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?