Economics 482 Game Theory and Economics - Problem Set 4 : Nash equilibrium
Economics 482 Spring 2021
Due Tuesday, February 23
Problem Set 4
1. Consider a Cournot game between two firms. The firms face an inverse demand function describedbytheequationP(Q)=α−QifQ≤α,P(Q)=0ifQ>α,whereP isthe price of output and Q is the total output produced by the two firms. Firm 1 produces its output q1 at a constant unit cost c1 (i.e, the total cost to firm 1 of producing q1 units of output is c1q1). Firm 2 produces output q2 at a constant unit cost c2.
1. (a) Solve for the Nash equilibrium output choices of each firm, (q1∗ , q2∗ ), as a function of α, c1, and c2. You may assume that both firms choose a positive quantity of output in the Nash equilibrium (this will be true as long as c1 and c2 are not too big, or not too far apart). Solve also for each firm’s profit, total output, and the Nash equilibrium price, also as functions of α, c1 and c2.
2. (b) Suppose firm 2 has an innovation that reduces its cost (that is, makes c2 smaller). What is the directional effect (up or down) on the Nash equilibrium outputs and profits of the firms, and on the Nash equilibrium price?
2. Consider the following Bertrand game between two firms, firm 1 and firm 2. As in the standard Bertrand game, each firm’s action is a choice of price (that is, some nonnegative real number). However, the firms produce goods that are not identical, and thus consumers do not use the rule of buying only from the firm that charges a lower price. Instead, the demand for each firm’s good is a continuous function of the prices they choose. In particular, demand for firm 1’s good as a function of p1 and p2 is 1 + p2 − p1, and demand for firm 2’s good as a function of p1 and p2 is 1+p1 −p2. Each firm has a constant unit cost of production c.
1. (a) Write down each firm’s profit as functions of c, p1, and p2.
2. (b) Solve for each firm’s best response function.
3. (c) Find the Nash equilibrium of the game (as a function of c).
4. (d) Are there any outcomes that yield both firms a higher payoff than the Nash equilibrium? If not, prove your answer. If so, give one example.
3.
1. (a) Suppose that there are four firms in the standard Hotelling game, in which each firm wishes to locate in a place that maximizes its market share. Find a Nash equilibrium in which each firm sells to 1/4 of the market. Demonstrate that each firm is using a best response in your Nash equilibrium.
2. (b) Consider the Hotelling game with three firms. Write down firm 1’s best response as a function of firm 2’s and firm 3’s choices of location, x2 and x3. A best response will not exist for some values of x2 and x3, but for others it will. Given your answer, can there be a Nash equilibrium in pure strategies of the Hotelling game with three firms? Explain.
