Consider a certain duopoly facing repeatedly over time (with no

prespecified end) a fixed demand function for its homogenous good of the form

where ˜ *θ**t *is an identically and independently distributed random variable in each

time period *t*. Both firms display a fixed cost function given by

Let *ν *≡ *pm**/**pc *where *pm **> *0 and *pc **> *0 are the prices prevailing in the (static)

Cournot-Nash equilibrium and the maximally collusive (symmetric) output profile,

respectively. Assume the random variable ˜ *θ**t *takes only two values: ˜ *θ**t *= 1*, ν, *each

arising with equal probability. In every period, firms choose their respective outputs

simultaneously, but they observe only the prevailing prices (i.e., they do not observe

the individual outputs chosen by other firms).

Model the situation as an infinitely repeated game with stage payoffs (profits)

being discounted at a common rate *δ *∈ (0*, *1). Suppose firms wish to sustain the

maximally collusive profile (*qm**, **qm*) through an equilibrium of the dichotomous

sort presented in Subsection 9.1.2 (i.e., an equilibrium that responds to “normal” and

“regressive” situations). Compute the optimal duration *T *∗ of the regressive phases

that achieve this objective. Furthermore, find how *T *∗ depends on the discount rate *δ*.