|Question||Two software companies 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. Let p1 be the price and x1 the quantity sold of product 1 and let p2 and x2 be the price and quantity sold of product 2. Then x1 = 1000 (90 – 1/2p1 + 1/4p2) and x2 = 1000 (90 – 1/2p2 + 1/4p1). Each company has incurred a fixed cost for designing their software and writing the programs, but the cost of selling to an extra user is zero. Therefore each company will maximize its profits by choosing the price that maximizes its total revenue.
(a) Write an expression for the total revenue of company 1, as a function of the its price p1 and the other company’s price p2.
b) Company 1’s best response function BR1(·) is defined so that BR1(p2) is the price for product 1 that maximizes company 1’s revenue given that the price of product 2 is p2. With the revenue functions we have specified, the best response function of company 1 is described by the formula BR1(p2) = __________
(c) Use a similar method to solve for company 2’s best response function __________
(d) Solve for the Nash equilibrium prices p1 = __________ and p2 = ___________
(e) Suppose that company 1 sets its price first. Company 2 knows the price p1 that company 1 has chosen and it knows that company 1 will not change this price. If company 2 sets its price so as to maximize its revenue given that company 1’s price is p1, then what price will company 2 choose? p2 = _____________ If company 1 is aware of how company 2 will react to its own choice of price, what price will company 1 choose? _________ Given this price for company 1, what price will company 2 choose? ___________