\documentclass{article}
\input{1402-preamble}
\psetnum{3}
\date{2009/03/20}

\begin{document}
\begin{pset}
  \begin{problem}
    \begin{subproblem}
      \textbf{True.} The market wage is the amount that employers are
      willing to pay for an employee. A worker's reservation wage is
      the wage below which the worker is indifferent to being employed
      or unemployed. Thus, if the reservation wage is above the market
      wage, the worker will not seek out a job.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} Increasing consumption (or, more generally,
      increasing output) causes unemployment to decrease, increasing
      nominal wages, and thus increasing the price level.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} If everyone could perfectly predict the future, then
      their expected price level $P^e$ would always be equal to the
      real price level $P$. Then, the economy would always be in
      equilibrium at the natural level of unemployment and natural
      level of output. In other words, output would be the same
      regardless of the price level --- giving a vertical AS curve.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True. }The natural level of unemployment is specified
      entirely by the labor market equilibrium condition, which is
      only a factor in the AS curve: it is the value of $u$ that
      satisfies $P=P^e (1+\mu)F(u,z)$ when $P=P^e$,
      i.e. $F(u,z)=\frac{1}{1+\mu}$. These variables don't factor into
      the AD curve, so a shift of the AD curve cannot affect the
      natural level of unemployment. The natural level of output is
      the level of output that puts the labor market in equilibrium at
      the natural level of unemployment, so it cannot be affected
      either.      
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} Fiscal policy can affect medium-run
      investment. For example, a deficit reduction will cause the
      output level, interest rate, and price level to drop in the
      short run. In the medium run, the price level decreases,
      allowing the economy to return to its natural level of
      output. However, the falling price level, combined with no
      monetary policy change, causes the real money supply to
      increase, the interest rate to drop, and investment to increase.
    \end{subproblem}

  \end{problem}

  \begin{problem}

    \begin{subproblem}
      \begin{align*}
        P &= (1+\mu)W\\
        &= (1+\mu)P^e(1+z-u)\\
        &= (1+\mu)P^e(1+z-(1 - \frac{N}{L})) = (1+\mu)P^e(z+\frac{Y}{L})
      \end{align*}

      The AS relation relates $P$ and $Y$; this also determines $W$,
      $Y$, and $u$. We see the price level $P$ is directly
      proportional to the expected price level $P^e$. This reflects
      the fact that increasing the expected price level causes wages
      to increase, and thus costs cause prices to increase.
    \end{subproblem}

    \begin{subproblem}
      At the natural level of output, $P = P^e$. So
      \begin{align*}
        P &= (1+\mu)P^e(z+\frac{Y}{L})\\
        1 &= (1+\mu)(z+\frac{Y_n}{L})\\
        z+\frac{Y_n}{L} &= \frac{1}{1+\mu}\\
        Y_n &= L \left(\frac{1}{1+\mu} - z\right)\\
      \end{align*}

      \vspace{2.5in}
    \end{subproblem}

    \begin{subproblem}
      IS:
      \begin{align*}
        Y &= C+I+G &= c_0 + c_1 Y - c_1 T + a - bi + G \\
        %Y &= \frac{c_0 - c_1 T + a - bi + G}{1 - c_1}
        i &= \frac{c_0 + c_1 Y - c_1 T + a + G - Y}{b}
      \end{align*}

      LM:
      \begin{align*}
        M &= P(d + eY - fi) \\
        %Y &= \frac{M - Pd + Pfi}{e}
        %M &= Pd + PeY - Pfi\\
        i &= \frac{d + eY - \frac{M}{P}}{f}
      \end{align*}

      AD: IS/LM in equilibrium
      \begin{align*}
        %\frac{c_0 - c_1 T + a - bi + G}{1 - c_1} &= \frac{M - Pd +
        %Pfi}{e}
        \frac{c_0 + c_1 Y - c_1 T + a + G - Y}{b} &= \frac{d + eY -
          \frac{M}{P}}{f}\\
        \frac{c_0 - c_1 T + a + G}{b} - \frac{1-c_1}{b} Y &=
        \frac{d}{f} + \frac{e}{f}Y - \frac{M}{Pf}\\
        \left(\frac{e}{f} + \frac{1-c_1}{b}\right) Y &= \frac{c_0 - c_1
          T + a + G}{b} - \frac{d}{f} + \frac{M}{P f}\\
        Y &= \frac{1}{\frac{e}{f} + \frac{1-c_1}{b}} \left(\frac{c_0 - c_1
          T + a + G}{b} - \frac{d}{f} + \frac{M}{P f}\right)
      \end{align*}

      The AD is downward-sloping: increases in price level decrease
      output. The interest rate is low when output is high and price
      level is low, and high when output is low and price level is
      high.
    \end{subproblem}

    \begin{subproblem}
      The decrease in expected prices causes the AS curve to move
      down: at every output level, prices are lower. The AD curve is
      unchanged, so a new equilibrium is reached in the short run with
      higher output level $Y_1$ and lower price level $P_1$. However,
      $P_1$ is still higher than the expected price level $P^e_1$

      The natural level of output is still given by $Y_n =
      L\left(\frac{1}{1+\mu}-z\right)$, as before. $L$, $\mu$, and $z$
      are unchanged, so $Y_n$ remains unchanged ($Y_n' = Y_n$), as
      does $P^e$. In the medium run, the AS curve shifts back up to
      its original value as wage setters revise their price level
      upward (because prices did not fall to their expected price),
      and the original equilibrium is restored.

      \vspace{2.5in}
    \end{subproblem}

    \begin{subproblem}
      With a corresponding decrease in demand, the AD curve will shift
      left (less output at every price level). Depending on the
      magnitude of this effect, it will either limit the amount output
      increases, or cause output to decrease.
    \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      In medium-run equilibrium, the unemployment rate is constant, so
      $u_t - u_{t-1} = 0$, and so by Okun's Law, $g_{yt} = 3\%$.

      By the aggregate demand relation, $g_{yt} = g_{mt} - \pi_t$, so
      $g_{mt} = g_{yt} + \pi_t = 7\% + 3\% = 10\%$.

      Assuming inflation remains constant (no intervention by the
      central bank), $\pi_t = \pi^e_t = \pi_{t-1} = 7\%$. So the
      Phillips curve implies that $u_t = 5\%$, the natural rate of
      unemployment.
    \end{subproblem}

    \begin{subproblem}
      Because the bank is not at all credible, the expected inflation
      rate ignores the announced rate; it is simply the previous
      year's inflation rate (7\%). The Phillips curve implies that
      decreasing the inflation rate must cause the unemployment rate
      to rise to 7\%:
      \begin{align*}
        \pi_t &= \pi^e_t (u_t - 5\%) \\
        5\% &= 7\% - (u_t - 5\%)\\
        u_t &= 7\%
      \end{align*}

      This new value of the inflation rate implies, by Okun's law,
      that the growth rate will fall to -1\%.
      \begin{align*}
        u_t - u_{t-1} &= -0.5\left(g_{yt} - 3\%\right)\\
        7\% - 5\% &= -0.5 \left(g_{yt} - 3\%\right)\\
        g_{yt} &= -1\%
      \end{align*}

      The aggregate demand equation gives the monetary policy that the
      bank must use to achieve this inflation rate. It must set a
      policy of 4\% money growth:
      \begin{align*}
        g_{yt} &= g_{mt} - \pi_t\\
        -1\% &= g_{mt} - 5\%\\
        g_{mt} &= 4\%
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Now the expected inflation rate takes into account both the
      previous year's inflation rate and the announced rate, in equal
      proportion:
      \[ \pi^e_t = \theta \pi_t^a + (1-\theta)\pi_{t-1}
      = \nicefrac{1}{2} \cdot 5\% + \nicefrac{1}{2} \cdot 7\% = 6\% \]
      The Phillips curve gives the unemployment rate:
      \begin{align*}
        \pi_t &= \pi^e_t (u_t - 5\%) \\
        5\% &= 6\% - (u_t - 5\%)\\
        u_t &= 6\%        
      \end{align*}
      and Okun's law the growth rate:
      \begin{align*}
        u_t - u_{t-1} &= -0.5\left(g_{yt} - 3\%\right)\\
        6\% - 5\% &= -0.5 \left(g_{yt} - 3\%\right)\\
        g_{yt} &= 1\%
      \end{align*}
      and the aggregate demand equation the necessary money growth
      rate:
      \begin{align*}
        g_{yt} &= g_{mt} - \pi_t\\
        1\% &= g_{mt} - 5\%\\
        g_{mt} &= 6\%
      \end{align*}      
    \end{subproblem}

    \begin{subproblem}
      Same as before, except that now the expected inflation rate will
      be the announced target rate: $\pi^e_t = \pi^a_t = 5\%$.

      The Phillips curve gives the unemployment rate:
      \begin{align*}
        \pi_t &= \pi^e_t (u_t - 5\%) \\
        5\% &= 5\% - (u_t - 5\%)\\
        u_t &= 5\%        
      \end{align*}
      and Okun's law the growth rate:
      \begin{align*}
        u_t - u_{t-1} &= -0.5\left(g_{yt} - 3\%\right)\\
        5\% - 5\% &= -0.5 \left(g_{yt} - 3\%\right)\\
        g_{yt} &= 3\%
      \end{align*}
      and the aggregate demand equation the necessary money growth
      rate:
      \begin{align*}
        g_{yt} &= g_{mt} - \pi_t\\
        3\% &= g_{mt} - 5\%\\
        g_{mt} &= 8\%
      \end{align*}      
      
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}