1. Suppose an industry produces an undifferentiated product with market demand given by Q = A - P. There are many potential producers for this product, each of whom has a cost function given by a fixed cost F and a constant marginal cost k. We imagine that firms decide whether to enter the industry under the supposition that, after all the firms that are going to enter have done so, competition will be according to the Cournot model. An equilibrium is achieved with N firms in the industry if each firm, having Cournot "conjectures", does no worse than break even and if another firm had entered and made this an N+1 firm Cournot oligopoly, all the firms would lose money.
(a) Compute N such that an equilibrium is achieved. Why is N finite?
(b) Repeat (a) under the assumption that firms have Bertrand "conjectures" throughout?
(c) Are the numbers of firms found in (a) and (b) different or not? Why?
2. Consider two manufacturers (indexed by i=1,2) of an identical product. Suppose the quantity that consumers demand from producer i is (a-pi) when pi < p-i (note: p-i is the price charged by the other manufacturer), zero when pi > p-i, and (a-pi)/2 when pi = p-i. Also assume fixed costs are zero and marginal cost is constant at c, where c < a.
(a) Show that if the two firms choose prices simultaneously, then the unique Nash equilibrium is that both firms charge the price c. (b) An (implicit) assumption behind the result in (a) is that each producer can meet any quantity demanded he or she faces (please convince yourself of this). Speculate as to what the equilibrium prices might be if both producers cannot produce beyond a certain level of output. Would you expect the equilibrium prices to be higher or not? Why?
(c) Another assumption behind the result in (a) is the belief each firm has on what the other firm is likely to do. Suppose one firm announces publicly that she will charge prices p = c + d, where d is positive and "small." What would you expect the other firm to do? What would you expect the (new) equilibrium prices to be? Why?
3. Luenberger. Chapter 8. Problem 5.
4. Imagine a duopoly in which firms produce an homogenous product for consumers who live along a 100-mile highway. We imagine that consumers are distribute uniformly along the highway at a density of ten consumers per mile. Each consumer has a reservation price of $100 for one unit of the good, and will buy either zero or one unit only. There is a transportation cost of getting to and from the store that must be added to the price of the good when comparing against the reservation price. The good cost each of the two firms $1 per unit to produce. If the firm is located d miles from a consumer, the consumer pays $(d/50)2. for the round trip from home to the store and back. The two firms locate their store (which serves to differentiate their products in the minds of consumers, unless located in the same place) and then compete with Bertrand conjectures.
(a) Suppose each store is located at opposite extremes of the highway. What are the equilibrium prices in this case?
(b) Suppose both stores are located right at mile 50. What are the equilibrium prices in this case?
(c) Suppose each firm decides first where to locate its store, and then compete in prices. What is the optimal location for each store? What are the equilibrium prices?
5. Consider a duopoly facing demand function P = A - Q. Each firm has a constant marginal cost ci, where ci < A. Fixed costs are zero. Suppose firm 2 is uncertain about the true value of c1. Moreover, firm 2 believes c1 can take only two values: c11 and c12, where c11 < c12, with probabilities p1 and p2. Firm 2 holds this probability assessment when she chooses its level of output. Firm 1 knows the true value of its marginal cost (which is either c11 and c12), and also knows the beliefs firm 2 hold. Suppose the two firms engage in a Cournot competition. Assume c1j and c2 are such that both firms always produce a positive quantity.
(a) What are the equilibrium quantities? An equilibrium in quantities is defined as follows: An output q2 for firm 2, and, for each value c1j, an output q1j (j=1,2). To compute q2, firm 2 maximizes the expected profits under the belief firm 1 will produce q1j with probability pj. And firm 1 will compute q1j assuming firm 2 will produce q2 and knowing its own marginal cost.
(b) How does q2 vary with p1? Explain.