Bertrand Competition
Consider a model in which two firms compete in goods that are essentially homogeneous, but one has a higher quality than the other. You can think of this as if there are different customers with valuation v for product 1 and valuation v+y for product 2, where v ist distributed according to the distribution function F(v), which is increasing on [vL,vH], with F(vH)=1 and F(vL)=0. Demand for product 1 (if it is the only one) is then given by D(p)=1†F(p), namely all customers for whom the price p is below their valuation. If the price is vH, no consumer is willing to buy. Demand for product 2 (if no other product is available) is then given by F(pâ€y). Clearly all customers buy product 1 if p1 < p2 – y and all buy product 2 if the opposite holds. Assume that all consumers buy the higher quality product 2 if p1 = p2 – y.
Question
a. Write the demand function, D2(p2,p1), for the firm that offers product 2 (the high quality product) as a function of the prices p1 and p2.
b. Draw a picture in (p1,p2) – space, indicating all the pairs of prices for which consumers buy from firm 1.
c. Use this picture to sketch a proof that the highest price that firm 2 will set in equilibrium is p2 = y.