\documentclass{article}
\input{1402-preamble}
\psetnum{5}
\date{2009/04/24}

\begin{document}
\begin{pset}
  \begin{problem}
    
    \begin{subproblem}
      \textbf{False}. The data show that investment is more volatile
      than consumption, but investment is a smaller fraction of the
      GDP, so volatility in consumption and investment have similar
      effects on the GDP.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} A monetary contraction causes an increase in the
      nominal interest rate (from the IS/LM model). With expected
      inflation constant, the real interest rate must also increase
      too, giving a decrease in investment.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True}. When an employee has a higher chance of losing
      his job, his total wealth declines (because his expected future
      earnings are lower), so current income plays a larger role in
      his consumption function.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} If the monetary expansion was expected, it would
      have already been reflected in the price of the stocks. And if a
      \emph{larger} monetary expansion was expected, the smaller
      actual one could cause prices to drop.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} Investors take risk into account in addition to
      expected value. So, for example, a risk-averse investor might
      prefer stock in a stable company to a high-risk junk bond, even
      if the prices are set so that they will have the same expected
      value.
    \end{subproblem}

  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The natural level of unemployment is given: $u_n = 0.5$. This
      specifies the natural level of output:
      \begin{align*}
        u_n &= 1 - \frac{Y_n}{L}\\
        Y_n &= L (1 - u_n)\\
         &= 2 (1 - 0.5) = 1
      \end{align*}

      The IS relation gives the natural real interest rate:
      \begin{align*}
        Y_n &= C + I = \frac{1}{2} Y_n + 1 - r_n\\
        \frac{Y_n}{2} &= 1 - r_n\\
        r_n &= 1 - \frac{Y_n}{2} = 0.5
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
     Here, we consider the LM equilibrium.
     \begin{align*}
       \frac{M^d}{P} &= Y_n(1 - i)\\
       \frac{1}{2} &= 1(1 - i)\\
       i &= 0.5
     \end{align*}
     So the nominal interest rate is $\nicefrac12$. The inflation
     level is zero, because at the medium-run equilibrium the
     inflation level is the level of money growth, and the money
     supply is fixed.
    \end{subproblem}

    \begin{subproblem}
      At time $t+1$, the money supply is $(1 + 10\%)M_t = 1.1$, and
      the expected inflation level is zero (because it was zero at
      time $t$). Moreover, the real and nominal interest rates are the
      same ($i_{t+1} = r_{t+1}$) because the expected inflation rate
      is zero. Then, the LM relation gives:
      \begin{align*}
        \frac{1.1}{P_{t+1}} &= Y_{t+1}(1 - i_{t+1}) \\
        i_{t+1} &=  1 - \frac{1.1}{P_{t+1}Y_{t+1}}
      \end{align*}

      The IS relation gives:
      \begin{align*}
        0.5 Y_{t+1} &= (1-r_{t+1}) = (1-i_{t+1})\\
        &= \frac{1.1}{P_{t+1} Y_{t+1}}
      \end{align*}

      From the definition of inflation, $\pi_{t+1} = \frac{P_{t+1} -
        P_t}{P_t} = \frac{P_{t+1} - 2}{2}$, and so $P_{t+1} =
      2(\pi_{t+1}+1)$. So
      \[0.5 Y_{t+1} = \frac{1.1}{2(\pi_{t+1}+1) Y_{t+1}}\]

      Also, from the Phillips curve (noting that expected inflation is
      zero and unemployment was previously at the natural rate)
      \begin{align*}
        \pi_{t+1} &= -2(u_{t+1} - u_n)\\
        &= -2\left(1-\frac{1}{2}Y_{t+1}\right) - \frac{1}{2})\\
        &= Y_{t+1} - 1
      \end{align*}

      Combining,
      \begin{align*}
        0.5 Y_{t+1} &= \frac{1.1}{2(\pi_{t+1} +1) Y_{t+1}}\\
        &= \frac{1.1}{2(Y_{t+1} -1 +1) Y_{t+1}}\\
        0.5 Y_{t+1}^3 - Y_{t+1}^2 + 0.55 &= 0\\
        Y_{t+1} &\approx 1.1147
      \end{align*}

      $Y_{t+1} = 2(1-r_{t+1})$, so \[i_{t+1} = r_{t+1} = 1 -
      \frac{1.1147}{2} \approx 0.443\]. And \[\pi_{t+1} = Y_{t+1} - 1
      \approx 0.1147\].
    \end{subproblem}

    \begin{subproblem}
      We can use essentially the same approach as above to compute the
      output/inflation/interest rates at any time $k$ from their
      values at $k-1$. The difference is that we must use the previous
      year's initial conditions, not those at time $t$. In particular,
      the expected inflation will be non-zero, and so the real and
      nominal interest rates will be different. Namely,
      \[ \pi^e_k = \pi_{k-1} \]
      \[ r_k = i_k - \pi^e_k = i_k - \pi_{k-1} \]

      The following relations specify the new levels of output,
      inflation, and interest:

      Definition of inflation
      \begin{align*}
        \pi_k &= \frac{P_k- P_{k-1}}{P_{k-1}}\\
        P_k &= P_{k-1} (\pi_k + 1)
      \end{align*}

      LM:
      \begin{align*}
        i_k &= 1- \frac{1.1 M_{k-1}}{P_k Y_k}\\
        &= 1- \frac{1.1 M_{k-1}}{P_{k-1} (\pi_k + 1) Y_k}
      \end{align*}

      IS:
      \begin{align*}
        0.5 Y_k &= 1 - r_k \\
        &= 1 - (i_k - \pi^e_k)\\
        &= 1 - i_k + \pi_{k-1}\\
        &= \frac{1.1 M_{k-1}}{P_{k-1} (\pi_k + 1) Y_k} + \pi_{k-1}
      \end{align*}

      Phillips curve:
      \begin{align*}
        \pi_k - \pi_{k-1} &= -2 (u_{k} - u_{k-1})\\
        &= -2 \left(\left(1-\frac{1}{2}Y_{k}\right) -
          \left(1-\frac{1}{2}Y_{k-1}\right)\right)\\
        &= Y_k - Y_{k-1}
      \end{align*}

      Combining the last two relations, one can solve for
      $Y_k,i_k,r_k,\pi_k$ from their values in the previous year. This
      process can be repeated iteratively.
    \end{subproblem}

    \begin{subproblem}
      The medium run level of output will return to the natural level
      of output (1). The inflation rate will be equal to the money growth
      rate (10\%). The real interest rate will be the natural real
      interest rate ($\nicefrac12$), but the nominal interest rate
      will be the natural real interest rate plus the money growth
      rate ($0.5 + 0.1 = 0.6$).

      The economy will evolve as follows: in the short run, both
      natural and real interest rates fall, output increases, and
      unemployment falls. Oer time, inflation will increase because
      the unemployment level is below its natural level, causing a
      contraction in the real money supply that decreases output and
      increases unemployment, until it reaches its medium-run
      equilibrium value.
    \end{subproblem}
  \end{problem}
\newpage
  \begin{problem}
    \begin{subproblem}
      The price of a one-year \$100 bond is
      \[ \frac{\$100}{1+i} = \frac{\$100}{1+0.5} \approx \$66.67\]
    \end{subproblem}

    \begin{subproblem}
      If the increase in money supply is unexpected, investors expect
      the current interest rate of 0.5 to continue for both years. So
      the two-year bond price is
      \[ \frac{\$100}{(1+i)(1+i)} = \frac{\$100}{2.25} \approx
      \$44.44\]
    \end{subproblem}

    \begin{subproblem}
      If the money supply increase is expected, investors can predict
      the interest rates at year $t$ and $t+1$: $i_t = 0.5$ as before,
      but $i_{t+1} \approx 0.443$ from above. So the price is
      \[ \frac{\$100}{(1+i_{t})(1+i_{t+1})} \approx
      \frac{\$100}{2.164} \approx \$46.2\]

      The price is higher because investors expect that at time $t+1$,
      purchasing a one-year bond would be more expensive because the
      monetary expansion caused the nominal interest rate to be
      lower.
    \end{subproblem}

    \begin{subproblem}
      The total price of the stock market is present value of the
      future dividend stream. Since we are considering the entire
      stock market, the dividends paid each year should be equal to
      total output, which, at equilibrium, will continue to be at its
      natural value of 1. The interest
      rate will be at its natural value of 0.5. So the price is
      \[ \frac{1}{(1+0.5)} + \frac{1}{(1+0.5)^2} + \frac{1}{(1+0.5)^3}
           + \cdots = \frac{1}{0.5} = 2 \]
    \end{subproblem}

    \begin{subproblem}
      The price will be given by the series
      \[ \frac{Y_{t+1}}{(1+{i_t})} +
      \frac{Y_{t+2}}{(1+{i_t})(1+{i_{t+1}})} + \cdots\]
      
      The price will be higher if the monetary expansion is
      expected. The monetary expansion increases output for all future
      time, which gives larger dividends each year, and it decreases
      the nominal interest rate, so the present value of each payment
      is higher.
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}