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A model of secret price cuts
The following model formalizes the discussion in the first part of Section 8.2.3
In order to help us concentrating on the main issues, we assume a simple demand structure,
namely an inelastic demand curve. Specifically, suppose that all consumers are willing to pay
u for the (homogeneous) product sold by two duopolists. In each period, demand can be high
(probability 1−α) or low (probability α). When demand is high, h = 1 units can be sold at
price u (or any lower price). When demand is low, only l = 0 < h units can be sold. The
probability that demand is high or low in each period is independent of what it was in the
previous period. Moreover, firms are unable to observe the state of market demand; all they
can observe is whether their own demand is high or low. Finally, for simplicity, assume that
production costs are zero.
This set of assumptions about what firms can and cannot observe implies that equilibria
of the sort considered before cannot be implemented. In fact, those equilibria assumed that
firms could detect (with certainty) that rivals cheated from setting the agreed-upon price,
whereas now we assume that firms cannot observe their rival’s price decisions. However, by
observing their own demand (and prices), firms can make inferences, imperfect as they might
be, about the rival’s past decisions. Is this sufficient to sustain collusion? The answer is, for
some parameter values, positive.
Consider the following equilibrium strategies. Firms start by setting p = u. If they receive
a positive demand (namely, 1/2 ), then they continue to set p = u, that is, they remain in the
“co-operative phase”. If however one of the firms (or both) receives zero demand, then both
firms enter into a “price war:” they set p = 0 during T periods and, after this period, go back
to p = u again (the co-operative phase).
The condition for this to be an equilibrium is, as usual, that the expected payoff from
playing the equilibrium strategy is greater then the payoff from deviating (by setting a slightly
lower price and taking all of the market demand). Let V be expected equilibrium discounted
profits starting in a period during the co-operative phase. Then
The first term on the right-hand side corresponds to the case when demand is high (probability
1−α), whereas the second term corresponds to the case of low demand (probability α). If
demand is high, then each firm receives current profits of u/2 . Moreover, beginning next period,
their continuation expected payoff is V , for there is no reason to start a price war. If, however,
demand is low (probability α), then it is common knowledge that at least one of the firms
receives zero demand, and that a price war will start in the next period. As a result, firms
receive zero profits today (because demand is zero) and zero profits in the next T periods
(because they engage in a price war). After these T periods, firms revert to the co-operative
phase, so that their continuation expected payoff from then on is V .
If a firm deviates (during the co-operative phase), then its expected discounted payoff is
In words: if demand turns out to be high (probability 1−α), then setting a slightly lower
price gives the deviator a current profit of u (as opposed to u/2 ). However, regardless of what
the state of demand is today, firms will certainly enter in a price war beginning in the next
period. In fact, regardless of the state of demand, the rival firm (the non-deviator) receives
zero demand today, the condition that triggers a price war. For this reason, expected future
discounted payoff is simply δT+1V .
The condition that the prescribed strategy constitutes an equilibrium is that V ≥ V . It
can be shown that this conditions simplifies into
If α < 1/2, this condition is equivalent to T ≥ T*, where T* is a positive number.
What is then the optimal equilibrium? In other words, what is the optimal value of T?
From (8.1), we can see that the equilibrium discounted payoff is decreasing in the value of T.
This is intuitive, for the greater the value of T the longer the price wars will be; and firms don’t
like price wars. Therefore, the optimal value of T is the lowest value such that the equilibrium
is stable, that is, the lowest value such that (8.3) holds, that is, T = T*.
To show that V ≥ V is equivalent to 8.3, we rearrange 8.1: V [1 − (1−α)δ −αδT+1] = (1−α)u/2.
[Note that this gives us a very clear expression for V which is the equilibrium payoff over the
entire length of the repeated oligopoly interaction. For example if α = 0.1 and δ = 0.9 and T = 5,
then V is approximately u6.3 (as a comparison if α = 0, then V =
= u10.]
1 
Let’s continue! Rearrange V ≥ V to obtain (1−α)δV−(1−α)δT+1V ≥ (1−α)u/2. Substitute for the
right hand side using the rearranged version of 8.1 from above:
(1−α)δV−(1−α)δT+1V ≥ V [1 − (1−α)δ −αδT+1], get rid of the V from both sides, rearrange to
obtain 2(1−α)δ − (1−2α)δT+1 ≥ 1.