Consider an industry in which N firms all produce an undifferentiated product. Demand for the product is given by X = A - P. Each producer is identical, having a constant average cost of k.
(a) Suppose all N producers have Cournot conjectures: Each conjectures that it can change the amount it produces and all its rivals will continue to produce the amounts they are producing. What is the symmetric equilibrium in this case? (An equilibrium is symmetric if all the firms are producing the same amount.) What happens to the equilibrium price as N approaches infinity?
(b) Suppose all N producers have Bertrand conjectures: Each conjectures that it can change the price it charges without any reaction from rivals in the prices they charge. What is the symmetric equilibrium in this case?
(c) Suppose one firm is a Stackelberg leader and all the rest have Cournot conjectures. That is, each of the N -1 followers believes that it can change its quantity without any response in quantities from any of the other firms. The one leader understands this and picks its quantity optimally. What equilibria can you find? What happens to price as N goes to infinity?
(d) Suppose the N firms are numbered 1, 2, ... , N, and 'they have the following sort of conjectures: Firm N has Cournot conjectures. Firm N -1 has Cournot conjectures about firms 1 through N-2 and believes that firm N works off its Cournot reaction curve. Firm n has Cournot conjectures about firms 1 through n -1 and believes that firms n+ 1 through N work off the reaction curves given by their conjectures. What equilibria can you find? What happens to price as N goes to infinity?
(e) Construct a model in the spirit of kinked demand for the setting of an N firm oligopoly.