\documentclass{article}
\input{1402-preamble}
\psetnum{6}
\date{2009/05/04}

\begin{document}
\begin{pset}
  \begin{problem}

    \begin{subproblem}
      \textbf{False}, at least in principle. The capital account and
      current account balances must sum to zero, so a current account
      deficit must be accompanied by a capital account
      surplus. However, statistical discrepancies in the way the
      accounts are measured may cause them to be slightly different.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True}. The multiplier is smaller in an open economy,
      because some of the increased demand goes towards imported goods
      rather than domestic goods.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False}. With perfect capital mobility, to maintain a
      fixed exchange rate, government must use monetary policy to set
      the interest rate that corresponds to the fixed exchange
      rate. Thus, it cannot adjust the interest rate to stimulate the
      economy without breaking the fixed exchange rate.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False} in general. This is true if the interest parity
      relation holds: then the exchange rate depends only on the
      domestic and foreign interest rates and the expected exchange
      rate. However, without perfect capital mobility, differences in
      transaction costs and risk can also affect the interest rates
      and exchange rate.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True}. A country that imports intermediate goods and
      exports final goods may have total exports greater than its GDP,
      because the total value of the goods is counted towards the
      total exports, but only the value added contributes to the GNP.
    \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      The condition that $0 < c_1 + d_1 < 1$ is a familiar one from
      the closed economy: it is the requirement that an increase in
      output corresponds to an increase in consumption and investment
      that is no larger than the increase in output. The condition
      that $0 < \frac{m_1}\epsilon < c_1+d_1$ reflects the fact that
      the goods imported are a subset of the goods required for the
      increase in consumption and investment, so the increase in value
      of the goods imported is less than the increase in
      consumption/investment.

      The Marshall-Lerner condition is that a currency depreciation
      increases the trade balance. The trade balance is given by
      \[ X - \frac{IM}\epsilon = x_1Y^{US} - \frac{m_1}\epsilon Y \]
      If the currency depreciates, $\epsilon$ decreases, and the trade
      balance \emph{decreases}. So the Marshall-Lerner condition does
      not hold. The reason is that both this economy and the US
      economy import a fixed fraction of their output, regardless of
      the exchange rate.
    \end{subproblem}

    \begin{subproblem}
      \begin{align*}
        Y &= C + I + G - \frac{IM}\epsilon + X\\
          &= c_0 + c_1 (Y - T) + d_0 + d_1 Y + G - \frac{m_1}\epsilon
          Y + x_1 Y^{US}\\
        Y - c_1 Y - d_1 Y + \frac{m_1}\epsilon Y &= c_0 - c_1 T + d_0
        + G + x_1 Y^{US} \\
        Y &= \frac{c_0 - c_1 T + d_0
        + G + x_1 Y^{US}}{1 - c_1 - d_1 + \frac{m_1}\epsilon}
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Total domestic output is now
      \[Y = \frac{c_0 - c_1 T + d_0
        + G + x_1 (Y^{US} - \Delta Y^{US})}{1 - c_1 - d_1 +
        \frac{m_1}\epsilon}\]
      \emph{i.e.} it decreases by the amount
      \[\frac{x_1}{1 - c_1 - d_1 +
        \frac{m_1}\epsilon} \Delta Y^{US} \]
    \end{subproblem}

    \begin{subproblem}
      The trade balance will increase: the 

      The amount of increase in the trade balance (net exports) is;
      \begin{align*}
        \Delta NX &= \Delta X - \Delta \frac{IM}\epsilon \\
        &= - x_1 \Delta Y^{US} - \frac{m_1}\epsilon \Delta Y\\
        &= - x_1 \Delta Y^{US} - \frac{m_1}{\epsilon}\frac{x_1}{1 - c_1 - d_1 +
        \frac{m_1}\epsilon} \Delta Y^{US}\\
      &= -\left(1 + \frac{m_1 x_1}{\epsilon(1 - c_1 - d_1 +
        \frac{m_1}\epsilon)} \right) x_1 \Delta Y^{US}
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      To offset the effect on output, an increase in government
      spending of $\Delta G$ is needed, where
      \[ \Delta G = x_1 \Delta Y^{US} \]
    \end{subproblem}

    \begin{subproblem}
      With output adjusted back to its original level, the increase in
      the trade balance is
      \begin{align*}
        \Delta NX &= \Delta X - \Delta \frac{IM}\epsilon \\
        &= - x_1 \Delta Y^{US} - \frac{m_1}\epsilon \Delta Y\\
        &= - x_1 \Delta Y^{US}
      \end{align*}
      Although output is at its original level, there is now a trade
      deficit because the demand for exports to the US has fallen, and
      the level of imported goods is the same (because output is the
      same).
    \end{subproblem}
  \end{problem}

  \newpage
  
  \begin{problem}

    \begin{subproblem}
      First, note that the domestic and foreign price levels are the
      same ($P = P^*$), so the nominal and real exchange rates are
      equal: $\epsilon = E$. Because expected inflation is zero, the
      nominal and real interest rates are equal.
      
      The LM relation gives us
      \[ i = \frac{Y-M}{e} \]
      and so the IS gives
      \begin{align*}
        Y &= C + I + NX\\
        &= cY + a - bi - \epsilon - dY\\
%        &= cY + a -\frac{b}{e}Y + \frac{b}{e}M - \epsilon - dY\\
%        Y &= \frac{1}{1 - c + \frac{b}{e} + d} \left( a + \frac{b}{e}M
%          - \epsilon \right)
      \end{align*}

      Interest parity requires that $E = \frac{1+i}{1+i^*}
      E^e$. Because $E^e = 1$, $i^* = 0$, and $\epsilon = E$ (from
      above),
      \[ \epsilon = (1 + i) \]

      Adding this to the IS:
      \begin{align*}
        Y &= cY + a - bi - (1 + i) - dY\\
         &= cY + a - (b+1)i -1 - dY\\
          &= cY + a -\frac{b+1}{e}Y + \frac{b+1}{e}M -1 - dY\\
          Y &= \frac{1}{1 - c + \frac{b+1}{e} + d} \left( a + \frac{b+1}{e}M
            - 1 \right)\\
          i &= \frac{\frac{1}{1 - c + \frac{b+1}{e} + d} \left( a + \frac{b+1}{e}M
            - 1 \right) - M}{e}
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      If $a$'s value drops to $a - \Delta a$, the new output and
      interest rate are given by the following (substituting $a -
      \Delta a$ in the results above:

      \begin{align*}
        Y &= \frac{1}{1 - c + \frac{b+1}{e} + d} \left( a-\Delta a + \frac{b+1}{e}M
          - 1 \right)\\
        \Delta Y &= \frac{- \Delta a}{1 - c + \frac{b+1}{e} + d}\\
        i &= \frac{\frac{1}{1 - c + \frac{b+1}{e} + d} \left( a-\Delta a + \frac{b+1}{e}M
            - 1 \right) - M}{e}\\
        \Delta i &= \frac{-\Delta a}{e\left(1 - c + \frac{b+1}{e} + d\right)}
      \end{align*}

      Thus, output and interest rates drop. The exchange rate (both
      nominal and real, since both price levels are fixed at 1 in the
      short run) is
      \begin{align*}
        E = \epsilon &= 1 + i\\
        &= 1 + \frac{\frac{1}{1 - c + \frac{b+1}{e} + d} \left(
            a-\Delta a + \frac{b+1}{e}M - 1 \right) - M}{e}
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      \vspace{3in}

      The exchange rate drops, because output and interest both
      drop. The result of the lower exchange rate is to increase
      exports (because domestic goods are more attractive to foreign
      purchasers), causing output to increase, and thus interest rates
      to increase. Thus, the change in exchange rate dampens the
      effects of the shock.
    \end{subproblem}

    \begin{subproblem}
      Now, to keep the exchange rate fixed at $E =1$, the interest rate must be
      fixed according to the interest parity condition:
      \[ E = \frac{1+i}{1+i*}E^e \],
      \emph{i.e.}
      \[ i = i^* = 0 \]
      Note that, again, the real and nominal exchange rates are the
      same, so $\epsilon = E = 1$.
      
      So the IS (before the shock) is simply
      \begin{align*}
        Y &= C + I + NX\\
        &= cY + a - 1 - dY\\
        Y &= \frac{a-1}{1-c+d}
      \end{align*}

      After the shock, the interest rate and exchange rate are still
      the same (because of the fixed exchange rate), and the new level
      of output is
      \begin{align*}
        Y &= \frac{a - \Delta a - 1}{1-c+d}\\
        \Delta Y &= \frac{-\Delta a}{1-c+d}
      \end{align*}

      The change in output is larger than before (the multiplier is
      larger: $\frac{1}{1-c+d}$ instead of
      $\frac{1}{1-c+\frac{b}{e}+d}$.
    \end{subproblem}

    \newpage
    \begin{subproblem}
      \vspace{3in}

      With the fixed exchange rate (and thus fixed interest rate), the
      drop in investment causes a drop in output. With the flexible
      exchange rate, the decrease in investment also causes output to
      decrease, but not as much as with the fixed exchange rate. There
      are two reasons for the dampening effect with the flexible
      exchange rate. First, the drop in output causes the exchange
      rate to drop, increasing exports. Second, the fall in the
      exchange rate causes the interest rate to drop (because of
      interest parity), encouraging investment. Neither occurs with
      the fixed exchange rate.
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}