4 a Why must firms which play an infinitely repeated Bertran

4. (a) Why must firms which play an infinitely repeated Bertrand game choose the Bertrand price each period if their discount factors are 0? [50%] (b) Why is there an equilibrium in which they earn their share of monopoly profits each period if their discount factors are close to 1? Are there any other equilibria? [50%] Explain your answers to each part.

Solution

a) If the discount factor d =0.

If firm cooperates they will get m/n ; where n is the number of firms and m is the monopoly profit.

So from cooperating firm gets (sub-script c is for co-operating and nc for not co-operating)

c = m/n + dm/n +(d)2 m/n+ ….

c = m/n(1-d)

From not co-operating firm will get

nc = d +0+0 ….

d> m/n. For d= 0

c = m/n. So profit from deviation are higher firms will not collude and play Bertrand price in each period.

b) If the discount factor d =1.

If firm cooperates they will get m/n ; where n is the number of firms and m is the monopoly profit.

So from cooperating firm gets

c = m/n + dm/n +(d)2 m/n+ ….

c = m/n(1-d)

From not co-operating firm will get

nc = d +0+0 ….

d> m/n. For d= 1

c = . So profit from deviation are not higher firms will collude and will not play Bertrand price in each period.

No there are no other equilibrium.