\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