205 econ the qquestion is complete

Exercise 2 (Cournot duopoly with asymmetric firms) In a market characterized by the following (inverse) demand function P = 40 Q two firms compete `a la Cournot. Firm A has production cost described by the cost function cA (qA ) = 20qA , while firm B's cost function is cB (qB ) = q 2 B . 1. Which firms has increasing marginal cost? Which one has constant marginal cost? 2. Define the reaction functions of the firms. 3. Compute the Cournot equilibrium quantities and price.

3. Cournot equilibrium is identified by the quantities that are mutually best responses for both firms; so, they are obtained by the solution of the following two-equation system: ( qA = 20qB 2 qB = 40qA 4 The equilibrium quantities are q ? A = 40 7 ; q ? B = 60 7 Q? = 100 7 and the equilibrium price is P(Q ? ) = 40 100 7 = 180 7 and firms' profits are: ? A = 180 7 q ? A 20q ? A = 1600 49 ? B = 180 7 q ? B q ?2 B = 7200 49.

Exercise 3 (Competition `a la Bertrand) Market demand is given by P (Q) = ( 100 Q if Q < 100 0 otherwise Suppose that two firms both have average variable cost c = $50. Assuming that firms compete in prices, then: 1. Define the reaction functions of the firms; 2. Find the Bertrand equilibrium; 3. Would your answer change if there were three firms? Why?

If two firms have the same constant marginal cost, they earn zero profits in the Bertrand equilibrium. This depends crucially on the feature that the goods involved are perfect substitutes. If products are differentiated instead, then the Bertrand equilibrium can lead to positive profits. The products are differentiated when consumers consider them only imperfect substitutes Whilst a consumer may be unwilling to buy the product of one producer, she will have the incentive to do this if the price of their favourite product becomes too high. To model this we allow the demand for each good to depend not only on its own price but also on the price of the other good. Assume for example that the demand for the good produced by F irm1, q1, and the demand for the good produced by F irm2, q2, are described by the following functions: q1 = 180 p1 (p1 p) q2 = 180 p2

If two firms have the same constant marginal cost, they earn zero profits in the Bertrand equilibrium. This depends crucially on the feature that the goods involved are perfect substitutes. If products are differentiated instead, then the Bertrand equilibrium can lead to positive profits. The products are differentiated when consumers consider them only imperfect substitutes Whilst a consumer may be unwilling to buy the product of one producer, she will have the incentive to do this if the price of their favourite product becomes too high. To model this we allow the demand for each good to depend not only on its own price but also on the price of the other good. Assume for example that the demand for the good produced by F irm1, q1, and the demand for the good produced by F irm2, q2, are described by the following functions: q1 = 180 p1 (p1 p) q2 = 180 p2 (p2 p)

where p is the average price that is taken over the prices of the two firms. Each firm has average (and marginal) cost c = 20. Suppose the firms can only choose between the three prices {94, 84, 74}. 1. Compute the profits of the firms under the 9 different price combinations that are possible in the model. 2. Using you answer to the previous point, construct the 3x3 payoff matrix for the normal form game where the payoffs are given by the profits of the firms 3. Find the (Bertrand-)Nash equilibrium of this game. What are the profits at this equilibrium?

Consider a horizontally differentiated product market in which two firms are located at any points l1 and l2 on the real line, respectively, with the notation l1 l2. Firms produce at marginal costs c. There is a continuum of consumers of mass 1 who are uniformly distributed on the unit interval. They have unit demand and have an outside utility of . A consumer located at x [0; 1] obtains indirect utility v = max (v1; v2) with v1 = r (x l1) 2 p1 if she buys one unit from firm 1 and v2 = r (l2 x) 2 p2 if she buys from firm 2. Firms have marginal costs equal to c.

a. Suppose that prices are regulated at pi = 2c. In the game in which firms simultaneously decide where to locate their product, characterize the Nash equilibrium.

b. Determine the demand function for each firm for each admissible price pair (p1; p2) given locations l1 and l2.

c. Suppose that the two firms simultaneously set prices. Determine the market equilibrium for all possible combinations of (l1; l2).

d. Suppose that the social planner chooses first-best optimal prices. Which price pairs would be socially optimal for the pair of locations l1 = 0 and l2 = 1/2?

e. Compare your results obtained in (c) and (d) for locations l1 = 0 and l2 = 1/2. Is the equilibrium socially efficient? Depending on your answer elaborate on the sources of the inefficiency or give the reason for efficiency.