game theory

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 products 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 – ¹/₂ p₁ +¹/₂ p₂)

                    X₂ = 1000(90 -  ¹/₂ p₂ + 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 maximise its profit by choosing the price that maximises its total revenue.

A). write an expression for the total revenue of company 1, as a function of the it’s 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. derive the best response function of company 1

c). similarly, derive company 2’s best response function BR₂(p₁).

d). solve for the Nash equilibrium prices.

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 set its price so as to maximise its revenue given that company 1’s price is p₁, then what price will company 2 choose?