Problem 4
This exercise extends the analysis of the Stackelberg duopoly game (from Chapter 15) to include fixed costs of production. The analysis produces a theory of limit quantity, which is a quantity the incumbent firm can produce that will induce the potential entrant to stay out of the market. Suppose two firms compete by selecting quantities \(q_{1}\) and \(q_{2}\), respectively, with the market price given by \(p=1000-3 q_{1}-3 q_{2}\). Firm 1 (the incumbent) is already in the market. Firm 2 (the potential entrant) must decide whether or not to enter and, if she enters, how much to produce. First the incumbent commits to its production level, \(q_{1}\). Then the potential entrant, having seen \(q_{1}\), decides whether to enter the industry. If firm 2 chooses to enter, then it selects its production level \(q_{2}\). Both firms have the cost function \(c\left(q_{i}\right)=100 q_{i}+F\), where \(F\) is a constant fixed cost. If firm 2 decides not to enter, then it obtains a payoff of 0 . Otherwise, it pays the cost of production, including the fixed cost. Note that firm \(i\) in the market earns a payoff of \(p q_{i}-c\left(q_{i}\right)\). (a) What is firm 2's optimal quantity as a function of \(q_{1}\), conditional on entry? (b) Suppose \(F=0\). Compute the subgame perfect Nash equilibrium of this game. Report equilibrium strategies as well as the outputs, profits, and price realized in equilibrium. This is the Stackelberg or entryaccommodating outcome. (c) Now suppose \(F>0\). Compute, as a function of \(F\), the level of \(q_{1}\) that would make entry unprofitable for firm 2. This is called the limit quantity. (d) Find the incumbent's optimal choice of output and the outcome of the game in the following cases: (i) \(F=18,723\), (ii) \(F=8,112\), (iii) \(F=1,728\), and (iv) \(F=108\). It will be easiest to use your answers from parts (b) and (c) here; in each case, compare firm 1's profit from limiting entry with its profit from accommodating entry.
Problem 6
Imagine a market setting with three firms. Firms 2 and 3 are already operating as monopolists in two different industries (they are not competitors). Firm 1 must decide whether to enter firm 2's industry and thus compete with firm 2 or enter firm 3's industry and thus compete with firm 3. Production in firm 2's industry occurs at zero cost, whereas the cost of production in firm 3's industry is 2 per unit. Demand in firm 2's industry is given by \(p=9-Q\), whereas demand in firm 3 's industry is given by \(p^{\prime}=14-Q^{\prime}\), where \(p\) and \(Q\) denote the price and total quantity in firm 2 's industry and \(p^{\prime}\) and \(Q^{\prime}\) denote the price and total quantity in firm 3 's industry. The game runs as follows: First, firm 1 chooses between \(\mathrm{E}^{2}\) and \(\mathrm{E}^{3}\). ( \(\mathrm{E}^{2}\) means "enter firm 2's industry" and \(\mathrm{E}^{3}\) means "enter firm 3's industry.") This choice is observed by firms 2 and 3 . Then, if firm 1 chooses \(\mathrm{E}^{2}\), firms 1 and 2 compete as Cournot duopolists, where they select quantities \(q_{1}\) and \(q_{2}\) simultaneously. In this case, firm 3 automatically gets the monopoly profit of 36 in its own industry. In contrast, if firm 1 chooses \(E^{3}\), then firms 1 and 3 compete as Cournot duopolists, where they select quantities \(q_{1}^{\prime}\) and \(q_{3}^{\prime}\) simultaneously; and in this case, firm 2 automatically gets its monopoly profit of \(81 / 4\). (a) Calculate and report the subgame perfect Nash equilibrium of this game. In the equilibrium, does firm 1 enter firm 2 's industry or firm 3 's industry? (b) Is there a Nash equilibrium (not necessarily subgame perfect) in which firm 1 selects \(E^{2}\) ? If so, describe it. If not, briefly explain why.
Problem 8
Consider the following market game: An incumbent firm, called firm 3 , is already in an industry. Two potential entrants, called firms 1 and 2, can each enter the industry by paying the entry cost of 10 . First, firm 1 decides whether to enter or not. Then, after observing firm 1's choice, firm 2 decides whether to enter or not. Every firm, including firm 3, observes the choices of firms 1 and 2. After this, all of the firms in the industry (including firm 3) compete in a Cournot oligopoly, where they simultaneously and independently select quantities. The price is determined by the inverse demand curve \(p=12-Q\), where \(Q\) is the total quantity produced in the industry. Assume that the firms produce at no cost in this Cournot game. Thus, if firm \(i\) is in the industry and produces \(q_{i}\), then it earns a gross profit of \((12-Q) q_{i}\) in the Cournot phase. (Remember that firms 1 and 2 have to pay the fixed cost 10 to enter.) (a) Compute the subgame perfect equilibrium of this market game. Do so by first finding the equilibrium quantities and profits in the Cournot subgames. Show your answer by designating optimal actions on the tree and writing the complete subgame perfect equilibrium strategy profile. [Hint: In an \(n\)-firm Cournot oligopoly with demand \(p=12-Q\) and 0 costs, the Nash equilibrium entails each firm producing the quantity \(q=12 /(n+1) .]\) (b) In the subgame perfect equilibrium, which firms (if any) enter the industry?
Problem 9
This exercise will help you think about the relation between inflation and output in the macroeconomy. Suppose that the government of Tritonland can fix the inflation level \(\dot{p}\) by an appropriate choice of monetary policy. The rate of nominal wage increase, \(W\), however, is set not by the government but by an employer-union federation known as the ASE. The ASE would like real wages to remain constant. That is, if it could, it would set \(\dot{W}=\dot{p} .\) Specifically, given \(\dot{W}\) and \(\dot{p}\), the payoff of the ASE is given by \(u(\dot{W}, \dot{p})=-(\dot{W}-\dot{p})^{2}\). Real output \(y\) in Tritonland is given by the equation \(y=30+(\dot{p}-\dot{W}) .\) The government, perhaps representing its electorate, likes output more than it dislikes inflation. Given \(y\) and \(\dot{p}\), the government's payoff is \(v(y, \dot{p})=y-\dot{p} / 2-30\). The government and the ASE interact as follows. First, the ASE selects the rate of nominal wage increase. Then the government chooses its monetary policy (and hence sets inflation) after observing the nominal wage increases set by the ASE. Assume that \(0 \leq \dot{W} \leq 10\) and \(0 \leq \dot{p} \leq 10\). (a) Use backward induction to find the level of inflation \(\dot{p}\), nominal wage growth \(\dot{W}\), and output \(y\) that will prevail in Tritonland. If you are familiar with macroeconomics, explain the relationship between backward induction and "rational expectations" here. (b) Suppose that the government could commit to a particular monetary policy (and hence inflation rate) ahead of time. What inflation rate would the government set? How would the utilities of the government and the ASE compare in this case with that in part (a)? (c) In the "real world," how have governments attempted to commit to particular monetary policies? What are the risks associated with fixing monetary policy before learning about important events, such as the outcomes of wage negotiations?
