\documentclass{article}
\input{1401-preamble}
\usepackage{../generic-preamble/tuple}

\begin{document}
\psetnum{8}
\date{2004/11/19}

\begin{pset}
  \begin{problem}
    \begin{subproblem}
      \textbf{False.} The utilitarian SWF is the sum of each
      individual's utility, so this allocation may be optimal; the
      distribution does not matter.
    \end{subproblem}
    
    \begin{subproblem}
      \textbf{False.} Suppose the two firms have linear PPFs with
      neither economies or diseconomies of scope. Then the firms are
      indifferent as to which combination of the two goods they will
      produce. So the total production in a Pareto-efficient
      allocation can be divided among the two firms differently.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} After making the trade, there may still be
      further trades that will make both parties better off.
    \end{subproblem}

    \begin{subproblem}
      \textbf{False.} Avoiding a deadweight loss may lead to increased
      efficiency, but may sacrifice equity. Depending on the social
      welfare function used, a redistribution involving a deadweight
      loss may be preferable.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The marginal product of labor is
      \[ MP_L = \partialder{Q}{L} = \frac{1}{2 \sqrt{L}} \]
      In a perfectly competitive market, the marginal revenue is equal
      to the price, so $MR = 2$
      \[ MRP_L = MR(MP_L) = \frac{1}{\sqrt{L}} \]
    \end{subproblem}

    \begin{subproblem}
      Suppose the market wage for labor is $\nicefrac{1}{2}$. This
      equals the $MRP_L$, so
      \[ MRP_L = \frac{1}{\sqrt{L}} = \frac{1}{2} \]
      \[ L = 4 \]

      If the price of hot dogs increases to 3, the $MRP_L$ becomes
      \[ MRP_L = 3\frac{1}{2\sqrt{L}} \]
      Then
      \[ MRP_L = \frac{3}{2\sqrt{L}}= \frac{1}{2} \]
      \[ L = 9 \]
    \end{subproblem}

    \begin{subproblem}
      The labor supply function is $L = 9w^2$, so
      \[ w = \sqrt{\frac{L}{9}} = \frac{\sqrt{L}}{3}\]
      We now have
      \[ ME = w(L) + L \partialder{w(L)}{L} = \frac{\sqrt{L}}{3} + L
      \frac{1}{2(3\sqrt{L})} = \frac{\sqrt{L}}{3} + \frac{\sqrt{L}}{6}
      = \frac{\sqrt{L}}{2}\]
      The price of hot dogs is $2$, so the $MRP_L =
      \frac{1}{\sqrt{L}}$ as before. So
      \[ MRP_L = \frac{1}{\sqrt{L}} = \frac{\sqrt{L}}{2} = ME \]
      \[ L = 2 \]
    \end{subproblem}

    \begin{subproblem}
      In a competitive labor market, a minimum wage generally
      increases unemployment, but this is not the case in a monopsony
      labor market. The marginal expenditure curve becomes horizontal
      at the minimum wage over the region where the minimum wage is
      binding. If the minimum wage is set at the competitive level, it
      will intersect the $MRP_L$ curve at the competitive equilibrium,
      thus increasing both the number of workers hired and their wage.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    The initial endowment is shown as point $e$. Bart's utility is
    \[ U^B = \min \left(60, \frac{80}{2}\right) = 40 \]
    so his indifference curve is the Leontief curve with corner at
    $\tup<40,80>$, and Homer's utility is
    \[ U^H = 4(40) + 3(120) = 160+360 = 520 \] so his indifference
    curve is the line between $\tup<100,40>$ and
    $\tup<0,\frac{520}{3}>$. This is shown in the Edgeworth box
    below.

    \begin{center}
      \includegraphics[width=4in]{ps8-3-1}
    \end{center}

    Both parties are better off with all bundles in region $R$.

    The set of Pareto-efficient allocations are those in which the two
    parties are unwilling to trade. Bart has a Leontief utility
    function, so he will always be willing to trade except at the
    corner points. Thus, the contract curve is the set of all corner
    points:

    \begin{center}
      \includegraphics[width=4in]{ps8-3-2}
    \end{center}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      Note that $H_B + H_S = 100$. Since $S = H_S$, $H_B = 100 - S$
      and so
      \[ B = 6 H_B^{\nicefrac{1}{2}} = 6 \left(100 -
        S\right)^{\nicefrac{1}{2}} \]
    \end{subproblem}

    \begin{subproblem}
      \begin{center}
        \includegraphics{ps8-4b}
      \end{center}
    \end{subproblem}

    \begin{subproblem}
      His utility is
      \[ U = 40B + BS = (40 + S)B = (40+S)\left(6 \sqrt{100-S}\right)
      \]
      At the maximum, marginal utility is zero:
      \[ \der{U}{S} = 6 \sqrt{100-S} - \frac{3(40+S)}{\sqrt{100-S}} =
      0\]
      \[ S = \frac{160}{3} \approx 53.333 \]
      \[ B = 6 \sqrt{100 - S} = 4 \sqrt{105} \approx 40.988 \]
    \end{subproblem}

    \begin{subproblem}
      The marginal rate of transformation is the slope of the PPF:
      \[ MRT = \abs{\der{B}{S}} = \frac{3}{\sqrt{100 - S}} =
      \frac{3}{\sqrt{100-\frac{160}{3}}} = \frac{3\sqrt{105}}{70} \approx
        .439 \]
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      Alice's initial utility is
      \[ U_A = 7^{\nicefrac{1}{3}} 1^{\nicefrac{2}{5}} \approx 1.912
      \]
      So her indifference curve is
      \begin{align*}  
      M_A^{\nicefrac{1}{3}} C_A^{\nicefrac{2}{5}} &= \sqrt[3]{7} \\
      M_A^{\nicefrac{1}{3}} &= \sqrt[3]{7} C_A^{-\nicefrac{2}{5}} \\
      M_A &= 7 C_A^{-\nicefrac{6}{5}}
      \end{align*}

      Brian's initial utility is
      \[ U_B = 3^{\nicefrac{2}{3}} 5^{\nicefrac{3}{5}} \approx 5.4634
      \]
      So his indifference curve is
      \begin{align*}
        M_B^{\nicefrac{2}{3}} C_B^{\nicefrac{3}{5}} &=
        3^{\nicefrac{2}{3}} 5^{\nicefrac{3}{5}} \\
        M_B^{\nicefrac{2}{3}}  &=
        3^{\nicefrac{2}{3}} 5^{\nicefrac{3}{5}} C_B^{-\nicefrac{3}{5}} \\
        M_B  &=
        3 (5^{\nicefrac{9}{10}}) C_B^{-\nicefrac{9}{10}} \approx 12.77
        C_B^{-\nicefrac{9}{10}}\\        
      \end{align*}

      This is shown in the Edgeworth box below:
      \begin{center}
        \includegraphics{ps8-5a}
      \end{center}
    \end{subproblem}

    \begin{subproblem}
      Alice's marginal utilities are
      \[ MU_{A,M} = \partialder{U_A}{M_A} = \frac{1}{3}
        C_A^{\nicefrac{2}{5}} M_A^{-\nicefrac{2}{3}} \]
      \[ MU_{A,C} = \partialder{U_A}{C_A} = \frac{2}{5}
        M_A^{\nicefrac{1}{3}} C_A^{-\nicefrac{3}{5}} \]
      so her MRS is
      \[ MRS_A = \frac{MU_{A,M}}{MU_{A,C}} = \frac{5 C_A}{6 M_A} \]

      Brian's marginal utilities are
      \[ MU_{B,M} = \partialder{U_B}{M_B} = \frac{2}{3}
         C_B^{\nicefrac{3}{5}} M_B^{-\nicefrac{1}{3}} \]
      \[ MU_{B,C} = \partialder{U_B}{C_B} = \frac{3}{5}
        M_B^{\nicefrac{2}{3}} C_B^{-\nicefrac{2}{5}} \]
      so his MRS is
      \[ MRS_B = \frac{MU_{B,M}}{MU_{B,C}} = \frac{10 C_B}{9 M_B} \]

      Recalling that Alice and Brian's bundles sum to 10 cups of milk
      and 6 cookies,
      \[ MRS_B = \frac{10 C_B}{9 M_B} = \frac{10 (6 - C_A)}{9 (10 -
        M_B)} \]
      On the contract curve, Alice and Brian's $MRS$es are equal.
      \[ MRS_A = \frac{5 C_A}{6 M_A} = MRS_B = \frac{10 (6 - C_A)}{9 (10 -
        M_B)} \]
      \[ M_A = \frac{30 C_A}{24 - C_A} \]
    \end{subproblem}

    \begin{subproblem}
      The allocations on the contract curve that represent mutual
      gains from trade are those points on the contract curve where
      utility is greater for both Alice and Brian. These are the
      points along the contract curve that are also in the region
      above Alice's indifference curve and below Brian's.
    \end{subproblem}

    \begin{subproblem}
      Now Alice's original indifference is
      \[ U_A = 6^{\nicefrac{1}{3}} 4^{\nicefrac{2}{5}}
      \]
      So her indifference curve is
      \begin{align*}  
        M_A^{\nicefrac{1}{3}} C_A^{\nicefrac{2}{5}} &=
        6^{\nicefrac{1}{3}} 4^{\nicefrac{2}{5}} \\
        M_A^{\nicefrac{1}{3}} &= 6^{\nicefrac{1}{3}}
        4^{\nicefrac{2}{5}} C_A^{-\nicefrac{2}{5}} \\
        M_A &= 6 4^{\nicefrac{6}{5}} 7 C_A^{-\nicefrac{6}{5}}
      \end{align*}

      and Brian's new initial utility is
      \[ U_B = 4^{\nicefrac{2}{3}} 2^{\nicefrac{3}{5}}
      \]
      So his indifference curve is
      \begin{align*}
        M_B^{\nicefrac{2}{3}} C_B^{\nicefrac{3}{5}} &=
        4^{\nicefrac{2}{3}} 2^{\nicefrac{3}{5}} \\
        M_B^{\nicefrac{2}{3}}  &=
        4^{\nicefrac{2}{3}} 2^{\nicefrac{3}{5}} C_B^{-\nicefrac{3}{5}} \\
        M_B  &=
        4 (2^{\nicefrac{9}{10}}) C_B^{-\nicefrac{9}{10}} 
      \end{align*}

      The contract curve does not depend on the initial endowment, so
      it remains the same.

      The new Edgeworth box looks like
      \begin{center}
        \includegraphics{ps8-5d}
      \end{center}

      There are no trades that will make both parties better off.
    \end{subproblem}

%     \begin{subproblem}
%       The social welfare is
%       \[ SW = U_A + U_B = M_A^{\nicefrac{1}{3}} C_A^{\nicefrac{2}{5}}
%       + M_B^{\nicefrac{2}{3}} C_B^{\nicefrac{3}{5}} \]
%       Recalling that Alice and Brian's bundles sum to 10 cups of milk
%       and 6 cookies,
%       \[ SW = M_A^{\nicefrac{1}{3}} C_A^{\nicefrac{2}{5}} +
%       \left(10-M_A\right)^{\nicefrac{2}{3}} (6-C_A)^{\nicefrac{3}{5}}
%       \]
%       Trading will lead to an allocation on the contract curve,
%       i.e. $ M_A = \frac{30 C_A}{24 - C_A}$. So
%       \[ SW = \left(\frac{30 C_A}{24 - C_A}\right)^{\nicefrac{1}{3}} C_A^{\nicefrac{2}{5}} +
%       \left(10-\left(\frac{30 C_A}{24 - C_A}\right)\right)^{\nicefrac{2}{3}} (6-C_A)^{\nicefrac{3}{5}}
%       \]
%     \end{subproblem}
  \end{problem}
\end{pset}
\end{document}