Exercise 4. [3 points] Two companies are selling software which are imperfect substitutes of each other. Let p; and x; denote the price and the quantity sold of software 1. Similarly, let p> and x; denote the price and the quantity sold of software 2 respectively. Demand for x, and x, are respectively given by P2 =) = 4+ % 33 P2 P g B g BL At 2 73 Each company has incurred 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 (which is same as its total profit in this case) (1) (0.5 points) Write payoff functions for the two companies [note: these will be in terms of p; and p;] (i1) (1 point) Suppose company 2 chooses p; = 90 with probability 0.5 and p> = 180 with probability 0.57 What is the best response for company 17 (i) The total revenue (profit) for each company is given by the product of the price and the quantity sold. The payoff functions for company 1 and company 2 are as follows: For company 1: 71 = P1 . x1 = P1 . (90 - 1 + For company 2: 712 = P2 . 202 = p2 . (90 - P2 + PL ) (ii) To find the best response for company 1 when company 2 chooses p2 = 90 with probability 0.5 and p2 = 180 with probability 0.5, we need to maximize the expected payoff for company 1. The expected payoff is the weighted sum of the payoffs corresponding to each possible price chosen by company 2. Expected 71 = 2 . 71(p2 = 90) + 2 . "1(p2 = 180) Substitute the expressions for 7 1 and 72 into the expected payoff expression and simplify to find the best response for company 1.Let's substitute the expressions for 7r; and 79 into the expected payoff expression and simplify: For company 1: m=pr- (90 5 + %) For company 2: =p2- (90 5 +%) Now, calculate the expected payoff for company 1when ps = 90 with probability 0.5 and pa = 180 with probability 0.5: Expected m = - mi(p2 = 90) + 5 - m1(p2 = 180) :%.[pl.(go_%%+%.[p1.(90_%+1_30)] Now, simplify and calculate the expression to find the expected payoff for company 1. Expected m; = [p1 - (90 B +30)] + 5 [p1 - (90 B + 60) ] Combine like terms: 1l (20 )] + 3 [ (150 - )] Combine the terms with ps: = 3 [120p1 gpi] + 3 [150p1 pi] Combine the two terms: % -120p1 g % 150p; Combine the coefficients: = 135p; ; So, the expected payoff for company 1is given by 135p; }'p