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\begin{document}
\section{\protect\bigskip \textbf{Box 1: Theory as Guide to the Empirical
Model}}
Assume that there are N domestic entrepreneurs, who are single mindfully
engaged in wealth accumulation (save only), and N foreign creditors, who
supply the credit necessary for domestic investment by the domestic
entrepreneurs. Let $I_{t}^{a}$ denote investment in capacity by an
individual entrepreneur, and let the leverage in finance be specified as $%
\lambda $ times \ the entrepreneur's net worth, W. Denote by $y_{t}$, $%
F_{t-1},$ and $p_{t},$\ the domestic output (produced by a standard
Cobb-Douglas technology with a capital input income share $\alpha ),$ the
initial debt, indexed to foreign goods, and the \textit{real exchange rate }%
(the relative price of foreign goods in terms of domestic goods),
respectively.A foreign lender imposes a limit on the entrepreneur
borrowings, so that the investment, $I_{t}^{a},$ is constrained by the
entrepreneur net worth and the leverage fraction:
$I_{t}^{a}\leq (1+\lambda )W_{t}$ \ ,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\bigskip
where, $W_{t}=\alpha y_{t}-p_{t}F_{t-1},$ is the entrepreneur's net worth.
\bigskip
The market clearing real exchange rate is a function of \textit{aggregate}
investment and aggregate output:
$p_{t}=\frac{[1-(1-\alpha )(1-\upsilon )]Y_{t}-(1-\upsilon )I_{t}}{\overset{%
\symbol{126}}{X_{t}}},$
where, I=N$I^{a,}Y=Ny,$denote the aggregate domestic investment and
aggregate output, respectively; the coefficient $\upsilon $ denotes the
marginal propensity to import, and $\overset{\symbol{126}}{X_{t}}$ denotes
the stochastic volume of exports, expressed in terms of foreign goods. Thus,
an increase the aggregate investment spending triggers real appreciation
through a \textquotedblright transfer problem\textquotedblright\ mechanism
(see Krugman (2000).
International differences in rates of return which induce foreign creditors
to extend loans to domestic entrepreneurs are given by the interest parity
condition:
(1+r$_{t})\frac{p_{t}}{p_{t+1}}\geq (1+r\ast ),$
where, r and r* denote the marginal productivity of capital and the foreign
interest rate, respectively. We start with perfect public information.
Figure B.1 shows the existence of at most three equilibrium outcomes
depending on the realization of exports, $\overset{\symbol{126}}{X_{t}}$.
With high exports, a unique equilibrium investment is governed by the
standard rate of return conditions. With low exports,because the
entrepreneur is insolvent and the credit constraint is binding, there exists
a unique equilibrium with zero investment . In an intermediate case there
are however multiple equilibrium- investment outcomes, due to a
expectations-coordination failure.
Now turn to the case of private information. A foreign creditor i receives a
private signal $\theta _{i}$ regarding $\overset{\symbol{126}}{X_{t}};$
$\theta _{i}=$ $\overset{\symbol{126}}{X_{t}}+\varepsilon _{ti}.$
The error term $\varepsilon _{ti}$ is assumed to be i.i.d. and uniformly
distributed over [-$\varepsilon ,\varepsilon $].
An individual foreign creditor's decision whether or not to extend credit
to the domestic entrepreneur crucially depends on her signal.
There exists a cut-off signal \ $\theta _{i}^{\ast }=$ $\overset{\symbol{126}%
}{X_{t}^{\ast }}+\varepsilon _{ti}^{\ast }$, so that
$\underset{N\symbol{126}U[0,1]}{E}[(1+r_{t})\frac{p_{(}\overset{\symbol{126}}%
{N}_{t}^{\ast },\overset{\symbol{126}}{X_{t}^{\ast }})}{p_{t+1}}]-(1+r\ast
)=0.$
The marginal individual creditor, who receives a threshold signal
$\theta _{i}^{\ast }=$ $\overset{\symbol{126}}{X_{t}^{\ast }}+\varepsilon
_{ti}^{\ast },$
must be indifferent between withholding, or extending the credit to the
domestic entrepreneur counterpart.
Observe that in the global game the market clearing real exchange rate, p$_{t%
\text{ \ }}$ ,is a decreasing function of $\overset{\symbol{126}}{N}$, $\ $%
the number of foreign creditors who decide to lend to the domestic
entrepreneurs, and a decreasing function of the fundamental which drives the
equilibrium outcome, $\overset{\symbol{126}}{X_{t}}$.
The export threshold, $\overset{\symbol{126}}{X_{t}^{\ast }}$ , therefore
determines a \textit{unique} equilibrium outcome which is a solution to the
global game. Below the threshold $\overset{\symbol{126}}{X_{t}^{\ast }}$ \
investment is equal to zero, because all foreign investors tend to withold
credit. Above the threshold $\overset{\symbol{126}}{X_{t}^{\ast }}$,
domestic investment is driven by the \textit{stndard} rate-of-return
consideration reaching a \textit{unique}\ level $\overset{\_}{I}_{t},$
because all foreign investors extend credit and interest parity prevails. \
This means that there is also a unique \textit{probability} of a sudden stop
in capital flow, denoted by G(X), where $G(.)$ is the exogenous cummulative
distribution function of export volumes:
Prob $\{I_{t}=0\}=G(\overset{\symbol{126}}{X_{t}^{\ast }}).$
Furthermore, the associated (expected) level of aggregate investment is
given by
$(\overset{\_}{I})(1-G(\overset{\symbol{126}}{X_{t}^{\ast }}))$.
Therefore, in this model the probability of sudden stops affects directly
the level of economic activity of the domestic economy.
\end{document}