\documentclass{article}
\input{1402-preamble}
\psetnum{4}
\date{2009/04/06}

\begin{document}
\begin{pset}
  \begin{problem}
    \begin{subproblem}
      \textbf{False.} The savings rate doesn't affect the level of
      growth in the long run. However, the absolute level of
      output at any point in steady-state is higher with a higher
      savings rate.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} Without population growth or technological
      progress, the capital stock converges to a fixed level
      (determined by the savings rate); with a production function
      consisting only of capital and labor, this fixes the level of
      output.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} If we assume a poor country has less education
      (less human capital) than a rich country, then diminishing
      returns to human capital mean that increasing the education
      level will have a greater effect in the poor country, measured
      as a fraction of its GDP. The absolute amount of the increase,
      however, could still be higher in the rich country depending on
      the production functions and the relative levels of capital,
      technology, and employment.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} The steady-state level of output per worker
      depends on the population growth rate. Because capital per
      worker declines as a result of increases in the number of
      workers, a given savings rate corresponds to a lower level of
      capital per worker when the growth rate is higher. Accordingly,
      the output level falls.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} Above the golden rule level of savings/capital,
      increasing the savings rate will cause capital to increase,
      causing output to increase. However, diminishing
      returns to capital mean that the output will not increase by as
      much as the increase in investment, \emph{i.e.}~that the amount
      of increase output per worker is less than the amount of
      increased savings. So consumption is lower.
    \end{subproblem}
  \end{problem}

  \begin{problem}

    \begin{subproblem}
      The function has constant returns to scale when considering
      employment, human capital, and physical capital all together:
      multiplying $K$, $N$, and $H$ by $k$ multiplies output by $k$
      because the exponents add to 1.  However, increasing any one or
      any two of $K$, $N$, and $H$ by $k$ multiplies output by less
      than $k$. Since $H$ is fixed, the function doesn't have constant
      returns to scale with respect to $K$ and $N$.
    \end{subproblem}

    \begin{subproblem}
      
      \begin{align*}
        Y_t &= K_t^\alpha H^\beta N^{1-\alpha-\beta}\\
        y_t = \frac{Y_t}{N} &= \frac{K_t^\alpha H^\beta
          N^{1-\alpha-\beta}}{N}\\
        y_t &= \frac{K_t^\alpha H^\beta}{N^{\alpha+\beta}} =
        \left(\frac{K_t}{N}\right)^\alpha \left(\frac{H}{N}\right)^\beta = k_t^\alpha h^\beta
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      The amount of capital at time $t+1$ is the capital stock at time
      $t$ minus the fraction that depreciated plus the amount of
      savings:
      \begin{align*}
        k_{t+1} &= k_t - \delta k_t + s Y_t\\
        k_{t+1} - k_t &= - \delta k_t + s k_t^\alpha h^\beta\\
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      At steady state, $k_{t+1} = k_{t} = k_{ss}$. So
      \begin{align*}
        \delta k_{ss} &= s k_{ss}^\alpha h^\beta\\
        k_{ss}^{1-\alpha} &= \frac{s}{\delta} \cdot h^\beta\\
        k_{ss} &= \left(\frac{s}{\delta} \cdot h^\beta\right)^
        {\frac{1}{1-\alpha}}
      \end{align*}
      The corresponding level of output is
      \begin{align*}
        y_{ss} &= k_{ss}^\alpha h^\beta\\
        y_{ss} &= \left(\frac{s}{\delta} \cdot h^\beta\right)^
        {\frac{\alpha}{1-\alpha}} h^\beta\\
        y_{ss} &=
        \left(\frac{s}{\delta}\right)^{\frac{\alpha}{1-\alpha}}
        h^{\frac{\beta}{1-\alpha}}\\
        &=
        \left(\frac{s^\alpha h^\beta}{\delta^\alpha}\right)^\frac{1}{1-\alpha}
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      Consumption per worker is the fraction of output per worker not
      saved:

      \begin{align*}
        c_{ss} &= (1-s) y_{ss} \\
        &= (1-s)
        \left(\frac{s^\alpha
            h^\beta}{\delta^\alpha}\right)^\frac{1}{1-\alpha}\\
        &=
        \left(\frac{h^\beta}{\delta^\alpha}\right)^\frac{1}{1-\alpha}
        \left(s^\frac{\alpha}{1-\alpha} - s^\frac{1}{1-\alpha}\right)
      \end{align*}
    \end{subproblem}

    \begin{subproblem}
      The steady-state level of consumption is maximized when its
      derivative with respect to $s$ is zero.

      \begin{align*}
        \der{c_{ss}}{s} &= 0\\
%        left(\frac{h^\beta}{\delta^\alpha}\right)^\frac{1}{1-\alpha}
%      \left(\frac{\alpha}{1-\alpha} s^\frac{
        \left(\frac{h^\beta}{\delta^\alpha}\right)^\frac{1}{1-\alpha}
        \left( -s^{\frac{\alpha}{1-\alpha}} + (1-s)
        {\frac{\alpha}{1-\alpha}} s^{\frac{\alpha}{1-\alpha}-1}
      \right) &= 0\\
        {\frac{\alpha}{1-\alpha}} s^{\frac{\alpha}{1-\alpha}-1} -
        {\frac{\alpha}{1-\alpha}} s^{\frac{\alpha}{1-\alpha}}
        -s^{\frac{\alpha}{1-\alpha}} &= 0\\
        s^{\frac{\alpha}{1-\alpha}-1} \left(
          {\frac{\alpha}{1-\alpha}} - \frac{\alpha}{1-\alpha}s -
          s\right) &= 0\\
        {\frac{\alpha}{1-\alpha}} - \frac{1}{1-\alpha}s &= 0\\
        \frac{\alpha}{1-\alpha} &= \frac{1}{1-\alpha}s\\
      s &= \alpha\\
      \end{align*}

      This savings rate doesn't depend on $h$, although the
      corresponding level of capital grows proportionally to
      $h^{\frac{\beta}{1-\alpha}}$. This is the golden-rule level of
      capital.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    
    \begin{subproblem}
      Capital per effective worker increases because of savings. It
      decreases because of depreciation, growth of the labor force
      (because the capital is now divided amongst more people), and
      growth of technology (because the effective labor force is
      larger).
    \end{subproblem}

    \begin{subproblem}
      The change in the capital stock is given by
      \[ \Delta K = I - \delta K - (g_A + g_N) K\]
      so the required level of investment to maintain a constant level
      of capital per effective worker is
      \[ (\delta + g_A + g_N) K \]
    \end{subproblem}

    \begin{subproblem}
      \[ i = \frac{I}{N} = \frac{(\delta + g_A + g_N) K}{N} = (\delta
      + g_A + g_N) k\]§ 
    \end{subproblem}
\newpage
    \begin{subproblem}
      Because the production function has constant returns to scale,
      we can divide through by $A_tN_T$, writing the production
      function as
      \[ y_t = f(\frac{K_t}{A_t N_t}) = f(k_t) \]
      The savings rate is a fraction $s$ of $y_{ss}$.

      At equilibrium, the savings ($s y_{ss}$) is equal to the
      required level of investment given above, $(\delta + g_A +
      g_N)k$.
      \vspace{3in}
    \end{subproblem}

    \begin{subproblem}
      The growth rate of total output at steady state is independent
      of the savings rate; it is simply the growth rate of effective
      labor: $g_N + g_A$.
    \end{subproblem}

    \begin{subproblem}
      The new levels of capital and output per effective worker
      fall. Because the savings rate is fixed, the equilibrium level
      of capital drops now that more capital is lost each year;
      consequently, the equilibrium level of output also falls.

      In the transition, capital per effective worker drops gradually
      from its old to new value, as the capital stock is depleted by a
      level of depreciation larger than the new capital from
      savings. The level of output falls similarly, although it is
      also growing over time because of technological progress.  The
      growth rate of total output in the new steady state remains the
      same.

      \vspace{3in}
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}