The Bijou Theater in Hermosa Beach, California, shows vintage movies. Customers arrive at the theater line at the rate of 100 per hour. The ticket seller averages 30 seconds per….
find how T ∗ depends on the discount rate δ.
Consider a certain duopoly facing repeatedly over time (with no
prespecified end) a fixed demand function for its homogenous good of the form
where ˜ θt 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 ˜ θt takes only two values: ˜ θt = 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.