Question

Recall the static Bertrand duopoly model (with homogenous products): the firms name prices simultaneously; demand for firm i's product is a - pi if pi < pj, is 0 if pi > pj, and is (a-pi)/2 if pi = pj; marginal costs are c < a. Consider the infinitely repeated game based on this stage game. Show that the firms can use trigger strategies (that switch forever to the stage-game Nash equilibrium after any deviation) to sustain the monopoly price level in a subgame-perfect Nash equilbriu if and only if ?? >= 1/2.

Answer 1

Consider the infinite stage repeated game with payoff function :

Pit = (pit - c)(a - pit)      if pit < pjt

Pit = (pit - c)(a - pit)/2    if pit = pjt

Pit = 0                       if pit > pjt

at each stage t.

Consider the following symmetric strategy :

Play pit = pjt = x > c in every period

If pit/pjt is not played then play pjs/pis = c for s = t+1, t+2 ,...

Assume that the cumulated payoff to each firm is :

sum{t=0 to infinity} of Pit*delta^t

On the equilibrium path this is:

.5(x-c)(a-x) * 1/(1-delta) > 0

Suppose there is a deviation by player i at time s. Instead of playing

x, they play y < x.

Their cumulated payoff is :

sum(t=0 to s-1) of .5(x-c)(a-x) * delta^t + (y-c)(a-y) * delta^s +

sum(t=s+1 to infinity) of 0

= .5(x-c)(a-x)*((1 - delta^s)/(1-delta)) + (y-c)(a-y) * delta^s

For this strategy to be an equilibrium, we must have that :

.5(x-c)(a-x) * 1/(1-delta) > .5(x-c)(a-x)*((1 - delta^s)/(1-delta)) +

(y-c)(a-y) * delta^s for all pis < x and for all s.

Rearranging :

(delta^s/(1-delta))*.5(x-c)(a-x) > (delta^s)*(y-c)(a-y)

or .5(x-c)(a-x) > (1-delta)*(y-c)(a-y)

This is only true for all x,y if 1-delta < .5 ie. if delta >= .5