4. Two companies, Firm 1 and Firm 2 sell the same product y. The market demand f

Question

4. Two companies, Firm 1 and Firm 2 sell the same product y. The market demand for the good y is given by y 200 -2p. Denote by c the marginal cost of Firm 1 and c2 the marginal cost of Firm 2. Let yi be the quantity produced by Firm 1, 3y/2 be the quantity produced by Firm 2, pi be the price of Firm 1, and p2 be the price of Firm 2. The firms compete à la Bertrand. (a) What are yi and y when pi p2 (b) Assume that c- c20. Find the equilibrium prices. (c) (Bonus question) Assume that C1-2 and C2-0. Find the equilibrium prices. Now suppose that they sell competing products. These products are substitutes, so that the number of units that either company sells is a decreasing function of its own price and an increasing function of the other product's price. The demand function for each product is given by 1 132-2p1 +p2 2102-2p2 +Pi For the following assume that c 2 0. (d) Write an expression for the total revenue of Firm 1, as a function of pi and p2. (e) Find the best response function of Firm 1. (Hint Find the revenue-maximizing Pi as a function of p2.) (f) Find the best response function of Firm 2 (g) (Bonus question) Find the equilibrium prices

Explanation / Answer

a) Since p1<p2 and given that te goods are homogeneous/same, this will lead consumers to go to the low charging firm thus the output of firm 2 y2 will be zero while y1 = 200-2p1. When p1>p2, we have y1 = 0 and y2 = 200-2p2.

b) At c1 = c2 = 0, the prices will be p1 = p2 = 0. This is because charging a higher price will bring down the market share of the firm charging higher price.

c) At c1 = 2 and c1 =0, the firm 1 will charge a price p2 = 2-e, e being very small. Firm 2 will charge c1 = 2. Charging any less will cause losses to firm 1 thus the firm 2 caters to all the demand in the market and y1 = 0.

d) Total revenue for firm 1 = p1*y1 = (132-2p1+p2)p1 = 132p1 -2p1^2 +p2*p1

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