Suppose that the demand functions in the Bertrand model are given by where 1 ≥ b > 0. This…

Suppose that the demand functions in the Bertrand model are given by

where 1 ≥ b > 0. This says that the quantity of gadgets sold by a firm will increase if the price set by the opposing firm is too high. Assume that both firms have a cost of production c ≤ min{ p1, p2}. Then, since profit is revenue minus costs the profit functions will be given by

(a) Using calculus, show that there is a unique Nash equilibrium at

(b) Find the profit functions at equilibrium.

 (c) Suppose the firms have different costs and sensitivities so that

 

 

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