\documentclass{article}
\input{1401-preamble}

\begin{document}
\date{2004/12/03}
\psetnum{9}

\begin{pset}
  \begin{problem}
    \begin{subproblem}
      \textbf{True.} For example, if either person has all 100 units,
      then there is no way to increase the other person's utility
      without decreasing the first person's.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} This is true only in a competitive labor market.
      If the labor market is a monopsony, a minimum wage may increase
      both the wage and the employment level.
    \end{subproblem}

    \begin{subproblem}
      \textbf{True.} He is risk-neutral, so he will accept the gamble
      if his expected gain is positive. The expected value is
      \[ \frac{1}{4}(200) + \frac{3}{4} (-60) = 5 \]
      which is indeed positive.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} It is possible that the cash flow will be
      extremely positive after the first three years, and that the
      utility due to this will outweigh the losses from the first
      three years. The exact amount required depends on the size of
      the losses in the first three years and the firm's utility
      function for wealth.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      No state is more Pareto-efficient than the others. In each of
      states $A$, $B$, and $C$, the other two states have less utility
      for at least one of the individuals.
    \end{subproblem}

    \begin{subproblem}
      Yes, $D$ is Pareto-superior to $A$ because the utilities for
      both are greater in $D$. It is not superior to $B$ and $C$
      because one of the two utilities is higher in $B$ and $C$ (1 in
      $B$, 2 in $C$).
    \end{subproblem}

    \begin{subproblem}
      States $B$, $C$, $D$, and $E$ are Pareto-optimal.
    \end{subproblem}

    \begin{subproblem}
      The social welfare is the sum of the individual's utilities:
      
      \begin{tabular}{| c | c c | c |}
        \hline
        State & $U_1$ & $U_2$ & $W$ \\
        \hline
        $A$ & 50 & 50 & 100 \\
        $B$ & 70 & 40 & 110 \\
        $C$ & 45 & 54 & 99 \\
        $D$ & 53 & 50.5 & 103.5 \\
        $E$ & 30 & 84 & 114 \\
        \hline
      \end{tabular}

      So $E$ is optimal.
    \end{subproblem}

    \begin{subproblem}
      This ratio is as small as possible when it equals $1$, which is
      the case in state $A$, where the utilities are equal.
    \end{subproblem}

    \begin{subproblem}
      The optimal state with a Rawlsian welfare function is $D$, since
      the worse-off person's utility is highest.
    \end{subproblem}

    \begin{subproblem}
      The government faces an equity-efficiency tradeoff. State $E$ is
      more efficient in that the total utility is highest, but is also
      the least equitable in that individual's 1 utility is so much
      lower than individual 2's.
    \end{subproblem}

    \begin{subproblem}
      Note that $E$ is the more efficient outcome, but $A$ is more
      equitable. The welfare in case $E$ is
      \[ W_A = \alpha(50) + (1-\alpha)(50) = 50 \]
      In outcome $E$, the welfare is
      \[ W_E = \alpha(30) + (1-\alpha)(84) = 84-54\alpha \]
      The less efficient outcome, outcome $A$, is chosen as long as
      $W_A > W_E$, i.e.
      \begin{align*}
        50 &> 84 - 54\alpha \\
        \alpha &> \frac{17}{27} \approx 0.63
      \end{align*}
      So if $0.5 \le \alpha < 0.63$, the more efficient outcome is
      chosen.
    \end{subproblem}

    \begin{subproblem}
      Individual 1 prefers state $B$, and individual 2 prefers state
      $A$. If individual 1 transfers 10 units to individual 2, then 2
      will be indifferent between $B$ and $A$.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    Sarah's MRS is
    \[ MRS_S = \frac{MU_{X,S}}{MU_{Y,S}} =
    \frac{\partialder{U_S}{X_S}}{\partialder{U_S}{Y_S}} = \frac{20 Y_S^{0.6}
      X_S^{-0.6}}{30 X_S^{0.4} Y_S^{-0.4}} = \frac{2}{3} \frac{Y_S}{X_S} \]
    and Jane's MRS is
    \[ MRS_J = \frac{MU_{X,J}}{MU_{Y,J}} =
    \frac{\partialder{U_J}{X_S}}{\partialder{U_J}{Y_S}} = \frac{50
      Y_J^{0.5} X_J^{-0.5}}{50 X_J^{0.5}Y_J^{-0.5}} = \frac{Y_J}{X_J}
    \]

    We also have the conditions
    \begin{align*}
      X_S + X_J &= 58 \\
      Y_S + Y_J &= 36
    \end{align*}

    At equilibrium, the MRSes are equal. So
    \begin{align*}
      MRS_S &= MRS_J \\
      \frac{2Y_S}{3X_S} &= \frac{Y_J}{X_J} \\
      \frac{2Y_S}{3X_S} &= \frac{36 - Y_S}{58 - X_S} \\
      Y_S &= \frac{108 X_S}{X_S + 116}
    \end{align*}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The initial allocation is $\tup<A,B,C> = \tup<400,100,16>$, so
      the initial social welfare is
      \[ W = \sqrt{F_A} + \sqrt{F_B} + \sqrt{F_C} = \sqrt{400} +
      \sqrt{100} + \sqrt{16} = 20 + 10 + 4 = 34 \]
      The initial individual utilities are
      \[ \tup<U_A,U_B,U_C> = \tup<20,10,4>\]
      After redistribution, the allocation is $\tup<A,B,C> = \tup<225,
      196, 16>$. So the social welfare is
      \[ W = \sqrt{F_A} + \sqrt{F_B} + \sqrt{F_C} = \sqrt{225} +
      \sqrt{196} + \sqrt{16} = 15+14+4 = 33\]
      The individual utilities are
      \[ \tup<U_A,U_B,U_C> = \tup<15,14,4>\]
      Social welfare decreases slightly. Intuitively, this is because
      the loss of food due to spoilage outweighs the greater utility
      that $B$ gains from the food compared to $A$.
    \end{subproblem}

    \begin{subproblem}
      Now the allocation is $\tup<A,B,C> = \tup<225, 100, 100>$. The
      social welfare is
      \[ W = \sqrt{F_A} + \sqrt{F_B} + \sqrt{F_C} = \sqrt{225} +
      \sqrt{100} + \sqrt{100} = 15+10+10 = 35\]
      The individual utilities are
      \[ \tup<U_A,U_B,U_C> = \tup<15,10,10>\]
    \end{subproblem}

    \begin{subproblem}
      The social welfare is higher in (b) than in the initial
      allocation and in the redistribution in (a), despite the fact
      that there is a greater loss due to spoilage in (c). This is
      because $C$ initially has much less food than $A$, $C$'s
      marginal utility is sufficiently higher than $A$'s that $C$'s
      increase in utility will outweigh both $A$'s decrease in utility
      and the loss due to spoilage.
    \end{subproblem}

    \begin{subproblem}
      For a fixed total amount of food, in this case utility is
      maximized when all three individuals divide the food
      equally. This is not true in general of utilitarian social
      welfare functions; it is only true here because each individual
      has the same utility function with diminishing returns.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      If the backpacker keeps all of her money, she has a
      $\nicefrac{1}{2}$ chance of having zero dollars and a
      $\nicefrac{1}{2}$ chance of having \$600. So her expected wealth
      is \$300.

      If she puts half of her money in the bush, she has a
      $\nicefrac{1}{4}$ chance of having zero dollars, a
      $\nicefrac{1}{2}$ chance of having \$300, and a
      $\nicefrac{1}{4}$ chance of having \$600. The expected wealth
      again is \$300.

      If the backpacker is risk neutral, her marginal utility of
      wealth is constant and she will choose based on expected
      value. So she is indifferent to the two options.
    \end{subproblem}

    \begin{subproblem}
      Now if she keeps all of her money, her expected utility is
      \[ \frac{1}{2} \sqrt{600} \approx 12.25 \]
      If she puts half in the bush, her expected utility is
      \[ \frac{1}{2} \sqrt{300} + \frac{1}{4} \sqrt{600} \approx 14.78
      \]
      So she will choose to put half of her money in the bush.
    \end{subproblem}

    \begin{subproblem}
      Her expected utility if she keeps all her money is
      \[ \frac{1}{2} 600^\alpha = 300^\alpha \left(\frac{2^\alpha}{2}\right)\]
      If she puts half in the bush, it is
      \[ \frac{1}{2} 300^\alpha + \frac{1}{4} 600^\alpha =
      300^\alpha\left(\frac{1}{2} + \frac{2^\alpha}{4}\right)\]

      She will only choose to keep all her money if
      \[ 300^\alpha \left(\frac{2^\alpha}{2}\right) >
      300^\alpha\left(\frac{1}{2} + \frac{2^\alpha}{4}\right) \]
      \[ 2^\alpha > 2 \]
      which is only true if $\alpha > 1$.

      (There is also the corner case where $\alpha = 0$. In this case,
      she has no utility of wealth; thus, she is utterly indifferent
      to how much money she has, and does not care which option she
      chooses.)
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The production possibility frontiers are shown below.
      \begin{center}
        \includegraphics{ps9-6a-1}
      \end{center}

      England has a comparative advantage in the cloth industry and
      Portugal has a comparative advantage in the port industry. The
      relative price of cloth in terms of wine is $\nicefrac{1}{3}$ in
      England and $4$ in Portugal.
    \end{subproblem}

    \begin{subproblem}
      England has the comparative advantage in the cloth industry, so
      it is better off when the relative price of cloth in terms of
      port is less than 4. Portugal is better off when the relative
      price is greater than $\nicefrac{1}{3}$. So only Portugal gains
      for prices between $0$ and $\nicefrac{1}{3}$, both gain for
      prices between $\nicefrac{1}{3}$ and $4$, and only England gains
      for prices above $4$.
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}