8. 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 and X1 be the price and quantity sold of product 1, and P2 and X2 the price and quantity sold of product 2. We have that X1 = 1,000 (90-P1+¹P2) and X2 = 1,000 (90-1/P2 + P1). Each company has incurred a fixed cost for designing their software and writing programmes, but the cost of selling to an extra user is zero. As the firms compete in prices, each company will choose a price that maximises its profits.

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8. 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 and X1 be the
price and quantity sold of product 1, and P2 and X2 the price and quantity sold of
product 2. We have that X1 = 1,000 (90 –P1 +P2) and X2 = 1,000 (90 –P2 +
P1). Each company has incurred a fixed cost for designing their software and writing
programmes, but the cost of selling to an extra user is zero. As the firms compete in
prices, each company will choose a price that maximises its profits.
(a) Explain why the price that maximises each company's profits is the same as the price
that maximises its total revenue.
(b) Write an expression for the total revenue of each company as a function of it its price
and the other company's price.
(c) Company's 1 best response function BR1(P2) is the price of product 1 that maximises
its profits given the price of product 2 is P2. Similarly, company's 2 best response
function BR2(P1) is the price of product 2 that maximises its profits given the price of
product 1. Using these functions, write the best response function of each company
and then calculate the Nash equilibrium prices and the total revenue of each company.
Show diagrammatically the BRs and the Nash equilibrium in prices.
Transcribed Image Text:8. 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 and X1 be the price and quantity sold of product 1, and P2 and X2 the price and quantity sold of product 2. We have that X1 = 1,000 (90 –P1 +P2) and X2 = 1,000 (90 –P2 + P1). Each company has incurred a fixed cost for designing their software and writing programmes, but the cost of selling to an extra user is zero. As the firms compete in prices, each company will choose a price that maximises its profits. (a) Explain why the price that maximises each company's profits is the same as the price that maximises its total revenue. (b) Write an expression for the total revenue of each company as a function of it its price and the other company's price. (c) Company's 1 best response function BR1(P2) is the price of product 1 that maximises its profits given the price of product 2 is P2. Similarly, company's 2 best response function BR2(P1) is the price of product 2 that maximises its profits given the price of product 1. Using these functions, write the best response function of each company and then calculate the Nash equilibrium prices and the total revenue of each company. Show diagrammatically the BRs and the Nash equilibrium in prices.
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