In the standard Bertrand model of oligopoly n firms each produce a differentiated product. The demand qi for the product of the ith firm depends on its own price and the price charged by all the other firms, that is,
qi = f(pi, p-i)
If each firm's production cost is measured by the cost function i, firm i's payoff function is
ui(pi, p-i) = pif (pi, p-i) - ci(f (pi, P-i))
In the simplest specification the demand functions are linear

with bi > 0 and the firm's produce at constant marginal cost i, so the payoff functions are
ui(pi, p-i) = (pi - i)f (pi, p-i)
Show that if the goods are gross substitutes (dij > 0 for every i, j), the Bertrand oligopoly model with linear demand and constant marginal costs is a super modular game.

Transcribed Image Text:

f(Pi: P-) = a, – b;P; +d; P, j+i