$. Since
all variables of the model (as functions of $\omega )$ typically take
identical value as long as the price is the same, without loss of generality
we will index goods by their underlying prices rather than the type of good $%
\omega $.

Under such reformulation, the preferences of the stand-in household from
country $n$ can be described by the following utility function: 
\begin{equation}
U_{n}=[\int_{0}^{\infty }c_{n}(p)^{\frac{\sigma -1}{\sigma }}dG_{n}(p)]^{%
\frac{\sigma }{\sigma -1}},  \label{Eq1}
\end{equation}%
where $\sigma $ denotes the elasticity of substitution between the goods ($%
\sigma >1$)$,$ $c_{n}(p)$ is the price-identical consumption level of goods
at price $p,$ and $dG_{n}(p)$ is the measure (weight/fraction) of goods at
price $p.$\footnote{%
Note that the above formulation restricts attention to allocations in which
the household chooses the same consumption of all goods that have the same
price. This would be the case endogenously, but here it is build into the
problem. This trick allows us to use the coarser indexation by price and
guarantee integrability.}

The problem of the household is thus to choose an integrable function $%
c_{n}(p)$ that maximizes (\ref{Eq1}) subject to the budget constraint:%
\begin{equation}
\int_{0}^{\infty }pc_{n}(p)dG_{n}(p)=w_{n}L_{n}+\Pi _{n},  \label{Eq1b}
\end{equation}%
where $\Pi _{n}$ are the profits paid out by the local firms (in equilibrium 
$\Pi _{n}$ will be zero), and $w_{n}L_{n}$ is compensation of labor $(L_{n}$
is endowment of labor).

From the household's problem, we calculate that the ideal CPI-price index is
given by:\ 
\begin{equation}
P_{n}=(\int_{0}^{\infty }p^{1-\sigma }dG_{n}(p))^{\frac{1}{1-\sigma }}.
\label{1Eq1}
\end{equation}

\subsubsection*{A Note on Optimization with Integrals}

Note that the utility maximization problem above involves a choice of an
optimal function that maximizes the integral. Taking first conditions in
such case may be confusing and requires some comments. For example, given
the Lagrangian to the household's problem,\ 
\begin{equation*}
\mathcal{L}=[\int_{0}^{\infty }c_{n}(p)^{\frac{\sigma -1}{\sigma }%
}dG_{n}(p)]^{\frac{\sigma }{\sigma -1}}-\lambda (\int_{0}^{\infty
}pc_{n}(p)dG_{n}(p)-...),
\end{equation*}%
we can no longer take the pointwise derivative over $c_{n}(p)$ (pointwise
derivative of an integral over $c_{n}(p)$ for any fixed $p$). Such
derivative is zero!

So, to make any progress, we need to think about the problem more generally,
and instead define the underlying variation (change) as an integrable
function of the price: $dc_{n}(p)$. Then, the necessary condition for $%
c_{n}(p)$ to solve the utility maximization problem would clearly be that
for such variation (integrable function) $dc_{n}(p),$ the distorted policy
function $c_{n}(p)+\varepsilon dc_{n}(p)$ (where $\varepsilon $ is some real
number) achieves a local extremum at $\varepsilon =0$. Because $\varepsilon $
is a real number, the necessary condition\ for this extremum can be
calculated using the standard calculus methods. This trick allows us to
translate the problem to a standard one.

So, given the Lagrangian,%
\begin{eqnarray*}
\mathcal{L}_{\varepsilon } &=&[\int_{0}^{\infty }\left[ c_{n}(p)+\varepsilon
dc_{n}(p)\right] ^{\frac{\sigma -1}{\sigma }}dG_{n}(p)]^{\frac{\sigma }{%
\sigma -1}} \\
&&-\lambda (\int_{0}^{\infty }p\left[ c_{n}(p)+\varepsilon dc_{n}(p)\right]
dG_{n}(p)-...),
\end{eqnarray*}%
we take the derivative wrt $\varepsilon $ and evaluate it at $\varepsilon
=0, $ to obtain:\ \ 
\begin{gather*}
\frac{d\mathcal{L}_{\varepsilon }}{d\varepsilon }|_{\varepsilon =0}\mathcal{=%
}[\int_{0}^{\infty }c_{n}(p)^{\frac{\sigma -1}{\sigma }}dG_{n}(p)]^{\frac{1}{%
\sigma -1}}\int_{0}^{\infty }c_{n}(p)^{\frac{-1}{\sigma }}dc_{n}(p)dG_{n}(p)
\\
-\lambda (\int_{0}^{\infty }pdc_{n}(p)dG_{n}(p))=0.
\end{gather*}%
Since the function $dc_{n}(p)$ is any arbitrary integrable function, we
observe that the above condition is equivalent to:

\begin{equation*}
\lbrack \int_{0}^{\infty }c_{n}(p)^{\frac{\sigma -1}{\sigma }}dG_{n}(p)]^{%
\frac{1}{\sigma -1}}c_{n}(p)^{\frac{-1}{\sigma }}=p,\text{ a.s.,}
\end{equation*}%
where `a.s.' symbol means that the relationship holds almost surely (for
almost all $p$ except sets of measure zero wrt to the measure induced by $%
G_{n}$). This is the first order condition we are looking for.

Note that the first order condition we have derived is analogous to the
first order condition we would have obtained, had we approximated the
integrals by summations. In the future, we will use this observation to
derive the first order condition quickly.

\begin{exercise}
\label{Exercise7}Given the above approach to solve the HH's problem that
involves and integral, derive the formula for the price index $P_{n}$ stated
above.
\end{exercise}

\subsubsection*{Firms}

In each country, the efficiency $z$ of producing good $\omega $ is assumed
to be Frechet distributed random variable (see (\ref{1Eq0})). Given the
realization of efficiency $z$ for a particular good type, the production
function takes the form%
\begin{equation}
y=zAl^{\beta }(\int_{0}^{\infty }q(p)^{\frac{\sigma -1}{\sigma }}dG_{n}(p))^{%
\frac{(1-\beta )\sigma }{\sigma -1}},  \label{1Eq2}
\end{equation}%
where $l$ is labor input, $q(\cdot )$ denotes intermediate inputs, and $A$
denotes a constant.

Competitive firms can use the above CRS\ technology\footnote{%
All firms will be subject to the same $z.$} to produce goods, and upon
paying the iceberg transportation cost, they can ship these goods to all
other countries in the world. Because production of the good is assumed to
be constant returns to scale, the number of firms is undetermined. The goal
of what follows is to impose conditions on the distribution of prices $%
G_{n}, $ so that it is consistent with the described above competitive
supply-side structure of the model.

Because of the constant returns to scale assumption, the marginal cost (or
per unit cost) $v_{ni}(\omega )$ of producing good $\omega $ in country $i$
for country $n$ is here a sufficient summary of the production process.
Conditional on the realization of $z$ for a given good, the marginal cost is
given by%
\begin{equation}
v_{ni}(z)=\frac{d_{ni}c_{i}}{z},  \label{1Eq3}
\end{equation}%
where 
\begin{equation}
c_{i}=w_{i}^{\beta }P_{i}^{1-\beta }  \label{1Eq4a}
\end{equation}%
is a per unit cost common across goods, and $A$ has been chosen to soak up
constants so that no constant appears in the formula $c_{i}=w_{i}^{\beta
}P_{i}^{1-\beta }.$

\begin{exercise}
\label{Exercise8}Derive the formula for $v_{ni}(z)$ stated above from the
underlying unit cost minimization problem, and calculate the value of $A$ so
that $c_{i}=w_{i}^{\beta }P_{i}^{1-\beta }.$
\end{exercise}

Now, letting $\mathcal{V}_{ni}$ denote the random variable describing the 
\textit{marginal cost of producing good } $\omega $ \textit{\ in country } $i$ 
\textit{\ for country }$n$, and letting $\mathcal{Z}_{i}$ denote the
underlying efficiency draw (in country $i$), the distribution of the random
variable $\mathcal{V}_{ni}$ can be derived as follows:\ \ \ 
\begin{eqnarray}
\Pr (\mathcal{V}_{ni} &\leq &v_{ni})=\Pr (\frac{d_{ni}c_{i}}{\mathcal{Z}_{i}}%
\leq v_{i})=  \label{1Eq5} \\
&=&\Pr (\mathcal{Z}_{i}\geq \frac{d_{ni}c_{i}}{v_{ni}})=1-F(\frac{d_{ni}c_{i}%
}{v_{ni}}).  \notag
\end{eqnarray}

The lowest price $\mathcal{P}_{n}$ of good $\omega $ from all possible
sources is a random variable linked to $\mathcal{V}_{ni}$, 
\begin{equation*}
\mathcal{P}_{n}=\min_{i=1..N}\{\mathcal{V}_{ni}\}.
\end{equation*}%
Using analogous steps to the proof of Property 2, we establish that the
distribution of the random variable $\mathcal{P}_{n}$ is given by 
\begin{equation}
G_{n}(p)=1-e^{-\Phi _{n}p^{\theta }},  \label{1Eq6}
\end{equation}%
where 
\begin{equation}
\Phi _{n}\equiv \sum_{i}T(c_{i}d_{ni})^{-\theta }.  \label{1Eq6a}
\end{equation}

\begin{exercise}
\label{Exercise9}Derive the formula for $G_{n}(p)$ stated above.
\end{exercise}

Finally, we analytically calculate the aggregate price index using (\ref%
{1Eq4a}). Plugging in the distribution function, we obtain\ 
\begin{eqnarray*}
P_{n}^{1-\sigma } &=&\int_{0}^{\infty }p^{1-\sigma }\Phi _{n}\theta
p^{\theta -1}e^{-\Phi _{n}p^{\theta }}dp \\
&=&\int_{0}^{\infty }\Phi _{n}p^{\theta -\sigma }e^{-\Phi _{n}p^{\theta }}dp.
\end{eqnarray*}%
Using substitution:%
\begin{equation*}
\Phi _{n}p^{\theta }=u;\text{ }dp=\frac{p^{1-\theta }}{\theta \Phi _{n}}du;%
\text{ }p=(\frac{u}{\Phi _{n}})^{\frac{1}{\theta }}
\end{equation*}%
we obtain%
\begin{equation*}
P_{n}^{1-\sigma }=\int_{0}^{\infty }(\frac{u}{\Phi _{n}})^{\frac{1-\sigma }{%
\theta }}e^{-u}du.
\end{equation*}%
From the definition of the gamma function $\Gamma $%
\begin{equation*}
\Gamma (z)\equiv \int_{0}^{\infty }t^{z-1}e^{-t}dt
\end{equation*}%
we get 
\begin{eqnarray*}
P_{n}^{1-\sigma } &=&(\Phi _{n}^{-\frac{1}{\theta }})^{1-\sigma
}\int_{0}^{\infty }u^{\frac{1-\sigma }{\theta }}e^{-u}du \\
&=&(\Phi _{n}^{-\frac{1}{\theta }})^{1-\sigma }\Gamma (\frac{\theta
+1-\sigma }{\theta }),
\end{eqnarray*}%
and%
\begin{equation}
P_{n}=\Phi _{n}^{-\frac{1}{\theta }}\left[ \Gamma (\frac{\theta +1-\sigma }{%
\theta })\right] ^{\frac{1}{1-\sigma }}=\gamma \Phi _{n}^{-\frac{1}{\theta }%
}.  \label{1Eq1a}
\end{equation}%
where $\gamma =\left[ \Gamma (\frac{\theta +1-\sigma }{\theta })\right] ^{%
\frac{1}{1-\sigma }}.$ (We need to assume $\theta +1-\sigma >0$; otherwise
the above integral is not be well defined.)

\begin{exercise}
\label{Exercise10}Show that the mean of the Frechet distribution is $T^{%
\frac{1}{\theta }}\Gamma (1-\frac{1}{\theta }).$
\end{exercise}

\subsubsection*{Market Clearing and Feasibility}

The aggregate resource constraint says that demand for labor in every
country equals the supply of labor in that country. This condition is
difficult to state because we miss the link between the realized
productivity $z$, type of good $\omega ,$ and the realized market price for
this good $p$.

To work around this problem, we instead note that since production function
is Cobb-Douglas, fraction $\beta $ of the expenditures of the entire world
on home goods is equal to the total compensation of labor producing these
goods (=compensation of labor in country $i$), and ($1-\beta $) fraction is
equal to payments to intermediate goods:%
\begin{eqnarray}
\text{(payments to labor) }w_{i}L_{i} &=&\beta \sum_{n}(\frac{X_{ni}}{X_{n}}%
)X_{n}.  \label{1Eq12} \\
\text{(payments for intermediate goods)} &\text{=}&(1-\beta )\sum_{n}(\frac{%
X_{ni}}{X_{n}})X_{n}  \notag
\end{eqnarray}%
Next, we note that by definition the total expenditures of country $n,$ $%
X_{n},$ are the total final expenditures of consumers on all goods, which
equal $w_{i}L_{i},$\textit{\ plus }total expenditures of home producers on
all intermediate goods (home and foreign). Thus, by equation (\ref{1Eq12}),
we have: 
\begin{eqnarray*}
X_{n} &=&w_{i}L_{i}+(1-\beta )\sum_{n}(\frac{X_{ni}}{X_{n}})X_{n} \\
&=&w_{i}L_{i}+\frac{(1-\beta )}{\beta }w_{i}L_{i} \\
&=&\frac{w_{i}L_{i}}{\beta }.
\end{eqnarray*}%
Finally, using the above, we derive:%
\begin{equation*}
L_{i}=\frac{w_{i}L_{i}}{w_{i}}=\frac{\beta \sum_{n}(\frac{X_{ni}}{X_{n}}%
)X_{n}}{w_{i}}=\frac{\sum_{n}(\frac{X_{ni}}{X_{n}})w_{n}L_{n}}{w_{i}}.
\end{equation*}

The above condition is not yet sufficient to define equilibrium because it
involves an endogenous term $\frac{X_{ni}}{X_{n}}$ that needs to be linked
to other equilibrium objects. The following lemma comes handy to fill this
gap:

\begin{lemma}
\begin{equation}
\frac{X_{ni}}{X_{n}}=\pi _{ni}=\frac{T_{i}\left( c_{i}d_{ni}\right)
^{-\theta }}{\sum_{i}T_{i}(c_{i}d_{ni})^{-\theta }},  \label{1Eq7}
\end{equation}
where $\pi _{ni}$ is the probability that the goods offered by country $i$
to country $n$ has the lowest price (and is sold to country $n$).
\end{lemma}

\begin{proof}
$\pi _{ni}$ can be calculated analogously to the proof of Property 2. Note
that 
\begin{equation*}
\pi _{ni}=\Pr (\mathcal{P}_{ni}\leq \min_{s\neq i}\mathcal{P}_{ns}).
\end{equation*}%
To prove $\frac{X_{ni}}{X_{n}}=\pi _{ni},$ we need to know the distribution
of the price of a good conditional on country $i$ selling this good in
country $n.$ It turns out that this distribution is independent from the
source, and is equal to the unconditional distribution $G_{n}.$ The result
is a consequence of Property 3, and the derivation of this fact is left as
and exercise (follows the proof of Property 2). The conditional distribution
of prices $\mathcal{G}_{ni}(p)$ is defined as follows:\ 
\begin{equation}
\mathcal{G}_{ni}(p)\equiv \frac{\int_{0}^{p}\Pi _{s\neq
i}[1-G_{ns}(q)]dG_{ni}(q)}{\pi _{ni}}  \label{1Eq8}
\end{equation}
This is enough to prove $\frac{X_{ni}}{X_{n}}=\pi _{ni}$. (Why?)
\end{proof}

By the above lemma, the final labor market clearing condition is thus given
by: 
\begin{equation}
L_{i}=\frac{\sum_{n}\frac{T_{i}\left( c_{i}d_{ni}\right) ^{-\theta }}{%
\sum_{i}T(c_{i}d_{ni})^{-\theta }}w_{n}L_{n}}{w_{i}}.  \label{1Eq9}
\end{equation}

\begin{exercise}
\label{Exercise11}Prove that $\mathcal{G}_{ni}(p)=G_{n}(p)$ and derive $\pi
_{ni}.$
\end{exercise}

\begin{exercise}
\label{Exercise12}Assume that there is a competitive sector that produces
non-tradable service goods using the following production function: 
\begin{equation*}
y=Al^{\beta }(\int_{0}^{\infty }q(p)^{\frac{\sigma -1}{\sigma }}dG_{n}(p))^{%
\frac{(1-\beta )\sigma }{\sigma -1}},
\end{equation*}%
and assume that the utility function of the household is Cobb-Douglas in
tradable and non-tradable components, i.e.:%
\begin{equation*}
U_{n}=C_{n}^{\alpha }[\int_{0}^{\infty }c_{n}(p)^{\frac{\sigma -1}{\sigma }%
}dG_{n}(p)]^{\frac{(1-\alpha )\sigma }{\sigma -1}},
\end{equation*}%
where $\alpha $ is the share of non-tradable goods, and $C_{n}$ is
consumption of the non-tradable good in country $n.$ Furthermore, assume
that labor is perfectly mobile across the two sectors producing tradable and
non-tradable good. Under this modification, derive the modified labor market
clearing condition. HINT: The formula is in the Eaton and Kortum paper. You
are asked to to derive it.
\end{exercise}

\subsection*{Equilibrium}

Having laid out the economy, we next define the equilibrium.

\begin{definition}
\textbf{Competitive equilibrium} in this economy is:\ 

\begin{itemize}
\item wages $(w_{n})_{i=1..N}$ and aggregate prices $(P_{n})_{n=1..N}$,
\end{itemize}

such that

\begin{itemize}
\item given (\ref{1Eq4a}) and (\ref{1Eq6}), $(w_{n})_{i=1..N}$ and $%
(P_{n})_{n=1..N}$ $\ $are consistent with (\ref{1Eq9}) and (\ref{1Eq1a}).
\end{itemize}
\end{definition}

Given the equilibrium wage vector and aggregate price vector, all other
equilibrium objects can be calculated from maximization of (\ref{Eq1})
subject to (\ref{1Eq1a}), and equations (\ref{1Eq7}) and (\ref{1Eq8}).

\subsection*{Computation, Existence and Uniqueness of Equilibrium}

As usually, the definition of equilibrium defines a fixed point problem. The
proof of uniqueness and existence thus requires to show that the fixed point
exists and is unique. For details, see Alvarez and Lucas (2007).

Similarly to the Armington model, we can construct here an iterative
numerical algorithm to solve for equilibrium. The sketch of the numerical
algorithm would be as follows:\ (i) Guess wages $w_{n}$ and aggregate prices 
$P_{n}$, (ii) Using (\ref{1Eq4a}), solve for wages from (\ref{1Eq9}) and
aggregate prices from (\ref{1Eq1a}), (iii) Iterate until convergence.%
\footnote{%
Use sluggish updating rule if neccessary: update = $\lambda $ $\times $ (new
value) + (1-$\lambda )$ $\times $ (old value).}.

\begin{exercise}
\label{Exercise13}Write a MATLAB\ code that mimics the setup from exercise
6, and solves the EK model numerically.
\end{exercise}

\subsection*{Predictions for Trade}

The following proposition summarizes the key predictions of the EK model for
trade. Interestingly enough, the model turns out to have isomorphic
predictions to the Armington model. The gravity equation is governed by
different parameters of the model, but it has exactly the same analytical
form after we relabel the parameters. Below, we study why and under which
conditions this is the case.

\begin{proposition}
In the Eaton-Kortum model, bilateral trade flows are governed by the
following gravity equation\ 
\begin{equation}
\frac{X_{ni}}{X_{n}}=X_{i}\frac{(\frac{d_{ni}}{P_{n}})^{-\theta }}{\sum_{n}(%
\frac{d_{ni}}{P_{n}})^{-\theta }X_{n}}.  \label{1Eq10}
\end{equation}%
In particular, when iceberg transportation cost is symmetric, i.e. $%
d_{ni}=d_{in},$ all $i,n=1$,..,$N,$ the gravity equation is given by 
\begin{equation}
X_{ni}=\frac{X_{n}X_{i}}{\sum_{n}X_{n}}(\frac{d_{ni}}{P_{i}P_{n}})^{-\theta
}.
\end{equation}%
\bigskip
\end{proposition}

\begin{proof}
Using equation (\ref{1Eq7}) and equation\ (\ref{1Eq1a}), the balanced trade
condition implies:\ \ 
\begin{equation*}
X_{i}=\sum_{n}X_{ni}=\sum_{n}\frac{X_{ni}}{X_{n}}X_{n}=T_{i}c_{i}^{-\theta
}\sum_{n}\frac{d_{ni}^{-\theta }X_{n}}{\Phi _{n}}=T_{i}c_{i}^{-\theta
}\sum_{n}(\frac{\gamma d_{ni}}{p_{n}})^{-\theta }X_{n},
\end{equation*}%
and thus 
\begin{equation*}
T_{i}c_{i}^{-\theta }=\frac{X_{i}}{\sum_{n}^{N}(\frac{\gamma d_{ni}}{p_{n}}%
)^{-\theta }X_{n}}.
\end{equation*}

Combining the above with (\ref{1Eq7}) and the definition of $\Phi _{n}$ (\ref%
{1Eq6a}), we derive:%
\begin{equation*}
\frac{X_{ni}}{X_{n}}=\frac{T_{i}(c_{i}d_{ni})^{-\theta }}{%
\sum_{i}T_{i}(c_{i}d_{ni})^{-\theta }}=\frac{\frac{1}{\sum_{n}(\frac{d_{ni}}{%
P_{n}})^{-\theta }X_{n}}(d_{ni})^{-\theta }}{P_{n}^{-\theta }}X_{i}=\frac{(%
\frac{d_{ni}}{P_{n}})^{-\theta }}{\sum_{n}(\frac{d_{ni}}{P_{n}})^{-\theta
}X_{n}}X_{i}.
\end{equation*}

Similarly to the Armington model, we can derive that 
\begin{eqnarray*}
P_{n} &=&\gamma (\sum_{i}T(c_{i}d_{ni})^{-\theta })^{-\frac{1}{\theta }} \\
P_{n} &=&(\sum_{i}\frac{(\gamma d_{ni})^{-\theta }}{\sum_{n}^{N}(\frac{%
\gamma d_{ni}}{P_{n}})^{-\theta }X_{n}}X_{i})^{-\frac{1}{\theta }}
\end{eqnarray*}%
and so%
\begin{equation*}
P_{n}=(\sum_{i}\left( \frac{\gamma d_{ni}}{\Theta _{i}}\right) ^{-\theta
}X_{i})^{-\frac{1}{\theta }}
\end{equation*}%
where 
\begin{equation*}
\Theta _{i}=(\sum_{n}\left( \frac{\gamma d_{ni}}{P_{n}}\right) ^{-\theta
}X_{n})^{-\frac{1}{\theta }}.
\end{equation*}%
Thus, in the special case of $d_{ni}=d_{in},$ we have%
\begin{equation*}
P_{n}(\sum_{n}X_{n})^{-\frac{1}{\theta }}=\Theta _{n},
\end{equation*}%
and 
\begin{equation*}
X_{ni}=\frac{X_{n}X_{i}}{\sum_{n}X_{n}}(\frac{d_{ni}}{P_{i}P_{n}})^{-\theta
}.
\end{equation*}%
\bigskip
\end{proof}

\subsection*{Discussion}

Even though equation (\ref{1Eq10}) is structurally identical to the one in
the Armington model, it is not the same. In the Armington model, the
critical parameter governing trade was the elasticity of substitution $%
\sigma $ between domestic and the foreign good. In contrast, in the EK
model, $\sigma $ is irrelevant, and the critical parameter is the dispersion
of technologies $\theta $. As a result, even though the model has the same
implications for trade, different assumptions on the physical environment
give rise to these predictions.\footnote{%
The important lesson is that the predictions of the Armington model and the
EK model can be reconciled once we think of $\sigma $ in the Armington model
as capturing also the substitution occurring at the production level. Under
such interpretation, EK model shows that such effect is important, but on
the aggregate level can be directly mapped onto elasticity without much loss
of generality.} Because of this peculiar isomorphism, the Armington model
can be thought of as a reduced form representation of the EK model.

The reason why dispersion $\theta $ turns out critical for trade in the EK
model can be understood as follows. When there is little dispersion in
productivities across goods, prices presented by all alternative sources of
country $n$ are very close to the current cheapest source $i$, and thus
country $n$ almost immediately switches to an alternative source when the
cheapest source becomes more expensive due to an increase in $d_{ni}.$ As a
result, trade flows between country $n$ and country $i$ fall drastically in
response to $d_{ni},$ and this sensitivity falls as the dispersion of
productivity and prices rises. 

The above reasoning explains why $\theta $ is important in the EK\ model,
but it does not explain why $\sigma $ does not matter at all (does not show
up in the formula). We will tackle this problem in the next paragraph by
considering a two-country version of the model.

\subsection*{Two-Country Case}

In this section, we simplify the model and assume that there are two
symmetric countries, i.e. $T_{1}=T_{2}=1,$ $d_{12}=d_{21}=d,$ $%
d_{11}=d_{22}=0,$ $\beta =1$ (labor is the only production factor).  The
goal is to understand better the workings of the EK model. The two country
case is much easier to analyze intuitively because it is possible to map the
probabilistic EK formulation onto the deterministic efficiency schedules a
la Dornbusch, Fisher and Samuelson (1977). 

To obtain the DFS ordering, we calculate the probability that the relative
productivity is smaller than some cutoff value $a$ and assume the
probability of this event to be the index value of the pivotal good ---
since goods are defined on a [0,1] interval, this probability does define a
total order\footnote{%
If X is totally ordered under $\leq $, then the following statements hold
for all a, b and c in X: (1) If a $\leq $ b and b $\leq $ a then a = b
(antisymmetry); (2) If a $\leq $ b and b $\leq $ c then a $\leq $ c
(transitivity); (3) a $\leq $ b or b $\leq $ a (totality). } in a
mathematical sense. Formally, a good index $\omega $ is defined by the
following relation: \ \ 
\begin{equation*}
\omega (a)\equiv \Pr (\frac{\mathcal{Z}_{2}}{\mathcal{Z}_{1}}\leq a)=\Pr (%
\mathcal{Z}_{2}\leq a\mathcal{Z}_{1}),
\end{equation*}%
which solves to%
\begin{equation*}
\omega (a)=\frac{1}{1+a^{-\theta }}.
\end{equation*}

Inverting the above expression, we obtain the mapping from the space of
indices to the corresponding relative productivities:%
\begin{equation*}
a(\omega )=(\frac{1-\omega }{\omega })^{-\frac{1}{\theta }}.
\end{equation*}

\begin{exercise}
\label{Exercise14}Formally derive the above expression for $\omega (a)$.
\end{exercise}

Given the $a(\omega )$ schedule, we next recover the underlying absolute
productivity schedules. From symmetry, consistency with the expression for $%
a(\omega )$ requires that:%
\begin{eqnarray*}
z_{1}(\omega ) &=&A\omega ^{-\frac{1}{\theta }}, \\
z_{2}(\omega ) &=&A(1-\omega )^{-\frac{1}{\theta }},
\end{eqnarray*}%
where $A$ is some constant of proportionality that we can calculate
explicitly\footnote{%
The mean of the Frechet distribution is $\mu =T^{\frac{1}{\theta }}\Gamma (1-%
\frac{1}{\theta })=\Gamma (1-\frac{1}{\theta }).$ Thus, we can calculate $A$
from $\int_{[0,1]}A\omega ^{-\frac{1}{\theta }}d\omega =\mu .$ Integrating,
we obtain $[A\frac{\theta }{\theta -1}\omega ^{\frac{\theta -1}{\theta }%
}]_{[0,1]}=A\frac{\theta }{\theta -1},$ and so $A=\frac{\theta -1}{\theta }%
\Gamma (1-\frac{1}{\theta }).$}.

Finally, given productivity schedules $z_{i}(\omega )$, we normalize wages
of country 1 and 2 to 1 (by numeraire assumption and symmetry), and derive
the competitive price schedules $p_{ni}(\omega )$ of each good $\omega $
presented by country $i$ to country $n$ (=marginal cost):%
\begin{eqnarray*}
p_{11}(\omega ) &=&\frac{\omega ^{\frac{1}{\theta }}}{A}, \\
p_{21}(\omega ) &=&d\frac{\omega ^{\frac{1}{\theta }}}{A}, \\
p_{22}(\omega ) &=&\frac{(1-\omega )^{\frac{1}{\theta }}}{A}, \\
p_{12}(\omega ) &=&d\frac{(1-\omega )^{\frac{1}{\theta }}}{A},
\end{eqnarray*}%
where $A=\frac{\theta -1}{\theta }\Gamma (1-\frac{1}{\theta })$.

In the EK model, the goods are always produced by the lowest cost supplier.
So, it must be true that country 1 is the sole producer of all the goods for
which $p_{21}(\omega )\geq p_{22}(\omega ),$ and country 2 is the sole
producer of goods for which $p_{12}(\omega )\geq p_{11}(\omega )$. All
remaining goods are produced by both countries, and are \textit{not} traded
internationally. Figure \ref{Fig3} illustrates the obtained this way ranges
of goods.

\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=8,scale=.44]{Figures.pdf} 
 \caption{Price schedules and
specialization pattern in the EK model.}%
\label{Fig3}
\end{figure}

%\FRAME{ftbpFU}{326.3333pt}{223.6667pt}{0pt}{\Qcb{Price schedules and
%specialization pattern in the EK model. }}{\Qlb{Fig3}}{Figure}{\special%
%{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
%TRUE;display "USEDEF";valid_file "T";width 326.3333pt;height
%223.6667pt;depth 0pt;original-width 731.3333pt;original-height
%500.0667pt;cropleft "0";croptop "1";cropright "1";cropbottom
%"0";tempfilename 'KD2WCH02.wmf';tempfile-properties "XPR";}}

As we can see from Figure 3, in order to fully characterize the
specialization pattern, we need to calculate two cutoff values. The first
cutoff determines the range of goods that are produced by country 1, and the
second cutoff determines the range of goods that are produced by country 2.
The goods which are between these cutoffs are produced by both countries,
but for the home market only. Using the formulas for prices, we can
calculate these cutoff values as follows\footnote{%
Note that $\frac{d^{-\theta }}{(1+d^{-\theta })}<\frac{1}{(1+d^{-\theta })}$
and so the range of goods that are not traded is non-empty as long as $d>1.$
The endogenous set of goods that are actually traded is one of the key
differences between the EK model and the Armington model.}:\ 
\begin{eqnarray*}
p_{21}(\bar{\omega}_{1}) &=&p_{22}(\bar{\omega}_{1}), \\
\bar{\omega}_{1} &=&\frac{d^{-\theta }}{1+d^{-\theta }}\text{ (first cutoff),%
}
\end{eqnarray*}%
\begin{eqnarray*}
p_{12}(\bar{\omega}_{2}) &=&p_{11}(\bar{\omega}_{2}), \\
\bar{\omega}_{2} &=&\frac{1}{1+d^{-\theta }}\text{ (second cutoff).}
\end{eqnarray*}

To better understand how the EK model differs from the Armington model
(Figure 4), it is instructive to derive the share of expenditures of country
1 on country 2 goods for both models. As illustrated in Figure 4, we can map
the Armington model onto the DFS framework by dividing the space of goods
into two equal parts . On each non-overlapping half of the $\omega $ domain
both countries present a finite and identical price $p$ at home and $pd$
abroad, and on the other half they present an infinite price. Clearly,
because the prices are constant on each interval, once we integrate over all
goods, we obtain this way an Armington model with two representative goods.
Let's now calculate the expenditure shares for each model.


\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=9,scale=.44]{Figures.pdf} 
 \caption{Price schedules and specialization
in the Armington model.}%
\label{Fig4}
\end{figure}

%\FRAME{ftbpFU}{337.2pt}{231pt}{0pt}{\Qcb{Price schedules and specialization
%in the Armington model.}}{\Qlb{Fig4}}{Figure}{\special{language "Scientific
%Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file
%"T";width 337.2pt;height 231pt;depth 0pt;original-width
%731.3333pt;original-height 500.0667pt;cropleft "0";croptop "1";cropright
%"1";cropbottom "0";tempfilename 'KD2WCH03.wmf';tempfile-properties "XPR";}}

In the Armington model, the expenditures on each good from the continuum are
given by 
\begin{equation}
x(\omega )=X_{1}(\frac{p(\omega )}{P})^{1-\sigma }.  \label{1Eq11}
\end{equation}%
If we normalize the wages in both countries to 1, by symmetry we have $%
p_{11}(\omega )=1,\ p_{21}(\omega )=d$, and so the price index is given by%
\begin{eqnarray*}
P_{1} &=&(\int_{0}^{1/2}1^{1-\sigma }+\int_{1/2}^{1}d^{1-\sigma })^{\frac{1}{%
1-\sigma }}= \\
&=&(\frac{1}{2})^{\frac{1}{1-\sigma }}(1+d^{1-\sigma })^{\frac{1}{1-\sigma }%
}.
\end{eqnarray*}%
Further normalizing the endowment vector $(L_{n})$ to 1, we can calculate
the share of expenditures of country 1 on the goods produced by country 2 by
integrating the demand function as follows\footnote{%
Notice that the result we derive below is consistent with the gravity
equation derived for the general case (see previous section).}:\ 
\begin{eqnarray*}
\frac{X_{12}}{X_{1}} &=&\frac{1}{P^{1-\sigma }}(\int_{1/2}^{1}d^{1-\sigma })=
\\
&=&\frac{1}{\frac{1}{2}(1+d^{1-\sigma })}\frac{1}{2}d^{1-\sigma }= \\
&=&\frac{d^{1-\sigma }}{1+d^{1-\sigma }}
\end{eqnarray*}

As we can see, in consistency with our earlier results, the parameter that
determines the effect of the iceberg transportation cost on trade in the
Armington model is the elasticity of substitution $\sigma $. Precisely, the
more substitutable the goods are, the more the share of expenditures $\frac{%
X_{12}}{X_{1}}$ is affected by the change in $d,$ as indicated by the
derivative evaluated at $d=1$:%
\begin{equation*}
\frac{d(\frac{X_{12}}{X_{1}})}{dd}=(1-\sigma ).
\end{equation*}

We next derive an analogous expression for the EK model. The calculations
are more complicated because we have to integrate over the price schedules
that aren't constant. In addition, the cutoffs determining the endogenous
set of imported goods depend on $d$.

Normalizing wages and endowments, the price index can be computed as follows:%
\begin{eqnarray*}
P_{1} &=&\left[ \int_{0}^{\frac{1}{1+d^{-\theta }}}p_{11}(\omega )^{1-\sigma
}d\omega +\int_{\frac{1}{1+d^{-\theta }}}^{1}p_{12}(\omega )^{1-\sigma
}d\omega \right] ^{\frac{1}{1-\sigma }}= \\
&=&\left[ \int_{0}^{\frac{1}{1+d^{-\theta }}}\left( \frac{\omega ^{\frac{1}{%
\theta }}}{A}\right) ^{1-\sigma }d\omega +\int_{\frac{1}{1+d^{-\theta }}%
}^{1}\left( d\frac{(1-\omega )^{\frac{1}{\theta }}}{A}\right) ^{1-\sigma
}d\omega \right] ^{\frac{1}{1-\sigma }}= \\
&=&\left[ (\ast )+(\ast \ast )\right] ^{\frac{1}{1-\sigma }}= \\
&=&\frac{(1-\sigma +\theta )^{\frac{1}{\sigma -1}}}{A\theta ^{\frac{1}{%
\sigma -1}}}\left[ (\frac{1}{1+d^{-\theta }})^{\frac{1-\sigma +\theta }{%
\theta }}+\frac{d^{-\theta }}{(1+d^{-\theta })^{\frac{1-\sigma +\theta }{%
\theta }}}\right] ^{\frac{1}{1-\sigma }}= \\
&=&A^{-1}(\frac{\theta }{1-\sigma +\theta })^{\frac{1}{1-\sigma }%
}(1+d^{-\theta })^{-\frac{1}{\theta }}
\end{eqnarray*}

where

\begin{eqnarray*}
(\ast ) &=&\int_{0}^{\frac{1}{1+d^{-\theta }}}\left( \frac{\omega ^{\frac{1}{%
\theta }}}{A}\right) ^{1-\sigma }d\omega = \\
&=&A^{\sigma -1}\int_{0}^{\frac{1}{1+d^{-\theta }}}\omega ^{\frac{1-\sigma }{%
\theta }}d\omega = \\
&=&A^{\sigma -1}[\frac{\theta }{1-\sigma +\theta }\omega ^{\frac{1-\sigma
+\theta }{\theta }}]_{0}^{\frac{1}{1+d^{-\theta }}}= \\
&=&\frac{A^{\sigma -1}\theta }{1-\sigma +\theta }(\frac{1}{1+d^{-\theta }})^{%
\frac{1-\sigma +\theta }{\theta }},
\end{eqnarray*}

\begin{eqnarray*}
(\ast \ast ) &=&\int_{\frac{1}{1+d^{-\theta }}}^{1}\left( d\frac{(1-\omega
)^{\frac{1}{\theta }}}{A}\right) ^{1-\sigma }d\omega = \\
&=&A^{\sigma -1}d^{1-\sigma }\int_{\frac{1}{1+d^{-\theta }}}^{1}(1-\omega )^{%
\frac{1-\sigma }{\theta }}d\omega = \\
&=&A^{\sigma -1}d^{1-\sigma }[-\frac{\theta }{1-\sigma +\theta }(1-\omega )^{%
\frac{1-\sigma +\theta }{\theta }}]_{\frac{1}{1+d^{-\theta }}}^{1}= \\
&=&\frac{A^{\sigma -1}\theta }{1-\sigma +\theta }\frac{d^{-\theta }}{%
(1+d^{-\theta })^{\frac{1-\sigma +\theta }{\theta }}}.
\end{eqnarray*}

Given the price index, the expenditure share is given by:

\begin{equation*}
\frac{X_{12}}{X_{1}}=\frac{1}{P^{1-\sigma }}(\int_{\frac{1}{1+d^{-\theta }}%
}^{1}p_{12}(\omega )^{1-\sigma }d\omega )=\frac{d^{-\theta }}{1+d^{-\theta }}%
.
\end{equation*}

\begin{exercise}
\label{Exercise15}By integrating over the price schedule, explicitly derive
the expression for $\frac{X_{12}}{X_{1}}$ stated above. (NOTE:\ You are not
allowed to use the gravity equation.)
\end{exercise}

In consistency with our earlier findings, the result is that even though $%
\sigma $ is critical for the demand for each good (as indicated by formula
for $p_{ni}(\omega ))$, the expression for the share of expenditures on
imported goods turns out to be independent from $\sigma $.

To understand why this happens intuitively, we next shut down the extensive
marginal (cutoff effect), and study the adjustment along the intensive
margin (how much of each good to purchase). Because $\sigma $ shows up in
the expression for the demand, our conjecture is that along the intensive
margin $\sigma $ is critical in a similar fashion as in the Armington model,
but it must be the extensive margin of changing cutoff values that offsets
it. Our goal is to formally confirm this intuition. 

To this end, we will start with the fixed cutoff values for the case $d=1$
(both cutoffs at $\frac{1}{2}),$ and increase $d$ by some $\Delta d$ while
keeping the cutoffs unchanged at $\frac{1}{2}$. As expected, we in fact
obtain an analogous expression to the Armington model: 
\begin{equation*}
\frac{X_{12}}{X_{1}}=\frac{\Delta d^{1-\sigma }}{1+\Delta d^{1-\sigma }}.
\end{equation*}%
This confirms our intuition that it is the offsetting effect of the
extensive margin that makes $\sigma $ drop out from the final expression,
and the extensive margin transforms the gravity equation to an isomorphic
form in which $\sigma $ plays no role. In a sense, the extensive margin
depends on $\sigma $ in an exactly offsetting effect so that in the overall $%
\sigma $ drops out. Clearly, this result hinges on the specific functional
forms for productivity distribution, and is not a general feature of the DFS
model. Nevertheless, it is comforting that in some plausible class of
parameterization Armington model can be thought of as the reduced form of
the EK model. 

\begin{exercise}
\label{Exercise16}Derive the above expression $\frac{X_{12}}{X_{1}}=\frac{%
\Delta d^{1-\sigma }}{1+\Delta d^{1-\sigma }}.$ HINT: Do not forget that you
also need to calculate the counterfactual price index $P$ corresponding
fixed cutoff values.
\end{exercise}

Last but not least, we should mention that Eaton and Kortum estimate the
parameter $\theta $ from the data on the dispersion of retail prices of
commodities across countries. Their result $\theta \simeq 8$ is consistent
with the high value of the elasticity parameter $\sigma $ that we needed in
the Armington model to account for trade patterns between Canada and US.

\section*{Endogenizing the Frechet Distribution}

The parameterization of the model using the Frechet distribution holds a
promise for a tractable integration of trade theory with the theory of
innovation and growth. Kortum (1997) and Eaton and Kortum (1999) show how a
process of innovation and diffusion can endogenously give rise to a Frechet
distribution, where $T_{i}$ reflects a country's stock of original (or
imported ideas). The appendix at the end provides necessary probability
theory background for this part.

\subsection*{Model of Innovation}

Time is continuous and an idea is a technique to produce a certain good. The
arrival of new ideas is governed by a Poisson process with time-varying
intensity $aR(t),$ where we interpret $R(t)$ as the flows of R\&D
expenditures at time $t$ and $a$ is productivity of research. $R(t)$ is
exogenous, but can, in principle, be endogenized\footnote{%
See for example Kortum and Klette (2004) and Eaton and Kortum (2001).}.

A technique (=idea) is summarized by a number $z,$ characterizing the labor
requirement to produce 1 unit of output. It is assumed that upon the arrival
of an idea, labor requirement $Z$ associated with this idea is a random
variable drawn from a Pareto distribution given by: 
\begin{equation*}
F(z)\equiv P(Z>z)=\left[ 
\begin{array}{c}
(\frac{z}{\bar{z}})^{-\theta }\text{ if }z\geq \bar{z} \\ 
1\text{ otherwise}%
\end{array}%
\right]
\end{equation*}%
where $\bar{z}$ and $\theta $ are parameters of the distribution.

Given that the arrival process is Poisson, the number of ideas of
productivity $Z>z$ discovered up to date $t$ is distributed Poisson with
parameter (see Appendix)%
\begin{equation*}
\lambda (t)=aT(t)(\frac{z}{\bar{z}})^{-\theta },\text{ }z\geq \bar{z}
\end{equation*}%
where $T$ denotes the cumulative research effort up to time $T$ 
\begin{equation*}
T(t)=\int_{0}^{t}aR(t)dt.
\end{equation*}%
In the part the follows, we will find it convenient to normalize $a$ so that 
$a\bar{z}^{\theta }=1,$ and think of $\bar{z}\rightarrow 0.$ This way we can
extend the domain, and have instead:%
\begin{equation*}
\lambda (t)=T(t)z^{-\theta },\text{ }z\in (0,\infty ).
\end{equation*}%
Finally, we note that Pareto distribution has a convenient property that the
conditional distribution on $Z\geq \hat{z},$ is also Pareto given by 
\begin{equation*}
F(z|\hat{z})\equiv P(Z>z)=(\frac{z}{\hat{z}})^{-\theta }.
\end{equation*}%
We will use this property in the proof below.

\subsubsection*{Distribution of the Best and the Second-Best Idea Up to Date}

Given all the techniques discovered up to period $t$, we can rank them from
the highest efficiency to the lowest, and define the underlying random
variables as follows: $Z^{1}\geq Z^{2}\geq ....$. Our goal is to
characterize the distribution of random variables $Z^{1}$ and $Z^{2}$ in
such ranking. For the EK model, it would be enough to characterize $Z^{1}$
only$.$ However, for later use we will derive a more general characterizing
the joint distribution of $Z^{1},Z^{2}.$

\begin{proposition}
The joint distribution function of the best and second best technique $Z^{1}$
and $Z^{2}$ is given by%
\begin{equation}
G(Z_{1}\leq z_{1},Z_{2}\leq z_{2})=(1+T(z_{2}^{-\theta }-z_{1}^{-\theta
}))e^{-Tz_{2}^{-\theta }},
\end{equation}%
where $z_{1}\geq z_{2}.$
\end{proposition}

From the above proposition, we note that the least restrictive condition on $%
z_{2}$ is that $z_{2}=z_{1}$ (as $z_{2}$ still has to be lower than $z_{1}$%
). Plugging in, we obtain the distribution function\ of the best draw that
we have used in the EK model: 
\begin{equation*}
e^{-Tz_{2}^{-\theta }}.
\end{equation*}%
We now turn to the proof of the above proposition.


\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=10,scale=.44]{Figures.pdf} 
 \caption{Decomposition of the distribution.}%
\label{Fig5}
\end{figure}

%\FRAME{ftbpFU}{364.2pt}{211.6pt}{0pt}{\Qcb{Decomposition of the distribution.%
%}}{\Qlb{Fig5}}{Figure}{\special{language "Scientific Word";type
%"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width
%364.2pt;height 211.6pt;depth 0pt;original-width 434pt;original-height
%251.2667pt;cropleft "0";croptop "1";cropright "1";cropbottom
%"0";tempfilename 'KD2WCH04.wmf';tempfile-properties "XPR";}}

\begin{proof}
(Courtesy of Yoichi Ueno -- who nicely simplified the original proof)
Suppose at period $t$ there are $n$ draws $(Z_{n})_{n}$ of $Z$ bounded by
some $\hat{z},$ with the following order$:$ $Z_{1}\geq Z_{2}\geq ....\geq
Z_{n}\geq \hat{z}.$ In the first step, conditional on $n$ draws and $%
Z_{1},...,Z_{n}\geq \hat{z},$ we will calculate the conditional probability
that $Z_{1}$ is above some cutoff value $z_{1}$ and $Z_{2}\ $is below some
cutoff value $z_{2}.$ Clearly, the conditional distribution of $Z$ is given
by $F(\zeta )=(\frac{z}{\hat{z}})^{-\theta }$, $z\geq \hat{z}.$ Since these
events are disjoint, the probability of the event described above is given
by the following Bernoulli trial formula:%
\begin{eqnarray*}
P(Z_{1} &\geq &z_{1},Z_{2}\leq z_{2}|n,\hat{z})=(%
\begin{array}{c}
n \\ 
1%
\end{array}%
)F(z_{1})(1-F(z_{2}))^{n-1}= \\
&=&n(\frac{z_{1}}{\hat{z}})^{-\theta }(1-(\frac{z_{2}}{\hat{z}})^{-\theta
})^{n-1}
\end{eqnarray*}%
Next, we note that since $n$ is distributed Poisson with parameter $T\hat{z}%
^{-\theta }$, the unconditional distribution must be given by 
\begin{eqnarray*}
P(Z_{1} &\geq &z_{1},Z_{2}\leq z_{2}|\hat{z})=\sum_{n=0}^{\infty }n(\frac{%
z_{1}}{\hat{z}})^{-\theta }(1-(\frac{z_{2}}{\hat{z}})^{-\theta })^{n-1}\frac{%
(T\hat{z}^{-\theta })^{n}}{n!}e^{-T\hat{z}^{-\theta }}= \\
&=&\sum_{n=1}^{\infty }(\frac{z_{1}}{\hat{z}})^{-\theta }(1-(\frac{z_{2}}{%
\hat{z}})^{-\theta })^{n-1}\frac{(T\hat{z}^{-\theta })^{n}}{(n-1)!}e^{-T\hat{%
z}^{-\theta }}.
\end{eqnarray*}%
Collecting terms to use (\ref{SPoisson}), we obtain:\ 
\begin{eqnarray*}
P(Z_{1} &\geq &z_{1},Z_{2}\leq z_{2}|\hat{z})=T\hat{z}^{-\theta }(\frac{z_{1}%
}{\hat{z}})^{-\theta }\sum_{n=1}^{\infty }\frac{(T\hat{z}^{-\theta
}-Tz_{2}^{-\theta })^{n-1}}{(n-1)!}e^{-T\hat{z}^{-\theta }}= \\
&=&T\hat{z}^{-\theta }(\frac{z_{1}}{\hat{z}})^{-\theta }e^{-Tz_{2}^{-\theta
}}\underset{=1}{\underbrace{\sum_{n=1}^{\infty }\frac{(T\hat{z}^{-\theta
}-Tz_{2}^{-\theta })^{n-1}}{(n-1)!}e^{-T(\hat{z}^{-\theta }-z_{2}^{-\theta
})}}}= \\
&=&T\hat{z}^{-\theta }(\frac{z_{1}}{\hat{z}})^{-\theta }e^{-Tz_{2}^{-\theta
}}=Tz_{1}^{-\theta }e^{-Tz_{1}^{-\theta }}.
\end{eqnarray*}%
Now, since the expression does not depend on $\hat{z},$ taking the limit $%
\hat{z}\rightarrow 0,\,$we obtain 
\begin{equation*}
P(Z_{1}\geq z_{1},Z_{2}\leq z_{2})=Tz_{1}^{-\theta }e^{-Tz_{2}^{-\theta }}.
\end{equation*}%
Similarly, we can derive:%
\begin{equation*}
P(Z_{1}\leq z_{2},Z_{2}\text{ anything)=}P(Z_{1}\leq z_{2},Z_{2}\leq z_{2}%
\text{)=}e^{-Tz_{2}^{-\theta }},\text{ }(\ast )
\end{equation*}%
Since by Figure \ref{Fig5}:%
\begin{eqnarray*}
P(Z_{1} &\leq &z_{1},Z_{2}\leq z_{2})=P(Z_{1}\leq z_{2},Z_{2}\leq z_{2})+ \\
P(Z_{1} &\geq &z_{2},Z_{2}\leq z_{2})-P(Z_{1}\geq z_{1},Z_{2}\leq z_{2}),
\end{eqnarray*}%
we have%
\begin{eqnarray*}
P(Z_{1} &\leq &z_{1},Z_{2}\leq z_{2})=e^{-Tz_{2}^{-\theta }}+Tz_{2}^{-\theta
}e^{-Tz_{2}^{-\theta }}-Tz_{1}^{-\theta }e^{-Tz_{2}^{-\theta }}= \\
&=&(1+T(z_{2}^{-\theta }-z_{1}^{-\theta }))e^{-Tz_{2}^{-\theta }}.
\end{eqnarray*}
\end{proof}

\begin{exercise}
(Optional) Derive the expression denoted by (*) in the proof above.
\end{exercise}

\subsection*{A Note on the Literature}

This part was based on   Eaton and Kortum (2002), as well as
the general equilibrium analysis of this model provided by Alvarez and Lucas
(2004). The material on link between the Frechet distribution and innovation
comes from the paper by Bernard, Eaton, Jensen and Kortum (2003) and Eaton
and Kortum textbook.

\section*{Appendix}

\subsection*{Poisson Process}

Poisson process is a jump process in the sense that a shock of a random
magnitude occurs at random times with same intensity (example: arrival of
phone calls or claims to a customer service office). The process $\pi (t)$
is Poisson with parameter $\lambda (t)$ (may depend on time or be constant)$%
, $ if it obeys the following probability conditions:

\noindent $(i)$ Pr(event occurs exactly once in time interval $(t,t+dt)$)=$%
\lambda (t)dt+o(dt),$

\noindent $(ii)$ Pr(event does not occur at all in time interval $(t,t+dt)$)=%
$1-\lambda (t)dt+o(dt),$

\noindent $(iii)$ Pr(event does occurs more than once in time interval $%
(t,t+dt)$)=$o(dt),$

\noindent where $dt$ is an infinitesimal time interval and $o(dt)$ refers to
terms sufficiently small relative to $dt\ $to be ignored (lim$%
_{dt\rightarrow 0}\frac{o(dt)}{dt}=0)$.

Conditions $(i)-(iii)$ imply the following distribution that the random
event occurs sometime before time $t$ is,%
\begin{equation*}
F(t)=1-e^{-\int_{0}^{t}\lambda (\tau )d\tau }.
\end{equation*}%
It can be derived as follows. Let $p(t)$ denote the probability that the
event does not occur up to time $t.$ Then, $p$ should obey the following
rule (by $i-iii$)%
\begin{equation*}
p(t+dt)=(1-\lambda (t)dt)p(t)+p(t)o(dt).
\end{equation*}%
This rule says that the probability that event does not occur to time $t+dt$
is given by the probability that it does not occur up to time $t$, $p(t),$
and does not occur in the interval ($t,t+dt$). Dividing both sides of the
above expression by $dt$ and taking limit $dt\rightarrow 0$, we obtain the
following differential equation%
\begin{equation*}
p^{\prime }(t)=-\lambda (t)p(t),
\end{equation*}%
which solves to 
\begin{equation*}
p(t)=e^{-\int_{0}^{t}\lambda (\tau )d\tau },
\end{equation*}%
given the initial condition $p(0)=1.$ Thus, the probability that event
occurs at least once sometime up to time $t$ is in fact given by%
\begin{equation*}
F(t)=1-e^{-\int_{0}^{t}\lambda (\tau )d\tau }.
\end{equation*}

A nice property of the Poisson process is that you can combine them
together. For example, if $X_{n}$ has Poisson distribution with parameter $%
\lambda _{n}(t)$, then $\sum_{n}^{N}X_{n}$ is also Poisson with parameter $%
\sum_{n}^{N}\lambda _{n}(t).$

\subsection*{Derivation of the Poisson Distribution}

Let the number of counts up to time $t$ be denoted by $N(t)$ (which is total
number of occurrence of certain "events" up to time $t),\,\ $e.g. arrival of
phone calls to a customer service center. The distribution of the number of
`counts' up to time $t$ is%
\begin{equation}
P(N=n;t)=\frac{(-\int_{0}^{t}\lambda (\tau )d\tau )^{n}}{n!}%
e^{-\int_{0}^{t}\lambda (\tau )d\tau }.  \label{Poisson}
\end{equation}%
(In particular, we note that because it is a probability distribution 
\begin{equation}
\sum_{n=0}^{\infty }\frac{(-\int_{0}^{t}\lambda (\tau )d\tau )^{n}}{n!}%
e^{-\int_{0}^{t}\lambda (t)dt}=1,  \label{SPoisson}
\end{equation}%
which is sometimes useful in the proofs.)

To derive the above, denote $p_{n}(t)=P(N=n;t),$ and note that by (i-iii)
the following recursive formula holds%
\begin{equation*}
p_{n}(t+dt)=p_{n}(t)(1-\lambda (t)dt-o(dt))+p_{n-1}(t)(\lambda (t)dt+o(dt)).
\end{equation*}%
Dividing by $dt$ and letting dt$\rightarrow 0$ (using $\frac{o(dt)}{dt}=0),$
we have%
\begin{equation*}
p_{n}^{\prime }(t)=p_{n}(t)(1-\lambda (t))+p_{n-1}(t)\lambda (t)dt.
\end{equation*}%
Next, we note that differential equation of the form 
\begin{equation*}
f(x)=Af(x)+g(x),\text{ (}x\geq 0\text{, }g\text{ continuous)}
\end{equation*}%
solves to%
\begin{equation*}
f(t)=e^{\int_{0}^{t}A(\tau )d\tau }\left[ f(0)+\int_{0}^{t}g(s)e^{-%
\int_{0}^{s}A(\tau )d\tau }ds\right] .
\end{equation*}%
Applying the above formula, we obtain%
\begin{equation*}
p_{n}(t)=e^{-\int \lambda (t)dt}\left[ p_{n}(0)+\lambda
(t)\int_{0}^{t}p_{n-1}(s)e^{-\int_{0}^{s}\lambda (t)dt}ds\right] .
\end{equation*}%
where $p_{n}(0)=1$ if $n>0,$ and $p_{n}(0)=0$ if $n=0.$ Using the above
equation, it is easy to show by induction that (\ref{Poisson}) holds.

\section{New Trade Theory}

The most primitive atom responsible for exporting, production and trade is a
firm. Thus, the question why countries trade naturally boils down to the
question why firms decide to export. This is the starting point of the new
trade theory, which takes the route of building a positive theory of
industry to directly model this decision.

The above approach to model trade has been first proposed by Krugman (1980),
and later developed in Helpman and Krugman (1985). Below, we first discuss
the original Krugman model, as laid out in the Helpman and Krugman (1985)
textbook, and then discuss the Melitz model, which merges Krugman's theory
with the closed economy Hopenhayn (1992) model of industry equilibrium.

\section{Krugman-Helpman Model}

Krugman\ model importantly departs from the traditional trade theory. In the
traditional trade theory, differences between countries give rise to trade.
In contrast, in the Krugman model countries are ex-ante identical, but still
decide to trade and specialize ex-post. The key driving force behind trade
and ex-post specialization is the combination of increasing returns and love
for variety by the consumers. In particular, the Krugman model takes a stand
what a firm is: it is an ownership right to a variety, that is imperfectly
substitutable with other varieties. In particular, exporting is a decision
of this entity.

\subsection*{Model Economy}

There are $N$ countries, and each country has access to a technology of
introducing a new variety (a new type of good). The space of varieties is
potentially infinite, and so there is zero probability that a newly
introduced variety will overlap with anything that already exists in the
world (always a finite measure).

Introducing a variety takes resources modeled by a sunk cost $\chi .$ Upon
introducing a variety, the entity that does so\ (called a firm), becomes a
monopolist in producing it. We will denote the space of existing varieties
in the world by $\Omega \subseteq \lbrack 0,\infty ).$ Note that this space
can be partitioned into disjoint subsets of varieties by the source country: 
$\Omega =\cup _{n=1,..,N}\Omega _{n},$ where $\Omega _{n}\cap \Omega
_{m}=\varnothing ,$ all $n,m=1,$...$,N.$

We model trade cost by the usual iceberg transportation cost.

\subsubsection*{Households}

Given the set of existing varieties, households' in country $n$ maximize
utility function given by the CES aggregator:

\begin{equation}
U_{n}=\sum_{i=1..N}(\int_{\Omega _{i}}c_{ni}(\omega )^{\frac{\sigma -1}{%
\sigma }}d\omega )^{\frac{\sigma }{\sigma -1}},  \label{2EqKHH}
\end{equation}%
subject to the budget constraint%
\begin{equation*}
\sum_{i=1..N}\int_{\Omega _{i}}p_{ni}(\omega )c_{ni}(\omega )=w_{n}L_{n}+\Pi
_{n}.
\end{equation*}

\subsubsection*{Firms}

Becoming a firm costs $\chi $ units of labor, and this cost is sunk. Upon
paying it, a new variety is born and the firm is a monopolist in producing
it. Given the demand function $c_{ni}(p,\omega )$ for this variety by
country $n$ households, the profit of a firm $\omega $ from country $i\ $is
selling variety $\omega $ to country $n$ is: 
\begin{equation}
\pi _{ni}(\omega )=\max_{p_{ni},l_{i}}[p_{ni}c_{ni}(p_{ni},\omega
)-w_{i}l_{i}],  \label{2EqKF}
\end{equation}%
where 
\begin{equation*}
\sum_{n=1..N}d_{ni}c_{ni}(p_{ni},\omega )=l_{i}.
\end{equation*}

Since there is fixed cost of introducing a variety, the total profits are
given by%
\begin{equation}
\Pi _{i}=\pi _{ni}-\chi w_{i}\,,\text{ all }i=1,\text{...},N  \label{2Eq0}
\end{equation}

\subsubsection*{Feasibility and Market Clearing}

Market clearing condition requires that the demand for labor equals the
supply of labor, and so 
\begin{equation}
\Omega _{n}(\chi +l_{n})=L_{n},\text{ all }n=1,\text{...},N.  \label{2Eq0a}
\end{equation}

The free entry condition implies that the profits from a marginal variety
are driven to zero:\ 
\begin{equation}
\Pi _{n}=0,\text{ all }n=1,\text{...},N.  \label{2Eq0b}
\end{equation}

\subsection*{Definition of Equilibrium}

Definition of equilibrium is as follows. 

\begin{definition}
Equilibrium in this economy is:

\begin{itemize}
\item prices $p_{ni}(\omega ),w_{i}$, and

\item demand functions\footnote{%
It is not neccessary to include demand function as part of the definition of
equilibrium. We include it for the sake of clarity.} $c_{ni}(p,\omega ),$

\item and allocation $\Omega _{n},l_{n},c_{ni}(\omega ),$
\end{itemize}

such that

\begin{itemize}
\item given prices, demand function $c_{ni}(p,\omega )$ is derived from the
household's problem given by (\ref{2EqKHH}),

\item given demand functions, $p_{ni}(\omega )$ and $l_{i}$ solve the firm's
maximization problem given by (\ref{2EqKF}),

\item equilibrium consumption $c_{ni}(\omega )$ is consistent with the
demand function and prices, i.e. $c_{ni}(\omega )=c_{ni}(p_{ni}(\omega
),\omega ),$

\item zero profit condition (\ref{2Eq0b}) holds, and

\item market clearing condition (\ref{2Eq0a}) is satisfied.
\end{itemize}
\end{definition}

\subsection*{Characterization of Equilibrium and Predictions for Trade}

The Krugman model predicts that the set of varieties introduced in
equilibrium is independent from trade costs. For example, a country that
adopts extreme protectionism will produce the same number of varieties as a
country that opens up to trade. As a result, its predictions for gravity are
identical to the Armington model. Below, we prove these two knife-edge
results.

\begin{proposition}
$\Omega _{i}$'s are independent on trade costs $d_{ni}$.
\end{proposition}

\begin{proof}
From the household problem, we can derive the demand for variety $\omega $
to be given by 
\begin{equation}
x_{ni}(\omega )=\left( \frac{p_{ni}(\omega )}{P_{n}}\right) ^{1-\sigma
}X_{n},  \label{2Eq1}
\end{equation}%
where $x_{ni}(\omega )$ denotes expenditures of HH from country $n$ on
variety $\omega $ coming from country $i.$ Given this demand function, it is
straightforward to show that profit maximization implies that the price will
optimally be set by firms as a constant markup on the marginal cost:\ \ 
\begin{equation}
p_{ni}(\omega )=\frac{\sigma }{\sigma -1}w_{i}d_{ni},  \label{2Eq2}
\end{equation}%
and profits are given by:%
\begin{equation*}
\pi _{i}(\omega )=\sum_{n=1..N}\frac{x_{ni}(\omega )}{\sigma }=\frac{1}{%
\sigma }\sum_{n=1..N}\left( \frac{\frac{\sigma }{\sigma -1}w_{i}d_{ni}}{P_{n}%
}\right) ^{1-\sigma }X_{n}.
\end{equation*}%
The zero profit condition says that%
\begin{equation*}
\pi _{i}(\omega )=\chi w_{i}.
\end{equation*}%
Using the formula for profit function $\pi _{i}(\omega )$ stated above, we
obtain%
\begin{equation*}
\pi _{i}(\omega )=\frac{1}{\sigma }\sum_{n=1}^{N}\left( \frac{\frac{\sigma }{%
\sigma -1}w_{i}d_{ni}}{P_{n}}\right) ^{1-\sigma }X_{n}=\chi w_{i}
\end{equation*}%
and thus%
\begin{equation}
\sum_{n=1}^{N}\left( \frac{\frac{\sigma }{\sigma -1}d_{ni}}{P_{n}}\right)
^{1-\sigma }X_{n}=\sigma w_{i}^{\sigma }\chi .  \label{C1}
\end{equation}%
Next, using symmetry (all firms in country $i$ are identical, and so $%
x_{ni}\equiv x_{ni}(\omega )=x_{ni}(\omega ^{\prime }),$ all $\omega ,\omega
^{\prime }$), we combine the above result with the fact that total spending
of each country on all differentiated goods are given by:\ 
\begin{equation}
\Omega _{i}\sum_{n=1}^{N}x_{ni}=w_{i}L_{i}.  \label{CC}
\end{equation}%
In addition, using the definition of expenditures, and equation (\ref{2Eq2}%
), we have%
\begin{equation*}
X_{i}\equiv \Omega _{i}\sum_{n=1}^{N}x_{ni}=\Omega _{i}\sum_{n=1}^{N}\left( 
\frac{\frac{\sigma }{\sigma -1}d_{ni}w_{i}}{P_{n}}\right) ^{1-\sigma }X_{n}.
\end{equation*}%
Combining the above with (\ref{CC}), we write%
\begin{equation}
\Omega _{i}\sum_{n=1}^{N}\left( \frac{\frac{\sigma }{\sigma -1}d_{ni}}{P_{n}}%
\right) ^{1-\sigma }X_{n}=w_{i}^{\sigma }L_{i}.  \label{C3}
\end{equation}%
Finally, combining (\ref{C1}) with (\ref{C3}), we establish%
\begin{eqnarray*}
\frac{w_{i}^{\sigma }L_{i}}{\Omega _{i}} &=&\sigma w_{i}^{\sigma }\chi , \\
\Omega _{i} &=&\frac{L_{i}}{\sigma \chi }.
\end{eqnarray*}
\end{proof}

\begin{proposition}
The gravity equation is the same as in the Armington model.
\end{proposition}

\begin{proof}
Using the definition of aggregate expenditures%
\begin{equation*}
X_{i}=\sum_{n}\int_{\Omega _{i}}x_{ni}(\omega ),
\end{equation*}%
symmetry 
\begin{equation*}
x_{ni}(\omega )=x_{ni}(\omega ^{\prime }),\text{ all }\omega ,\omega
^{\prime }\in \Omega _{i},
\end{equation*}%
and constant markup pricing (\ref{2Eq2}), we obtain 
\begin{equation*}
X_{i}=w_{i}^{1-\sigma }\Omega _{i}\sum_{n=1}^{N}\left( \frac{\frac{\sigma }{%
\sigma -1}d_{ni}}{P_{n}}\right) ^{1-\sigma }X_{n}
\end{equation*}%
and thus%
\begin{equation}
w_{i}^{1-\sigma }\Omega _{i}=\frac{X_{i}}{\sum_{n=1}^{N}\left( \frac{\frac{%
\sigma }{\sigma -1}d_{ni}}{P_{n}}\right) ^{1-\sigma }X_{n}}  \label{C4}
\end{equation}%
Using (\ref{2Eq1}),%
\begin{equation*}
x_{ni}(\omega )=\left( \frac{p_{ni}(\omega )}{P_{n}}\right) ^{1-\sigma
}X_{n},
\end{equation*}%
and combining with (\ref{2Eq2}), we obtain%
\begin{equation*}
\frac{\Omega _{i}x_{ni}}{X_{n}}=\Omega _{i}\left( \frac{\frac{\sigma }{%
\sigma -1}d_{ni}w_{i}}{P_{n}}\right) ^{1-\sigma }
\end{equation*}%
and thus%
\begin{equation}
\frac{X_{ni}}{X_{n}}=\Omega _{i}w_{i}^{1-\sigma }\left( \frac{\frac{\sigma }{%
\sigma -1}d_{ni}}{P_{n}}\right) ^{1-\sigma }.  \label{C5}
\end{equation}%
Plugging in from (\ref{C4}) to (\ref{C5}), we obtain%
\begin{equation}
\frac{X_{ni}}{X_{n}}=\frac{X_{i}}{\sum_{n=1}^{N}\left( \frac{\frac{\sigma }{%
\sigma -1}d_{ni}}{P_{n}}\right) ^{1-\sigma }X_{n}}\left( \frac{\frac{\sigma 
}{\sigma -1}d_{ni}}{P_{n}}\right) ^{1-\sigma }  \label{C6}
\end{equation}%
Finally, following the same steps as in Armington model, it is easy to show
that (\ref{C6}) simplifies to\ 
\begin{equation*}
X_{ni}=\frac{X_{n}X_{i}}{\sum_{n}X_{n}}\left( \frac{\frac{\sigma }{\sigma -1}%
d_{ni}}{P_{n}P_{i}}\right) ^{1-\sigma }.
\end{equation*}
\end{proof}

\begin{exercise}
\label{Exercise17}Consider a 2-country version of the Krugman model, with
one large country called North and one small country called South. Size is
modeled by labor endowment, and so North has a larger endowment of labor
than South. Except for a different labor endowment, the countries are
identical.

In addition, assume that each country produces a homogeneous agricultural
tradable good A, and consumers have Cobb-Douglas preferences between the
agricultural good and the composite of all differentiated products C, i.e.
preferences in country $n=N,S$ are given by: 
\begin{equation*}
U_{n}=A_{n}^{\mu }C_{n}^{1-\mu },
\end{equation*}%
where 
\begin{equation*}
C_{n}=(\sum_{i=S,N}\int_{\Omega _{i}}c_{ni}(\omega )^{\frac{\sigma -1}{%
\sigma }}d\omega )^{\frac{\sigma }{\sigma -1}}.
\end{equation*}%
The homogeneous good can be trade frictionlessly ($d=1)$, but the
differentiated good is still subject to potentially positive iceberg
transportation cost $d>1.$ \newline

Derive analytically how the fraction of varieties produced by the South $%
\zeta =\frac{|\Omega _{S}|}{|\Omega _{S}|+|\Omega _{N}|}$ depends on the
relative size of south $s=\frac{L_{S}}{L_{N}+L_{S}},$ where here  $|\Omega _{i}|$ denotes the mass of differentiated varieties produced
by $i.$ HINT: The existence of a homogeneous and tradable agricultural good
that can be trade frictionlessly implies that that wages in both countries
must be equal (despite size differences), thus by numeraire normalization $%
w_{N}=w_{S}=1.$ Cobb-Douglas preferences imply that households spend a
constant fraction $\mu $ on the agricultural good and rest on the
differentiated goods.
\end{exercise}

\section{Melitz Model}

The Krugman model falls short in two important respects. On one hand, it
introduces firms as primitive atoms of trade---which is good---but on the
other hand, it is grossly at odds with the firm-level data. In the data,
most firms do not export, and in the Krugman model they all do (or none of
them does). In addition, despite anecdotal evidence, trade liberalizations
or protectionism have no bearing on the industry structure. So, even though
the model nicely illustrates a new motive for trade and models firms as the
primitive atom of trade, it is not useful as a quantitative basis for
applied work.

These shortcomings motivated Melitz (2003) to extend the simple setup by
introducing fixed costs of entry and heterogeneity in the firm level
productivity. In the Melitz model, it is no longer true that all firms
export. Precisely, there is a cutoff level of productivity above which a
firm decides to enter foreign markets, and below which it decides to sell
only at home. This cutoff endogenously changes in response to trade costs,
and thereby trade policies naturally lead to industry level reallocation.
Also, unlike in the Krugman model, only some firms export.

By being consistent with basic industry and firm level facts, the Melitz
model has recently become a basis for quantitative analysis in trade. An
alternative to the Melitz model is a brilliant work by Bernard, Eaton,
Jensen and Kortum (2003), and the unified framework by Eaton, Kortum and
Kramartz (2006) (EKK hereafter). EKK model nests a specialized version of an
asymmetric Melitz model due to Chaney (2008), BEJK model and the EK model.
In what follows, we will talk about the original Melitz model and touch upon
the BEJK (2003). I encourage you to read the EKK paper as well. (In the
exercise at the end of the chapter, you will be asked to review the key firm
level facts from the trade literature.)

\subsection*{Model Economy}

Time is discrete and horizon infinite. There are $N+1$ symmetric countries
in the world. Symmetry is built into notation, and so all variables pertain
to either a representative foreign country, denoted by $f,$ or the domestic
country, denoted by no subscript or $d$ depending on the context (domestic
and foreign country is identical).

Similarly as in the Krugman model, varieties are non-overlapping, and so the
total mass $\Omega \in R_{+}$ of goods available in the domestic country is $%
\Omega =\Omega _{d}+N\Omega _{f}$, where $\Omega _{d}\ $is the mass of
varieties produced at home, and $\Omega _{f}\ $is the mass of varieties
produced by a representative foreign country and available at home.\footnote{%
Unlike in the Krugman model, here $\Omega $ is not a set -- it is a positive
real number describing the measure of varities.} The iceberg transportation
cost is denoted by $d$.

\subsubsection*{Households}

Following the same approach as in the EK model, we will index goods by
prices rather the type of the good $\omega $. Since in the Melitz model no
two varieties of goods overlap, we must multiply the underlying integrals by
the measure of goods over which the distribution is calculated (probability
measures normalize everything to 1).

Assuming that the price distribution of home varieties is given by $P$ and
of foreign varieties (available domestically) by $P_{f},$ in the stationary
equilibrium the households' problem can be described by the following
Bellman equation:

\begin{equation}
J=\max_{c_{d}(\cdot ),c_{f}(\cdot )}\left[ \left( \Omega
_{d}\int_{0}^{\infty }c(p)^{\frac{\sigma -1}{\sigma }}dP_{d}+N\Omega
_{f}\int_{0}^{\infty }c_{f}(p)^{\frac{\sigma -1}{\sigma }}dP_{f}\right) ^{%
\frac{\sigma }{\sigma -1}}+\beta J\right] ,  \label{2EqHHBE}
\end{equation}%
subject to 
\begin{equation*}
\Omega _{d}\int_{0}^{\infty }pc(p)dP+N\Omega _{f}\int_{0}^{\infty
}pc_{f}(p)dP_{f}=wL+\Pi .
\end{equation*}%
The equations should be self explanatory. Note that integrals are multiplied
by the mass of goods $\Omega _{d}$ and $N\Omega _{f},$ as mentioned above.

\subsubsection*{Firms}

Firms can freely enter into production by paying a sunk startup cost $\chi
_{e}$ (denominated in labor units)$.$ All existing firms are assumed to be
subject to an exogenous destruction rate $\delta ,$ which will induce here
an endogenous turnover of firms (process of continual entry and exit).

The timing of entry and production is as follows. Upon startup, a
productivity draw $\phi $ is assigned to a new entrant (from distribution $G$%
), and the entrant decides whether to enter into production or not. The cost
of acquiring the draw of $\phi $ is sunk, and costs $\chi _{e}$ units of
labor. If the the entrant decides to enter, it then pays an a production
setup cost $\chi $ to sell at home. If, in addition, the entrant wants to
become an exporter, it pays another setup cost $\chi _{x}$ abroad---in each
of the foreign countries it intends to sell to.

It is assumed that the cost of exporting $\chi _{x}$ is large enough, so
that in equilibrium there are firms which decide to sell at home only. Such
firms will be referred to as home or domestic producers. Other firms will be
referred to as exporters.

After entry (i.e. after paying $\chi _{e}$ and obtaining $\phi )$, firm's
problem can be described by the following Bellman equation:%
\begin{equation}
\Pi (\phi )=\max \{\Pi _{d}(\phi )-\chi ,\Pi _{d}(\phi )-\chi +N(\Pi
_{x}(\phi )-\chi _{x})\},  \label{2EqF1BE}
\end{equation}%
where

\begin{equation}
\Pi _{d}(\phi )=\max_{p_{d},l_{d}}\left[ pc(p)-wl_{d}+\beta (1-\delta )\Pi
_{d}(\phi )\right] ,  \label{2EqF2BE}
\end{equation}%
subject to

\begin{equation*}
c(p)=\phi l,
\end{equation*}%
and 
\begin{equation}
\Pi _{x}(\phi )=\max_{p_{f},l_{x}}\left[ p_{f}c_{f}(p_{f})-wl_{x}+\beta
(1-\delta )\Pi _{x}(\phi )\right] ,  \label{2EqF3BE}
\end{equation}%
subject to 
\begin{equation*}
dc_{f}(p_{f})=\phi l_{x},
\end{equation*}%
where $c_{d}\left( p\right) $ is a demand function for domestic variety and $%
c_{f}\left( p\right) $ for a foreign variety.

The above Bellman equation can be understood as follows. The first equation
says that upon paying the entry cost $\chi _{e},$ and seeing the assigned
productivity draw $\phi ,$ the value of the firm (=present discounted value
of profits) $\Pi $ is determined by the max of two alternatives: (i)\
production for the home market only: $\Pi _{d}(\phi )-\chi ,$ or (ii)
production for both the home and the foreign market: $\Pi _{d}(\phi )-\chi
+N(\Pi _{x}(\phi )-\chi _{x}).$\footnote{%
Note that we built into the problem the fact that the fixed costs are such
that the firm never finds it optimal to export and not to sell at home. We
will later make this assumption explicit by imposing a condition on the
fixed cost $\chi _{d}$ and $\chi _{f}.$} It is easy to prove the following
property:\ 

\begin{lemma}
Entry decision and exporting decision takes the form of a cutoff rule, in
which all firms with a productivity draw above $\phi _{d}$ decide to produce
at least at home, and all firms above $\phi _{x}$ decide to export.
Moreover, $\phi _{x}>\phi _{d}$ iff $\chi d^{1-\sigma }<\chi _{x}.$
\end{lemma}

\begin{exercise}
\label{Exercise18}Prove the above lemma.
\end{exercise}

In what follows, we will directly build the above Lemma into notation.

\subsection*{Feasibility and Market Clearing}

Aggregate feasibility requires that total demand for labor used in
production (by all operating firms), setup cost, and startup of firms is
equal to the total supply of labor:\ \ 
\begin{eqnarray}
L &=&\chi _{e}\frac{\delta \Omega _{d}}{1-G(\phi _{d})}\text{ (startups)}+
\label{2EqMC} \\
&&+\chi \delta \Omega _{d}\text{ (production setup at home)}+  \notag \\
&&+N\chi _{x}\delta \Omega _{d}\frac{1-G(\phi _{x})}{1-G(\phi _{d})}\text{%
(production setup abroad)}+  \notag \\
&&+\frac{\Omega _{d}}{1-G(\phi _{d})}\int_{\phi _{d}}l_{d}(\phi )dG(\phi )%
\text{ (labor demand by domestic producers)}+  \notag \\
&&+\frac{\Omega _{f}}{1-G(\phi _{x})}\int_{\phi _{x}}(l_{d}(\phi
)+Nl_{x}(\phi ))dG(\phi )\text{ (labor demand by exporters).}  \notag
\end{eqnarray}

To write down this condition formally, we have used the fact that the mass
of varieties is equal to the number of firms. So, $\Omega _{d}$ is the total
mass (measure) of firms that produce both at home and export, and $\Omega
_{f}$ is the mass of firms that also export (which by symmetry applies to
the domestic country and to any foreign country).

Since in the stationary equilibrium (steady state), $\delta \Omega _{d}$ of
firms must be replaced every period by new entrants, there must be $\frac{%
\delta \Omega _{d}}{1-G(\phi _{d})}$ startups (draws of $\phi )$---as only
fraction $1-G(\phi _{d})$ of startups eventually decides to produce.

Moreover, given fraction $\delta \Omega _{d}$ of new entrants (firms that
stay and produce), the fraction of new entrants that also decides to export
something is given by the conditional probability that $\phi $ falls above
the exporting cutoff $\phi _{x},\ $conditioned on $\phi $ being already
above the entry cutoff $\phi _{d}$. This probability is equal to $\frac{%
1-G(\phi _{x})}{1-G(\phi _{d})},$ and thus the expression $\delta \Omega _{d}%
\frac{1-G(\phi _{x})}{1-G(\phi _{d})}.$ Since all firms which find it
profitable to export will export to all markets or not at all, we must
multiply this term by $N$.

The last two terms are the variable demand for labor by domestic producers\
(firms that sell at home only) and exporters (firms that sell both at home
and abroad).

The definitions of the cutoff values $\phi _{d}$,$\phi _{x}\ $imply that the
firm at the cutoff level must be indifferent between the two decisions, i.e.%
\begin{equation}
\Pi (\phi _{d})=0,  \label{2Eq3}
\end{equation}%
and 
\begin{equation}
\Pi _{d}(\phi _{x})-\chi w=\Pi _{d}(\phi _{x})-\chi w+N(\Pi _{x}(\phi
_{x})-\chi _{x}w).  \label{2Eq4}
\end{equation}%
What's more, under symmetry the process of entry and exit implies 
\begin{equation}
\Omega _{f}=\frac{1-G(\phi _{x})}{1-G(\phi _{d})}\Omega _{d},  \label{omega}
\end{equation}%
and the free entry and exit condition implies%
\begin{equation}
\Pi \equiv E\Pi (\phi )-\chi _{e}w=0.  \label{2Eq5}
\end{equation}

Finally, the price distribution $P$ can be linked to the other equilibrium
by the optimal policy function of firms: $p(\phi )$ and $p_{x}(\phi )$ Given
these functions, $P$ (and $P_{f})$ is derived as follows:

\begin{equation}
P(p)=\Pr (\text{p}\leq p)=\Pr (p(\phi )\leq p)=\Pr (\phi \leq
p^{-1}(p))=G(p^{-1}(p)),  \label{2Eq6}
\end{equation}%
and similarly 
\begin{equation}
P_{f}(p)=\Pr (\text{p}_{f}\leq p)=\Pr (p_{x}(\phi )\leq p)=\Pr (\phi \leq
p_{x}^{-1}(p))=G(p_{x}^{-1}(p)).  \label{2Eq7}
\end{equation}

\subsection*{Equilibrium}

\begin{definition}
\textbf{Symmetric stationary equilibrium} in this economy is:

\begin{itemize}
\item value functions $J,\Pi (\phi ),\Pi _{d}(\phi ),\Pi _{x}(\phi )$,

\item policy functions $c(p),c_{f}(p),p(\phi ),p_{f}(\phi ),l_{d}(\phi
),l_{x}(\phi )$

\item distribution functions $P(p),P_{f}(p),$

\item measures $\Omega _{d},\Omega _{f},$

\item entry cutoffs $\phi _{d}$, $\phi _{x},$

\item aggregate profits $\Pi ,$

\item and wage $w$
\end{itemize}

such that

\begin{itemize}
\item value function $J$ and policy functions $c(p),c_{f}(p)$ are derived
from (\ref{2EqHHBE}),

\item value functions $\Pi (\phi ),\Pi _{d}(\phi ),\Pi _{x}(\phi )$ and
policy functions $p(\phi ),p_{f}(\phi ),l_{d}(\phi ),l_{x}(\phi )$ are
derived from (\ref{2EqF1BE}), (\ref{2EqF2BE}) and (\ref{2EqF3BE}),

\item $P_{d}(p),P_{f}(p)$ are consistent with (\ref{2Eq6}) and (\ref{2Eq7})

\item $p(\phi ),p_{f}(\phi )$ are invertible functions,

\item cutoff values $\phi _{d},\phi _{x},$ and measures $\Omega _{d},\Omega
_{f}$ are consistent with (\ref{2Eq3}), (\ref{2Eq4}) and (\ref{omega}),

\item zero profit condition (\ref{2Eq5}) is satisfied,

\item and market clearing condition (\ref{2EqMC}) holds.
\end{itemize}
\end{definition}

\begin{proposition}
The above symmetric equilibrium exists and is unique.
\end{proposition}

\begin{proof}
The strategy is to show that the zero profit conditions can be used to solve
for the cutoffs $\phi _{d}$ and $\phi _{x},$ and once we find the cutoffs,
the remaining equilibrium objects follow. Before we proceed, we note the
following. Since this a monopolistic competition model, the formula for
prices is given by:\ \ 
\begin{eqnarray*}
p(\phi ) &=&\frac{\sigma }{\sigma -1}\frac{1}{\phi }, \\
p_{x}(\phi ) &=&\frac{\sigma }{\sigma -1}\frac{d}{\phi },
\end{eqnarray*}%
and expenditures on individual goods by:%
\begin{eqnarray}
x\left( \phi \right) &=&X(\frac{p\left( \phi \right) }{P})^{1-\sigma }=X(%
\frac{\sigma -1}{\sigma }P\phi )^{\sigma -1},  \label{XX} \\
x_{x}(\phi ) &=&X(\frac{p_{f}\left( \phi \right) }{P})^{1-\sigma
}=d^{1-\sigma }X(\frac{\sigma -1}{\sigma }P\phi )^{\sigma -1}=d^{1-\sigma
}x\left( \phi \right) .  \notag
\end{eqnarray}%
(Note that, in fact, prices are an invertible functions of $\phi $ as
required by the definition of equilibrium.) In particular, from (\ref{XX}),
we obtain%
\begin{eqnarray}
\frac{x\left( \phi \right) }{x_{x}(\phi ^{\prime })} &=&d^{\sigma -1}(\frac{%
\phi }{\phi ^{\prime }})^{\sigma -1},  \label{XXX} \\
\frac{x\left( \phi \right) }{x(\phi ^{\prime })} &=&(\frac{\phi }{\phi
^{\prime }})^{\sigma -1},  \notag \\
\frac{x_{x}\left( \phi \right) }{x_{x}(\phi ^{\prime })} &=&(\frac{\phi }{%
\phi ^{\prime }})^{\sigma -1},  \notag
\end{eqnarray}%
for any $\phi ,\phi ^{\prime }.$

In the first step, we decompose the expected profits to profits earned at
home and abroad:%
\begin{equation}
E[\pi (\phi )]=(1-G(\phi _{d}))E_{\phi >\phi _{d}}[\pi _{d}(\phi
)]+N(1-G(\phi _{x}))E_{\phi >\phi _{x}}[\pi _{x}(\phi )],
\label{expectedprofits}
\end{equation}%
and calculate the conditional profits as follows:\ 
\begin{eqnarray}
E_{\phi >\phi _{d}}[\pi _{d}(\phi )] &=&\frac{E_{\phi >\phi _{d}}[x(\phi )]}{%
\sigma }-\chi ,  \label{PI} \\
E_{\phi >\phi _{x}}[\pi _{x}(\phi )] &=&\frac{E_{\phi >\phi _{x}}[x_{x}(\phi
)]}{\sigma }-\chi _{x}.  \notag
\end{eqnarray}%
Given (\ref{XXX}), and using the above expressions, we obtain%
\begin{eqnarray}
E_{\phi >\phi _{d}}[x(\phi )] &\equiv &x(\phi _{d})E_{\phi >\phi _{d}}[\frac{%
x(\phi )}{x(\phi _{d})}]\equiv \frac{x(\phi _{d})}{\phi _{d}^{\sigma -1}}%
\int_{\phi _{d}}^{\infty }\frac{\phi ^{\sigma -1}g(\phi )}{1-G(\phi _{d})}%
d\phi ,  \label{1} \\
E_{\phi >\phi _{x}}[x_{x}(\phi )] &\equiv &x_{x}(\phi _{x})E_{\phi >\phi
_{x}}[\frac{x_{x}(\phi )}{x_{x}(\phi _{x})}]\equiv \frac{x_{x}(\phi _{x})}{%
\phi _{x}^{\sigma -1}}\int_{\phi _{x}}^{\infty }\frac{\phi ^{\sigma
-1}g(\phi )}{1-G(\phi _{x})}d\phi ,  \notag
\end{eqnarray}%
where ~$g(\cdot )$ denotes the density function corresponding to $G(\cdot )$%
. Furthermore, assuming $\phi _{d}<\phi _{x}$ (which holds iff $d^{\sigma
-1}\chi _{x}>\chi $), we use (\ref{PI}) and the definition of the cutoff
productivity $\phi _{d},\ $to derive%
\begin{equation*}
\pi (\phi _{d})=\pi _{d}(\phi _{d})=\frac{x(\phi _{d})}{\sigma }-\chi =0,
\end{equation*}%
and%
\begin{equation}
x(\phi _{d})=\sigma \chi .  \label{2}
\end{equation}%
Substituting from (\ref{1}) and (\ref{2}) into (\ref{PI}), we have%
\begin{eqnarray*}
E_{\phi >\phi ^{\ast }}[\pi _{d}(\phi )] &=&\frac{x(\phi _{d})}{\sigma (\phi
_{d})^{\sigma -1}}\int_{\phi _{d}}^{\infty }\frac{\phi ^{\sigma -1}g(\phi )}{%
1-G(\phi _{d})}d\phi -\chi = \\
&=&\chi \lbrack (\frac{\widetilde{\phi }(\phi _{d})}{\phi _{d}})^{\sigma
-1}-1],
\end{eqnarray*}%
where%
\begin{equation*}
\widetilde{\phi }(\phi ^{\ast })\equiv \lbrack \int_{\phi ^{\ast }}^{\infty }%
\frac{\phi ^{\sigma -1}g(\phi )}{1-G(\phi ^{\ast })}d\phi ]^{\frac{1}{\sigma
-1}}.
\end{equation*}%
The analogous expression for exporters is 
\begin{equation*}
E_{\phi >\phi _{x}}[\pi _{x}(\phi )]=\chi _{x}[(\frac{\widetilde{\phi }(\phi
_{x}^{\ast })}{\phi _{x}^{\ast }})^{\sigma -1}-1].
\end{equation*}%
Next, we express the ex-ante zero profit condition in terms of the cutoff
values by substituting out the above equations for conditional profits into (%
\ref{expectedprofits}) :%
\begin{gather}
E[\pi (\phi )]=\chi (1-G(\phi _{d}))[(\frac{\widetilde{\phi }(\phi _{d})}{%
\phi _{d}})^{\sigma -1}-1]+  \label{fp1a} \\
+N\chi _{x}(1-G(\phi _{x}))[(\frac{\widetilde{\phi }(\phi _{x})}{\phi _{x}}%
)^{\sigma -1}-1]=\chi _{e}.  \notag
\end{gather}%
Furthermore, we express the exporting cutoff $\phi _{x}$ in terms of the
entry cutoff. Using the fact that%
\begin{eqnarray}
\pi _{d}(\phi _{d}) &=&\frac{x(\phi _{d})}{\sigma }-\chi =0,  \label{pid} \\
\pi _{x}(\phi _{x}) &=&\frac{x(\phi _{x})}{\sigma }-\chi _{x}=0,  \notag
\end{eqnarray}%
we derive%
\begin{eqnarray}
\frac{x_{x}(\phi _{x})}{x(\phi _{d})} &=&d^{1-\sigma }(\frac{\phi _{x}}{\phi
_{d}})^{\sigma -1}=\frac{\chi _{x}}{\chi },  \label{fp2} \\
\phi _{x} &=&\phi _{d}(\frac{\chi _{x}}{\chi })^{\frac{1}{\sigma -1}}d. 
\notag
\end{eqnarray}%
Since, $\phi _{x}$ is a linear function of $\phi _{d}$, with a slope
coefficient $\zeta =(\frac{\chi _{x}}{\chi })^{\frac{1}{\sigma -1}}d,$ we
reduce the equilibrium fixed point problem to 1 equation in 1 unknown:%
\begin{equation}
\chi j(\phi _{d})+N\chi _{x}j(\zeta \phi _{d})=\chi _{e},  \label{FixedPoint}
\end{equation}%
where 
\begin{equation}
j(\phi _{d})\equiv (1-G(\phi _{d}))[(\frac{\widetilde{\phi }(\phi _{d})}{%
\phi _{d}})^{\sigma -1}-1].
\end{equation}%
To goal is to show that there is a solution (\ref{FixedPoint}), and that it
is unique. Given continuity of $j(\cdot )$, it is sufficient to prove that $%
j(\cdot )$ strictly decreases from $\infty $ to 0 for $\phi _{d}\in
(0,\infty ).$ To this end, we note the following properties of $j(\cdot )$:
(i) $j(\phi _{d})>0$ on $\phi _{d}\in (0,\infty ),$(ii) $\phi
_{d}\rightarrow 0,$ $j(\phi _{d})\rightarrow +\infty $ (follows because $%
\widetilde{\phi }(\phi _{d})\rightarrow \infty ),$ and (iii) $\frac{%
j^{\prime }(\phi _{d})\phi _{d}}{j(\phi _{d})}<-(\sigma -1).$ The only
nontrivial property is (iii). We show it as follows. Consider $f(\phi
_{d})\equiv \log j(\phi _{d})\ ($this is possible because $j(\phi _{d})$ is
positive valued function). Note that by monotonicity of log, $(\phi
_{d}\rightarrow \infty ,$ $j(\phi _{d})\rightarrow 0)$ iff $(\phi
_{d}\rightarrow \infty ,f(\phi _{d})\rightarrow -\infty ).$ Since from (ii)
we know $f^{\prime }(\phi _{d})<-\frac{(\sigma -1)}{\phi _{d}},$ we use the
Fundamental Theorem of Calculus to link this fact to the function itself: 
\begin{eqnarray}
f(b)-f(a) &=&\int_{a}^{b}f^{\prime }(\phi _{d})<\int_{a}^{b}-\frac{(\sigma
-1)}{\phi _{d}}=[-\frac{1}{2}(\sigma -1)\log \phi _{d}]_{a}^{b}= \\
&=&\frac{1}{2}(\sigma -1)(\log a-\log b).  \notag
\end{eqnarray}%
Taking $b\rightarrow \infty $, we have $f\left( b\right) \rightarrow -\infty
.$

The remaining part of the proof is straightforward and is omitted. It
requires to show that given the cutoffs, we can unambiguously determine all
other equilibrium objects.
\end{proof}

\subsection*{Comparative Statics Results}

The central result of the Melitz paper is that trade liberalizations lead to
the industry level reallocations. Intuitively, in the Melitz model, when $d$
goes up, consumers shift their spending onto new goods (in equation (\ref{XX}%
) P falls). This lowers their spending on the goods that they were
purchasing so far, and unless the firm is exporting, it necessarily faces a
decline in spending and profits (by equation (\ref{pid})). Consequently, the
cutoff $\phi _{d}$ goes up. Moreover, since entry cutoff goes up, entry
becomes more costly. As indicated by equation (\ref{2Eq5}), the only way a
firm can break even in expectation is that profits at the top of the
distribution compensate for the losses at the bottom. This is the key result
that Melitz proves and illustrates in Figure 2 (reproduced below).


\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=11,scale=.44]{Figures.pdf} 
 \caption{Industry level relocations in
 the Melitz model.}%
\label{FigureMelitz1}
\end{figure}


%\FRAME{dtbpFU}{202.6pt}{119.0667pt}{0pt}{\Qcb{Industry level relocations in
%the Melitz model. }}{\Qlb{FigureMelitz1}}{Figure}{\special{language
%"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
%"USEDEF";valid_file "T";width 202.6pt;height 119.0667pt;depth
%0pt;original-width 293.875pt;original-height 171.75pt;cropleft "0";croptop
%"1";cropright "1";cropbottom "0";tempfilename
%'KD2WCH05.wmf';tempfile-properties "XPR";}}

We next frame this result into a formal proposition.

\begin{proposition}
In the equilibrium of the Melitz model, $\frac{d\phi _{d}}{dd}<0,\frac{d\phi
_{x}}{dd}>0,$ and there exists $\bar{\phi}$ such that for all $\phi >\bar{%
\phi},\frac{d\pi }{dd}<0.$
\end{proposition}

\begin{exercise}
\label{Exercise19}Prove the above proposition. HINT: For the first part
consider study the expression (\ref{FixedPoint}) and (\ref{fp2}) using
Implicit Function Theorem. For the second part, show 
\begin{equation*}
x_{d}(\phi )+Nx_{x}(\phi )=(1+Nd^{1-\sigma })x_{d}(\phi )
\end{equation*}%
using (\ref{XXX}), and show 
\begin{equation*}
x_{d}(\phi )=(\frac{\phi }{\phi _{d}})^{\sigma -1}\sigma \chi ,\text{ all }%
\phi \geq \phi _{d},
\end{equation*}%
using similar arguments as in (\ref{pid}). Next, consider the change in
profits of a firm that exports before and after reduction of $d$ by some $%
\Delta d.$ Use the fact that 
\begin{equation*}
\pi (\phi )=\frac{x(\phi )}{\sigma }-\chi -N\chi _{x},
\end{equation*}%
where $x(\phi )=x_{d}(\phi )+Nx_{x}(\phi ).$
\end{exercise}

\subsection*{A Note on the Literature}

Melitz model is based on the model of industry equilibrium by Hopenhayn
(1992). The original Melitz setup has been further extended to a
multi-country asymmetric framework by Chaney (2006). Unlike the Melitz model,
Chaney's model is in partial equilibrium. The main contribution of his paper
is to derive analytically a gravity equation. Finally, Eaton, Kortum and
Kramartz (2006) propose a framework that flexibly nests all 3 models: EK
model, BEJK\ model and Chaney's model.

\section{Bernard, Eaton, Jensen and Kortum Model}

A competing model with monopolistic producers is the model by Bernard,
Eaton, Jensen and Kortum (2003). The paper extends the original framework
originally due to Eaton and Kortum (2002), and introduces imperfect
competition. The key idea is that a producer has an exclusive right to the
technology draw he gets (unlike in the EK model), and the distribution of
efficiency level in producing every good is given by the formula we have
derived in the previous section from a model of endogenous innovation: 
\begin{equation*}
P(Z_{1}\leq z_{1},Z_{2}\leq z_{2})=(1+T(z_{2}^{-\theta }-z_{1}^{-\theta
}))e^{-Tz_{2}^{-\theta }},
\end{equation*}%
where $Z_{1}$ is the r.v. denoting the draw of the most efficient producer
and $Z_{2}$ is the r.v. denoting the draw of the second most efficient
producer. The producers are Bertrand competitors, so the highest efficiency
(lowest cost) producer sets the price as high as possible, but low enough so
that it precludes second lowest cost producer from entry. This, of course,
leads to endogenous markups, which crucially depend on the distance in
efficiency between the first and second lowest cost producers. We should
note that in this model innovation is confined to factor saving innovation,
and does not involve product innovation as the Melitz model.

Interestingly enough, the model is set up in such a way that all
simplifications that were valid in the EK model go through in this extended
framework. But, unlike using the EK model, here we can start talking about
firm-level facts in addition to the aggregate facts that EK model has
predictions on. 

BEJK\ document in their paper key characteristics of an `exporting plant' (%
\textit{relative} \textit{to} `non-exporting plant') based on the data from
1992 US\ Census of Manufactures (200,000 plants in the sample). The key
exporter facts they focus on are:\ 

\begin{enumerate}
\item Very few plants report exporting anything -- about 21\%

\item Those that do, still sell mostly at home -- 2/3 of plants in the
sample export less than 10\% of their total output

\item Exporting firms are on average larger (ship 5.6 more output), and
appear to be about 9\% more `productive' (or more accurately profitable)%
\footnote{\textit{Productivity} is defined here as the ratio of total value
added of the plant relative to the total payroll bill of production workers
(after controlling for capital/skill intensity of a given plant).}
\end{enumerate}

They show that their model can account for all the above facts
qualitatively, and in many respects comes close at accounting for them
quantitatively. The amazing thing is that it does so without any fixed costs
of exporting. Below, you will find a short overview of the key features of
the theory:

Market Structure and Ownership

\begin{itemize}
\item Bertrand competition between the producers within same variety (each
producer owns one technology)

\item Similarly to EK'02, each market is captured by the low cost supplier,
but unlike in EK'02 the markup can be positive:

\begin{itemize}
\item the lowest cost supplier to country $n$ is constrained not to charge
more than the second-lowest cost supplier potentially entering from any
country in the world%
\begin{equation*}
c_{2n}(j)=\min \left\{ c_{2nl}(j),\min_{i\neq l}\{c_{1ni}(j)\right\}
\end{equation*}%
where $c_{2n}(j)$ is the second lowest cost supplier of commodity $j$ to
country $n,$ and $l$ is the country of origin of the lowest-cost supplier (=
min\{second lowest cost supplier from the same country as the lowest cost
supplier, min of the first lowest cost supplier to country n from all other
countries\}).
\end{itemize}
\end{itemize}

Prices

\begin{itemize}
\item The price of good $j$ in country $n$ is given by%
\begin{equation*}
P_{n}(j)=\min \{c_{2n}(j),\frac{\sigma }{\sigma -1}c_{1n}(j)\}.
\end{equation*}

\item The markup is the maximal feasible markup as long as it is not higher
than the MC\ optimal markup
\end{itemize}

Probabilistic Formulation of Technology

\begin{itemize}
\item To cover all possibilities, need to know the highest and
second-highest efficiency draw $z_{1i}(j),z_{2i}(j)$ in each country

\item Similarly to EK model, the efficiency levels are realizations of a
random variable drawn from a carefully chosen distribution%
\begin{equation*}
F_{i}(z_{1},z_{2})=\Pr \left[ Z_{1i}\leq z_{1},Z_{2i}\leq z_{2}\right] =%
\left[ 1+T_{i}(z_{2}^{-\theta }-z_{1}^{-\theta })\right] e^{-T_{i}z_{2}^{-%
\theta }},
\end{equation*}%
for $0\leq z_{2}\leq z_{1},$ drawn independently across countries $i$ and
goods $j$ (see derivation of this function from a process of endogenous
innovation in the previous sections)
\end{itemize}

Cost Functions

\begin{itemize}
\item The cost is a realization of the following two random variables: 
\begin{eqnarray*}
c_{1ni}(j) &=&\frac{w_{i}}{Z_{1i}(j)}d_{ni} \\
c_{2ni}(j) &=&\frac{w_{i}}{Z_{2i}(j)}d_{ni}
\end{eqnarray*}

\item Given distribution of efficiency can obtain distribution of first and
second lowest cost as follows:\ 
\begin{eqnarray*}
G_{ni}^{c}(C_{1} &>&c_{1},C_{2}>c_{2})=\Pr \left[ C_{1}>c_{1},C_{2}>c_{2}%
\right] \\
&=&\Pr \left[ Z_{1}\leq \frac{w_{i}d_{ni}}{c_{1}},Z_{2}\leq \frac{w_{i}d_{ni}%
}{c_{2}}\right] \\
&=&G(\frac{w_{i}d_{ni}}{c_{1}},\frac{w_{i}d_{ni}}{c_{2}}).
\end{eqnarray*}%
which solves to 
\begin{equation*}
G_{ni}^{c}(c_{1},c_{2})=\left[ 1+T_{i}[w_{i}d_{ni}]^{-\theta }(c_{2}^{\theta
}-c_{1}^{\theta })\right] e^{-T_{i}[w_{i}d_{ni}]^{-\theta }c_{2}^{\theta }}
\end{equation*}

\item The complementary distribution of the lowest and second-lowest cost
regardless the source (the probability that the lowest and second-lowest
cost in all countries is above $c_{2}+$ probability that in one of the
countries the lowest cost is between $c_{1}$ and $c_{2},$ second-lowest is
above $c_{2}$ and in all other countries both lowest and second-lowest is
above $c_{2})$ is 
\begin{eqnarray*}
G_{n}^{c}(c_{1},c_{2}) &=&\Pi _{i=1}^{N}G_{ni}^{c}(c_{2},c_{2})+ \\
&&+\sum_{i=1}^{N}[G_{ni}^{c}(c_{1},c_{2})-G_{ni}^{c}(c_{2},c_{2})]\Pi
_{k\neq i}G_{nk}^{c}(c_{2},c_{2}) \\
&=&\left[ 1+\Phi _{n}(c_{2}^{\theta }-c_{1}^{\theta })\right] e^{-\Phi
_{n}c_{2}^{\theta }}
\end{eqnarray*}%
where 
\begin{equation*}
\Phi _{n}=\sum_{i}T_{i}[w_{i}d_{ni}]^{-\theta }.
\end{equation*}

\item The cost distribution is 
\begin{equation*}
G_{n}(c_{1},c_{2})=1-G_{n}^{c}(0,c_{2})-G_{n}^{c}(c_{1},0)+G_{n}^{c}(c_{1},c_{2})
\end{equation*}

\item The distribution of the lowest cost regardless second lowest cost can
be obtained by taking the limit $c_{2}\rightarrow \infty $ of the expression
above
\end{itemize}

Key Aggregate Results

\begin{enumerate}
\item The probability $\pi _{ni}$ that country $i$ is the lowest cost
supplier to $n$ is the same as in EK: 
\begin{equation*}
\pi _{ni}=\int_{0}^{\infty }\Pi _{k\neq i}[1-G_{1nk}(c)]dG_{1ni}(c)=\frac{%
T_{i}(w_{i}d_{ni})^{-\theta }}{\Phi _{n}}.
\end{equation*}

\item The joint distribution of the lowest and second lowest cost of
supplying country $n,$ conditional on country $i$ being the low cost
supplier is independent on the source as in EK%
\begin{equation*}
G_{n}^{c}(c_{1},c_{2}|i)=G_{n}^{c}(c_{1},c_{2})=\left[ 1+\Phi
_{n}(c_{2}^{\theta }-c_{1}^{\theta })\right] e^{-\Phi _{n}c_{2}^{\theta }}.
\end{equation*}%
Unlike in the EK model, this result does not translate yet to gravity
equation because there are markups. Need to know that distribution of
markups is independent on the source.

\item The markup $M_{n}(j)=P_{n}(j)/C_{1n}(j)$ is the realization of a
random variable $M_{n}$. Conditional on $C_{2}=c_{2}$, the markup is Pareto
distributed and independent on $c_{2}$. Thus, the distribution of markups
conditioned on being the source is also Pareto, and in particular,
independent on the source.

\item Given the above, the share that country $n$ spends on goods from
country $i$ is $\pi _{ni},$ and gravity equation is analogous to the EK model
\end{enumerate}

Firm-Level Results

\begin{enumerate}
\item A plant with higher efficiency is likely to have a higher markup%
\begin{equation*}
H_{n}(m|z)=1-\exp (-\Phi _{n}w_{n}^{\theta }z^{-\theta }(m^{\theta }-1))%
\text{, }1\leq m\leq \bar{m},
\end{equation*}%
(a plant unusually efficient relative to other producing plants tend to be
unusually efficient relative to its latent competitors as well, so charges a
higher markup)

\item Greater efficiency makes the producer more likely to export and to be
big, explaining the correlations between size and export status that we see
in the data.
\end{enumerate}

\begin{exercise}
\label{Exercise20}Read data sections of the following papers: (i) Das,
Sanghamitra \& Mark J. Roberts \& James R. Tybout, 2001. \textquotedblleft
Market Entry Costs, Producer Heterogeneity, and Export
Dynamics,\textquotedblright\ NBER Working Papers 8629, National Bureau of
Economic Research, (ii)\ \ Eaton, Jonathan \& Samuel Kortum \& Francis
Kramarz, 2004. \textquotedblleft Dissecting Trade: Firms, Industries, and
Export Destinations,\textquotedblright\ NBER Working Papers 10344, National
Bureau of Economic Research, (iii)\ Bernard, Andrew B. \& Bradford J.
Jensen, 1999. \textquotedblleft Exceptional Exporter Performance: Cause,
Effect or Both?\textquotedblright , Journal of International Economics,
February 1999, 47(1), pp. 1-25, and (iv) Ruhl, Kim J. \& Willis Jonathan ,
2007. \textquotedblleft Convexities, Nonconvexities, and Firm Export
Behavior\textquotedblright . The last paper you will find under the
following link:
http://editorialexpress.com/conference/MWM2008/program/MWM2008.html (see
session 20 in the conference schedule). Summarize and briefly describe the
key producer-level facts about exporting and trade that emerge from this
literature.
\end{exercise}

\chapter{International Business Cycle}

\textit{\textquotedblleft In modern developed economies, goods and assets
are traded across national borders, with the result that events in one
country generally have economic repercussions in others. International
business cycle research focuses on the economic connections among countries
and on the impact these connections have on the transmission of aggregate
fluctuations. In academic studies this focus is expressed in terms of the
volatility and comovement of international time series data. Examples
include the volatility of fluctuation in the balance of trade, and
correlation of the trade balance with output, the correlation of output and
consumption across countries, and the volatility of prices of foreign and
domestic goods.\textquotedblright , Backus, Kehoe and Kydland, 1995.}

\section{Introduction}

In this chapter, we extend the time horizon of our analysis to short-run and
medium-run. What occurs at these frequencies, and our long-run analysis has
so far abstracted from, are business cycle fluctuations. This phenomenon
brings in an additional set of facts about the comovement and volatility of
aggregate quantities and prices, and introduces a new motive of
international borrowing and lending to share business cycle risk and
reallocate production over the business cycle.

\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=12,scale=.44]{Figures.pdf} 
 \caption{Time-horizon of the
 business cycle analysis.}%
\label{Fig_IRBC}
\end{figure}


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The research in the international business cycle literature has been focused
on a development of a workhorse model (laboratory) that is useful for
broad-based policy analysis. Since this objective coincides with the agenda
of the closed economy business cycle modeling, its open economy counterpart
is often referred to as open economy macro. The central questions of the
open economy macro are largely, but not only, centered around the extent to
which predictions of the successful closed economy macro models hold in the
open economy environments, and the extent to which these models are
consistent with the additional evidence such analysis brings in (e.g.
international comovement, international prices, current account dynamics).

The international business cycle literature is organized into two major
branches: (i) International Real Business Cycle, and (ii) New Open Economy
Models. These two classes of models are not disjoint. In fact, NOEM builds
on the IRBC by adding sticky prices to the RBC model, and monetary shocks
(liquidity shocks). Since in the RBC models money is neutral,\ without
further modifications of the theory, such shocks have no bite\footnote{%
There are RBC models with real financial frictions that involve monetary
shocks and money is not neutral. For example, see Atkeson and Kehoe (2000),
\textquotedblleft Money, Interest Rates, and Exchange Rates With
Endogenously Segmented Asset Markets\textquotedblright , WP 605, Minneapolis
Fed.}.

In this course, we will talk about IRBC and occasionally touch upon NOEM (to
be discussed in Econ 872). Both types of models are based on shocks backed
out from the data in a disciplined manner. In the IRBC theory, the shocks
are modeled as a stochastic shift of the production function, and are backed
out from the cyclical properties of the Solow residual in the data. The
NOEM\ models add monetary shocks (liquidity shocks) that are backed out from
the data on interest rates or money supply.

The shocks in the RBC models are often referred to as technology shocks.
However, one should not interpret them literally, but rather think of them
as capturing all sorts of unmodeled distortions that boil down to a shift of
a production function in a reduced form model. For a theoretical foundation
of such view, see the paper by Chari, Kehoe and McGrattan, \textquotedblleft 
\textit{Business Cycle Accounting}\textquotedblright , Econometrica 2007,
Vol 75(3).\footnote{%
Under such view RBC theory no longer implies that there is no space for
broadly defined government intervention to stabilize business cycle. Such
conclusion is only true if we literally interpret technology shocks as
coming from some \textit{exogenous} shifts in the production function (like
oil shocks or hurricanes). In addition, such interpretation implies that
much less can be said about the source of business cycle fluctuations.
Nevertheless, the theory is still useful in providing a coherent and
disciplined framework to study the effects of the business cycle on various
aspect of economic activity. }

A standard reference to RBC theory is the textbook (collection of RBC
papers): \textquotedblleft Frontiers of Business Cycle
Research\textquotedblright\ by Thomas Cooley (editor)---which is a bit out
of date at this point.\footnote{%
Especially in terms of solution methods.} As an introduction to RBC, I
encourage you to read the Nobel lecture by Prescott: \textquotedblleft Nobel
Lecture, The Transformation (...)\textquotedblright , JPE, 2006, Vol114(2).

\section{Business Cycle Measurement}

Here, we develop measurement tools that allow us to summarize the facts
about business cycle fluctuation. Specifically, we develop methods that
isolate the long-run properties (low-frequency) of the time-series from the
short-run properties (high frequency).

The business cycle literature typically defines the business cycle as a
phenomenon occurring at the frequencies 2 quarters to 8 years. To narrow
down the focus to this domain, we must find an appropriate detrending method
that will allow us to focus attention on this particular frequency domain.

The standard technique in the literature has been to use an HP\ filter. HP
filter can flexibly and quickly remove lower frequencies that we choose to
remove.\footnote{%
An alterantive method would be to use the band-pass filter or linear
detrending. The advantage of the band-pass filter is that the frequency
domain can be precisely defined, and in the limit the filter exactly cuts
off the frequencies we want to abstract from. However, since this is only a
limiting result, the transparency of the HP\ filter smoothing still makes a
preferred choice. For more details on band-pass filter, see Lawrence J.
Christiano \& Terry J. Fitzgerald, 1999. "The Band Pass Filter," NBER
Working Papers 7257, National Bureau of Economic Research, Inc.}

\paragraph*{HP filter\protect\footnote{%
You can find a free HP\ filter for MATLAB at:
http://dge.repec.org/codes/izvorski/hpfilter.m}}

The reasoning behind the HP filter is as follows. Let $y_{t}=\log x_{t}$,
for $t=$1, 2 , ...,T, denote the logarithms of a time series variable $x_{t}$%
. The series $x_{t}$, is made up of a trend component, denoted by $\bar{y}%
_{t}=\log \bar{x}_{t}$, and a cyclical component, denoted by $c_{t}$, such
that 
\begin{equation*}
x_{t}=\bar{y}_{t}+c_{t}.
\end{equation*}

The log in the above formulation allows us to focus attention on the
normalized percentage deviations from trend, rather than less meaningful
absolute deviations. It follows from the following approximation using the
Taylor's theorem:%
\begin{eqnarray*}
c_{t} &\equiv &y_{t}-\bar{y}_{t}\equiv \log x_{t}-\log \bar{x}_{t}= \\
&=&\frac{x_{t}-\bar{x}_{t}}{\bar{x}_{t}}+o(x_{t}-\bar{x}_{t}),
\end{eqnarray*}%
where $\bar{x}_{t}\equiv \exp (\bar{y}_{t})$.

Given an adequately chosen positive value of $\lambda $, the HP-trend
component $\bar{y}_{t}$ solves

\begin{equation*}
\min_{\{\bar{y}_{t}\}_{t}^{T}}\sum_{t}^{T}(y_{t}-\bar{y}_{t})^{2}+\lambda
\sum_{t}^{T-1}\left[ (\bar{y}_{t+1}-\bar{y}_{t})-(\bar{y}_{t}-\bar{y}_{t-1})%
\right] ^{2}.
\end{equation*}

The above objective function can be understood as follows. The first term is
the sum of the squared deviations $y_{t}-\bar{y}_{t}$ which penalizes the
cyclical component. So, the trend component should be as close as possible
to the actual series. The second term is a multiple $\lambda $ of the sum of
the squares of the trend component's second differences. This term penalizes
variations in the growth rate of the trend component. By putting the two
terms together, the objective\ function trades off the smoothness of the
trend (second component) with the objective of tracking closely the actual
time-series$.$Clearly, the larger the value of $\lambda $, the more weight
we put on trend being smooth. Hodrick and Prescott advise that, for
quarterly data, a value of $\lambda =1600$ is a reasonable. For annual data,
a value of 6.25 is recommended\footnote{%
Ravn, Morten O. \& Uhlig, Harald, 2001. "On Adjusting the HP-Filter for the
Frequency of Observations," CEPR Discussion Papers 2858, C.E.P.R. Discussion
Papers.}, although in the case of annual data researchers often use values
significantly higher than that.

From now one, unless otherwise noted, all statistics we talk about refer to
variables that are first logged and the HP filtered to recover the cyclical
component. Occasionally, we will use linear detrending to preserve the
information about persistence of the underlying time-series.

\section{Prototype International Business Cycle Model}

In this section, we set up a simple two-country endowment model (along the
lines of Lucas (1982)). The model will adopt the Armington framework as the
underlying trade model, and in line with the RBC literature assume that the
business cycles are driven by an assumed stochastic process. Agents in the
model will be aware of the properties of this process and how the economy
works.\footnote{%
Models departing from the assumption of rational expectations are not policy
invariant and subject to Lucas critique. See Lucas (1976), \textquotedblleft
Econometric Policy Evaluation: A Critique\textquotedblright ,
Carnegie-Rochester Conference Series on Public Policy.}

Our simple model will allow us to focus attention on the workings of the
demand-side of business cycle models (including some NOEM models), and for
the time being will ignore the supply side. This will be a very useful
exercise to develop intuition for later.

\subsection*{Model Economy}

There are 2 ex-ante symmetric countries labeled domestic country, and
foreign country. Households in the domestic country own an endowment tree
that stochastically pays off in the domestic goods $d$, and households in
the foreign country own a similar endowment tree that pays off in the
foreign good $f.$ Both types of goods, $d$ and $f,$ are used for
consumption, are not perfectly substitutable, and are tradable. Trade is
frictionless, but preferences are biased towards the local good.\footnote{%
Here we build trade friction directly into preferences as a home bias
parameter $\omega .$ An almost equivalent formulation would be to assume an
iceberg transportation cost instead.}

\subsubsection*{Uncertainty}

In any period $t,$ the world economy experiences one of the finitely many
stochastic events $s_{t}\in S.$ The history of such events up to and
including period $t$ is denoted by\footnote{%
In particular, the realization of the exogenous endowment is a function of
the history of events, i.e. $y(s^{t}),y^{\ast }(s^{t})$.} $s^{t},$ where $%
s^{t}=(s_{0},s_{1},...,s_{t}).$ The product probability of each history is
known and denoted by $\pi (s^{t}).$

For later use, note that our history dependent notation implies that it is
the same to write $(s^{t-1},s_{t})$ as $s^{t}$ or $(s_{0},s_{1},...,s_{t}).$

\subsubsection{Asset Market}

In each country, there is a separate asset market in which a one-period
forward state contingent bond is traded. At home, this bond is denominated
in the home country numeraire, and abroad this bond is denominated in the
foreign country numeraire. Bonds allow households to frictionlessly transfer
wealth across all dates and states.

The asset market is assumed to be internationally integrated, in the sense
that households in each country can trade bonds of both types bond.
Specifically, there is a state contingent price of the foreign numeraire in
terms of domestic numeraire that allows the households to trade foreign
numeraire good for the domestic one. This price is denoted by $x$. H%
\footnote{%
Two types of bonds are introduced to make the model symmetric. One state
contingent bond would be sufficient (equivalent).}

In terms of notation, all variables that have a foreign country analog, are
distinguished by an asterisk, and the setup is symmetric between the
domestic and the foreign country. By symmetry we mean ex-ante symmetry,
meaning that the same probability laws that govern the stochastic
realization of endowment in each country.

\subsubsection*{Households}

Households supply goods to the market, trade financial assets, and purchase
final consumption goods. At each history $s^{t},$ they choose their
allocation, 
\begin{equation*}
c(s^{t}),\text{ }d(s^{t}),\text{ }f(s^{t}),%
\{B_{d}(s_{t+1},s^{t}),B_{f}(s_{t+1},s^{t})\}_{s_{t+1}\in S},
\end{equation*}%
to maximize the expected present discounted utility%
\begin{equation}
U=\max \sum_{t}^{\infty }\sum_{s^{t}\in S^{t}}\beta ^{t}\pi
(s^{t})u(c(s^{t}))  \label{3Eq1}
\end{equation}

\begin{equation*}
c(s^{t})=G(d(s^{t}),f(s^{t})),\text{ all }s^{t}\in S^{t}\text{,}
\end{equation*}%
where G($\cdot $) is given by CES aggregator parameterized by elasticity $%
\sigma $ and home-bias parameter $\omega :$%
\begin{equation*}
G(d,f)=\left\{ 
\begin{array}{c}
d^{\omega }f^{1-\omega }\text{ if }\sigma =1 \\ 
(\omega d^{\frac{\sigma -1}{\sigma }}+(1-\omega )f^{\frac{\sigma -1}{\sigma }%
})^{\frac{\sigma }{\sigma -1}}\ \text{if }\sigma \neq 1%
\end{array}%
\right. ,
\end{equation*}%
and the utility function is parameterized by risk aversion parameter $\theta 
$:\ 
\begin{equation*}
u(c)=\left\{ 
\begin{array}{c}
\log (c)\text{ if }\theta =1 \\ 
\frac{c^{1-\theta }}{1-\theta }\text{ if }\theta \neq 1%
\end{array}%
\right. .
\end{equation*}

Household's utility maximization is subject to:\ (i)\ non-Ponzi condition $%
B_{d}(s_{t+1},s^{t})\geq -B,B_{f}(s_{t+1},s^{t})\geq -B,$ where $B$ is some
arbitrarily large constant, (ii)\ non-negativity for all variables except
for bond holdings, and (ii) the budget constraint,\ 
\begin{eqnarray}
&&p_{d}(s^{t})d(s^{t})+p_{f}(s^{t})f(s^{t})+  \label{3Eq2} \\
&&+\sum_{s_{t+1}\in S}Q(s_{t+1}|s^{t})B_{d}(s_{t+1},s^{t})+ \\
&&+\sum_{s_{t+1}\in S}x(s^{t})Q^{\ast }(s_{t+1}|s^{t})B_{f}(s_{t+1},s^{t}) \\
&=&B_{d}(s^{t})+x(s^{t})B_{f}(s^{t})+p_{d}(s^{t})y(s^{t}),\text{ all }s^{t}.
\notag
\end{eqnarray}

The budget constraint reads from left to right: (i)\ expenditures on
consumption $p_{d}(s^{t})d(s^{t})+p_{f}(s^{t})f(s^{t}),$ (ii) purchases of
state contingent domestic bonds $B_{d}(s_{t+1},s^{t})$ at price $%
Q(s_{t+1}|s^{t}),$(ii) purchases of a set of state contingent foreign bonds $%
B_{f}(s_{t+1},s^{t})$ at price $x(s^{t})Q^{\ast }(s_{t+1}|s^{t})$ ($Q^{\ast
} $ is denominated in the foreign numeraire units and must be translated to
domestic numeraire units through $x$), (iii) income from maturing domestic
bonds that have been purchased at $s^{t-1}$ for contingency $s_{t},$(iii)
income from maturing foreign bonds purchased at state $s^{t-1}$ for
contingency $s_{t},$ and (iv) the endowment income $p_{d}(s^{t})y(s^{t}).$

The foreign households solve an analogous problem, which we state below to
clarify the notation: 
\begin{equation}
U^{\ast }=\max \sum_{t}^{\infty }\sum_{s^{t}\in S^{t}}\beta ^{t}\pi
(s^{t})u(c^{\ast }(s^{t}))  \label{3Eq3}
\end{equation}

\begin{equation*}
c^{\ast }(s^{t})=G(f^{\ast }(s^{t}),d^{\ast }(s^{t})),\text{ all }s^{t}\in
S^{t}\text{ }
\end{equation*}%
subject to 
\begin{eqnarray}
&&p_{d}^{\ast }(s^{t})d^{\ast }(s^{t})+p_{f}^{\ast }(s^{t})f^{\ast }(s^{t})+
\label{3Eq4} \\
&&+\sum_{s_{t+1}\in S}Q^{\ast }(s_{t+1}|s^{t})B_{f}^{\ast }(s_{t+1},s^{t})+ 
\notag \\
&&+\sum_{s_{t+1}\in S}\frac{Q(s_{t+1}|s^{t})}{x(s^{t})}B_{d}^{\ast
}(s_{t+1},s^{t})  \notag \\
&=&\frac{B_{d}^{\ast }(s^{t})}{x(s^{t})}+B_{f}^{\ast }(s^{t})+p_{f}^{\ast
}(s^{t})y^{\ast }(s^{t}),\text{ all }s^{t}\in S^{t}  \notag
\end{eqnarray}

The implied price index for the domestic country is given by 
\begin{equation}
P(s^{t})=\min_{G(d(s^{t}),f(s^{t}))=1}\left[
p_{d}(s^{t})d(s^{t})+p_{f}(s^{t})f(s^{t})\right]  \label{3Eq5a}
\end{equation}%
and for the foreign country by 
\begin{equation}
P^{\ast }(s^{t})=\min_{G(f^{\ast }(s^{t}),d^{\ast }(s^{t}))=1}\left[
p_{d}^{\ast }(s^{t})d^{\ast }(s^{t})+p_{f}^{\ast }(s^{t})f^{\ast }(s^{t})%
\right] .  \label{3Eq6}
\end{equation}%
Later, we will find it convenient to normalize the prices by assuming that
the composite consumption baskets in each country are the numeraire
(composite consumption basket will serve as numeraire in each country), i.e.%
\begin{equation}
P(s^{t})=P^{\ast }(s^{t})=1,\text{ all }s^{t}.  \label{3Eq7}
\end{equation}

Under such numeraire normalization, by definition, the domestic bond pays
off in the domestic composite consumption, and the foreign bond pays off in
the foreign composite consumption. The relative price of the foreign
numeraire in terms of the domestic numeraire $x(s^{t})$ is then, again by
definition, the ideal\footnote{%
Ideal (sometimes also called welfare-based) means that the basket with
respect to which CPI is measured is optimal on a period by period basis.}
real exchange rate. \textit{Real exchange rate is defined as the price of
the foreign consumption basket in terms of the domestic consumption basket}.
In the data, the real exchange rate is measured by the ratio of foreign CPI
to domestic CPI measured in common unit, $x=\frac{eCPI^{\ast }}{CPI},$ where 
$eCPI^{\ast }$ is foreign CPI converted to home currency units using nominal
exchange rate $e.$ Because in the model weights of the CPI\ price index are
optimal, we refer to it as \textit{ideal} CPI, and \textit{ideal} real
exchange rate. Sometimes this distinction of theoretical real exchange rate
and a corresponding data object makes a difference (like in the model by
Ghironi and Melitz (2005)). Here, the ideal CPI and fixed-weights CPI are
almost the same thing, and we will not make an explicit distinction.

\subsubsection*{Feasibility and Market Clearing}

Finally, the allocation must fulfil several market clearing and feasibility
conditions. First, frictionless trade in the goods market requires that the
law of one price holds:\footnote{%
We could have made the above condition endogenous by building in a choice
about location of sales into the household's utility maximization problem.}%
\begin{eqnarray}
p_{d}(s^{t}) &=&x(s^{t})p_{d}^{\ast }(s^{t}),  \label{3Eq8} \\
p_{f}(s^{t}) &=&x(s^{t})p_{f}^{\ast }(s^{t}),\text{ all }s^{t}.  \notag
\end{eqnarray}

Second, supply of goods must equal demand\ 
\begin{eqnarray}
d(s^{t})+d^{\ast }(s^{t}) &=&y(s^{t}),  \label{3Eq9} \\
f(s^{t})+f^{\ast }(s^{t}) &=&y^{\ast }(s^{t}),\text{ all }s^{t}.  \notag
\end{eqnarray}

Third, assets must be in zero-net supply:\ 
\begin{eqnarray}
B_{d}(s_{t+1},s^{t})+B_{d}^{\ast }(s_{t+1},s^{t}) &=&0,  \label{3Eq10} \\
B_{f}(s_{t+1},s^{t})+B_{f}^{\ast }(s_{t+1},s^{t}) &=&0,\text{ all }s^{t}. 
\notag
\end{eqnarray}%
\bigskip

\subsection*{Definition of Equilibrium}

Having formally laid out the model economy, we are now ready to define the
equilibrium.

\begin{definition}
By \textbf{competitive equilibrium} in this economy, we mean:

\begin{itemize}
\item prices $p_{d}(s^{t}),p_{f}(s^{t}),p_{d}^{\ast }(s^{t}),p_{f}^{\ast
}(s^{t}),Q(s_{t+1}|s^{t}),Q^{\ast }(s_{t+1}|s^{t}),x(s^{t}),$

\item and allocation $d(s^{t}),f(s^{t}),d^{\ast }(s^{t}),f^{\ast
}(s^{t}),c(s^{t}),c^{\ast }(s^{t}),B_{d}(s_{t+1}|s^{t}),$

$B_{d}^{\ast }(s_{t+1}|s^{t}),B_{f}(s_{t+1}|s^{t}),B_{f}^{\ast
}(s_{t+1}|s^{t})$
\end{itemize}

such that

\begin{itemize}
\item given prices, allocation solves household's problem given by (\ref%
{3Eq1}) and (\ref{3Eq3}),

\item law of one price (\ref{3Eq8}) is satisfied,

\item and all markets clear.
\end{itemize}
\end{definition}

\subsection*{Recursive Formulation}

[to be completed]

\subsection*{Characterization of the Equilibrium}

Note that because asset markets are complete, and the households can
independently transfer wealth between any states and dates, the 1st Welfare
Theorem applies. Consequently, the equilibrium allocation is Pareto optimal,
and thus solves a planning problem.

Given that the households are ex-ante symmetric, and so their ex-ante wealth
is the same, the planning problem for this economy is to choose the
allocation to maximize: 
\begin{equation*}
\max \left[ \sum_{t}^{\infty }\sum_{s^{t}\in S^{t}}\beta ^{t}\pi
(s^{t})u(c(s^{t}))+\sum_{t}^{\infty }\sum_{s^{t}\in S^{t}}\beta ^{t}\pi
(s^{t})u(c^{\ast }(s^{t}))\right]
\end{equation*}%
subject to aggregation constraints 
\begin{eqnarray*}
c(s^{t}) &=&G(d(s^{t}),f(s^{t})), \\
c^{\ast }(s^{t}) &=&f^{\ast }(s^{t})^{\omega }d^{\ast }(s^{t})^{1-\omega },%
\text{ all }s^{t}\in S^{t}\text{ }
\end{eqnarray*}%
and and feasibility constraints%
\begin{eqnarray*}
d(s^{t})+d^{\ast }(s^{t}) &=&y(s^{t}), \\
f(s^{t})+f^{\ast }(s^{t}) &=&y^{\ast }(s^{t}),\text{ all }s^{t},
\end{eqnarray*}

Since in the above problem, periods are not physically connected through the
objective function of the constraint set (no state variables, objective
function is time-separable), wlog, we can recast the above dynamic problem
as a sequence of static planning problems given by:%
\begin{equation*}
\max_{c,c^{\ast },d,f}\left[ u(c(s^{t}))+u(c^{\ast }(s^{t}))\right]
\end{equation*}%
subject to 
\begin{eqnarray*}
c(s^{t}) &=&G(d(s^{t}),f(s^{t})), \\
c^{\ast }(s^{t}) &=&G(f^{\ast }(s^{t}),d^{\ast }(s^{t})),\text{ }
\end{eqnarray*}%
and%
\begin{eqnarray*}
d(s^{t})+d^{\ast }(s^{t}) &=&y(s^{t}), \\
f(s^{t})+f^{\ast }(s^{t}) &=&y^{\ast }(s^{t}),\text{ all }s^{t}.
\end{eqnarray*}

Since the above planning problem has a unique solution, the 2nd Welfare
Theorem implies that the CE\ allocation not only exists, but it unique.

\begin{exercise}
\label{Exercise21}Prove the equivalence between dynamic problem and a
sequence of static problems formally. HINT:\ You are asked to show that if
an allocation solves one problem, it solves the other.
\end{exercise}

\begin{exercise}
\label{3solution}Assume $\theta =1$ and $\sigma =1.$ Show that the solution
to the above planning problem is of the form:\ 
\begin{eqnarray*}
c(s^{t}) &=&\omega ^{\omega }(1-\omega )^{1-\omega }y(s^{t})^{\omega
}y^{\ast }(s^{t})^{1-\omega }, \\
c^{\ast }(s^{t}) &=&\omega ^{\omega }(1-\omega )^{1-\omega }y^{\ast
}(s^{t})^{\omega }y(s^{t})^{1-\omega } \\
d(s^{t}) &=&\omega y(s^{t}), \\
d^{\ast }(s^{t}) &=&(1-\omega )y(s^{t}), \\
f(s^{t}) &=&(1-\omega )y^{\ast }(s^{t}), \\
f^{\ast }(s^{t}) &=&\omega y^{\ast }(s^{t}),\text{ all }s^{t}.
\end{eqnarray*}
\end{exercise}

\begin{exercise}
Set $\theta =1$ and $\sigma =1,$ and consider financial autarky (no
borrowing and lending possible), i.e. 
\begin{equation*}
B_{d}(s_{t+1}|s^{t})=B_{d}^{\ast }(s_{t+1}|s^{t})=B_{f}^{\ast
}(s_{t+1}|s^{t})=B_{f}(s_{t+1}|s^{t})=0,\text{ all }s^{t},s_{t+1}.
\end{equation*}%
Show that the allocation under financial autarky is exactly the same as in
the complete asset market economy. Interpret your findings. In particular,
answer what it implies about the dynamics of the trade balance in this
economy.
\end{exercise}

We next proceed to find the supporting prices and complete the
characterization of the equilibrium.

\subsubsection*{Prices}

Let $\lambda (s^{t})$ be the multiplier on budget constraint, and $\mu
(s^{t})$ be the multiplier on the aggregation constraint.

The first order conditions of the domestic household are given by%
\begin{eqnarray*}
c(s^{t}) &:&\beta \pi (s^{t})u^{\prime }(c(s^{t}))-\mu (s^{t})=0, \\
B(s_{t+1}|s^{t}) &:&-\lambda (s^{t})Q(s_{t+1}|s^{t})+\lambda (s^{t+1})=0, \\
B^{\ast }(s_{t+1}|s^{t}) &:&-\lambda (s^{t})x(s^{t})Q^{\ast
}(s_{t+1}|s^{t})+\lambda (s^{t+1})x(s^{t+1})=0, \\
d(s^{t}) &:&-\mu (s^{t})G_{d}(d(s^{t}),f(s^{t}))-\lambda
(s^{t})p_{d}(s^{t})=0, \\
f(s^{t}) &:&-\mu (s^{t})G_{f}(d(s^{t}),f(s^{t}))-\lambda
(s^{t})p_{f}(s^{t})=0.
\end{eqnarray*}%
and the first order conditions of the foreign household are

\begin{eqnarray*}
c(s^{t}) &:&\beta \pi (s^{t})u^{\prime }(c^{\ast }(s^{t}))-\mu ^{\ast
}(s^{t})=0, \\
B_{f}^{\ast }(s_{t+1}|s^{t}) &:&-\lambda ^{\ast }(s^{t})Q^{\ast
}(s_{t+1}|s^{t})+\lambda ^{\ast }(s^{t+1})=0, \\
B_{d}^{\ast }(s_{t+1}|s^{t}) &:&-\lambda ^{\ast }(s^{t})\frac{%
Q(s_{t+1}|s^{t})}{x(s^{t})}+\frac{\lambda ^{\ast }(s^{t+1})}{x(s^{t+1})}=0,
\\
d(s^{t}) &:&-\mu ^{\ast }(s^{t})G_{d}(f(s^{t}),d(s^{t}))-\lambda ^{\ast
}(s^{t})p_{d}^{\ast }(s^{t})=0, \\
f(s^{t}) &:&-\mu ^{\ast }(s^{t})G_{f}(f(s^{t}),d(s^{t}))-\lambda ^{\ast
}(s^{t})p_{f}^{\ast }(s^{t})=0.
\end{eqnarray*}

From the first set of equations, we derive 
\begin{eqnarray}
(i) &:&Q(s_{t+1}|s^{t})=\beta \pi (s_{t+1}|s^{t})\frac{u^{\prime
}(c(s^{t+1}))}{u^{\prime }(c(s^{t}))},  \label{3FOC1} \\
(ii) &:&\frac{x(s^{t})}{x(s^{t+1})}Q^{\ast }(s_{t+1}|s^{t})=\beta \pi
(s_{t+1}|s^{t})\frac{u^{\prime }(c(s^{t+1}))}{u^{\prime }(c(s^{t}))},  \notag
\\
(iii) &:&p_{d}(s^{t})=G_{d}(d(s^{t}),f(s^{t})),  \notag \\
(iv) &:&p_{f}(s^{t})=G_{f}(d(s^{t}),f(s^{t})).  \notag
\end{eqnarray}%
where $\pi (s_{t+1}|s^{t})\equiv \frac{\pi (s^{t+1})}{\pi (s^{t})}$ is the
conditional probability of state $s_{t+1}$ conditional on $s^{t},$ and from
the second set, we have: 
\begin{eqnarray}
(v) &:&Q^{\ast }(s_{t+1}|s^{t})=\beta \pi (s_{t+1}|s^{t})\frac{u^{\prime
}(c^{\ast }(s^{t+1}))}{u^{\prime }(c^{\ast }(s^{t}))},  \label{3FOC2} \\
(vi) &:&\frac{x(s^{t+1})}{x(s^{t})}Q(s_{t+1}|s^{t})=\beta \pi (s_{t+1}|s^{t})%
\frac{u^{\prime }(c^{\ast }(s^{t+1}))}{u^{\prime }(c^{\ast }(s^{t}))}, 
\notag \\
(vii) &:&p_{d}^{\ast }(s^{t})=G_{d}(f^{\ast }(s^{t}),d^{\ast }(s^{t})), 
\notag \\
(viii) &:&p_{f}(s^{t})=G_{f}(f^{\ast }(s^{t}),d^{\ast }(s^{t})).  \notag
\end{eqnarray}

Note that if prices are normalized so that $P\equiv P^{\ast }\equiv 1$, we
must have%
\begin{equation*}
\lambda (s^{t})=\mu (s^{t}).
\end{equation*}

\begin{exercise}
\label{Exercise23}Show that $\lambda (s^{t})=\mu (s^{t}).$ HINT: Use the
fact that at the optimal solution, by definition of the price index, we have:%
\begin{equation*}
p_{d}(s^{t})d(s^{t})+p_{f}(s^{t})f(s^{t})=P(s^{t})c(s^{t})=P(s^{t})d(s^{t})^{\omega }f(s^{t})^{1-\omega }.
\end{equation*}%
Exploit the equality to show $\mu =\lambda $.
\end{exercise}

\paragraph*{Uncovered Interest Rate Parity}

In the first step, we combine equations (i) and (ii) to obtain a
state-by-state non-arbitrage condition on bond prices:%
\begin{equation}
\frac{x(s^{t+1})}{x(s^{t})}=\frac{Q^{\ast }(s_{t+1}|s^{t})}{Q(s_{t+1}|s^{t})}%
.  \label{3bnac}
\end{equation}%
Intuitively, this condition says that there are no pure profits taking a
short position on one bond in some states and an offsetting long position on
the other bond.

The above condition is closely related to the so called \textit{uncovered
interest rate parity. }The \textit{uncovered interest rate parity} states
that the domestic risk-free interest rate must be equal to the foreign risk
free interest rate augmented by the expected change of the exchange rate
between the two periods. In real terms, if we let $r$ denote an effective%
\footnote{%
The definition of an effective return of a bond is $r=-logP$, where $P$ is
the per-unit price of the bond. The formula comes from the idea that
compounding is continuous, and so the future value of the price of the bond
under continous compounding must be equal to the promised payoff of the bond
(=\$1): 1\$=lim $_{n\rightarrow \infty }$P(1+$\frac{r}{n}$)$^{n}=P\exp (r),$
and so $r=-logP.$} risk free domestic interest rate on domestic bonds and $%
r^{\ast }$ denote the effective risk-free interest rate on foreign bonds,
the uncovered interest rate parity says:

\begin{equation*}
\exp (r(s^{t}))=E[\exp (r^{\ast }(s^{t}))\frac{x(s^{t+1})}{x(s^{t})}],
\end{equation*}%
which we can rewrite in logs as%
\begin{equation*}
r(s^{t})=r^{\ast }(s^{t})+\log E(\frac{x(s^{t+1})}{x(s^{t})}|s^{t}).
\end{equation*}

Since UIP equalizes the expected payoff from the two alternative investment
strategies that shift wealth from today to tomorrow, when agents are risk
neutral and rational, it should be clear why such condition must hold. In
particular, in such case, it follows directly from condition (\ref{3bnac})
listed above---but not all the way around.

However, when the agents are risk averse, the state-by-state non-arbitrage
condition turns out to be not equivalent to the UIP condition. The reason is
that investment in the home risk-free bond is risk free for the home
households, but investment in the foreign bond is not due to stochastic
movements of the real exchange rate that may be potentially correlated with
consumption. Thus, in general, the model may not or may predict that UIP
should hold. In what follows, we will investigate to what extent it does.

Before we proceed, it is convenient to adopt the standard language of
finance and define the conditional pricing kernels\footnote{%
Pricing kernel is a price of one unit of payoff in state $s_{t+1}$
(following history $s^{t})$ under an abstract assumption that state $s_{t+1}$
occurs with probability 1.} (called also stochastic discount factor SDF)
that isolate the probabilities from the state contingent prices $Q$ and $%
Q^{\ast }$. In our model, they are given by 
\begin{eqnarray*}
M(s_{t+1}|s^{t}) &=&\frac{Q(s_{t+1}|s^{t})}{\pi (s_{t+1}|s^{t})}, \\
M^{\ast }(s_{t+1}|s^{t}) &=&\frac{Q^{\ast }(s_{t+1}|s^{t})}{\pi
(s_{t+1}|s^{t})}.
\end{eqnarray*}

The reason why it is more convenient is because using the conditional
pricing kernels we can simply price assets using an expectation operator,
and that simplifies notation.\footnote{%
It also makes the applications of probabilistic calculus more
straightforward (without going through the integrals, we can use the
formulas readily available).} For example, in the case of one-period forward
assets, we can price it as follows:\ 
\begin{equation*}
(\text{Price of asset})\text{ = }E[M(s_{t+1}|s^{t})\times (\text{payoff of
asset at }s^{t+1})|s^{t}],
\end{equation*}%
which can be written as%
\begin{equation*}
(\text{Price of asset})\text{ = }E_{t}[M\times (\text{payoff of asset at }%
t+1)].
\end{equation*}

Moreover, dividing both sides of the above equation by the price of the
asset, we can write

\begin{equation*}
1\text{= }E_{t}[M\times (\text{implied return on asset at }t+1)].
\end{equation*}%
In equilibrium, such condition must hold for all assets that are traded
(with no frictions). Otherwise, the risk averse agent, with kernel $M,$
could profit from arbitrage.

Going back to our model, we let $r\ $be the effective return on a one period
forward risk-free domestic bond denominated in domestic composite
consumption, and we let $r^{\ast }$ to be the effective return on a one
period forward risk free foreign bond (denominated in foreign consumption).
Given pricing kernels defined above, in equilibrium, the domestic household
must be indifferent whether to invest in any of the two assets, and so: 
\begin{eqnarray*}
1 &=&E_{t}[M\times \frac{x_{t+1}\exp (r_{t}^{\ast })}{x_{t}}], \\
1 &=&E_{t}[M\times \exp (r_{t})].
\end{eqnarray*}%
(The above conditions would be endogenously implied by FOC\ if the budget
constraint additionally included risk-free assets in the household problem.)

Combining the above asset pricing equations, we obtain 
\begin{equation*}
E_{t}[M\times \exp (r_{t})]=E_{t}[M\times \frac{x_{t+1}exp(r_{t}^{\ast })}{%
x_{t}}],
\end{equation*}%
which in log terms implies%
\begin{equation*}
r_{t}-r_{t}^{\ast }=\log E_{t}[M\times \frac{x_{t+1}}{x_{t}}]-\log E_{t}[M].
\end{equation*}

Using (\ref{3bnac}), we next note that%
\begin{equation*}
\log E_{t}[M\times \frac{x_{t+1}}{x_{t}}]=\log E_{t}[M^{\ast }].
\end{equation*}%
Substituting into the previous expression, we derive%
\begin{equation}
r_{t}-r_{t}^{\ast }=\log E_{t}[M^{\ast }]-\log E_{t}[M],  \label{3interim}
\end{equation}%
which in combination with the expression derived from taking expectation of
the log of equation (\ref{3bnac}), 
\begin{equation}
E_{t}\log \frac{x(s^{t+1})}{x(s^{t})}=E_{t}[\log M^{\ast }]-E_{t}[\log M],
\label{3interim1}
\end{equation}%
gives 
\begin{eqnarray*}
r_{t}-r_{t}^{\ast } &=&\log E_{t}[M^{\ast }]-\log E_{t}[M]= \\
&=&\log E_{t}[M^{\ast }]-\log E_{t}[M]+ \\
&&+\left[ E_{t}[\log \frac{x(s^{t+1})}{x(s^{t})}]-E_{t}[\log M^{\ast
}]+E_{t}[\log M]\right] .
\end{eqnarray*}

Defining the residual as \textit{risk premium} $\mathcal{P}_{t}$ (risk
premium essentially means residual) 
\begin{eqnarray*}
\mathcal{P}_{t} &\equiv &\left[ \log E_{t}[M^{\ast }]-E_{t}[\log M^{\ast }]%
\right] -\left[ \log E_{t}[M]-E_{t}[\log M]\right] + \\
&&+E_{t}\log \frac{x(s^{t+1})}{x(s^{t})}-\log E_{t}\frac{x(s^{t+1})}{x(s^{t})%
}
\end{eqnarray*}%
we can write the UIP\ condition implied by the the model as 
\begin{equation*}
r_{t}=r_{t}^{\ast }+\log E_{t}\frac{x(s^{t+1})}{x(s^{t})}+\mathcal{P}_{t}.
\end{equation*}

To understand the intuition behind the above equation, it is instructive to
consider a special case of pricing kernels and real exchange rate growth
rate that are log-normally distributed. The trick here is that in the case
of a lognormally distributed random variable, we can easily evaluate the
expectation of this variable by exploiting the following fact:

\begin{eqnarray}
X &\sim &\log \text{normal}  \label{3LogNormal} \\
E(X) &=&e^{\mu +\sigma ^{2}/2},  \notag
\end{eqnarray}%
where $\mu $ is the mean of $\log X$ (which is normally distributed), and $%
\sigma ^{2}$ is the variance of $\log X$.

\begin{exercise}
Prove the above property by integrating over the normal distribution.
\end{exercise}

So, let's assume that $\log M$, $\log M^{\ast }$ and $\log \frac{x_{t+1}}{%
x_{t}}$ are both normally distributed. In such case, from (\ref{3LogNormal}%
),\ we have 
\begin{eqnarray*}
\log E_{t}[M] &=&\log e^{E_{t}[\log M]+Var_{t}\left[ \log M\right] /2}, \\
\log E_{t}[M^{\ast }] &=&\log e^{E_{t}\left[ \log M^{\ast }\right] +Var_{t}%
\left[ \log M^{\ast }\right] /2}, \\
\log E_{t}[\frac{x_{t+1}}{x_{t}}] &=&\log e^{E_{t}\left[ \log \frac{x_{t+1}}{%
x_{t}}\right] +Var_{t}\left[ \log \frac{x_{t+1}}{x_{t}}\right] /2}
\end{eqnarray*}%
thus%
\begin{equation*}
\mathcal{P}_{t}=\frac{Var_{t}\left[ \log M^{\ast }\right] }{2}-\frac{Var_{t}%
\left[ \log M\right] }{2}-\frac{Var_{t}\left[ \log \frac{x_{t+1}}{x_{t}}%
\right] }{2},
\end{equation*}%
and 
\begin{equation}
r_{t}=r_{t}^{\ast }+\log E_{t}[\frac{x(s^{t+1})}{x(s^{t})}]+\mathcal{P}_{t},
\label{3ModelUIP}
\end{equation}%
where E$_{t}$ and Var$_{t}$ denote conditional expectation based on the
information available at $t$ and conditional variance based on the
information available at $t$, respectively.

The above simplified equation is much easier to interpret and understand. It
simply says that the return on the domestic risk free bond adds a premium
(potentially time-varying)\ whenever the difference between the conditional
volatility of the foreign pricing kernel and domestic pricing kernel
differs, or alternatively conditional variance of the real exchange rate
growth differs.

This above result can be understood as follows. Since the depreciation of
the real exchange rate is correlated with the domestic pricing kernel, we
should expect that either long- or short position in the foreign bond market
is a good hedge for domestic households against their consumption risk. For
the sake of argument, say that a long position on the foreign bond hedges
home households.\footnote{%
What this means is that the pricing kernel $M$ positively covaries with the
payoff of the bond $\frac{x_{t+1}}{x_{t}}r^{\ast }$, and so the bond pays
off exactly when the household needs more consumption (is hungry).}
Consequently, in order to hedge consumption risk, the domestic country
household could take a short position on the domestic country bond and a
long position on the foreign country bond. Similarly, the foreign country
household could take a long position on the domestic country bond and a
short position on the foreign country bond. Now, because these position
would be exactly offsetting, as long as there is symmetry in consumption
risk across countries, the market for bonds can clear, and neither bond is
traded at a premium. However, there is a flip side to this argument.
Whenever symmetry is distorted in some way, so that one of the countries
faces higher consumption risk as measured by $var_{t}(M)$, the demand and
supply for the bonds will no longer be balanced. As a result, one of the
bonds will have to be traded at a premium for the market to clear. This is
exactly what the risk premium term $\mathcal{P}_{t}$ captures.

(The second `real exchange rate' term is just a mechanical implication of
Jensen inequality and volatility of real exchange rate, and doesn't play a
major role in deviations from UIP in the data.)

\begin{exercise}
\label{Exercise24}Assume, you know the state contingent effective rate of
inflation both at home and abroad. Let the notation for inflation be $\Pi
(s^{t}),$ and $\Pi ^{\ast }(s^{t}),$ to distinguish inflation from the
state-contingent probabilities. Assuming log-normality whenever needed,
derive the deviations from the nominal version of the UIP in our model.
HINT:\ Real exchange rate is linked to nominal exchange rate $e$ by the
following relation $x=\frac{eP^{\ast }}{P},$ and so $\frac{x_{t+1}}{x_{t}}=%
\frac{e_{t+1}}{e_{t}}\frac{\exp (\Pi ^{\ast }(s^{t}))}{\exp (\Pi (s^{t}))}.$
\end{exercise}

\paragraph{Forward Premium Puzzle}

Despite its theoretical appeal and simplicity, there is little evidence
supporting the uncovered interest rate parity in the data. The UIP\ relation
has been tested widely by running a regression of interest rate premia $%
r-r^{\ast }$ on the real exchange rate changes (called Fama regression).
According to the UIP hypothesis, the implied regression coefficient should
be positive, and close to 1. The puzzling finding is that it is
significantly negative and its value is around\footnote{%
See the survey by Engel (1996).} -3, which is referred to as the \textit{%
forward premium anomaly or the UIP\ puzzle. }

Clearly, from the perspective of a risk neutral agent, the failure of the
UIP\ relation implies that there are excess returns from taking a short
position on the low interest rate currencies (like Japan), and by taking an
offsetting long position on the high interest rate currencies (like Poland).
This strategy is called `carry trade', and it is actually exploited by
investors (and they do make money!). Of course, this does not necessarily
imply that the `carry trade' strategy is a good deal for a risk averse
agent, as the historically observed premium may be simply a compensation for
risk. The literature has tried to identify the risk factor associated with
the forward premium, and has so far failed to identify one. 

The question is whether our model, which does imply some deviations from
UIP, can account for the negative regression coefficient. Unfortunately, the
answer is negative. In our model, the term $\mathit{P}_{t}$ is not going to
move much, and UIP\ will approximately hold.\footnote{%
It is called forward premium puzzle because the interest rate differential
can be expressed as a difference between the forward exchange rate and spot
exchange rate, $f_{t}-s_{t}=r-r^{\ast }.$ This is a completely risk free
arbitrage condition, as here the investor needs to buy a foreign bond and at
the same time a future contract for the exchange rate. This relation, called
covered interest rate parity, holds in the data. The term $f_{t}-s_{t}$ is
called forward premium.} \ Our next task is to demonstrate this property of
the model. 

To this end, let's take a look at the analytical case with log utility and
Cobb-Douglas utility function. In this case, we know that

\begin{eqnarray*}
Var_{t}\left[ \log M\right]  &=&Var_{t}\left[ \log \beta \frac{c_{t+1}}{c_{t}%
}\right] = \\
&=&Var_{t}\left[ \log \beta \omega ^{\omega }(1-\omega )^{1-\omega }\frac{%
y_{t+1}}{y_{t}}^{\omega }\frac{y_{t+1}^{\ast }}{y_{t}^{\ast }}^{1-\omega }%
\right] = \\
&=&Var_{t}\left[ \omega \log \frac{y_{t+1}}{y_{t}}+(1-\omega )\log \frac{%
y_{t+1}^{\ast }}{y_{t}^{\ast }}\right] .
\end{eqnarray*}%
Assuming the standard AR(1) stochastic process for income,%
\begin{equation*}
\log y_{t+1}=\rho \log y_{t}+\varepsilon _{t},
\end{equation*}%
we calculate (note:\ variance is conditional on period $t$ and so E$_{t}\log
y_{t}$ is just a known constant here)%
\begin{gather*}
Var_{t}\left[ \log M\right] =Var_{t}\left[ \omega \log \frac{y_{t+1}}{y_{t}}%
+(1-\omega )\log \frac{y_{t+1}^{\ast }}{y_{t}^{\ast }}\right] = \\
=\omega ^{2}Var_{t}[\log y_{t+1}]+(1-\omega )^{2}Var[\log y_{t+1}^{\ast }]+
\\
2\omega (1-\omega )Cov_{t}[\log y_{t+1}^{\ast },\log y_{t+1}]= \\
=\omega ^{2}Var_{t}[\varepsilon _{t+1}]+(1-\omega )^{2}Var[\varepsilon
_{t+1}^{\ast }]+ \\
+2\omega (1-\omega )Cov_{t}[\varepsilon _{t+1}^{\ast },\varepsilon _{t+1}]
\end{gather*}%
and thus under symmetry (i.e. $Var_{t}[\varepsilon
_{t+1}]=Var_{t}[\varepsilon _{t+1}^{\ast }]):$%
\begin{equation*}
\frac{Var_{t}\left[ \log M^{\ast }\right] }{2}-\frac{Var_{t}\left[ \log M%
\right] }{2}=0.
\end{equation*}

The above finding is not just a property of the particular log/Cob-Douglas
case. A similar property holds more generally. In fact, it is not an easy
task to generate sensibly looking fluctuations of the risk premium in this
class of models. The problem seems to be the conditioning on the information
from period $t$ that shows up in the formula for risk premium $\mathcal{P}%
_{t}.$ To have any time-varying fluctuations, the model has to generate
heteroscedasticity in either the uncertainty structure or the sensitivity of
the pricing kernels to uncertainty. One of the features that can give rise
to the latter are market segmentation and habit formation. This has been
documented in the following papers: Atkeson, Kehoe and Alvarez (2008):
\textquotedblleft Time-Varying Risk, Interest Rates, and Exchange Rates in
General Equilibrium\textquotedblright , Minneapolis FED Staff Report 371,
September, and Verdelhan (2008): \textquotedblleft A Habit-Based Explanation
of the Exchange Rate Risk Premium\textquotedblright , Journal of Finance,
forthcoming.\footnote{%
In the first paper mentioned above, the pricing kernel $M$ and $M^{\ast }$
is the pricing kernel of only active traders in a given moment of time, and
exhibits time-varying sensitivity to shocks. In the second paper, the
conditional variance of the kernel is time-varying due to time-varying risk
aversion in the habit model. An alternative way to go, more difficult to
discipline, is to play with the uncertainty structure of the model and
introduce time-varying exposure to shocks. Some work along these lines you
can find in the most recent paper by Lustig, Roussanov, Verdelhan (2008).}
Some authors also try to account for this fact by models exhibiting
ambiguity aversion (robust control). 

\paragraph*{Real Exchange Rate and Risk Sharing}

In this environment, real exchange rates are tightly linked to relative
consumption. Combining equations (i) and (vi) to substitute out for $%
Q(s_{t+1},s^{t}),$ a recursive equation for the real exchange rate can be
obtained:%
\begin{equation*}
x(s^{t+1})=x(s^{t})\frac{u^{\prime }(c(s^{t}))}{u^{\prime }(c(s^{t+1}))}%
\frac{u^{\prime }(c^{\ast }(s^{t+1}))}{u^{\prime }(c^{\ast }(s^{t}))}.
\end{equation*}

Using the fact that in state $s^{0}$ is deterministic, under the assumption
of ex-ante symmetry, one can collapse the above recursive law by solving it
backwards, and write instead%
\begin{equation}
x(s^{t})=\frac{u^{\prime }(c^{\ast }(s^{t}))}{u^{\prime }(c(s^{t}))}.
\label{3PRC}
\end{equation}

The above equation is probably the most famous equation in international
economics. It says that households trade assets to equalize the MRS of
consumption abroad to consumption at home so that it aligns with the
relative price of consumption abroad to consumption at home. A a result, to
the first approximation, a household consumes more in a given state and data
if and only if its consumption is cheaper in this state and data.

The above condition would not be that surprising, if the MRS pertained to
the same household. It would then say that the household optimally trades
off consumption of $c^{\ast }$ and $c$ --- a standard optimality condition
to maximize total utility. However, the MRS here pertains to two marginal
utility of two separate households in two different countries. Nevertheless,
the model predicts that these household will trade assets, and effectively
act as if they were a family maximizing the joint utility. In this sense, we
can say that this equation represents \textit{perfect risk sharing}, and
refer to this condition as \textit{perfect risk sharing condition. }

\paragraph{Implications of Perfect Risk Sharing for Real Exchange Rates}

It is not difficult to see that our model may have a hard time to match the
real exchange rate and consumption data simultaneously. The real exchange
rates in the data are volatile, persistent (almost indistinguishable from
random walks in short time-series), and not very correlated with anything
else.\footnote{%
Real exchange rates in the data closely track nominal exchange rates.}
Consumption is persistent,\ but the least volatile among aggregate variables
due to consumption smoothing.

For example, a simple model with log utility and Cob-Douglas aggregator
would fall short at least by a factor of 4 in terms of volatility, presuming
it accounts for the consumption data. To see this, evaluate the standard
deviation of both sides of (\ref{3PRC})\ 
\begin{equation*}
var(\log x)=var(\log (c)-\log (c^{\ast })).
\end{equation*}%
Using variance decomposition (denote $var(\log x)\equiv var_{x}),$%
\begin{equation*}
var_{x}=(var_{c}+var_{c^{\ast }}-2corr_{c,c^{\ast }}std_{c^{\ast }}std_{c}),
\end{equation*}%
under symmetry ($std_{c}=std_{c^{\ast }}),$ we have: 
\begin{equation*}
\frac{std_{x}}{std_{y}}=\frac{std_{c}}{std_{y}}\sqrt{2(1-corr_{c,c^{\ast }})}%
.
\end{equation*}

Now, for the US\ data (versus rest of the world)\footnote{%
See Drozd and Nosal (2008), Table 7.}, we roughly have:$\ \frac{std_{x}}{%
std_{y}}=3,$ $\frac{std_{c}}{std_{y}}=\frac{3}{4},$ and $corr_{c,c^{\ast }}=%
\frac{1}{4}.$ Plugging in these moments to the above equation, we obtain%
\begin{equation}
\frac{std_{x}}{std_{y}}=\frac{3}{4}\sqrt{2(1-\frac{1}{4})}=\frac{3}{4}\sqrt{%
\frac{6}{4}}=\frac{3\sqrt{6}}{8}=0.92.  \label{3compare1}
\end{equation}%
This is about 3 times less than the value in the data. 

At this point, you might be tempted to think that a mere departure from log
utility to CRRA utility $u(c)=\frac{c^{1-\theta }}{1-\theta }$ should fix
it. For CRRA, the analog of the above condition would say: 
\begin{equation*}
var(\log x)=\sigma ^{2}var(\log (c)-\log (c^{\ast })),
\end{equation*}%
and then%
\begin{equation*}
\frac{std_{x}}{std_{y}}=\sigma \frac{std_{c}}{std_{y}}\sqrt{%
2(1-corr_{c,c^{\ast }})}.
\end{equation*}

So, superficially, it seems that matching the data in terms of volatility,
should be just a matter of picking the right value of $\sigma .$ For
example, for the statistics we listed above, we would need to pick%
\begin{equation*}
\sigma \simeq 3.
\end{equation*}

Unfortunately, the problem is much deeper than that. The problem is that
models will also typically underpredict the volatility of $\frac{std_{c}}{%
std_{y}}$ and overstate the correlation $corr_{c,c^{\ast }}.$ As a result,
the same calculation but using the values implied by the models will look
much worse. Moreover, in the models with explicitly set up supply-side
(physical capital accumulation; labor/leisure choice), a higher value of $%
\sigma $ will create an additional incentive to smooth consumption using the
supply-side channels, and the smoother consumption will additionally offset
the effect of an increased value of $\sigma $ through these statistics.
Consequently, the resulting models with high $\sigma $ might neither match
consumption data nor the real exchange rate data. The literature refers to
this problem as the \textit{real exchange rate volatility puzzle.\footnote{%
Obstfeld and Rogoff (2000) give a nice overview of the 6 main puzzles of
international economics. } }

To some extent, our simple model also suffers from the problem described
above. Using the solution from exercise (\ref{3solution}), we can plug in to
the perfect risk sharing equation to derive: 
\begin{equation*}
x=(\frac{y}{y^{\ast }})^{(2\omega -1)}.
\end{equation*}%
Since our domestic country is US, to match the share of imports in US\ GDP, $%
\omega $ would have to be something around .85 (this is the share of
expenditures on foreign goods in total expenditures), and we would have 
\begin{equation*}
\log x=\frac{1}{2}(\log (y)-\log (y^{\ast })).
\end{equation*}%
Given that output is highly correlated across countries (about .4), under
symmetry, we thus must have%
\begin{equation*}
\frac{std_{x}}{std_{y}}=1-corr_{y,y^{\ast }}=0.6.
\end{equation*}

Comparing the above result to (\ref{3compare1}), we see that the model
performs much worse than it potentially could according to (\ref{3compare1}%
). The problem is that consumption is counterfactually more correlated than
output (in the data the opposite is true). Consequently, for our model to
match the data, we need to make the real exchange rate at least 5 times more
volatile, which is way more than required according to (\ref{3compare1}).

In the endowment economy, it is still possible to match real exchange rate
volatility by increasing $\sigma $ to 6 or 7. Nevertheless, also in this
problem the relationship between $\sigma $ and real exchange volatility will
be far from\ 1 to 1. The problem is that a high value of $\sigma $ creates
and additional incentive to trade more intensively to further improve on
consumption smoothing. (We would have to simultaneously set $\gamma $ low
and $\sigma $ high---this can help mechanically.)

In terms of the correlation of the real exchange rate with the consumption
ratio $\frac{c}{c^{\ast }}$, the model fares as bad as in terms of
volatility. Namely, according to equation (\ref{3PRC}) implies that the
correlation of the real exchange rate with consumption ratio should be one.
In the data, this correlation is negative (around $-0.2$). The problem with
correlation is known in the literature as \textit{Backus-Smith puzzle}.%
\footnote{%
The orginal reference is Backus and Smith (1993). See also Corsetti, Dedola
and Leduc (2007).}

\paragraph*{Terms of Trade and Export-Import Prices}

The model has predictions on 3 other basic international prices: (i)\ the
real export price $x(s^{t})p_{d}^{\ast }(s^{t})=p_{d}(s^{t})$, (ii)\ the
real import price $p_{f}(s^{t})$, and (iii) the terms of trade $\frac{%
p_{f}(s^{t})}{x(s^{t})p_{d}^{\ast }(s^{t})}$ (price of imports in terms of
exports).

In the data, we can construct these prices by looking at the aggregator
deflator price of imports or exports, and divide it by the CPI to make it
real.\footnote{%
In the data, weights in the CPI are kept constant for a couple of years and
do not represent optimal wights at all times. Fixing weights to measure CPI
in the model has almost no effect on the time-series of the CPI.} The
deflator price is defined as a ratio of the nominal value of a variable
divided by the constant price value of the variable. Terms of trade is
defined as the ratio of the deflator price of imports relative to the price
of exports.

According to the model, all these three prices should be closely related to
the terms of trade, which we will illustrate using our simple
log/Cobb-Douglas case. These properties, however, hold more generally.

In the Cobb-Douglas case, the ideal CPI can be expressed as a weighted
average of the prices of good $d$ and $f,$ i.e.\footnote{%
Precisely, $P=\omega ^{-\omega }(1-\omega )^{-(1-\omega )}p_{d}^{\omega
}p_{f}^{1-\omega }$} 
\begin{equation*}
P=p_{d}^{\omega }p_{f}^{1-\omega }.
\end{equation*}%
and abroad 
\begin{equation*}
P^{\ast }=p_{f}^{\ast \omega }p_{d}^{\ast 1-\omega }.
\end{equation*}%
Given that the CPI is a numeraire here ($P\equiv 1)$, we can derive the real
export price from the following evaluation%
\begin{equation*}
p_{x}\equiv \frac{xp_{d}^{\ast }}{P}=\frac{p_{d}}{p_{d}^{\omega
}p_{f}^{1-\omega }}=(\frac{p_{f}}{p_{d}})^{\omega -1},
\end{equation*}%
and the real import price from%
\begin{equation*}
p_{m}\equiv \frac{p_{f}}{P}=\frac{p_{f}}{p_{d}^{\omega }p_{f}^{1-\omega }}=(%
\frac{p_{f}}{p_{d}})^{\omega }.
\end{equation*}

As we can see, both prices are tightly linked to the terms of trade, which
is defined as:%
\begin{equation*}
p\equiv \frac{p_{f}}{xp_{d}^{\ast }}=\frac{p_{f}}{p_{d}}.
\end{equation*}

Furthermore, the real exchange in this environment is also intimately linked
to the terms of trade, and thus to the other prices. By definition, the real
exchange rate is given by the ratio of the foreign CPI\ to the domestic CPI
measured in a common unit. By the law of one price and assumed by us
numeraire normalization ($P^{\ast }\equiv P\equiv 1),$ we obtain the real
exchange rate from the following evaluation:\ 
\begin{equation*}
x\equiv \frac{xP^{\ast }}{P}=\frac{x(p_{f}^{\ast \omega }p_{d}^{\ast
1-\omega })}{p_{d}^{\omega }p_{f}^{1-\omega }}=\frac{p_{f}^{\omega
}p_{d}^{1-\omega }}{p_{d}^{\omega }p_{f}^{1-\omega }}=(\frac{p_{f}}{p_{d}}%
)^{2\omega -1}.
\end{equation*}

On the basis of the above formulas, we thus conclude that under the
conditions of home-bias (i.e. $\frac{1}{2}<\omega <\frac{1}{2}$), the model
has very sharp predictions how these 3 aggregate price should move. First,
the correlation between the real export price and the real import price
should be -1, second, the correlation of real export price with the real
exchange rate should be -1, and third, the terms of trade should be \textit{%
more} volatile than the real exchange rate.
 
\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=13,scale=.44]{Figures.pdf} 
 \caption{Comparison of real exchange
 rate with terms of trade (linearly detrended data). }%
\label{Fig_totrer}
\end{figure}

%
%\FRAME{ftbpFU}{324.7333pt}{222.4667pt}{0pt}{\Qcb{Comparison of real exchange
%rate with terms of trade (linearly detrended data). }}{\Qlb{Fig_totrer}}{%
%Figure}{\special{language "Scientific Word";type
%"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width
%324.7333pt;height 222.4667pt;depth 0pt;original-width
%731.3333pt;original-height 500.0667pt;cropleft "0";croptop "1";cropright
%"1";cropbottom "0";tempfilename 'KD2WCH07.wmf';tempfile-properties "XPR";}}

\paragraph{Export-Import Price Correlation Puzzle}

%TCIMACRO{\TeXButton{B}{\begin{table}[t] \centering}}%
%BeginExpansion
\begin{table}[t] \centering%
%EndExpansion
\caption{Real export and real import prices.s$^{a}$}

%TCIMACRO{%
%\TeXButton{DATA Tab1-4}{\begin{minipage}[b]{0.94\linewidth} 
%\begin{tabular}{p{0.9in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}}  
%\toprule  
%& \multicolumn{4}{l}{Correlation} & \multicolumn{3}{l}{Relative volatility$^{b}$ to $x$ (\%)}\\  
%\cmidrule(r){2-5}  
%\cmidrule(r){6-8}  
%Country & $p_{x},p_{m}$ & $p_{x},x$ & $p_{m},x$ & $p,x$ & $p_{x}$ & $p_{m}$ & $p$\\  
%\midrule  
%\small Belgium & \ \small 0.93& \ \small 0.71& \ \small 0.73& \ \small 0.43& \ \small 105.3& \ \small 131.7& \ \small 50.5\\  
%\small Canada & \ \small 0.77& \ \small 0.55& \ \small 0.90& \ \small 0.56& \ \small 75.8& \ \small 79.4& \ \small 52.8\\  
%\small Switzerland & \ \small 0.62& \ \small 0.50& \ \small 0.85& \ \small 0.77& \ \small 38.2& \ \small 83.0& \ \small 66.5\\  
%\small France & \ \small 0.90& \ \small 0.63& \ \small 0.66& \ \small 0.59& \ \small 74.5& \ \small 149.9& \ \small 89.2\\  
%\small Germany & \ \small 0.58& \ \small 0.44& \ \small 0.85& \ \small 0.83& \ \small 33.1& \ \small 101.4& \ \small 86.6\\  
%\small Italy & \ \small 0.87& \ \small 0.65& \ \small 0.69& \ \small 0.60& \ \small 55.9& \ \small 116.5& \ \small 73.3\\  
%\small Japan & \ \small 0.88& \ \small 0.92& \ \small 0.88& \ \small 0.72& \ \small 38.8& \ \small 86.1& \ \small 55.3\\  
%\small Netherlands & \ \small 0.94& \ \small 0.77& \ \small 0.81& \ \small 0.15& \ \small 130.9& \ \small 134.1& \ \small 47.0\\  
%\midrule  
%\small US & \ \small 0.74& \ \small 0.39& \ \small 0.71& \ \small 0.67& \ \small 37.0& \ \small 58.6& \ \small 40.0\\  
%\midrule  
%\small Australia & \ \small 0.45& \ \small 0.32& \ \small 0.94& \ \small 0.83& \ \small 42.1& \ \small 70.4& \ \small 63.8\\  
%\small Sweden & \ \small 0.88& \ \small 0.58& \ \small 0.71& \ \small 0.55& \ \small 58.7& \ \small 76.1& \ \small 37.1\\  
%\small UK & \ \small 0.90& \ \small 0.59& \ \small 0.78& \ \small 0.64& \ \small 58.4& \ \small 69.1& \ \small 30.6\\  
%\midrule 
%\sc Median&\ \small 0.87&\ \small 0.58&\ \small 0.80&\ \small 0.62&\ \small 57.1&\ \small 84.6&\ \small 54.1\\  
%\bottomrule \addlinespace\end{tabular}  
%\newline  
%\scriptsize $^{a}p_{x},p_{m}$ denote real export and import prices, $p\equiv \frac{p_{m}}{p_{x}}$ denotes terms of trade, $x$ trade weighted real exchange rate. Statistics based on logged \& HP filtered quarterly series for the period 1980:1-2000:1. Data sources listed in the Appendix. \newline $^{b}$Relative volatility is the standard deviation relatively to the standard deviation of the country's real exchange rate 
%\end{minipage} 
%
%}}%
%BeginExpansion
\begin{minipage}[b]{0.94\linewidth} 
\begin{tabular}{p{0.9in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}p{0.5in}}  
\toprule  
& \multicolumn{4}{l}{Correlation} & \multicolumn{3}{l}{Relative volatility$^{b}$ to $x$ (\%)}\\  
\cmidrule(r){2-5}  
\cmidrule(r){6-8}  
Country & $p_{x},p_{m}$ & $p_{x},x$ & $p_{m},x$ & $p,x$ & $p_{x}$ & $p_{m}$ & $p$\\  
\midrule  
\small Belgium & \ \small 0.93& \ \small 0.71& \ \small 0.73& \ \small 0.43& \ \small 105.3& \ \small 131.7& \ \small 50.5\\  
\small Canada & \ \small 0.77& \ \small 0.55& \ \small 0.90& \ \small 0.56& \ \small 75.8& \ \small 79.4& \ \small 52.8\\  
\small Switzerland & \ \small 0.62& \ \small 0.50& \ \small 0.85& \ \small 0.77& \ \small 38.2& \ \small 83.0& \ \small 66.5\\  
\small France & \ \small 0.90& \ \small 0.63& \ \small 0.66& \ \small 0.59& \ \small 74.5& \ \small 149.9& \ \small 89.2\\  
\small Germany & \ \small 0.58& \ \small 0.44& \ \small 0.85& \ \small 0.83& \ \small 33.1& \ \small 101.4& \ \small 86.6\\  
\small Italy & \ \small 0.87& \ \small 0.65& \ \small 0.69& \ \small 0.60& \ \small 55.9& \ \small 116.5& \ \small 73.3\\  
\small Japan & \ \small 0.88& \ \small 0.92& \ \small 0.88& \ \small 0.72& \ \small 38.8& \ \small 86.1& \ \small 55.3\\  
\small Netherlands & \ \small 0.94& \ \small 0.77& \ \small 0.81& \ \small 0.15& \ \small 130.9& \ \small 134.1& \ \small 47.0\\  
\midrule  
\small US & \ \small 0.74& \ \small 0.39& \ \small 0.71& \ \small 0.67& \ \small 37.0& \ \small 58.6& \ \small 40.0\\  
\midrule  
\small Australia & \ \small 0.45& \ \small 0.32& \ \small 0.94& \ \small 0.83& \ \small 42.1& \ \small 70.4& \ \small 63.8\\  
\small Sweden & \ \small 0.88& \ \small 0.58& \ \small 0.71& \ \small 0.55& \ \small 58.7& \ \small 76.1& \ \small 37.1\\  
\small UK & \ \small 0.90& \ \small 0.59& \ \small 0.78& \ \small 0.64& \ \small 58.4& \ \small 69.1& \ \small 30.6\\  
\midrule 
\sc Median&\ \small 0.87&\ \small 0.58&\ \small 0.80&\ \small 0.62&\ \small 57.1&\ \small 84.6&\ \small 54.1\\  
\bottomrule \addlinespace\end{tabular}  
\newline  
\scriptsize $^{a}p_{x},p_{m}$ denote real export and import prices, $p\equiv \frac{p_{m}}{p_{x}}$ denotes terms of trade, $x$ trade weighted real exchange rate. Statistics based on logged \& HP filtered quarterly series for the period 1980:1-2000:1. Data sources listed in the Appendix. \newline $^{b}$Relative volatility is the standard deviation relatively to the standard deviation of the country's real exchange rate 
\end{minipage} 

%
%EndExpansion
\label{Tab_pxpm}%
%TCIMACRO{\TeXButton{E}{\end{table}}}%
%BeginExpansion
\end{table}%
%EndExpansion

%TCIMACRO{\TeXButton{B}{\begin{table}[t] \centering}}%
%BeginExpansion
\begin{table}[t] \centering%
%EndExpansion
\caption{Relative volatility of the terms of trade.}

%TCIMACRO{%
%\TeXButton{DATA Tab1-4}{\begin{minipage}[b]{0.94\linewidth}
%\begin{tabular}{p{1.55in}p{1.20in}p{1.1in}p{1.1in}}
%\toprule
%& \multicolumn{3}{l}{\small Volatility of $p$ relative to $x$ (in \%)}\\
%\cmidrule(r){2-4}
%& \multicolumn{3}{l}{\small Price index used to construct$^{a}$ $x$}\\
%\small Country & \footnotesize CPI all-items & \footnotesize WPI or PPI & \footnotesize None (nominal)\\
%\midrule
%\small Australia & \  \small  0.51& \   \small 0.54& \   \small 0.60\\
%\small Belgium & \   \small 0.57& \   \small 0.70& \   \small 0.47\\
%\small Canada & \   \small 0.56& \   \small 0.76& \   \small 0.61\\
%\small France & \   \small 0.80& \   \small 0.74& \   \small 0.73\\
%\small Germany & \  \small  0.83& \   \small 0.81& \   \small 0.80\\
%\small Italy & \ \small   0.75& \  \small  0.79& \  \small  0.77\\
%\small Japan & \  \small  0.52& \  \small  0.54& \  \small  0.55\\
%\small Netherlands & \  \small   0.52& \   \small 0.49& \  \small  0.44\\
%\small Sweden & \   \small 0.21& \   \small 0.21& \  \small  0.37\\
%\small Switzerland & \  \small  0.71& \   \small 0.68& \   \small 0.67\\
%\small UK & \   \small 0.30& \   \small 0.32& \   \small 0.37\\
%\small US & \  \small  0.31& \  \small   0.33& \   \small 0.28\\
%\midrule
%\small MEDIAN&\   \small 0.54&\   \small 0.61&\   \small 0.57\\
%\bottomrule \addlinespace\end{tabular}
%\newline
%\scriptsize Notes: We have constructed trade-weighted exchange rates
%using weights and bilateral exchange rates for the set of 11 fixed
%trading partners for each country. The trading partners included in
%the sample are the countries listed in this table. Statistics are
%computed from logged and H-P-filtered quarterly time-series for the
%time period 1980:1-2000.01 ($\lambda=$1600). Data sources are listed
%at the end of the paper. \newline $^{a}$RER constructed these indices instead of the CPI.
%\end{minipage}}}%
%BeginExpansion
\begin{minipage}[b]{0.94\linewidth}
\begin{tabular}{p{1.55in}p{1.20in}p{1.1in}p{1.1in}}
\toprule
& \multicolumn{3}{l}{\small Volatility of $p$ relative to $x$ (in \%)}\\
\cmidrule(r){2-4}
& \multicolumn{3}{l}{\small Price index used to construct$^{a}$ $x$}\\
\small Country & \footnotesize CPI all-items & \footnotesize WPI or PPI & \footnotesize None (nominal)\\
\midrule
\small Australia & \  \small  0.51& \   \small 0.54& \   \small 0.60\\
\small Belgium & \   \small 0.57& \   \small 0.70& \   \small 0.47\\
\small Canada & \   \small 0.56& \   \small 0.76& \   \small 0.61\\
\small France & \   \small 0.80& \   \small 0.74& \   \small 0.73\\
\small Germany & \  \small  0.83& \   \small 0.81& \   \small 0.80\\
\small Italy & \ \small   0.75& \  \small  0.79& \  \small  0.77\\
\small Japan & \  \small  0.52& \  \small  0.54& \  \small  0.55\\
\small Netherlands & \  \small   0.52& \   \small 0.49& \  \small  0.44\\
\small Sweden & \   \small 0.21& \   \small 0.21& \  \small  0.37\\
\small Switzerland & \  \small  0.71& \   \small 0.68& \   \small 0.67\\
\small UK & \   \small 0.30& \   \small 0.32& \   \small 0.37\\
\small US & \  \small  0.31& \  \small   0.33& \   \small 0.28\\
\midrule
\small MEDIAN&\   \small 0.54&\   \small 0.61&\   \small 0.57\\
\bottomrule \addlinespace\end{tabular}
\newline
\scriptsize Notes: We have constructed trade-weighted exchange rates
using weights and bilateral exchange rates for the set of 11 fixed
trading partners for each country. The trading partners included in
the sample are the countries listed in this table. Statistics are
computed from logged and H-P-filtered quarterly time-series for the
time period 1980:1-2000.01 ($\lambda=$1600). Data sources are listed
at the end of the paper. \newline $^{a}$RER constructed these indices instead of the CPI.
\end{minipage}%
%EndExpansion
\label{Tab_totrer}%
%TCIMACRO{\TeXButton{E}{\end{table}}}%
%BeginExpansion
\end{table}%
%EndExpansion

As we can see from Tables \ref{Tab_pxpm}-\ref{Tab_totrer} and Figure \ref%
{Fig_totrer}), none of these predictions listed above are consistent with
the data for any of the 12 countries in our sample. In fact, the opposite is
true in the data. The real export and the real import prices are highly
positively correlated, and the terms of trade is much \textit{less} volatile
than the real exchange rate.\footnote{%
There are reasons to actually argue that the terms of trade in the model
severly understantes the volatility of the terms of trade in the data. In
the data, crude oil enters asymmetrically into import price, and by being
very volatile price, increases volatility of the terms of trade. Since there
is no crude oil in the model, we should remove it from the data and then
compare the volatilities. In such case,\ the volatiltiy of the terms of
trade relative to the real exchange rate falls further by about 50\% wrt to
Table X.} All prices, pretty much move with the real exchange rate.
Following Drozd and Nosal (2008), we refer to these problems as: \textit{(i)
export-import price correlation puzzle and (ii) terms of trade relative
volatility puzzle}.

\paragraph*{Summary}

We summarize all our findings for prices in Table \ref{Tab_summary}. (We
should stress that these are all predictions that the corresponding closed
economy version of the model would not have.)

%TCIMACRO{\TeXButton{B}{\begin{table}[t] \centering}}%
%BeginExpansion
\begin{table}[t] \centering%
%EndExpansion
\caption{Summary of the puzzles on prices.}%
\begin{equation*}
\begin{tabular}{lll}
\hline
Statistic & Data & Model \\ \hline
{\small UIP coefficient} & {\small around }$-3$ & {\small around }$1$ \\ 
${\small std(x)/std(y)}$ & {\small between }$3\ ${\small and }$6$ & {\small %
less than }$1$ \\ 
${\small corr(x,}\frac{c}{c^{\ast }}{\small )}$ & {\small negative} & $%
{\small +1}$ \\ 
${\small corr(p}_{x}{\small ,p}_{m}{\small )}$ & {\small highly positive} & $%
{\small -1}$ \\ 
${\small corr(p}_{x}{\small ,x)}$ & {\small highly positive} & ${\small -1}$
\\ 
${\small std(p)/std(x)}$ & {\small less than 1} & {\small more than 1} \\ 
\hline
\end{tabular}%
\end{equation*}%
\label{Tab_summary}%
%TCIMACRO{\TeXButton{E}{\end{table}}}%
%BeginExpansion
\end{table}%
%EndExpansion

\section{Basic Supply-Side Extensions}

This section extends the basic setup we have discussed above to incorporate
production and labor-leisure choice along the lines of Backus, Kehoe and
Kydland (1995) model (BKK hereafter).

At this point, it is important to stress that capital accumulation does not
invalidate any of the findings from the previous section. Capital
accumulation and production both pertain to the upstream structure of the
model, and since we have not relied upon the particular properties of the
endowment process, our results still stand. (Labor-leisure choice may
invalidate some of our finding, but it does not. We will demonstrate it
later quantitatively.)

\subsubsection*{Physical Capital Accumulation}

To incorporate investment, capital and production to our economy, we must
modify the household's problem to include capital accumulation decision and
split up income into various factor payments. The modified problem of the
household is given by:

\begin{equation}
U=\max \sum_{t}^{\infty }\sum_{s^{t}\in S^{t}}\beta ^{t}\pi
(s^{t})u(c(s^{t}))  \label{3Eq1}
\end{equation}%
subject to

\begin{eqnarray*}
c(s^{t})+i(s^{t}) &=&G(d(s^{t}),f(s^{t})),\text{ } \\
gk(s^{t+1}) &=&(1-\delta )k(s^{t})+i(s^{t})\text{,}
\end{eqnarray*}%
\ and%
\begin{eqnarray*}
&&p_{d}(s^{t})d(s^{t})+p_{f}(s^{t})f(s^{t})+ \\
&&+\sum_{s_{t+1}\in S}Q(s_{t+1}|s^{t})B_{d}(s_{t+1},s^{t})+ \\
&&+\sum_{s_{t+1}\in S}x(s^{t})Q^{\ast }(s_{t+1}|s^{t})B_{f}(s_{t+1},s^{t}) \\
&=&B_{d}(s^{t})+x(s^{t})B_{f}(s^{t})+w(s^{t})+r(s^{t})k(s^{t})+\Pi (s^{t}),
\end{eqnarray*}%
where $g$ is some constant that will turn out helpful later. 

As we can see, the representative household uses the composite good for both
consumption and investment, and its income includes labor income, capital
income and profits paid by firms (equal to zero in equilibrium).

In addition, we introduce a competitive firm that by maximizing profits
makes the decision how to optimally combine capital and labor to produce
output. Its objective is formalized by the choice of allocation 
\begin{equation*}
D(s^{t}),D^{\ast }(s^{t}),k(s^{t}),l(s^{t}),
\end{equation*}%
to maximize%
\begin{equation}
\Pi (s^{t})=p_{d}(s^{t})D(s^{t})+x(s^{t})p_{d}^{\ast }(s^{t})D^{\ast
}(s^{t})-w(s^{t})l(s^{t})-r(s^{t})k(s^{t}),
\end{equation}%
subject to the production constraints 
\begin{equation*}
D(s^{t})+D^{\ast }(s^{t})=k(s^{t})^{\alpha }(A(s^{t})l(s^{t}))^{1-\alpha }.
\end{equation*}

Market clearing in the extended setup additionally requires that the supply
of each type of good by the firms equals the demand:\ 
\begin{eqnarray}
d(s^{t}) &=&D(s^{t}), \\
d^{\ast }(s^{t}) &=&D^{\ast }(s^{t}),  \notag \\
f(s^{t}) &=&F(s^{t}),  \notag \\
f^{\ast }(s^{t}) &=&F^{\ast }(s^{t}),  \notag
\end{eqnarray}%
and that the labor market clears: 
\begin{eqnarray}
l(s^{t}) &=&1, \\
l^{\ast }(s^{t}) &=&1.  \notag
\end{eqnarray}

\begin{remark}
The particular parameterization\ of the production function using
Cobb-Douglas utility function is justified by the fact that share of
payments to labor in output is roughly constant in the data (one of the
Kaldor's growth facts)---suggesting a unit elasticity between labor and
capital in the aggregate production function.
\end{remark}

\begin{exercise}
\label{RMK1}Demonstrate that the model with capital laid out above is
equivalent to a combination of our prototype endowment model with an
additional problem solved by a representative international firm given by:%
\begin{equation*}
\max \sum_{t}\sum_{s^{t}}\beta ^{t}\hat{Q}%
(s^{t})[p_{d}y(s^{t})+p_{f}(s^{t})y^{\ast }(s^{t})]
\end{equation*}%
subject to 
\begin{eqnarray*}
y(s^{t}) &=&A(s^{t})^{1-\alpha }k(s^{t})^{\alpha }-I_{d}(s^{t})-I_{d}^{\ast
}(s^{t}) \\
y^{\ast }(s^{t}) &=&A^{\ast }(s^{t})^{1-\alpha }k^{\ast }(s^{t})^{\alpha
}-I_{f}(s^{t})-I_{f}^{\ast }(s^{t}) \\
i(s^{t}) &=&G(I_{d}(s^{t}),I_{f}(s^{t})), \\
i^{\ast }(s^{t}) &=&G(I_{f}^{\ast }(s^{t}),I_{d}^{\ast }(s^{t})), \\
gk(s^{t+1}) &=&(1-\delta )k(s^{t})+i(s^{t}), \\
gk^{\ast }(s^{t+1}) &=&(1-\delta )k^{\ast }(s^{t})+i^{\ast }(s^{t}),
\end{eqnarray*}%
where $g$ is some constant (can be 1), and $\hat{Q}(s^{t})$ is defined
recursively as $\hat{Q}(s^{t})\equiv \hat{Q}(s^{t-1})Q(s_{t}|s^{t}).$ HINT:\
Use the fact that first order conditions are necessary and sufficient,
equilibrium exists and is unique.
\end{exercise}

\subsubsection*{Labor/Leisure Choice}

In the baseline model, households inelastically supply all the labor. To
incorporate labor/leisure choice, we will use the following utility
function: 
\begin{equation*}
u(c,l)=\frac{(c^{\eta }(1-l)^{1-\eta })^{1-\theta }}{1-\theta }.
\end{equation*}

(Note that with labor-leisure choice, the existence of balanced growth path
requires technological progress to be labor augmenting, i.e. $y=k^{\alpha
}(Al)^{1-\alpha }.$)

\begin{remark}
The choice of this utility function is justified by the fact that per capita
leisure in the post-war period is roughly constant. At the same time, real
wages have been increasing steadily. Taken together, these two observations
suggest a unit elasticity between consumption and leisure, which is assumed
above.
\end{remark}

To characterize the equilibrium, in the set of our first order conditions,
we would need to additionally include the following equations: (i)\ Euler
equation for capital:

\begin{equation}
u_{c}\left( s^{t}\right) g=\beta E_{s^{t}}[u_{c}\left( s^{t+1}\right) \left(
\left( 1-\delta \right) +r\left( s^{t+1}\right) \right) ],
\end{equation}

(ii)\ labor leisure choice condition:%
\begin{equation}
\frac{u_{l}\left( s^{t}\right) }{u_{c}\left( s^{t}\right) }=-w\left(
s^{t}\right) ,
\end{equation}

and (iv) factor prices: 
\begin{eqnarray*}
r(s^{t}) &=&\alpha p_{d}(s^{t})k(s^{t})^{\alpha
-1}(A(s^{t})l(s^{t}))^{1-\alpha }, \\
w(s^{t}) &=&(1-\alpha )A(s^{t})p_{d}(s^{t})k(s^{t})^{\alpha
}(A(s^{t})l(s^{t}))^{-\alpha },
\end{eqnarray*}%
where analogous conditions for the foreign country apply.

\paragraph*{Final remarks on the complete asset market assumption?}

At this point, you may wonder to what extent it makes sense to assume
completeness of asset market? To what extent the results discussed above
carry through to economies that restrict the asset span?

Surprisingly, the answer is that the assumption completeness of asset
markets is almost without loss of generality in this particular environment.
Specifically, it turns out that the allocation in an analogous frictionless
model with incomplete markets\footnote{%
By incomplete markets I\ mean a symmetric setup in which either two state
uncontingent bonds are traded, domestic and foreign, or households can hold
domestic or foreign equity.} exactly coincides with the allocation of the
complete asset market economy. Concluding, if we one believes that the
allocation in the data is far away from a complete asset market allocation,
one should rethink the assumptions that make this `equivalence result' hold,
which is obviously way more difficult.

The key thing that makes the restriction of asset space not so relevant in
this class of models is that under incomplete markets, a portfolio of
different types of bonds can deliver almost full risk sharing. The key
reason is that when, for example, the exchange rate appreciates during
recessions, by simply taking a short position on the home-bond and a long
position on the foreign bond, the household can hedge its consumption risk
and obtain this state of the world a transfer from abroad. A similar
completion of markets can be achieved more generally using a set of bonds
but with a different maturity. These results have been established in the
series of papers by Lucas (1982), Gourinchas and Coeurdacier (2008),
Heathcote and Perri (2007), and under sticky prices, Engel and Matsumoto
(2008). Typically, to get any action from incompleteness of market, the
literature considers the asymmetric case of only one state uncontingent bond
(denominated in, say, foreign consumption). Also, with more types of shocks,
two bonds may not in general be sufficient to provided full risk sharing,
and again restriction of asset span may matter. It is not obvious, however,
how to depart from the complete markets in a way that is sensible and at the
same time gives some action. 

To setup an incomplete markets economy, we typically restrict attention to
state uncontingent asset span of payoffs in the domestic numeraire at home
and in the foreign numeraire abroad. This restriction imposes the following
feasibility on bond trade: 
\begin{eqnarray*}
B_{d}(s_{t+1}|s^{t}) &=&B_{d}(s_{t+1}^{\prime }|s^{t}),\text{ } \\
B_{f}(s_{t+1}|s^{t}) &=&B_{f}(s_{t+1}^{\prime }|s^{t}),\text{ } \\
B_{d}^{\ast }(s_{t+1}|s^{t}) &=&B_{d}^{\ast }(s_{t+1}^{\prime }|s^{t}), \\
B_{f}^{\ast }(s_{t+1}|s^{t}) &=&B_{f}^{\ast }(s_{t+1}^{\prime }|s^{t}),\text{%
all }s_{t+1},s_{t+1}^{\prime },s^{t}.
\end{eqnarray*}%
It can be imposed either directly on the household's problem (i.e. built
into notation), or as a feasibility restriction.\footnote{%
Sometimes researcher restrict trade to just one bond. In such case, risk
sharing can be hindered. However, this friction seems to be somewhat
arbitrary.}

\section{Calibration}

Our model economy is parameterized by the following 5 parameters $(\sigma
,\theta ,\omega ,\eta ,\alpha ),$ and the stochastic process that governs
the technology shocks: $A$ and $A^{\ast }$. Since it is difficult and also
pointless to explore the implications of the model for all possible
combinations of the parameters, real business cycle literature calibrates
the values of the parameters so that the model is broadly consistent with
the facts that do not pertain to the business cycle directly, where
calibrating means \textit{to standardize as a measuring instrument}.

The basic idea underlying calibration is to exploit the fact that business
cycle models also have long-run and cross-sectional predictions (first
moments), and it is reasonable to require that the parameter values are such
that the model is consistent with these observations (e.g. share of leisure
in time endowment, or depreciation of capital etc...).\footnote{%
In other words, the calibration starts from the premise that the same model
should be used to account for both the business cycle and the long-run
observations, as the theory bundles these aspects together. In dynamic
macroeconomics, the distinction between a growth model and business cycle
model is artificial---this is the implication of the theory.}

To calibrate the model, we identify the domestic country with the US, and
the foreign country with an aggregate of 15 major European countries \textit{%
plus} Switzerland, Japan, Canada, and Australia. Whenever possible, we
assume symmetry and use the US\ data to calibrate a parameter, and only when
necessary use the data from rest of the world.

Since there are many parameters to calibrate, it is convenient to separate
our analysis to the choice of parameters that are common to the open economy
and the underlying closed economy, and then consider separately the
parameters specific to the open economy.

\subsubsection{Parameters common to open economy and closed economy}

In this section, we calibrate the values of the parameters that are common
to open economy model and the underlying closed economy model. To calibrate
their values, we require that the balanced growth path implications of the
model are consistent with the long-run mean values in the US\ data. This
group includes: $\alpha ,\beta ,\delta ,\theta .$

The references for the procedure we follow below are: (1) the classic paper%
\footnote{%
The book is a bit out of date, especially in terms of methods. Nevertheless,
I encourage you to read this paper and the BKK 95 paper included in chapter
11.} by\ Cooley and Prescott, \textquotedblleft Economic Growth and Business
Cycle,\textquotedblright\ in Thomas Cooley, ed., "Frontiers of Business
Cycle Research,", Princeton, NJ: Princeton University Press, 1995, pp.1-38,
and two more recent papers by (2)\ Gomme, Paul \& Peter Rupert (2007):\
\textquotedblleft Theory, Measurement and Calibration of Macroeconomic
Models\textquotedblright , Journal of Monetary Economics, Volume 54, Issue
2, March 2007, Pages 460-497, and (3) Gomme, Rupert, Ravikumar (2006):
\textquotedblleft Returns to Capital and the Business
Cycles,\textquotedblright\ Federal Reserve Bank of Cleveland WP06-03, Feb.
With respect to these papers, our analysis below will be simplified. For
example, we will ignore an explicit consideration of taxes in the model,
home production, or durable consumption. (It won't change much the values of
parameters, though.)

Under the assumption of symmetry, it is not difficult to see that the
following standard neoclassical growth model describes the balanced growth
path of our two country model (each country separately, as well as the
entire world): 
\begin{equation}
\max \sum_{t}^{\infty }\hat{\beta}^{t}\frac{(C_{t}^{\eta
}(N_{t}h-L_{t})^{1-\eta })^{1-\theta }}{1-\theta }  \label{3Neoclassical}
\end{equation}%
subject to 
\begin{eqnarray*}
C_{t}+I_{t} &=&Y_{t}, \\
Y_{t} &=&K_{t}^{\alpha }(\gamma ^{t}L_{t})^{1-\alpha } \\
K_{t+1} &=&(1-\delta )K_{t}+I_{t},
\end{eqnarray*}%
where capital letters stand for aggregate variables, $\gamma $ is labor
augmenting technological progress and $N_{t}$ is the size of the working age
population, and $h$ is the total time endowment of non-sleeping hours per
working age person (assumed 7 x 15h per day = 105h per week).

The working age population size is assumed to be equal to the product of a
constant share of working age population in total population \textit{times}
the size of the total population $P_{t}$ that grows at some constant rate $%
\zeta .$ These assumptions, up to the fluctuations in the share due to WWII
baby-bust and post-war baby-boom, are consistent with the data, as
illustrated in Figure \ref{Fig_population}.


 
\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
\subfloat{\includegraphics[scale=.45, page=14]{Figures.pdf}} \\  
\subfloat{\includegraphics[scale=.45, page=2]{Figures.pdf}}  
\caption{Share of working age population and population growth in the US.}
\label{Fig_population}
\end{figure}
%
%%TCIMACRO{\TeXButton{B}{\begin{figure}[tbp] \centering}}%
%%BeginExpansion
%\begin{figure}[tbp] \centering%
%%EndExpansion
%\begin{equation*}
%\begin{array}{c}
%\FRAME{itbpF}{365.8182pt}{250.7988pt}{0in}{}{}{Figure}{\special{language
%"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
%"USEDEF";valid_file "T";width 365.8182pt;height 250.7988pt;depth
%0in;original-width 658.1968pt;original-height 450.0578pt;cropleft
%"0";croptop "1";cropright "1";cropbottom "0";tempfilename
%'KD2WCH08.wmf';tempfile-properties "XPR";}} \\ 
%\FRAME{itbpF}{368.2182pt}{252.5388pt}{0in}{}{}{Figure}{\special{language
%"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
%"USEDEF";valid_file "T";width 368.2182pt;height 252.5388pt;depth
%0in;original-width 658.1968pt;original-height 450.0578pt;cropleft
%"0";croptop "1";cropright "1";cropbottom "0";tempfilename
%'KD2WCH09.wmf';tempfile-properties "XPR";}}%
%\end{array}%
%\end{equation*}%
%\caption{Share of working age population and population growth in the US.}%
%\label{Fig_population}%
%%TCIMACRO{\TeXButton{E}{\end{figure}}}%
%%BeginExpansion
%\end{figure}%
%%EndExpansion

Since the share of working age population in total population is roughly
constant, we may normalize $\frac{N_{t}h}{P_{t}}$ to 1 and translate the
model to per capita terms by defining the corresponding per capita variables
as follows: $C_{t}=\zeta ^{t}c_{t},$ $K_{t}=\zeta ^{t}k_{t},$ $%
L_{t}=lhN_{t}=l\zeta ^{t},N_{t}=\zeta ^{t}/h,$ $P_{t}=\zeta ^{t}$ (where $%
\zeta =\frac{P_{t+1}}{P_{t}}$ denotes population grow)$.$ Substituting out
and simplifying whenever possible, we obtain%
\begin{equation}
\max \sum_{t}^{\infty }\beta ^{t}\frac{(c_{t}^{\eta }(1-l_{t})^{1-\eta
})^{1-\theta }}{1-\theta }  \label{3Neoclassical1}
\end{equation}%
subject to 
\begin{eqnarray*}
c_{t}+i_{t} &=&y_{t}, \\
y_{t} &=&k_{t}^{\alpha }(\gamma ^{t}l_{t})^{1-\alpha } \\
\zeta k_{t+1} &=&(1-\delta )k_{t}+i_{t},
\end{eqnarray*}%
where $\beta \equiv \hat{\beta}\zeta .$

The first order conditions to the maximization problem stated above can be
derived from the following Lagrangian%
\begin{equation*}
\mathcal{L}=\sum_{t}\beta ^{t}\frac{(c_{t}^{\eta }(1-l_{t})^{1-\eta
})^{1-\theta }}{1-\theta }-\sum_{t}\lambda _{t}(c_{t}+\zeta
k_{t+1}-(1-\delta )k_{t}-k_{t}^{\alpha }(\gamma ^{t}l_{t})^{1-\alpha }).
\end{equation*}%
The first order conditions are: 
\begin{eqnarray*}
\beta ^{t}\eta c_{t}^{\eta -1}(1-l_{t})^{1-\eta }(c_{t}^{\eta
}(1-l_{t})^{1-\eta })^{-\theta } &=&\lambda _{t}, \\
\beta ^{t}(1-\eta )c_{t}^{\eta }(1-l_{t})^{-\eta }(c_{t}^{\eta
}(1-l_{t})^{1-\eta })^{-\theta } &=&\lambda _{t}\gamma ^{t}(1-\alpha
)k_{t}^{\alpha }(\gamma ^{t}l_{t})^{-\alpha }, \\
\lambda _{t}\zeta &=&\lambda _{t+1}[(1-\delta )+\alpha k_{t+1}^{\alpha
-1}(\gamma ^{t+1}l_{t+1})^{1-\alpha }],
\end{eqnarray*}%
and can be compactly summarized by the following 3 conditions: (i) the
familiar Euler equation 
\begin{equation}
\zeta =\frac{\beta c_{t+1}^{-1}(c_{t+1}^{\eta }(1-l_{t+1})^{1-\eta
})^{1-\theta }}{c_{t}^{-1}(c_{t}^{\eta }(1-l_{t})^{1-\eta })^{1-\theta }}%
[(1-\delta )+\alpha k_{t+1}^{\alpha -1}(\gamma ^{t+1}l_{t+1})^{1-\alpha }],
\label{3NEE}
\end{equation}%
(ii)\ labor/leisure choice condition%
\begin{equation}
\frac{c_{t}}{1-l_{t}}\frac{1-\eta }{\eta }=\gamma ^{t}(1-\alpha
)k_{t}^{\alpha }(\gamma ^{t}l_{t})^{-\alpha },  \label{3NLLC}
\end{equation}%
and (iii)\ aggregate feasibility condition 
\begin{equation}
c_{t}+\zeta k_{t+1}-(1-\delta )k_{t}=k_{t}^{\alpha }(\gamma
^{t}l_{t})^{1-\alpha }.  \label{3NF}
\end{equation}

The supporting prices (gross rental price of capital and the wage rate) can
be calculated as follows: 
\begin{eqnarray}
r_{t} &=&\alpha k_{t}^{\alpha -1}(\gamma ^{t}l_{t})^{1-\alpha },  \label{3NP}
\\
w_{t} &=&(1-\alpha )\gamma ^{t}k_{t}^{\alpha }(\gamma ^{t}l_{t})^{-\alpha }.
\notag
\end{eqnarray}

\paragraph{Balanced Growth Path}

Here, we solve for the balanced growth path (BGP) of this model. By
definition, the balanced growth path is given by:\ 
\begin{eqnarray}
c_{t} &=&\gamma _{c}^{t}\hat{c},  \label{3BGP} \\
k_{t} &=&\gamma _{k}^{t}\hat{k},  \notag \\
i_{t} &=&\gamma _{i}^{t}\hat{\imath},  \notag \\
y_{t} &=&\gamma _{y}^{t}\hat{y},  \notag \\
l_{t} &=&\gamma _{l}^{t}\hat{l},  \notag
\end{eqnarray}%
where $\gamma _{j}$ denotes the growth rate of the corresponding variable $j$%
, and $\hat{\cdot}$ are the initial values of the variables.

Our task is to find the growth rates $\gamma _{c},\gamma _{k},\gamma
_{i},\gamma _{y}$,$\gamma _{l}$ and initial values $\hat{c},\hat{k},\hat{%
\imath},\hat{y},\hat{l}\ $so that the implied balanced growth path sequence $%
\{c_{t},k_{t},i_{t},y_{t},l_{t}\}_{t}$ solved the planning problem stated
above. The proposition below establishes that in this model all variable
except stationary leisure grow at the same rate $\gamma $---a property that
is broadly consistent with the growth experience of the industrial countries.

\begin{proposition}
The model given by (\ref{3Neoclassical1}) has a unique balanced growth path
(BGP) that satisfies 
\begin{equation*}
\gamma _{c}=\gamma _{k}=\gamma _{i}=\gamma _{y}=\gamma _{l}=\gamma ,\gamma
_{l}=1,
\end{equation*}%
and the initial values $\hat{c},\hat{k},\hat{\imath},\hat{y},\hat{l}\ $ are
given by the solution to the following system: (i) Euler's equation\ 
\begin{equation}
\zeta =\beta \gamma ^{\eta (1-\theta )-1}[(1-\delta )+\alpha \frac{\hat{y}}{%
\hat{k}}],  \label{3P1a}
\end{equation}%
(ii)\ labor/leisure choice condition%
\begin{equation}
\frac{\hat{c}}{1-\hat{l}}\frac{1-\eta }{\eta }=(1-\alpha )\frac{\hat{y}}{%
\hat{l}},  \label{3Pqb}
\end{equation}%
and (iii)\ aggregate feasibility condition 
\begin{eqnarray}
\hat{c}+\gamma \zeta \hat{k}-(1-\delta )\hat{k} &=&\hat{y},  \label{3Pqc} \\
\hat{y} &=&\hat{k}^{\alpha }\hat{l}^{1-\alpha },  \notag \\
\hat{\imath} &=&\gamma \zeta \hat{k}-(1-\delta )\hat{k}.  \notag
\end{eqnarray}

The supporting equilibrium prices can be found from:%
\begin{eqnarray}
r &=&\alpha \frac{\hat{y}}{\hat{k}},  \label{3Pqd} \\
w_{t} &=&\gamma ^{t}(1-\alpha )\frac{\hat{y}}{\hat{l}}.  \notag
\end{eqnarray}
\end{proposition}

\begin{proof}
Since we know that the solution to the planning problem stated in (\ref%
{3Neoclassical1}) is unique up to the given value of initial capital, it is
sufficient to show that the proposed balanced growth path (\ref{3BGP})
solves the model, and the conditions (i)-(iii) uniquely pin down the values
of $\hat{c},\hat{k},\hat{\imath},\hat{y},\hat{l}.$

Clearly, equations (\ref{3P1a})-(\ref{3Pqd}) have been obtained by
substituting out for allocation from the balance growth path into the first
order conditions given by (\ref{3NEE})-(\ref{3NP}). To show that they imply
a unique solution for $\hat{c},\hat{k},\hat{\imath},\hat{y},\hat{l}.$, we
will solve for this solution explicitly. To this end, observe that from the
first equation, we can find the value of $\frac{\hat{y}}{\hat{k}}.$ Given
this value, we can divide equation (\ref{3Pqc}) by $\hat{k}$ to find $\frac{%
\hat{c}}{\hat{k}}.$ Finally, we can rewrite \ref{3Pqb} as%
\begin{equation*}
\frac{\hat{l}}{1-\hat{l}}\frac{1-\eta }{\eta }=(1-\alpha )(\frac{\hat{c}}{%
\hat{k}})^{-1}\frac{\hat{y}}{\hat{k}},
\end{equation*}%
and obtain the value of $\frac{\hat{l}}{1-\hat{l}}.$ After solving for $\hat{%
l},$ we can recover the values $\hat{c},\hat{k},\hat{\imath},\hat{y}$.

We conclude that the balanced growth path is uniquely pinned down by
conditions (i)--(iii), and it satisfies the first order conditions to the
planning problem (\ref{3Neoclassical1}). Since first order conditions are
also necessary and sufficient, the proposition follows.
\end{proof}

\begin{exercise}
Establish the connection between the balanced growth path for the planning
problem stated in (\ref{3Neoclassical1}) and the balanced growth path in our
original two-country model. Specifically, include $d,f,d^{\ast },f^{\ast }$
in the definition of the balanced growth path, and show that if these
variables also growth at rate $\gamma $, such `extended balanced growth'
path solves the corresponding two-country planning problem that is given
by:\ 
\begin{equation*}
\max \left[ \sum_{t}^{\infty }\beta ^{t}\sum_{s^{t}\in S^{t}}[\pi
(s^{t})u(c(s^{t}))+\sum_{s^{t}\in S^{t}}\beta ^{t}\pi (s^{t})u(c^{\ast
}(s^{t}))]\right]
\end{equation*}%
subject to 
\begin{eqnarray*}
c(s^{t})+i(s^{t}) &=&\xi G(d(s^{t}),f(s^{t})), \\
c^{\ast }(s^{t})+i^{\ast }(s^{t}) &=&\xi G(f^{\ast }(s^{t}),d^{\ast }(s^{t}))%
\text{ }
\end{eqnarray*}%
and%
\begin{eqnarray*}
d(s^{t})+d^{\ast }(s^{t}) &=&y(s^{t}), \\
f(s^{t})+f^{\ast }(s^{t}) &=&y^{\ast }(s^{t}),\text{ } \\
y(s^{t}) &=&k(s^{t})^{\alpha }(\gamma ^{t}l(s^{t}))^{1-\alpha }, \\
y^{\ast }(s^{t}) &=&k^{\ast }(s^{t})^{\alpha }(\gamma ^{t}l^{\ast
}(s^{t}))^{1-\alpha }, \\
\zeta k(s^{t+1}) &=&(1-\delta )k(s^{t})+i(s^{t}), \\
\zeta k^{\ast }(s^{t+1}) &=&(1-\delta )k^{\ast }(s^{t})+i^{\ast }(s^{t}),%
\text{ all }s^{t}\in S^{t},
\end{eqnarray*}%
where G($\cdot $) is hod of degree 1 and $\xi $ is some arbitrary constant.
Argue that, in fact, when the value of $\xi $ is properly chosen (say what
it must be), on the balanced growth path this model boils down to a simple
neoclassical growth model.
\end{exercise}

Our next task is to use data to calibrate the parameters of the closed
economy model. The data comes from the Economic Report of the President
(available online),\footnote{%
See http://www.gpoaccess.gov/eop/tables08.html.} with the source tables
denoted by B-XX. Economic Report of the President follows the layout of NIPA
tables, but for our purposes it is tabulated more conveniently than NIPA. On
my website you will find an Excel file with the data and the calibration
discussed below.

\paragraph{Calibrating $\protect\alpha $}

In the first step, we calibrate the value of $\alpha $. According to the
model, $\alpha $ is the share of payments to labor: 
\begin{equation*}
\frac{w_{t}L_{t}}{Y_{t}}=\frac{w_{t}l_{t}N_{t}}{y_{t}N_{t}}=\frac{w_{t}l_{t}%
}{y_{t}}=1-\alpha ,
\end{equation*}

To calculate the\ above share, we need to have total factor payments to
domestic factors, and payments to labor. To obtain these values we look at
the GDI NIPA\ accounts (GDP calculated from factor payments), which is
illustrated in Figure \ref{Fig_NIPA0}. GDI breaks down GDP\ on the income
side,\footnote{%
GDI\ (gross domestic income) is essentially GDP measured from income side up
to a statistical discrepancy. For further information about NIPA, refer to
the handbook of NIPA available at:
http://www.bea.gov/national/pdf/NIPAhandbookch1-4.pdf.} and it is the right
measure. GDP, by definition, is production by factors that are within US\
borders---exactly what we are looking for.



\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=18,scale=.44]{Figures.pdf} 
 \caption{Internal structure of NIPAs.}%
\label{Fig_NIPA0}
\end{figure}

%\FRAME{ftbpFU}{372.5982pt}{279.9586pt}{0pt}{\Qcb{Structure of NIPA\
%Accounts. }}{\Qlb{Fig_NIPA0}}{Figure}{\special{language "Scientific
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The problem is that a big chunk of GDI are excise and sales taxes (taxes on
production and imports less subsidies). These taxes, by inflating prices,
inflate GDP, and are not related to any factor payments. Another thing we
have to check, is whether the black-box item called operating surplus has
only non-labor income in it. To find out, we look at the corresponding NIPA
table for GDI---which is illustrated in Figure \ref{Fig_NIPA2} (see BEA's
website).\footnote{%
Note that corporate taxes are not a problem, because these are taxes imposed
directly on capital. To see this write down a simple firm problem and impose
a tax on rental price of capital --- it will not distort the share of total
payments to capital relative to labor.}


\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=19,scale=.44]{Figures.pdf} 
 \caption{GDI (=GDP +\ statistica
 discrepancy) breaken down by income type.}%
\label{Fig_NIPA2}
\end{figure}

%\FRAME{ftbpFU}{445.8pt}{264.9333pt}{0pt}{\Qcb{GDI (=GDP +\ statistica
%discrepancy) breaken down by income type.}}{\Qlb{Fig_NIPA2}}{Figure}{\special%
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%'KD2WCH0B.wmf';tempfile-properties "XPR";}}

From this table, we see that, in fact, some of the labor income may be
buried under item 15 (proprietor's income). To deal with this problem, we
assume, in consistency with our production function, that $1-\alpha $
fraction of proprietor's income is labor income. We also subtract excise and
sales taxes from GDP to obtain the `pure' total factor payments. Our
adjustments give:\ 

\begin{itemize}
\item Total factor payments to capital and labor = (GDP $Y_{t}$ B-1) -
(taxes on production and imports \textit{less} subsidies B-27).

\item Total payments to labor = (compensation of employees B28)+($1-\alpha $
fraction of the proprietor's income B28).
\end{itemize}

The value of $\alpha $ can now be obtained from the following calculation%
\begin{equation*}
1-\alpha =\frac{\text{{\small Compensation of employees+}}{\small (1-\alpha )%
}\text{{\small Proprietor's income}}}{\text{{\small GDP -Taxes on production
and imports less subsidies}}},
\end{equation*}%
which gives\footnote{%
Prescott in his measure of capital included consumer durables, his $\alpha $
was a bit higher. It is an issue how to deal with consumer durables,
certainly one way is to lump it all into capital and assume consumption is
consumption of non-durables + services from durables.}%
\begin{equation*}
\alpha =1-\frac{\text{{\small Compensation of employees}}}{\text{{\small GDP
- Taxes (..)\ - Proprietor's income}}}.
\end{equation*}

The calibrated value of $\alpha $ for the period 1959-2005 is about 1/3, as
expected. This value is slightly less than the one obtained by Cooley and
Prescott. The difference is that Cooley and Prescott, in consistency with
the model, included durable consumption goods in their measure of broadly
defined capital. To keep things simple, we \ omitted this consideration. Figure \ref{Fig_alpha} illustrates the time-series of the calibrated value
for each year. As required, the parameter is approximately constant over time.  

\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=3,scale=.44]{Figures.pdf} 
 \caption{Calibrated values of $\alpha ,$ 1959-2005.}%
\label{Fig_alpha}
\end{figure}

%\FRAME{ftbpFU}{332.6667pt}{228.2pt}{0pt}{\Qcb{Calibrated values of $\protect%
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%'KD2WCH0C.wmf';tempfile-properties "XPR";}}

\paragraph{Calibrating $\delta $}

To calibrate the depreciation rate of capital, we use the information on
investment in capital and consumption of fixed capital (i.e. the estimate of
depreciation of capital by NIPA). In this respect, we obtain

\begin{itemize}
\item Gross investment $\equiv K_{t+1}-(1-\delta )K_{t}$ = (gross private
investment B-1) + (gross government investment B-20),

\item Depreciation$\equiv \delta K_{t}=($consumption of fixed capital B-26),
and thus

\item Net investment$\equiv K_{t+1}-K_{t}=[K_{t+1}-(1-\delta )K_{t}]-\delta
K_{t}.$
\end{itemize}

Having these time-series, we use the fact that on the balance growth path in
our model, we have%
\begin{equation*}
\frac{(K_{t+1}-K_{t})}{Y_{t}}=\frac{(k_{t+1}\frac{P_{t+1}}{P_{t}}-k_{t})P_{t}%
}{y_{t}P_{t}}=(\gamma \zeta -1)\frac{\hat{k}}{\hat{y}},
\end{equation*}%
which implies: 
\begin{equation}
\frac{\hat{y}}{\hat{k}}=\frac{(\gamma \zeta -1)Y_{t}}{K_{t+1}-K_{t}},
\label{3yk}
\end{equation}

Having the above ratio, we can calculate $\delta $ from the following
identity ($K_{t},Y_{t}$ both grow at the same rate $\gamma \zeta )$: 
\begin{equation*}
\delta \equiv \frac{\delta K_{t}}{Y_{t}}\frac{Y_{t}}{K_{t}}=\frac{\delta
K_{t}}{Y_{t}}\frac{\hat{y}\gamma ^{t}P_{t}}{\hat{k}\gamma ^{t}P_{t}}=\frac{%
\delta K_{t}}{Y_{t}}\frac{\hat{y}}{\hat{k}}.
\end{equation*}

To calculate $\delta $, we have used micro-level information about
depreciation of capital from NIPA. The whole calculation was just correcting
for the fact that part of the investment must be used to augment capital in
consistency with the BGP---rest followed from the aggregate estimates of
capital depreciation by BEA.

The average value of $\delta $ for the period 1959-2005 is about 4.5\% per
annum (1.14\% per quarter). Figure \ref{Fig_delta} illustrates the
underlying time-series of the calibrated values for each year. As we can
see, it is not as nice and stationary as $\alpha $, but the secular trend is
small. It also fluctuates quite a bit due to the high volatility of
investment over the business cycle---which is nothing to be worried about.
We know that we are not looking at the `pure' balanced growth path in the
data. The secular trend is more problematic. It suggests that the economy
may not be exactly on the balanced growth path the way our model looks at it.

\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=4,scale=.44]{Figures.pdf} 
 \caption{Calibrated values of $\delta ,$ 1959-2006.}%
\label{Fig_delta}
\end{figure}

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%0pt;original-width 731.3333pt;original-height 500.0667pt;cropleft
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%'KD2WCH0D.wmf';tempfile-properties "XPR";}}

\paragraph{Calibrating $\protect\beta $}

On the balanced growth path, Euler's equation implies

\begin{equation*}
\zeta =\beta \gamma ^{-1}\gamma ^{a(1-\theta )-1}[(1-\delta )+\alpha
k_{t+1}^{\alpha -1}(\gamma ^{t+1}l_{t+1})^{1-\alpha }],
\end{equation*}
and thus%
\begin{equation}
\beta =\frac{\zeta \gamma ^{\eta (\theta -1)+1}}{\alpha \frac{\hat{y}}{\hat{k%
}}+1-\delta }.
\end{equation}

Since we lack the values of $\theta $ and $\eta ,$ and can not compute $%
\beta $ yet. Prescott chose $\theta =1$ on the basis that such value is
roughly consistent with the spread between the real rates of return in
countries with the highest and lowest consumption growth. In such case, $%
\beta $ can be calculated right away because $\eta $ cancels out. Later,
more evidence become available to pin down $\theta $. Experiments of risk
aversion point to values between 1-3 (see info in Mehra and Prescott
(1985)), and time series analysis by Eichenbaum, Hansen and Singleton (1988)
suggests values centered around 2, that we should also consider. (The value
of $\theta =2$ is widely used in the literature; Prescott uses $\theta =1$.)

When $\theta =2,$ we need to pin down $\eta $ first$.$ The value of $\eta $
can be obtained by calculating the ratio of work in total time endowment,
which gives $\hat{l}$. We calculate it by evaluating the ratio of the total
number of workers to size of total working age population, and multiply it
by the ratio of the average number of hours worked per average worker to the
total endowment of non-sleep hours; assumed to be 105h per week (15h of
non-sleeping time per day).\footnote{%
See formulas in the Excel file on my website for more detail.} The obtained
this way value of $\hat{l}$ oscillates at around 1/3, and implies the value
of $\frac{1-\hat{l}}{\hat{l}}$ equal to about $2.$ (The data on hours comes
from CPS census\footnote{%
Source: Cociuba, Ueberfeld and Prescott (2007).}, and the number is
consistent with other micro-level studies pointing to slightly lower number
of 30\% (e.g. Juster and Stafford (1991), \textquotedblleft The Allocation
of Time ...\textquotedblright . Journal of Economic Literature, 29:471:522)

Having calculated the share of market activities in total time endowment,
from the first order condition on labor/leisure choice, we thus derive:

\begin{equation*}
\frac{\hat{c}}{1-\hat{l}}\frac{1-\eta }{\eta }=(1-\alpha )\frac{\hat{y}}{%
\hat{l}}
\end{equation*}%
and obtain the value of $\eta $ from 
\begin{eqnarray*}
\frac{1-\eta }{\eta }\frac{\hat{l}}{1-\hat{l}} &=&(1-\alpha )\frac{\hat{y}}{%
\hat{c}}=(1-\alpha )\frac{Y_{t}}{C_{t}}, \\
\frac{1-\eta }{\eta } &=&\frac{1-\hat{l}}{\hat{l}}(1-\alpha )\frac{Y_{t}}{%
C_{t}}, \\
\eta  &=&\frac{1}{\frac{1-\hat{l}}{\hat{l}}(1-\alpha )\frac{Y_{t}}{C_{t}}+1}.
\end{eqnarray*}

(To calculate $\frac{Y_{t}}{C_{t}}$, we correct for the impact of the large
negative NX, and calculate $\frac{Y_{t}}{C_{t}}$ by evaluating: $\frac{Y_{t}%
}{C_{t}}=\frac{Y_{t}+NX}{Y_{t}+NX-[K_{t+1}-(1-\delta )K_{t}]}$.\footnote{%
In consistency with theory, we are treating here NX as part of domestic
output. This is the total that in our model is split by households into
consumption and investment.} The correction does not make much difference
(see Excel file posted online).)

The average value of $\eta $ calculated this way is .$37,$ and having $\eta ,
$ we can calculate $\beta $ from the formula:%
\begin{equation*}
\beta =\gamma ^{\eta (\theta -1)+1}[(1-\delta )+\alpha \frac{\hat{y}}{\hat{k}%
}]^{-1}.
\end{equation*}

Figures \ref{Fig_eta}-\ref{Fig_beta} illustrate the time-series for the
calibrated values of $\eta $ and $\beta $ for each year$.$ As required, both
are roughly stationary.

\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=5,scale=.44]{Figures.pdf} 
 \caption{Calibrated values of $\eta ,$ 1959-2006.}%
\label{Fig_eta}
\end{figure}

\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=6,scale=.44]{Figures.pdf} 
 \caption{Calibrated values of $\beta $ for $\theta =2,$ 1959-2006.} 
\label{Fig_beta}
\end{figure}


%\FRAME{ftbpFU}{331.6pt}{227.2pt}{0pt}{\Qcb{Calibrated values of $\protect%
%\eta ,$ 1959-2006.}}{\Qlb{Fig_eta}}{Figure}{\special{language "Scientific
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%\FRAME{ftbpFU}{334.8pt}{229.4pt}{0pt}{\Qcb{Calibrated values of $\protect%
%\beta $ for $\protect\theta =2,$ 1959-2006. }}{\Qlb{Fig_beta}}{Figure}{%
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%'KD2WCH0F.wmf';tempfile-properties "XPR";}}\bigskip

Last but not least, we have to translate our model to quarterly frequency
(our estimates are annual). In order to do this, we calculate implied
quarterly depreciation rate and discount rate from the formula for
compounded change as follows: $\delta _{q}=1-(1-\delta _{a})^{\frac{1}{4}},$
and $\beta _{q}=\beta _{a}^{\frac{1}{4}}.$ The summary of the obtained this
way parameter values for a quarterly model is given in Table \ref%
{Tab_calibration1}.

%TCIMACRO{\TeXButton{B}{\begin{table}[t] \centering}}%
%BeginExpansion
\begin{table}[t] \centering%
%EndExpansion
\caption{Parameter values for the quarterly model.}

\begin{minipage}[htb]{0.94\linewidth}
\begin{tabular}{p{2.65in}p{1.20in}p{1.20in}}
\toprule
Parameter & \multicolumn{2}{l}{Parameter value}\\
\cmidrule(r){2-3}
\small  &  \small $\theta=2$ &  \small $\theta=1$\\
\midrule
\small $\gamma$& \  \small 1.021& \  \small 1.021\\
\small $\zeta$ & \  \small 1.011& \  \small 1.011\\
\small $\alpha$ & \  \small 0.325& \  \small 0.325\\
\small $\delta$ & \  \small 0.014& \  \small 0.014  \\
\small $\eta$ & \  \small 0.367& \  \small 0.367\\
\small $\beta$ & \  \small 0.988& \  \small 0.99\\
\bottomrule \addlinespace\end{tabular} 
\newline
\scriptsize Notes: Based on mean values estimated from annual US data (1959-2006) and translated to quarterly frequency. 
\end{minipage}\label{Tab_calibration1}%
%TCIMACRO{\TeXButton{E}{\end{table}}}%
%BeginExpansion
\end{table}%
%EndExpansion

\paragraph{Pre-tax real net return on capital}

Given the values of the calibrated parameters, using Euler's equation, we
next calculate the net return on capital (note that this return is risky
when there is business cycle):\footnote{%
To use the analogy to our earlier analysis, return $r$ must satisfy along
the balance growth path the equation $1=\beta \frac{u^{\prime }(c_{t+1})}{%
u^{\prime }(c_{t})}(1+r).$}%
\begin{equation*}
r-\delta =\alpha \frac{\hat{y}}{\hat{k}}-\delta =\frac{\gamma ^{\eta (\theta
-1)+1}}{\beta }-1+\delta -\delta .
\end{equation*}

In our model, it is on average equal to 8.2\%. This is not too bad, but
slightly too high comparing to the pre-tax real return calculated directly
from NIPA data of about 6\% (see the paper by Ravikumar et al.).

\begin{exercise}
Using data from National Accounts available at www.sourceOECD.org and data
available from http://www.statistik.at/web\_en/statistics/index.html,
calibrate the parameters $\alpha ,\delta $,$\eta ,\beta $ for Austria (AUT).
How do these values compare with the values listed above for the US?
\end{exercise}

\subsubsection{Calibrating parameters specific to the open economy}

Our open economy model adds two additional parameters that we need to
discipline. These parameters are the elasticity of substitution between
domestic and foreign goods $\sigma $, and the home-bias parameter $\omega .$

The most reasonable thing to do would be to adopt the value of $\sigma $
consistent with long-run oriented trade literature. In the previous chapter,
we have argued that trade literature suggests values centered around 5-15.
This value followed, for example, from the cross-sectional estimate of $%
\theta $ in Eaton and Kortum (2002) (recall $\theta -1$ is isomorphic to $%
\sigma )$, or the studies based on the impact of tariff reductions on trade
(e.g. Head and Ries (2001): \textit{\textquotedblleft Increasing Returns
(...),\textquotedblright }\ American Economic Review, 91(4), pp. 858-976).
Also, the analysis by Anderson and Wincoop (2003) pointed us to such values.

Given the value of $\sigma $, finding $\omega $ is an easy task. It is
sufficient to require that the theory is consistent with the observed
imports to GDP ratio. In the symmetric steady state, the two country model
implies 
\begin{equation*}
f_{t}=(\frac{\omega }{1-\omega })^{-\sigma }d_{t},
\end{equation*}%
which we can use to back out $\omega $. To see this, define the import share
as 
\begin{equation*}
is=\frac{f_{t}}{d_{t}+f_{t}},
\end{equation*}%
and note that import share can be measured from the data by imports/GDP
ratio. Next, rewrite the above condition to calculate: 
\begin{equation}
\omega =\frac{(\frac{1-is}{is})^{\frac{1}{\sigma }}}{(\frac{1-is}{is})^{%
\frac{1}{\sigma }}+1}.  \label{3om}
\end{equation}%
As long as we have $\sigma ,$ we have $\omega .$

\paragraph{Long-run versus short-run elasticity puzzle}

Since firm evidence on a reasonable value for $\sigma $ came later, early
business cycle literature used an alternative strategy to pin down this
parameter$.$ According to the theory, both approaches should give the same
answer. However, the puzzle is that they do not, which has been later
labeled \textit{long-run versus short-run elasticity puzzle}.

The alternative strategy of measuring elasticity $\sigma $ is motivated by
the fact that in the business cycle models, the demand for domestic and
foreign good is modeled by a CES aggregator. In such case, it is
straightforward to show that the import ratio is tied to the relative price
of domestic and imported goods by (see equations \ref{3FOC1}, (iii)-(iv)) 
\begin{equation}
\log \frac{f_{t}}{d_{t}}=\sigma \log \frac{p_{d,t}}{p_{f,t}}+\log \frac{%
\omega _{t}}{1-\omega _{t}}.  \label{d_f}
\end{equation}%
(To be more general, we are allowing here for $\omega $ to be time-varying.)

Under normal conditions (i.e., when the supply curve is an upward-sloping
function of the price and the supply shocks are not correlated with the $%
\omega _{t}$-demand shocks), we should expect the correlation between $\log 
\frac{\omega _{t}}{1-\omega _{t}}$ and $\log \frac{p_{d,t}}{p_{f,t}}$ to be
positive. But then, the \textit{volatility ratio} defined by 
\begin{equation}
VR\equiv std(\log \frac{f_{t}}{d_{t}})/std(\log \frac{p_{d,t}}{p_{f,t}})
\end{equation}%
places an upper bound on the value of the intrinsic price elasticity of
trade flows $\sigma $, as implied by the following evaluation of (\ref{d_f})
and variance decomposition: 
\begin{eqnarray}
\sigma  &=&std(\log \frac{f_{t}}{d_{t}})/std(\log \frac{p_{d,t}}{p_{f,t}}+%
\frac{1}{\sigma }\log \frac{\omega _{t}}{1-\omega _{t}})\leq  \\
&\leq &std(\log \frac{f_{t}}{d_{t}})/std(\log \frac{p_{d,t}}{p_{f,t}})=VR.
\end{eqnarray}%
In particular, in the Armington model with $\omega $ assumed constant, the
volatility ratio is exactly equal to the elasticity of substitution $\sigma $%
.

This is the measurement of short-run elasticity that I used with
Jaromir in our customer capital paper. It avoids the use of time-series regressions. Regression
in this context may require  to specify a model with error
correction. This gives the upper bound of the regression coefficient. In most applications the upper bound is enough for the analysis at hand. 

The computed values of the volatility ratio for the data are shown in Table %
\ref{Tab_VR}. As we can see, these values are very low and grossly at odds
with the value of the parameter $\sigma $ implied by the trade literature.%
\footnote{%
Similar results, using different method, are obtained by Wilson (2001) or
Reinert et al. (1992).} At business cycle frequencies, the median value of
the volatility ratio is as low as $0.7$ for both HP-filtered and linearly
detrended data.

%TCIMACRO{\TeXButton{B}{\begin{table}[t] \centering}}%
%BeginExpansion
\begin{table}[t] \centering%
%EndExpansion
\caption{Volatility Ratio in a Cross-Section of
Countries.}

\begin{minipage}[htb]{0.94\linewidth}
\begin{tabular}{p{2.65in}p{1.20in}p{1.10in}}
\toprule
& \multicolumn{2}{l}{Detrending method}\\
\cmidrule(r){2-3}
\small Country &  \small HP-1600 &  \small Linear$^{a}$\\
\midrule
\small Australia & \  \small 0.94& \  \small 0.93\\
\small Belgium & \  \small 0.57& \  \small 0.50\\
\small Canada & \  \small 1.27& \  \small 0.64\\
\small France & \  \small 0.54& \  \small 0.73\\
\small Germany & \  \small 0.90& \  \small 1.16\\
\small Italy & \  \small 0.69& \  \small 0.46\\
\small Japan & \  \small 0.60& \  \small 0.43\\
\small Netherlands & \  \small 0.44& \  \small 0.72\\
\small Switzerland & \  \small 0.71& \  \small 1.16\\
\small Sweden & \  \small 0.95& \  \small 0.95\\
\small UK & \  \small 0.65& \  \small 0.61\\
\small US$^{b}$ & \  \small 1.23& \  \small 1.02\\
\midrule
\small MEDIAN  &\  \small 0.71&\  \small 0.73\\
\bottomrule \addlinespace\end{tabular}\label{Ch1T3}
\newline
\scriptsize Notes: Based on quarterly time-series, $1980:1-2000:1$.
Data sources are listed in Drozd and Nosal (2008). \newline
$^{a}$Linear trend subtracted from logged time series.
\newline $^{b}$For the entire postwar period ($1959:3-2004:2$) this
ratio in U.S. is 0.88.
\end{minipage}\label{Tab_VR}%
%TCIMACRO{\TeXButton{E}{\end{table}}}%
%BeginExpansion
\end{table}%
%EndExpansion

Because of this discrepancy, in our quantitative analysis of the model, we
will report the results for both low and high values of $\sigma $. Namely,
we will consider the values based on the estimates from Head and Ries of $%
\sigma =7.9,$ as well as the value $\sigma =.73$ consistent with low value
of the volatility ratio in the data.\footnote{%
We choose the value 1 instead of .7 because we can solve this Cobb-Douglas
case analytically when $\theta =1,$ and $\eta =1$. It will be helpful later
to understand the intuition.} As we will later see, the model is going to
perform much better in the latter case.

\paragraph{Estimating the forcing process}

The prediction of the model crucially depends on the properties of the
stochastic process for output and technology $A$. Below, we first back out
the process for output that we use in conjunction with the endowment economy
model, and then we back out the process for the Solow residuals that we use
in conjunction with the extended model. The details are available from the
Excel file posted on my website.

\subparagraph{Endowment shocks}

To back out endowment process from the data, we must calculate the aggregate
output of the rest of the world. The difficulty is that the real GDP is
expressed in units that can not be readily compared (local currency units).

To adjust the units of real GDP, we use a PPP adjusted GDP for a chosen
year, say year 2000, and normalize real GDP of each country so that it is 1
in year 2000. Next, we multiply each country series by the value of PPP\
adjusted GDP in year 2000 pulled out from the Penn World Tables. Note that
we can follow a similar procedure to aggregate virtually any aggregate
time-series. For example, if we want to aggregate investment, we must know
the share of investment in GDP in year 2000 in each country, and after
normalizing investment series in each country so that it is 1 in year 2000,
to obtain series that we can add up, it suffices to multiply each individual
series by the PPP adjusted GDP in year 2000 and the share of investment in
GDP for the year 2000. (The choice of the baseline year, here year 2000,
matters in general, but does not change the results drastically.)

From the data, we need the real GDP series and population series for
countries that we define as the rest of the world, and the US. Then, we
divide output by population to translate it to per capita terms. The source
of the aggregate data is OECD (www.sourceOECD.org) and Penn World Tables for
population (smoothly extrapolated from the growth rate of population). Using
this data, we estimate the process of the form:%
\begin{eqnarray*}
\log y(s^{t+1}) &=&\xi _{1}+\gamma t+\log \hat{y}(s^{t+1}) \\
\log y^{\ast }(s^{t+1}) &=&\xi _{2}+\gamma t+\log \hat{y}^{\ast }(s^{t+1}) \\
\log \hat{y}(s^{t+1}) &=&\rho \log \hat{y}(s^{t})+\phi \log y^{\ast
}(s^{t})+\varepsilon (s^{t+1}), \\
\log y(s^{t+1}) &=&\rho \log y^{\ast }(s^{t})+\phi \log \hat{y}%
(s^{t})+\varepsilon ^{\ast }(s^{t+1}),
\end{eqnarray*}%
where $\gamma $ is a common growth parameter, $\rho $ is persistence
parameter, $\phi $ is spillover parameters, and $\varepsilon ,\varepsilon
^{\ast }$ are i.i.d. normally distributed random variables with a symmetric
variance-covariance matrix $\Sigma $.

To estimate this system we proceed as follows. We estimate the model using
SUR (seemingly unrelated regression) method with symmetry restrictions. We
check if the spillover parameter $\phi $ is statistically different from
zero, and if not, we reestimate the model with a restriction $\phi =0.$
Using the estimated parameters, we then back out the regression error vector 
$(\hat{\varepsilon},\hat{\varepsilon}^{\ast }),$ and fit a two-dimensional
Normal distribution with a symmetry restriction imposed. The results of the
estimation give:\ 
\begin{eqnarray*}
\rho  &=&.95\text{ }(0.02) \\
\phi  &=&0.0, \\
\sum  &=&\left[ 
\begin{array}{cc}
3.56 & 1.17 \\ 
1.17 & 3.56%
\end{array}%
\right] \times 10^{-5}.
\end{eqnarray*}%
The variance-covariance matrix implies that correlation of residuals $%
\varepsilon ,\varepsilon ^{\ast }$ is .33, and the standard deviation is
about .006. The 5\% confidence interval on $\rho $ is $[.90,.99]$.

\subparagraph{Technology shocks}

To recover Solow residuals, we use the following formula (based on log of $%
Y_{t}=K_{t}^{\alpha }(A_{t}L_{t})^{1-\alpha })$: 
\begin{equation*}
\log A_{t}=\frac{\log Y_{t}-\alpha \log K_{t}+(1-\alpha )\log L_{t}}{%
1-\alpha },
\end{equation*}%
where $Y_{t}$ is real GDP, $K_{t}$ is total stock of capital backed out from
real investment series using perpetual inventory method\footnote{%
The basic idea behind perpetual inventory method is to cumulate capital
using equation: $K_{t+1}=(1-\delta _{t})K_{t}+I_{t},$ where $I_{t}$ are the
series for real gross investment, and $\delta _{t}$ is our calibrated value
for each year (or a constant mean value $\delta )$. The initial capital $%
K_{0}$ is obtained from an educated guess that requires that the average
growth rate of capital over the first 10 years is the same as in the initial
year, i.e. 
\begin{equation*}
\frac{K_{1}-K_{0}}{K_{0}}=\frac{1}{10}\sum_{i=1,..,10}\frac{K_{i+1}-K_{i}}{%
K_{i}}.
\end{equation*}%
}, and $L_{t}$ are total hours worked for US\ and total civilian employment
for the rest of the world. Our underlying data is quarterly and pertains to
the time period 1980-2005.

Given series for $A_{t}$ for US\ and $A_{t}^{\ast }$ for the rest of the
world, following the same procedure as with output, we estimate the process:%
\begin{eqnarray}
\log A(s^{t+1}) &=&\xi _{1}+\gamma t+\log \hat{A}(s^{t+1}),  \label{3SolowA}
\\
\log A^{\ast }(s^{t+1}) &=&\xi _{2}+\gamma t+\log \hat{A}^{\ast }(s^{t+1}), 
\notag \\
\log \hat{A}(s^{t+1}) &=&\rho \log \hat{A}(s^{t})+\phi \log \hat{A}^{\ast
}(s^{t})+\varepsilon (s^{t+1}),  \notag \\
\log \hat{A}^{\ast }(s^{t+1}) &=&\rho \log \hat{A}^{\ast }(s^{t})+\phi \log 
\hat{A}(s^{t})+\varepsilon ^{\ast }(s^{t+1}),  \notag
\end{eqnarray}%
where as before $\gamma $ is a common growth parameter, $\rho $ is
persistence parameter, $\phi $ is spillover parameters, and $\varepsilon
,\varepsilon $ are $i.i.d.$ normally distributed symmetric innovations with
a variance-covariance matrix $\Sigma $.

The results are:%
\begin{eqnarray*}
\rho &=&.91\text{ }(0.025) \\
\phi &=&0.0, \\
\sum &=&\left[ 
\begin{array}{cc}
5.12 & 1.20 \\ 
1.20 & 5.12%
\end{array}%
\right] \times 10^{-5},
\end{eqnarray*}%
where the variance-covariance matrix implies that correlation of residuals $%
\varepsilon ,\varepsilon ^{\ast }$ is .23, and standard deviation about
.0071. The 5\% confidence interval on $\rho $ is $[.86,.96])$.

Lastly, as a consistency check, having the series for capital, we can look
up K/Y ratio in the data. According to the model, it should exhibit no
trend. In Figure \ref{Fig_KY}, we illustrate the actual time series for K/Y
in the data (with initial point set equal to 3.5) over the sample period
1980-2005. As we can see, it exhibits no trend. You will find these
time-series in the Excel file posted online.
 


\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=17,scale=.44]{Figures.pdf} 
 \caption{Comparison of K/Y ratio between
 model (BGP path)\ and US\ data (initial value matched by construction).} 
\label{Fig_KY}
\end{figure} 
 
%\FRAME{ftbpFU}{454pt}{311.0667pt}{0pt}{\Qcb{Comparison of K/Y ratio between
%model (BGP path)\ and US\ data (initial value matched by construction). }}{%
%\Qlb{Fig_KY}}{Figure}{\special{language "Scientific Word";type
%"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "T";width
%454pt;height 311.0667pt;depth 0pt;original-width 731.3333pt;original-height
%500.0667pt;cropleft "0";croptop "1";cropright "1";cropbottom
%"0";tempfilename 'KD2WCH0G.wmf';tempfile-properties "XPR";}}

\subsubsection*{Solving the Model}

Having the parameter values, we next solve the model numerically. This will
allow us to generate artificial data, and compare it to the actual data. Our
comparison will focus on the basic statistics measuring volatility and
comovement of time-series. These are not the only characteristics of
time-series that can be compared, but they are first order statistics, and
we will focus on them.

Our model is given by the following planning problem: 
\begin{equation*}
\max \left[ \sum_{t}^{\infty }\sum_{s^{t}\in S^{t}}\beta ^{t}\pi
(s^{t})u(c(s^{t}))+\sum_{t}^{\infty }\sum_{s^{t}\in S^{t}}\beta ^{t}\pi
(s^{t})u(c^{\ast }(s^{t}))\right]
\end{equation*}%
subject to 
\begin{eqnarray*}
c(s^{t})+i(s^{t}) &=&G(d(s^{t}),f(s^{t})), \\
c^{\ast }(s^{t})+i^{\ast }(s^{t}) &=&G(f^{\ast }(s^{t}),d^{\ast }(s^{t}))%
\text{ }
\end{eqnarray*}%
and%
\begin{eqnarray*}
d(s^{t})+d^{\ast }(s^{t}) &=&y(s^{t}), \\
f(s^{t})+f^{\ast }(s^{t}) &=&y^{\ast }(s^{t}),\text{ } \\
y(s^{t}) &=&k(s^{t})^{\alpha }(A(s^{t})l(s^{t}))^{1-\alpha }, \\
y^{\ast }(s^{t}) &=&k^{\ast }(s^{t})^{\alpha }(A^{\ast }(s^{t})l^{\ast
}(s^{t}))^{1-\alpha }, \\
\zeta k(s^{t+1}) &=&(1-\delta )k(s^{t})+i(s^{t}), \\
\zeta k^{\ast }(s^{t+1}) &=&(1-\delta )k^{\ast }(s^{t})+i^{\ast }(s^{t}),%
\text{ all }s^{t}\in S^{t},
\end{eqnarray*}%
where the process for $A$ and $A^{\ast }$ is given by (\ref{3SolowA}).

To solve this model, we redefine all the variable by dividing them by the
corresponding growth rate of the economy (by analogy for foreign country
variables): 
\begin{eqnarray*}
\hat{c}_{t} &\equiv &\frac{c_{t}}{\gamma ^{t}}, \\
\hat{k}_{t} &\equiv &\frac{k_{t}}{\gamma ^{t}}, \\
\hat{\imath}_{t} &\equiv &\frac{i_{t}}{\gamma ^{t}}, \\
\hat{y}_{t} &\equiv &\frac{y_{t}}{\gamma ^{t}}, \\
\hat{l}_{t} &\equiv &l_{t}, \\
\hat{d}_{t} &\equiv &\frac{y_{t}}{\gamma ^{t}}, \\
\hat{f}_{t} &\equiv &\frac{y_{t}}{\gamma ^{t}},
\end{eqnarray*}%
and by plugging in these values, we rewrite the original problem in a
stationary form: 

\begin{equation*}
\max \left[ \sum_{t}^{\infty }(\beta \gamma ^{\eta (1-\theta
)})^{t}[\sum_{s^{t}\in S^{t}}\pi (s^{t})u(\hat{c}(s^{t}))+\sum_{s^{t}\in
S^{t}}\pi (s^{t})u(\hat{c}^{\ast }(s^{t}))]\right]
\end{equation*}%
subject to 
\begin{eqnarray*}
\hat{c}(s^{t})+\hat{\imath}(s^{t}) &=&G(\hat{d}(s^{t}),\hat{f}(s^{t})), \\
\hat{c}^{\ast }(s^{t})+\hat{\imath}^{\ast }(s^{t}) &=&G(\hat{f}^{\ast
}(s^{t}),\hat{d}^{\ast }(s^{t}))\text{ }
\end{eqnarray*}%
and%
\begin{eqnarray*}
\hat{d}(s^{t})+\hat{d}^{\ast }(s^{t}) &=&\hat{y}(s^{t}), \\
\hat{f}(s^{t})+f^{\ast }(s^{t}) &=&\hat{y}^{\ast }(s^{t}),\text{ } \\
\hat{y}(s^{t}) &=&\hat{k}(s^{t})^{\alpha }(\hat{A}(s^{t})\hat{l}%
(s^{t}))^{1-\alpha }, \\
\hat{y}^{\ast }(s^{t}) &=&\hat{k}^{\ast }(s^{t})^{\alpha }(\hat{A}^{\ast
}(s^{t})\hat{l}^{\ast }(s^{t}))^{1-\alpha }, \\
\gamma \zeta \hat{k}(s^{t+1}) &=&(1-\delta )\hat{k}(s^{t})+\hat{\imath}%
(s^{t}), \\
\gamma \zeta \hat{k}^{\ast }(s^{t+1}) &=&(1-\delta )\hat{k}^{\ast }(s^{t})+%
\hat{\imath}^{\ast }(s^{t}),\text{ all }s^{t}\in S^{t},
\end{eqnarray*}%
where the process for $\hat{A}$ and $\hat{A}^{\ast }$ is given by (\ref%
{3SolowA}).

In what follows, we substitute the value for adjusted discount $(\beta
\gamma ^{\eta (1-\theta )})^{t}$ and instead simply write $\beta $. If $%
\theta =2,$ such adjusted value is given by $\beta \equiv (\beta \gamma
^{\eta (1-\theta )})^{t}=.983.$ We also drop the notation with `hats', and
write $g$ instead of $\gamma \zeta $. After these simplifications, the
resulting stationary planning problem is: 

\begin{equation*}
\max \left[ \sum_{t}^{\infty }\beta ^{t}[\sum_{s^{t}\in S^{t}}\pi
(s^{t})u(c(s^{t}))+\sum_{s^{t}\in S^{t}}\pi (s^{t})u(c^{\ast }(s^{t}))]%
\right]
\end{equation*}%
subject to 
\begin{eqnarray*}
c(s^{t})+i(s^{t}) &=&G(d(s^{t}),f(s^{t})), \\
c^{\ast }(s^{t})+i^{\ast }(s^{t}) &=&G(f^{\ast }(s^{t}),d^{\ast }(s^{t}))%
\text{ }
\end{eqnarray*}%
and%
\begin{eqnarray*}
d(s^{t})+d^{\ast }(s^{t}) &=&y(s^{t}), \\
f(s^{t})+f^{\ast }(s^{t}) &=&y^{\ast }(s^{t}),\text{ } \\
y(s^{t}) &=&k(s^{t})^{\alpha }(A(s^{t})l(s^{t}))^{1-\alpha }, \\
y^{\ast }(s^{t}) &=&k^{\ast }(s^{t})^{\alpha }(A^{\ast }(s^{t})l^{\ast
}(s^{t}))^{1-\alpha }, \\
gk(s^{t+1}) &=&(1-\delta )k(s^{t})+i(s^{t}), \\
gk^{\ast }(s^{t+1}) &=&(1-\delta )k^{\ast }(s^{t})+i^{\ast }(s^{t}),\text{
all }s^{t}\in S^{t},
\end{eqnarray*}

The competitive equilibrium to this planning solution corresponds exactly to
the setup we have described in previous sections (we even have introduced a
constant g to take growth into account). So, below, instead of taking the
first order conditions to this planning problem, we instead use the
equilibrium conditions from decentralized competitive equilibrium that we
have stated before. To summarize, these conditions are:\ 

(i) Demand equations (note that this equation embeds numeraire
normalization, see derivation of FOC\ for our prototype model)

\begin{eqnarray}
p_{d}(s^{t}) &=&G_{d}(d(s^{t}),f(s^{t})), \\
p_{f}(s^{t}) &=&G_{f}(d(s^{t}),f(s^{t})), \\
p_{f}^{\ast }(s^{t}) &=&G_{f}(f(s^{t}),d(s^{t})), \\
p_{d}^{\ast }(s^{t}) &=&G_{d}(f(s^{t}),d(s^{t})),
\end{eqnarray}

(ii)\ labor/leisure choice:%
\begin{eqnarray}
\frac{u_{l}\left( s^{t}\right) }{u_{c}\left( s^{t}\right) } &=&-w\left(
s^{t}\right) , \\
\frac{u_{l}^{\ast }\left( s^{t}\right) }{u_{c}^{\ast }\left( s^{t}\right) }
&=&-w^{\ast }\left( s^{t}\right) ,
\end{eqnarray}

(iii) perfect risk-sharing

\begin{equation}
x(s^{t})=\frac{u_{c}(c^{\ast }(s^{t}))}{u_{c}(c(s^{t}))}
\end{equation}

(iv) Euler equations: 
\begin{eqnarray}
u_{c}\left( s^{t}\right) g &=&\beta E_{s^{t}}[u_{c}\left( s^{t+1}\right)
\left( \left( 1-\delta \right) +r\left( s^{t+1}\right) \right) ], \\
u_{c^{\ast }}^{\ast }\left( s^{t}\right) g &=&\beta E_{s^{t}}[u_{c}^{\ast
}\left( s^{t+1}\right) \left( \left( 1-\delta \right) +r^{\ast }\left(
s^{t+1}\right) \right) ],
\end{eqnarray}

(v) law of one price 
\begin{eqnarray}
p_{d}(s^{t}) &=&x(s^{t})p_{d}^{\ast }(s^{t}), \\
p_{f}(s^{t}) &=&x(s^{t})p_{f}^{\ast }(s^{t}),
\end{eqnarray}

(vi) factor prices: 
\begin{eqnarray}
r(s^{t}) &=&\alpha p_{d}(s^{t})k(s^{t})^{\alpha
-1}(A(s^{t})l(s^{t}))^{1-\alpha }, \\
r^{\ast }(s^{t}) &=&\alpha p_{d}^{\ast }(s^{t})k^{\ast }(s^{t})^{\alpha
-1}(A^{\ast }(s^{t})l^{\ast }(s^{t}))^{1-\alpha }, \\
w(s^{t}) &=&(1-\alpha )A(s^{t})p_{d}(s^{t})k(s^{t})^{\alpha
}(A(s^{t})l(s^{t}))^{-\alpha }, \\
w^{\ast }(s^{t}) &=&(1-\alpha )A^{\ast }(s^{t})p_{d}^{\ast }(s^{t})k^{\ast
}(s^{t})^{\alpha }(A^{\ast }(s^{t})l^{\ast }(s^{t}))^{-\alpha },
\end{eqnarray}

(vii) feasibility and market clearing

\begin{eqnarray}
c(s^{t})+i(s^{t}) &=&G(d(s^{t}),f(s^{t})), \\
c^{\ast }(s^{t})+i^{\ast }(s^{t}) &=&G(f^{\ast }(s^{t}),d^{\ast }(s^{t}))%
\text{ }
\end{eqnarray}%
\begin{eqnarray}
d(s^{t})+d^{\ast }(s^{t}) &=&y(s^{t}), \\
f(s^{t})+f^{\ast }(s^{t}) &=&y^{\ast }(s^{t}),\text{ } \\
y(s^{t}) &=&k(s^{t})^{\alpha }(A(s^{t})l(s^{t}))^{1-\alpha }, \\
y^{\ast }(s^{t}) &=&k^{\ast }(s^{t})^{\alpha }(A^{\ast }(s^{t})l^{\ast
}(s^{t}))^{1-\alpha }, \\
gk(s^{t+1}) &=&(1-\delta )k(s^{t})+i(s^{t}), \\
gk^{\ast }(s^{t+1}) &=&(1-\delta )k^{\ast }(s^{t})+i^{\ast }(s^{t}),\text{
all }s^{t}\in S^{t}.
\end{eqnarray}

(viii) technology shocks%
\begin{eqnarray}
\log A(s^{t+1}) &=&\rho \log A(s^{t})+\varepsilon (s^{t+1}), \\
\log A^{\ast }(s^{t+1}) &=&\rho \log A^{\ast }(s^{t})+\varepsilon ^{\ast
}(s^{t+1}).
\end{eqnarray}

As we can see, in the system above, we have 12 variables we should count
twice, $c,i,y,k,l,A,d,f,r,w,p_{d},p_{f},$ and 1 variable that we should
count once, $x.$ Together, it gives us 25 variables. Since we have 25
equations, unless we have mistakenly restated same equilibrium conditions
twice (not the case), we can proceed setting up the model on the computer. 

To solve the model, we implement the perturbation method using the package
Dynare\footnote{%
See http://www.cepremap.cnrs.fr/dynare/.}. Dynare will locally approximate
the solution around the deterministic steady state, which we next calculate
analytically.

To calculate the steady state, it is convenient to unwind some of the steps
used in the calibration. Specifically, we proceed as follows:\ 

Step 1: Under symmetry, $p_{f}=p_{d},$ and from demand equations (i), we have

\begin{equation*}
f=(\frac{\omega }{1-\omega })^{-\sigma }d.
\end{equation*}%
From the definition of the import ratio, we obtain%
\begin{equation*}
is\equiv \frac{f}{d+f}=\frac{(\frac{\omega }{1-\omega })^{-\sigma }d}{(\frac{%
\omega }{1-\omega })^{-\sigma }d+d}=\frac{(\frac{\omega }{1-\omega }%
)^{-\sigma }}{(\frac{\omega }{1-\omega })^{-\sigma }+1},
\end{equation*}%
and calculate 
\begin{equation}
\omega =\frac{(\frac{1-is}{is})^{\frac{1}{\sigma }}}{(\frac{1-is}{is})^{%
\frac{1}{\sigma }}+1}.
\end{equation}%
Using Euler's law, we next solve for prices $p_{d}=p_{f}$ (equal by
symmetry). By hod 1 of $G,$ we have%
\begin{equation*}
p_{d}d+p_{f}f=G(d,f),
\end{equation*}%
and thus%
\begin{equation}
p_{d}=p_{f}=\frac{G(d,f)}{d+f}=(\omega \times (1-is)^{\frac{\sigma -1}{%
\sigma }}+(1-\omega )\times is^{\frac{\sigma -1}{\sigma }})^{\frac{\sigma }{%
\sigma -1}},
\end{equation}%
where $is$ denotes the import share that we have used in calibration as one
of the data targets.

Step 2:\ Since the share of labor in time endowment of has been calculated
to be $l=.329$, knowing that by Euler's equation (iv) 
\begin{equation}
r=\frac{g}{\beta }-(1-\delta ),
\end{equation}%
we can use factor price equation for rental price of capital (ii)%
\begin{equation*}
r=\alpha p_{d}(\frac{k}{l})^{\alpha -1}
\end{equation*}%
to find steady state level of capital\footnote{%
We could have alternatively solved for steady state in the closed economy
model first, and used the quantities from there to plug in here. This
approach would require some adjustment so that $G(d,f)=y$ for steady state
values.}%
\begin{equation}
k=l(\frac{\alpha p_{d}}{r})^{\frac{1}{1-\alpha }}.
\end{equation}

Step 3: By definition of $is$, from feasibility conditions (vii), we have 
\begin{align}
d& =(1-is)\times k^{\alpha }l^{1-\alpha }, \\
f& =is\times k^{\alpha }l^{1-\alpha }.
\end{align}

Step 4:\ The remaining steady state variables can be calculated from (vi),
(vi) and (iii):%
\begin{align}
c& =(\omega d^{\frac{\sigma -1}{\sigma }}+(1-\omega )f^{\frac{\sigma -1}{%
\sigma }})^{\frac{\sigma }{\sigma -1}}-(g-1+\delta )k, \\
w& =(1-\alpha )p_{d}k^{\alpha }l^{-\alpha },  \notag \\
x& =1\text{ (by symmetry)}.
\end{align}

Step 5: The implied values of the deep parameter $\eta $ can be retrieved
from the assumed value of $l\ $the same way as we have done to calibrate the
model. Labor-leisure choice (vi) implies%
\begin{eqnarray*}
w &=&-\frac{u_{l}}{u_{c}}=\frac{c}{1-l}\frac{1-\eta }{\eta }, \\
\frac{1-\eta }{\eta } &=&\frac{w(1-l)}{c}
\end{eqnarray*}%
and thus 
\begin{equation}
\eta =\frac{1}{1+\frac{w(1-l)}{c}}.
\end{equation}%
The numbered equations fully characterize the deterministic steady state we
set out to find.

We next proceed with the implementation of the Dynare code\footnote{%
http://www.cepremap.cnrs.fr/juillard/mambo/download/manual/Dynare\_UserGuide%
\_WebBeta.pdf} to solve the model using perturbation method. The codes can
be downloaded from my website.

\begin{exercise}
The example will illustrate (in an abstract way)\ how to use the
perturbation method to solve portfolio choice problems under uncertainty. In
the deterministic steady state, any portfolio that gives the same wealth
distribution is equivalent. However, this equivalence breaks down when we
add uncertainty (shocks), because stochastic payoffs of different portfolios
will typically correlate differently with the stochastic consumption. To
solve the optimal portfolio problem using perturbation method, the idea is
to start with an arbitrary portfolio, solve for the policy for assets with a
penalty function imposed (2nd order approximation at least), and use this
policy to obtain the next guess. For an implementation of such method, see,
for example, Heathcote and Perri (2008): \textquotedblleft The International
Diversification Puzzle is Not as Bad as You Think\textquotedblright ,
Minneapolis Fed Staff Report 389. Using an abstract example, we will
demonstrate how it works. 

Consider the following problem%
\begin{equation*}
\max_{x\geq 0}[x^{\rho }+(1-x)^{\rho }].
\end{equation*}%
Clearly, when $0<\rho <1,$ there is a unique solution $x=\frac{1}{2},$ and
when $\rho =1,$ any solution $x\in \lbrack 0,5]$ solves the problem.

Suppose now that we do not know the true solution when $0<\rho <1,$ but we
do know that any value of x satisfies the problem for $\rho =1.$ We will
exploit this fact by using the perturbation method solve for the case when $%
\rho <1$.

\noindent a. Take as a starting guess $x_{guess}=0.25$ and proceed as
follows. Impose a convex penalty on the objective function and consider the
following artificial problem: 
\begin{equation*}
\max_{x\geq 0}[x^{\rho }+(5-x)^{\rho }-\phi (x_{guess}-x)^{2}].
\end{equation*}%
Using the idea of perturbation method, implement on Matlab the following
scheme. Choose appropriate value of $\phi $ that is large enough, but not
too small, and take 2nd order approximation of the solution to the
artificial problem for $\rho =.95$ with $x_{guess}=0.25$ and perturbation
wrt $\rho $ (evaluated at $\rho =1)$. After you solve for a new approximate
value, take this value as the next guess $x_{guess},$ and solve it again.
Iterate until convergence.

\noindent b. Have you obtained the true solution $x=.5$? Briefly explain
what makes this scheme work, and why we needed to impose penalty $\phi $
(what would go wrong?)?
\end{exercise}

\section{Quantitative Comparison of Theory and Data}

Our goal here is to obtain the simulated time-series from the models, and
after treating this `artificial data' the same way as the actual data, to
compare the statistics pertaining to the properties of the business cycle
fluctuations. Below, we calculate business cycle statistics implied by 4
different versions of our two-country model (all for $\theta =2)$:

\begin{enumerate}
\item Model with capital and labor/leisure choice (model 1) with high
elasticity of substitution $\sigma =.73,$

\item Model with capital and labor/leisure choice (model 1) with low
elasticity of substitution $\sigma =8.$

\item Prototype endowment model (model 2) with a high elasticity of
substitution $\sigma =.73,$

\item Prototype endowment model (model 2) with a low elasticity of
substitution $\sigma =8$
\end{enumerate}

In our comparison with data, we will rely on a casual notion of the quality
of fit. We should stress that in order to focus on the qualitative economic
mechanisms, we have deliberately setup a model that is mispecified in many
dimensions. Fitting such model to the data using statistical estimation
would likely result in a biased inference about parameter values.\footnote{%
If the omitted variables turn out correlated with the error term, this
creates a problem for statistical methods that try to fit the data directly.}
Of course, with more complete models, it is a good idea to try to estimate
them too. To learn about the methods of estimating large-scale DSGE models,
check out the website of Frank Schorfheide. On the website of Dynare you
will also find helpful examples applying Bayesian techniques to estimate a
macro model (see the link to examples).

\subsubsection*{Properties of the Data}

Before we proceed any further, we should first characterize how economic
activity behaves over the business cycle. Without it, we will not be able to
say much about the performance of our model. To characterize properties of
economic activity over the business cycle, we will focus attention on two
first order aspects of the data:\ comovement and volatility.

Figure \ref{Fig_BCQ} lists key summary statistics pertaining to 5 basic
measures of aggregate activity over the business cycles for 3 major economic
regions of the world:\ US, Japan and Europe (aggregate of EU15 countries).
The 5 measures are: output, consumption expenditures, investment, employment
and net exports (borrowing from rest of the world). All measures are real
and do not depend on prices. The data is quarterly, and has been first
logged and the HP-filtered (1600). $NX$ has been calculated as the ratio of
the difference between nominal exports and nominal imports to nominal GDP
(results are the same if CPI is used to deflate NX or trend part of nominal
GDP). 
 
 
\begin{figure}[ptb]
\centering
%\captionsetup[subfigure]{labelformat=empty}
 \includegraphics[page=16,scale=.44]{Figures.pdf} 
 \caption{Basic Properties of International
 Business Cycles: Volatility and Comovement Patterns.} 
\label{Fig_BCQ}
\end{figure} 

%\FRAME{ftbpFU}{419pt}{387.0667pt}{0pt}{\Qcb{Properties of International
%Business Cycles: Volatility and Comovement. }}{\Qlb{Fig_BCQ}}{Figure}{%
%\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
%TRUE;display "USEDEF";valid_file "T";width 419pt;height 387.0667pt;depth
%0pt;original-width 447.1333pt;original-height 412.8667pt;cropleft
%"0";croptop "1";cropright "1";cropbottom "0";tempfilename
%'KD2WCH0H.wmf';tempfile-properties "XPR";}}\bigskip

As we can see, even though statistics are widely dispersed, several patterns
emerge. First, in all three cases the ranking of relative volatilities is
similar. Investment is clearly the most volatile, output and Solow residual
are second, and consumption is the least volatile. Solow residual is less
volatile in US and EU15 than output and more in Japan.\footnote{%
At this point we should mention that it is rather standard in the literature
to call $A^{1-\alpha }$ a Solow residual. Residual defined in such a way
would be less volatile than output in all three economic blocks (not
reported here).} 

In terms of international comovement, in all three blocks economic activity
tends to positively comove with the rest of the world. In Japan,
international comovement is the weakest, but Europe positively comoves with
the rest of the world very strongly. The US is somewhat inbetween. As far as
ranking of comovement across our 4 aggregate measures, the data shows the
following pattern. Output, investment, labor and Solow residuals all
positively comove, and in particular, they comove more than consumption
does. Also, output is most correlated internationally, in particular, more
so than the Solow residuals. The low positive comovement of consumption
relative to output is somewhat puzzling, because one should expect that
consumption risk sharing should make consumption to be most strongly
correlated internationally. It actually turns out a wrong intuition when the
elasticity of substitution between the domestic and the foreign goods is
very low, and there is home-bias (to understand why think about the Leontief
case). 

Another clear pattern is the countercyclicality of net exports (NX). It
turns out that a country that has a boom tends to import more and borrow
from the rest of the world---again the opposite to what simple a logic about
risk sharing would suggest. Interestingly enough, there is little evidence
that Solow residuals are negatively correlated with NX. It is capital and
labor that seem to be behind this negative correlation between output and
NX. 

All these properties, perhaps except the last one, are robust across
countries, and have been widely documented in the literature. Finally, we
should also mention that all variables are highly persistent, which we do
not report here.

Having characterized how economic activity behaves over the business cycle,
we next study the predictions of the model. The goal is intuitively
understand the economic forces behind the implications of the theory, and
organize the findings in a set of discrepancies between the theory and the
data. Since our theory is the most basic frictionless environment one could
think of, we will refer to these discrepancies as \textit{puzzles wrt to
standard theory, }which\textit{\ }helps us organize further work to improve
upon this theory.

\subsubsection*{Predictions of the Models}

To map the model onto the data, we first have to define consistent with the
data system of measurement. In this respect, we identify the following
objects in the model with their corresponding counterparts in the data:

\begin{itemize}
\item Real GDP$=p_{d}^{ss}d+p_{f}^{ss}d+x^{ss}p_{d}^{ss}d^{\ast
}-p_{f}^{ss}f,$

\item GDP in current prices $=p_{d}d+p_{f}d+xp_{d}d^{\ast }-p_{f}f,$

\item Consumption$=\frac{c}{G(d,f)}(p_{d}^{ss}d+p_{f}^{ss}),$

\item Investment$=\frac{i}{G(d,f)}(p_{d}^{ss}d+p_{f}^{ss}),$

\item Net exports (in current prices)$=(xp_{d}d^{\ast }-p_{f}f)/GDP,$

\item Real net exports$=(xp_{d}^{ss}d^{\ast }-p_{f}^{ss}f)/Real$ $GDP$

\item Real export price$=xp_{d}^{\ast },$

\item Real import price$=p_{f},$

\item Terms of trade=$p,$

\item Real exchange rate=$x,$
\end{itemize}

\noindent where $\cdot ^{ss}$denotes steady state prices. (To be fully
consistent with the data, we should actually measure prices like real
exchange rate using fixed weights CPI rather than the ideal one---just like
in the data. This would not matter in this model, and so we omit this
distinction.)

%TCIMACRO{\TeXButton{B}{\begin{table}[h] \centering}}%
%BeginExpansion
\begin{table}[h] \centering%
%EndExpansion
\caption{Comparison of Models with Data.}

\begin{minipage}[b]{0.97\linewidth} 
\begin{tabular}{p{1.45in}p{0.65in}p{0.75in}p{0.75in}p{0.75in}p{0.75in}}  
\toprule  
&  & \multicolumn{4}{l}{\small Models}\\  
\cmidrule(r){3-6}  
&  & \footnotesize Model 1 & \footnotesize Model 1 & \footnotesize Model 2 & \footnotesize Model 2\\  
&  & \footnotesize with Low  & \footnotesize with High & \footnotesize with Low & \footnotesize with High\\  
\small Statistic & \small Data & \footnotesize Elasticity & \footnotesize Elasticity & \footnotesize Elasticity & \footnotesize Elasticity\\  
\midrule  
\multicolumn{5}{l}{International Prices} \\  
\multicolumn{5}{l}{\textit{\small A. Correlations}} \\  
\small \qquad $p_{x},p_{m}$ & \ \small 0.75& \small -1.00& \small -1.00& \small -1.00& \small -1.00\\  
\small \qquad $p_{x},x$ & \ \small 0.46& \small -1.00& \small -1.00& \small -1.00& \small -1.00\\  
\small \qquad $p_{m},x$ & \ \small 0.69& \ \small 1.00& \ \small 1.00& \ \small 1.00& \ \small 1.00\\  
\small \qquad $p,x$ & \ \small 0.61& \ \small 1.00& \ \small 1.00& \ \small 1.00& \ \small 1.00\\  
\multicolumn{5}{l}{\textit{\small B. Standard deviation}} \\  
\small \qquad $x$ & \ \small 3.60& \ \small 0.52& \ \small 0.15& \ \small 1.12& \ \small 0.18\\  
\multicolumn{5}{l}{\quad \textit{\small - Relative to $x$}} \\  
\small \qquad $p_{x}$ & \ \small 0.37& \ \small 0.17& \ \small 0.17& \ \small 0.17& \ \small 0.17\\  
\small \qquad $p_{m}$ & \ \small 0.61& \ \small 1.17& \ \small 1.17& \ \small 1.17& \ \small 1.17\\  
\small \qquad $p$ & \ \small 0.27& \ \small 1.33& \ \small 1.33& \ \small 1.33& \ \small 1.33\\  
%\small \qquad $xp^{\ast}_{d}/p_{d}$ & \ \small 0.70& \ \small 0.00& \ \small 0.00& \ \small 0.00& \ \small 0.00\\  
\midrule  
\multicolumn{5}{l}{Quantities} \\  
\multicolumn{5}{l}{\textit{\small A. Correlations}} \\  
\multicolumn{5}{l}{\quad \textit{\small - Domestic with foreign}} \\  
\small \qquad \small Solow Res. & \ \small 0.30& \ \small 0.26& \ \small 0.26& \ \small n.a. & \ \small n.a.\\  
\small \qquad \small GDP & \ \small 0.40& \ \small 0.34& \small -0.02& \ \small 0.36& \ \small 0.36\\  
\small \qquad \small Consumption & \ \small 0.25& \ \small 0.37& \ \small 0.74& \ \small 0.69 & \ \small 0.99\\  
\small \qquad \small Employment & \ \small 0.21& \ \small 0.55& \small -0.34& \ \small n.a & \ \small n.a\\  
\small \qquad \small Investment & \ \small 0.23& \ \small 0.24& \small -0.46& \ \small n.a. & \ \small n.a.\\  
\multicolumn{5}{l}{\quad \textit{\small - GDP with}} \\  
\small \qquad \small Consumption & \ \small 0.83& \ \small 0.95& \ \small 0.86& \ \small 0.98& \ \small 0.86\\  
\small \qquad \small Employment & \ \small 0.85& \ \small 0.94& \ \small 0.97& \ \small n.a.& \ \small n.a. \\  
\small \qquad \small Investment & \ \small 0.93& \ \small 0.64& \ \small 0.47& \ \small n.a.& \ \small n.a. \\  
\small \qquad \small Net exports & \small -0.49& \small -0.57& \small -0.08& \small -0.57& \ \small 0.57\\  
\multicolumn{5}{l}{\quad \textit{\small - Terms of trade with}} \\  
\small \qquad \small Net exports & \small -0.17& \small -0.83& \ \small 0.85& \small -1.00& \ \small 1.00\\  
\multicolumn{5}{l}{\textit{\small B. Standard deviations}} \\  
\small \qquad \small GDP & \ \small 1.33& \ \small 0.87& \ \small 1.01& \ \small 0.80& \ \small 0.80\\  
\multicolumn{5}{l}{\quad \textit{\small - Relative to GDP$^{**}$}} \\  
\small \qquad \small Consumption & \ \small 0.74& \ \small 0.45& \ \small 0.34& \ \small 0.89& \ \small 0.81\\  
\small \qquad \small Investment & \ \small 2.79& \ \small 3.03& \ \small 3.97& \ \small n.a.& \ \small n.a. \\  
\small \qquad \small Employment & \ \small 0.81& \ \small 0.37& \ \small 0.48& \ \small n.a.&  \ \small n.a. \\  
\small \qquad \small Net exports & \ \small 0.29& \ \small 0.13& \ \small 0.40& \ \small 0.02& \ \small 0.47\\  
\bottomrule \addlinespace\end{tabular}  
\newline  
\scriptsize Statistics based on logged and Hodrick-Prescott filtered time series (with $\lambda=1600$). Data column refers to US data for the time period 1980:1-2004:1. \newline $^{*}$Ratio of corresponding standard deviation to the standard deviation of $x$. \newline $^{**}$Ratio of corresponding standard deviation to the standard deviation of $GDP$.  
\end{minipage}\label{Tab_Results}%
%TCIMACRO{\TeXButton{E}{\end{table}}}%
%BeginExpansion
\end{table}%
%EndExpansion

Table \ref{Tab_Results} illustrates the results implied by the models. As we
can see, in terms of prices, all 4 models exhibit the patterns we have
discussed. This should not surprise, as the supply-side extensions that we
have considered are irrelevant for these facts. Specifically, real exchange
rate is about 4 times less volatile than in the data, and it barely moves
when elasticity is high. This is to be expected, since real exchange rate
movements come from relative price movements (terms of trade). When goods
are closely substitutable, this relative price does not move much. Export
and import prices are negative correlated, and terms of trade is more
volatile than the real exchange rate. Pretty much everything is the opposite
of what it should be.

In terms of quantities, the models fare much better, but only with low
elasticity of substitution between goods. The high elasticity case is a
disaster. In the case of low elasticity, statistics are in the neighborhood
of what they should be. The models predict positive international comovement
of economic activity. Of course, statistics do not match up exactly, but we
know that there are additional tweaks on the model can further improve the
fit (e.g. home production, convex adjustment cost on capital, non-tradable
sector, see exercise below).

In all \ models, there is clearly too little propagation---absolute
volatility of GDP is too low. Also, consumption comoves internationally too
much, in particular, it is more correlated internationally than output. The
volatility of GDP and consumption falls short in terms of the data.
Investment, on the other hand, is way too volatile, but just like in the
data, it is the most volatile time series and highly procyclical.

As already mentioned, the problems on the quantity side can be fixed by
incorporating additional features, especially convex adjustment cost on
consumption and home production. With these two features the model can
replicate quantities in most dimensions, except for excess international
comovement of consumption. However, we should stress that consumption in the
data measures consumption expenditures, and does not take into account that
durable consumption can be way more volatile. One should thus be careful
with the interpretation of statistics pertaining to consumption data.%
\footnote{%
See, for example, the paper by Charles Engel \& Jian Wang, 2008.
\textquotedblleft International Trade in Durable Goods: Understanding
Volatility, Cyclicality, and Elasticities,\textquotedblright\ NBER Working
Papers 13814, National Bureau of Economic Research, Inc.}

Lastly, note that the model with capital quite successfully predicts
countercyclicality of NX. In fact, our model goes against simple intuition
that during booms, due to risk sharing, country should export more than
import and lend to the rest of the world, which requires some explanation.
As we can see, two factors are important:\ low elasticity of substitution
(even when there is no capital), and capital.

The reason why capital (and labor leisure choice) plays a role is the
production efficiency motive. This motive dictates that economic activity
should move to the country with highest productivity. In the words of Backus
et al., in these models \textit{\textquotedblleft one wants to grow hey
where the sun shines\textquotedblright }. As a result, during booms, even
though there is a risk sharing motive to ship goods abroad, there is also an
offsetting motive to invest in capital at home to take advantage of higher
productivity. This results in NX\ in `consumption goods' going into surplus,
but NX in `investment goods' going into deficit. If the second effect
dominates, the NX becomes countercyclical. To formalize this idea, recall
the setup from exercise (\ref{RMK1}). In this setup, we can directly
decompose net export into `consumption component of NX' and `investment
component of NX' 
\begin{equation*}
NX=\underset{NX_{c}}{\underbrace{(p_{d}D^{\ast }-p_{f}F)}}+\underset{NX_{I}}{%
\underbrace{(I_{d}^{\ast }-I_{f})}}.
\end{equation*}%
By the equivalence result you proved in this exercise, such decomposition
also applies to our benchmark economy---it is just less trasparent in such
case. Through the lens of this decomposition, consumption risk sharing
motive makes $NX_{c}\ $typically go up, but production efficiency motives
offsets these movements through $NX_{I}$.

To understand why elasticity is so important, we should look at our
endowment economy. In this economy, we can analytically show that when
elasticity is 1 NX\ is actually zero---which also goes against the common
wisdom of risk sharing. Why is that? The following calculation, works out
this case, which gives a clear intuition:\ 

\begin{eqnarray*}
NX &=&exports-imports= \\
&=&xp_{d}^{\ast }(y-\omega y)-p_{f}(1-\omega )y^{\ast }= \\
&=&xp_{d}^{\ast }[(y-\omega y)-\frac{p_{f}}{xp_{d}^{\ast }}(1-\omega
)y^{\ast }],
\end{eqnarray*}%
where%
\begin{eqnarray*}
\frac{p_{f}}{xp_{d}^{\ast }} &=&\frac{1-\omega }{\omega }\frac{d^{\omega
}f^{-\omega }}{d^{\omega -1}f^{1-\omega }}=\frac{1-\omega }{\omega }\frac{f}{%
d}= \\
&=&\frac{1-\omega }{\omega }\frac{\omega y}{(1-\omega )y^{\ast }}=\frac{y}{%
y^{\ast }},
\end{eqnarray*}%
and thus

\begin{eqnarray*}
NX &=&xp_{d}^{\ast }[y-\omega y-\frac{y}{y^{\ast }}(1-\omega )y^{\ast }]= \\
&=&xp_{d}^{\ast }[y-\omega y-(1-\omega )y]=0.
\end{eqnarray*}

Namely, these are not quantities that are countercyclical. During booms, the
country ships more goods abroad in terms of physical units (because $d^{\ast
}=(1-\omega )y$ and $f=(1-\omega )y^{\ast })$, it is the value of what is
exported in terms of what is important that offsets these movements (terms
of trade movements). \footnote{%
It is of interest to look at constant price NX\ in the data. It has been
done, and the model performs much worse in such case. Still, by including
physical capital, the model can account for the data.} When the elasticity
is low, these movements cause wealth effects that level off NX, and for $%
\sigma $ below unity, actually turn it negative.

\bigskip 

[ADD IMPULSE RESPONSE FUNCTIONS HERE]

\bigskip 

\begin{exercise}
Implement in Dynare a simple closed economy model with analogous parameter
setting. Compare the statistics with the corresponding open economy model.
\end{exercise}

\begin{exercise}
Download the codes from class website. Solve all 4 models under the
assumption of financial autarky (i.e. add equation NX=0 instead of perfect
risk sharing equation). Compare results to the complete market economy, and
discuss them.
\end{exercise}

\begin{exercise}
(Optional) Implement in Dynare an extended model with a convex adjustment
cost on capital. Document the effect of the convex adjustment cost on the
statistics discussed above.
\end{exercise}

\section{Drozd and Nosal (2008)}

Drozd and Nosal (2008) propose a simple way to reconcile the quantity side
of dynamic IRBC\ theory with a high long-run elasticity, and show that such
frictions can also successfully account for the correlations of
international prices. The basic idea is that firms, in order sell their
output, need to build the demand and marketing infrastructure. The modeling
tool is search theory.

Drozd and Nosal adopt the baseline BKK model and embed their model of
marketing into this basic structure. Their model generates, low measured
volatility ratio that is consistent with high assumed long-run elasticity $%
\sigma $, and high measured long-run elasticity. At the same time,
quantities behave exactly as in the standard model with low elasticity of
substitution.

[to be completed]

\section{Other Extensions}

\subsection*{Exotic Elasticities}

[to be completed]

\subsection*{Non-tradable Goods}

[to be completed]

\subsection*{Home Production}

[to be completed]

\subsection*{Durable Consumption}

[to be completed]

\subsection*{Habit formation}

[to be completed]

\chapter{Papers for Student Presentations}

Suggested papers:

\begin{itemize}
\item Atkeson, Andrew \& Patrick Kehoe \& Fernando Alvarez (2008):
\textquotedblleft Time-Varying Risk, Interest Rates, and Exchange Rates in
General Equilibrium\textquotedblright , Minneapolis FED Staff Report 371,
September

\item Perri, Fabrizio \& Jonathan Heathcote (2007): \textquotedblleft The
International Diversification Puzzle Is Not as Bad as You
Think\textquotedblright , Staff Report 398, October

\item Michael Waugh (2007): \textquotedblleft International Trade and Income
Differences\textquotedblright , University of Iowa, unpublished manuscript

\item Arellano, Cristina (2007): \textquotedblleft Default Risk and Income
Fluctuations in Emerging Economies\textquotedblright , American Economic

\item Klette, Tor Jakob \& Samuel Kortum (2004): \textquotedblleft
Innovating Firms and Aggregate Innovation,\textquotedblright\ Journal of
Political Economy, University of Chicago Press, vol. 112(5), pages 986-1018,
October

\item Mark Aguiar \& Manuel Amador \& Gita Gopinath, 2005. \textquotedblleft
Efficient Fiscal Policy and Amplification,\textquotedblright\ NBER Working
Papers 11490, National Bureau of Economic Research, Inc.

\item Mark Aguiar \& Gita Gopinath, 2007. \textquotedblleft Emerging Market
Business Cycles: The Cycle Is the Trend,\textquotedblright\ Journal of
Political Economy, University of Chicago Press, vol. 115, pages 69-102.
\end{itemize}

\end{document}
