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

\begin{document}
\psetnum{7}
\date{2004/11/12}

\begin{pset}
  \begin{problem}
    \textbf{False.} The firm that produces first will have the
    advantage because it knows what the second-producing firm's
    profit-maximizing output will be for any decision it makes. It can
    therefore choose its output to maximize its own profits, taking
    into account this information.
  \end{problem}

  \begin{problem}
    \textbf{False.} For example, the perfectly competitive outcome can
    be forced, by government price regulation.
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      \begin{center}
        \begin{tabular}{|r|cc|}
          \hline
          & advertise & don't \\
          \hline
          advertise & 10,5 & 16,0 \\
          don't & 6,8 & 10,2 \\
          \hline
        \end{tabular}
      \end{center}

      If Firm A chooses to advertise, then Firm B will also choose to
      advertise, giving a payoff of 5. If Firm A chooses not to
      advertise, then Firm B will choose to advertise, giving a payoff
      of 8. So advertising is a dominant strategy for Firm B;
      whatever decision A makes, advertising will lead to more profit
      for B.

      If Firm B chooses to advertise, then Firm A will choose to
      advertise, giving a payoff of 10. If Firm B chooses not to
      advertise, Firm A will choose to advertise, giving a payoff of
      16. So advertising is also a dominant strategy for firm B.
    \end{subproblem}

    \begin{subproblem}
      Each player's choices holding the player's strategy constant are
      shown in bold below:
      
      \begin{center}
        \begin{tabular}{|r|cc|}
          \hline
          & advertise & don't \\
          \hline
          advertise & \textbf{10},\textbf{5} & \textbf{16},0 \\
          don't & 6,\textbf{8} & 10,2 \\
          \hline
        \end{tabular}
      \end{center}
      
      The Nash equilibrium is where both players choose to
      advertise.
    \end{subproblem}

    \begin{subproblem}
      The payoff matrix now becomes:
      \begin{center}
        \begin{tabular}{|r|cc|}
          \hline
          & advertise & don't \\
          \hline
          advertise & \textbf{10},\textbf{5} & 16,0 \\
          don't & 6,\textbf{8} & \textbf{20},2 \\
          \hline
        \end{tabular}
      \end{center}

      Firm B's dominant strategy is still to advertise. Firm A has no
      dominant strategy. The Nash equilibrium is still where both
      players choose to advertise.
    \end{subproblem}  
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The market demand for widgets is $Q = 100 - \frac{p}{4}$. So the
      residual demand for firm A is $q_A = 100 - \frac{p}{4} - q_B$,
      and so $p = 400 - 4q_A - 4q_B$ and its marginal revenue curve is
      \[ MR = 400 - 8 q_A - 4 q_B \]
      At the profit-maximizing point, the marginal revenue equals the
      marginal cost, which is constant at 100. So
      \[ 400 - 8q_A - 4 q_B = 100 \]
      \[ q_A = 37.5 - \frac{q_B}{2} \]

      By symmetry,
      \[ q_B = 37.5 - \frac{q_A}{2} \]

      And so
      \[ q_A = 37.5 - \frac{37.5 - \frac{q_A}{2}}{2} = 37.5 - 18.75 +
      \frac{q_A}{4} \]
      \[ \frac{3}{4} q_A = 18.75 \]
      \[ q_A = 25 = q_B \]

      The equilibrium price is
      \[ p = 400 - 4q_A - 4q_B = 200 \]

      The profits for each firm are
      \[ \pi_A = \pi_B = 200(25) - 100(25) = 2500 \]
    \end{subproblem}

    \begin{subproblem}
      This is a Stackelberg model, with Firm A as the leader. Firm B's
      best response will be
      \[ q_B = 37.5 - \frac{q_A}{2} \]
      as before. Firm A thus faces a residual demand function of
      \[ p = 400 - 4 q_A - 4 q_B = 400 - 4q_A - 4(37.5 -
      \frac{q_A}{2}) = 400 - 4q_A - 150 + 2q_A = 250 - 2q_A\]

      A's marginal revenue is 
      \[ MR = p'(q_A)q_A + p(q_A) = 250 - 4q_A \]
      To maximize profit, this equals the marginal cost of 100,

      \[ 250 - 4q_A = 100 \]
      \[ q_A = 37.5 \]
      
%       \[ 250 - \frac{q_A}{2}  = 100 \]
%       \[ q_A = 300 \]
      and so
      \[ q_B = 37.5 - \frac{37.5}{2} = 18.75 \]
      and
      \[ p = 400 - 4(37.5) - 4(18.75) = 400 - 150 - 75 = 175 \]
%       \[ q_B = 0 \]
%       and
%       \[ p = 400 - 300 - 0 = 100 \]

      Firm A's profits are
      \[ \pi_A = p q_A - 100 q_A = 2812.5\]
      and Firm B's profits are
      \[ \pi_A = p q_B - 100 q_B = 1406.25 \]
    \end{subproblem}

    \begin{subproblem}
      The firms have the same marginal cost, so if they combine they
      will still have $MC = 100$. The marginal revenue is as in a
      monopoly situation
      \[ MR = p'(q)q + p(q) = 400 - 8Q_J \]
      \[ 400 - 8Q_J = 100\]
      \[ Q_J = \frac{300}{8} = 37.5 \]
      \[ p = 400 - 4Q_J = 250 \]
      \[ \pi = pQ_J - 100Q_J= 9375 -  3750 = 5625 \]
    \end{subproblem}

    \begin{subproblem}
      The total quantity to be sold to maximize joint profits is
      \[ Q_J = 37.5 \] from above, at a price $p = 250$. Firm A will
      give Firm B a market share of $\beta$, so B will produce $\beta
      37.5$. Its profits will therefore be
      \[ (p - 100)Q_B = (250-100)\beta 37.5 = \beta 5625 \]
      This must be greater than the profits it would make in Cournot
      equilibrium for this to be preferable for B, so
      \[ \beta 5625 > 2500 \]
      \[ \beta > \frac{2500}{5625} \approx 0.4444\]
      So A must offer B at least 44.4\% market share.
    \end{subproblem}

    \begin{subproblem}
      The price is 250, and the total quantity produced is 37.5. A
      produces 20.83 units and B 16.67 units. The profit for $A$ is
      3125, and the profit for B is 2500; the total profit is
      5625. Thus B does not gain anything from the deal (at this
      market share), and A gains $3125-2500 = 625$; 625 is the total
      the firms gain jointly.
      
      The consumer surplus without collusion is $\int_0^50 400 - 4Q -
      200 \; dQ = 5000$. With the collusion, it is $\int_0^37.5 400 -
      4Q - 250 \; dQ = 2812.5$, so consumers lose 2187.5 from the
      collusion.
      
    \end{subproblem}

    \begin{subproblem}
      Firm B will produce exactly $\frac{37.5}{2} = 18.75$ units. Thus
      the residual demand function for $A$ is
      \[ p_A = 400 - 4q_A - 4q_B = 400 - 4q_A - 75 = 325 - 4 q_A \]

      A's marginal revenue will be
      \[ MR = 325 - 8 q_A \]
      Setting this equal to marginal cost gives
      \[ 325 - 8 q_A = 100 \]
      \[ q_A = 28.125 \]
      \[ p_A = 212.5 \]
    \end{subproblem}

    \begin{subproblem}
      With Firm B in the market, Firm A makes a profit of 2500.
      Without B, A can make the full monopoly profit of 5625. So A
      will be willing to pay up $5625-2500=3125$ for B. Consumers
      would lose 2187.5 from the deal.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      Firm $k$ faces a market demand curve of
      \[ p = a - b \sum_{i \neq k} q_i - b q_k \]
      Its marginal revenue is
      \[ MR = a - b \sum_{i \neq k} q_i - 2b q)k \]
      At the profit-maximizing point, this equals the marginal cost
      $c$
      \[ a - b \sum_{i \neq k} q_i - 2b q_k = c \]
    \end{subproblem}

    \begin{subproblem}
      Firm $k$'s reaction function is given using the maximization
      shown above
      \[ a - b \sum_{i \neq k} q_i - 2b q_k = c \]
      \[ a - b \sum_{i \neq k} q_i - c = 2b q_k \]
      \[ q_k = \frac{a - b \sum_{i \neq k} q_i - c}{2b} \]
      \[ q_k = \frac{a - c}{2b} - \frac{\sum_{i \neq k} q_i}{2} \]
    \end{subproblem}

    \begin{subproblem}
      Since the firms have identical marginal costs, they will have
      the same quantity strategies for a Nash equilibrium. So
      \[ q_k = \frac{a - c}{2b} - \frac{\sum_{i \neq k} q_k}{2} \]
      \[ q_k = \frac{a - c}{2b} - \frac{n-1}{2} q_k \]
      \[ (n+1) q_k = \frac{a - c}{b}\]
      \[ q_k = \frac{a - c}{b(n+1)} \]
    \end{subproblem}

    \begin{subproblem}
      \[ p = a - b q = a - b \sum_i q_i = a - bn q_k \]
      \[ p = a - bn \frac{a-c}{b(n+1)} \]
      \[ p = a - \frac{n}{n+1}(a-c) \]
    \end{subproblem}

    \begin{subproblem}
      With free entry, the market price is the limit as $n$ goes to
      infinity.
      \[ p = \lim_{n \to \infty} a - \frac{n}{n+1}(a-c) = c \]

      The market price approaches the marginal cost, which is the
      price in a competitive equilibrium. This is the expected result,
      because when many firms can enter the market, the price-setting
      power of each individual firm is reduced.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The firms produce identical products, so their Bertrand
      equilibrium is where price is equal to marginal cost:
       $p = MC = 1$.
      So the market demand is
      $Q = 2 - p = 1$

      This is split between the two firms: $q_1 = q_2 =
      \nicefrac{1}{2}$. Neither firm makes a profit. The total welfare
      is the consumer surplus, which is $\int_0^1 2 - Q - 1 \; dQ =
      \nicefrac{1}{2}$. 
    \end{subproblem}

    \begin{subproblem}
      Suppose now that firm 1 commits to its price first. Let $p$ be
      the price it chooses. Then there are three cases:

      \begin{enumerate}
      \item $p < 1$. This case will not happen: since the price is
        below the average variable cost, firm 1 will shut down
        instead of producing at this price.
      \item $p = 1$. In this case, firm 2 will set its price to 1 as
        well (if they set a higher price, they have no sales; if they
        set a lower price, they take a loss.) So the two firms split
        the market with a price of $1$ and a total quantity of $1$,
        just as before.
      \item $p > 1$. Then firm 2 will set its price just below $p$ and
        take all of the sales. Firm 1 makes no profit.
      \end{enumerate}

      Since the second case is the only one in which firm 1 makes a
      profit, it will always choose $p=1$, and the output is the
      same as part a.
    \end{subproblem}

    \begin{subproblem}
      Suppose firm 2's cost falls to $0.5 q_2$. Note that firm 1 still
      has cost of $1 q_1$, so it will shut down for prices below $1$.
      Thus, at any price below 1, firm 1 will shut down, and firm 2
      will sell the entire market demand. So firm 2 sets its price
      just below 1 ($1 - \epsilon$ for some $\epsilon$ which we can
      make arbitrarily small, so in the limit we treat it as zero).

      The quantity it sells is 1, as before, but now firm 2 makes the
      entire profit, and firm 1 produces nothing. Firm 1 now makes a
      profit of $\nicefrac{1}{2}$, and the consumer surplus is still
      $\nicefrac{1}{2}$, so total welfare is 1.
    \end{subproblem}
  \end{problem}

  \begin{problem}
    \begin{subproblem}
      The marginal product of labor is
      \[ MP_L = \partialder{Q}{L} = \frac{1}{4}L^{-\nicefrac{1}{2}} \]
      so the marginal revenue product of labor is
      \[ MPR_L = MR( MP_L) = p MP_l = \frac{1}{4}L^{-\nicefrac{1}{2}} \]
      This is graphed below:

      \begin{center}
        \includegraphics{ps7-7a}
      \end{center}
    \end{subproblem}

    \begin{subproblem}
      Since the wage as a function of labor is $w(L) =
      \nicefrac{L}{2}$,

      \[ ME = w(L) + L \partialder{w(L)}{L} = \frac{L}{2} + L
      \frac{1}{2} = L \]
        This equals the
       marginal revenue product of labor:
       \[ L = \frac{1}{4} L^{-\nicefrac{1}{2}} \] 
       \[ L^{\nicefrac{3}{2}} = \frac{1}{4} \]
       \[ L = \left(\frac{1}{4}\right)^{\nicefrac{2}{3}} \approx 0.397
       \]

       and so
       \[ w = \frac{L}{2} \approx 0.198 \]
       % \[L^{-\nicefrac{1}{2}} = 16w \]
%       \[L = \frac{1}{256w^2} \]
%       Substituting the labor supply equation,
%       \[ 2w = \frac{1}{256w^2} \]
%       \[ w^3 = \frac{1}{512} \]
%       \[ w = \frac{1}{8} \]
%       and so
%       \[ L = \frac{1}{256\left(\frac{1}{8}\right)^2} = \frac{1}{4}\]
    \end{subproblem}

    \begin{subproblem}
      The wage is now fixed at $w$, so the marginal expenditure of
      labor is $ME = w$. Now this equals the MRPL:
      \[ w = \frac{1}{4} L^{-\nicefrac{1}{2}} \]
      \[ L_A = \frac{1}{16 w^2} \]

      This is the amount of labor used by firm A. The other three
      firms use the same amount of labor, so the total labor demand is
      four times this:
      \[ L = \frac{1}{4 w^2} \]

      The labor market equilibrium will have $L = 2w$, so
      \[ 2w = \frac{1}{4 w^2} \]
      \[ w = \frac{1}{2} \]

      Wages increase because there is now competition among the four
      firms for labor.
    \end{subproblem}
  \end{problem}
\end{pset}
\end{document}

% LocalWords:  Stackelberg
