Collusion under monitoring of sales.

Collusion under imperfect monitoring is explored when firms' prices are private information and their quantities are public information; such an information structure is consistent with several recent price-fixing cartels, such as those in lysine and vitamins. For a class of symmetric oligopoly games, it is shown that symmetric equilibrium punishments cannot sustain any collusion. An asymmetric punishment is characterized that does sustain collusion and it has firms whose sales exceed their quotas compensating those firms with sales below their quotas. In practice, cartels could have performed such transfers through sales among the cartel members.

... if I'm assured that I'm gonna get 67,000 tons [of lysine sales] by the year's end, we're gonna sell it at the prices we agreed to and l frankly don't care what you sell it for.

Terrance Wilson of Archer Daniels Midland from the March 10, 1994, meeting of the lysine cartel.

And that total for us for the year, calendar year is 68,000; 68,334, 68,334 and our target was 67,000 plus alpha. Almost on target.

Mark Whitacre of Archer Daniels Midland from the January 18, 1995, meeting of the lysine cartel. (1)

1. Introduction

* Many, if not most, price-fixing cartels involve firms selling to industrial buyers, with the lysine cartel being a notable example. As price can be settled through private negotiation, it is not typically observable. In such cases, compliance with the collusive agreement is often based on firms' sales. Indeed, cartels can go to great lengths to ensure that sales are public information among the cartel members. In the citric acid cartel, for example, firms hired an international accounting firm to independently audit sales reports (Connor, 2001). The objective of this article is to explore collusion in an imperfect monitoring setting in which prices are private information and firms' quantities are public information.

In spite of such a monitoring environment being applicable to many market settings, there is relatively little work with such a structure even though, interestingly enough, it was the one described by Stigler (1964) when he originally raised the issue of imperfect monitoring. There are, of course, many articles using the classical monitoring setting of Green and Porter (1984), in which firms' quantities are private information and the market price is publicly observed. In the context of repeated auctions, Blume and Heidhues (2003) and Skrzypacz and Hopenhayn (2004) assume price is private information, while who won the auction is known. However, the assumption of one unit per period makes the model unsuitable for many markets and, pertinent to the issue at hand, constrains the monitoring of collusion through sales (a point we elaborate on later). Tirole (1988), Bagwell and Wolinsky (2002) and Campbell, Ray, and Muhanna (2005) allow for multiunit demand in the context of the infinitely repeated Bertrand price model with imperfect monitoring. The assumption that firm demand is discontinuous is obviously extreme and, furthermore, plays an important role in sustaining collusion. Our model is the first to consider collusion when prices are private information and monitoring occurs with respect to sales, while making standard and fairly general demand assumptions: demand is multiunit and expected firm demand is everywhere continuous.

Our first main finding is a surprising impossibility result. For a general class of symmetric demand structures with inelastic market demand, no collusion can be sustained by symmetric punishments. By way of example, one such demand structure is when the probability distribution of demand depends only on the difference in firms' prices, as is true with the discrete-choice model. The rough intuition for our result can be conveyed as follows for the duopoly case. To begin, one would expect punishment to occur when market shares are sufficiently skewed. Suppose, for example, punishment occurs when the market share of one of the firms exceeds 70%. A firm that considers charging a price below the collusive price raises the probability that its market share exceeds 70%--which makes punishment more likely--but lowers the probability that the other firm's market share exceeds 70%--which makes punishment less likely. What we show is that, for small price cuts, these two effects exactly offset each other, which implies that a firm's continuation payoff is unaffected by its price. Therefore, an equilibrium price for the infinite horizon game must be the same as that for the stage game. Though shown for the extreme case of fixed market demand, robustness prevails when market demand is stochastic and sensitive to firms' prices. Specifically, if market demand is very insensitive to firms' prices, then collusive prices are very close to noncollusive prices.

The conclusion we draw from this result is not that firms cannot collude but rather of the importance of treating apparent deviators differently from apparent nondeviators. The second main result is showing that collusion can be sustained with asymmetric punishments involving transfers in which firms that sold too much compensate those who sold too little. In fact, some price-fixing cartels, such as those in citric acid (Arbault et al., 2002) and sodium gluconate (European Commission, 2002), did indeed deploy asymmetric punishments through the use of interfirm sales, which can act as transfers. The main message of this article is that, if we are to understand the actual practices of some cartels, it is essential that we take account of imperfect monitoring with respect to prices and the role of asymmetric punishments that condition on sales.

After the model is described in Section 2, the inability of symmetric punishments to sustain collusion is established in Section 3. Some robustness issues are explored in Section, 4, while a characterization of asymmetric equilibria that sustain collusion is provided in Section 5. We relate these results to the literature in Section 6 and briefly conclude in Section 7.

2. Model

* Consider an infinitely repeated game in which n [greater than or equal to] 2 firms make simultaneous price decisions. Cost functions are common and linear and, without loss of generality, cost is zero. Demand is fixed at m discrete units. (2) We often refer to there being m customers (with unit demands). Though total demand is fixed, firm demand is stochastic. Letting [q.sub.i] denote the quantity of firm i, the set of feasible quantity vectors is

[DELTA][equivalent to]{(q.sub.1],...,[q.sub.n])[member of][{0, 1,..., m}.sup.n]:[n.summation over (i=1][q.sub.i]=m).

Define [psi]([q.bar]; [p.bar]) [DELTA] x [R.sup.n] [right arrow] [0, 1] as the probability of realizing quantity vector [q.bar] given price vector [p.bar]. As regards the stochastic nature of firm demand, one can imagine that products are differentiated or that they are homogeneous but buyer-specific shocks, which may be independent or correlated, that influence demand in each period. We describe some examples below.

We make three assumptions on the probability distribution on firm demand.

Assumption 1. [psi] is continuously differentiable with respect to [p.sub.i] for all i.

Assumption 2. [psi]([q.bar]; [p.bar]) = [psi]([omega]([q.bar]; i, j); [omega]([p.bar]; i, j)), for all i, j and all ([q.bar]; [p.bar]), where [omega]([q.bar]; i, j) is the vector [q.bar] when elements i and j are exchanged.

Assumption 3. [[summation].sup.n.sub.i=1]([partial derivative][psi]([q.bar];(p,...,p))/ [partial derivative] [p.sub.i]0, for all ([q.bar], p).

Assumption 1 is standard and Assumption 2 imposes symmetry in that permuting the price vector permutes the probability function. Assumption 3 is the key restriction, though it is satisfied in many models. Assumption 3 implies, that if we start at equal prices, then the distribution of demand remains unchanged if firms make small, identical price changes.

When n = 2, Assumption 3 holds if the demand distribution depends solely on the difference in prices; in that case, equal changes in price do not affect the difference. For general n, a sufficient condition for Assumption 3 to be true is that [psi] depends only the price differences for all pairs of firms. To show this explicitly, consider n = 3 and suppose [there exists][xi]:[DELTA] x [R.sup.3] [right arrow] [0, 1] such that

[psi]([q.bar];[p.bar]) = [xi]([q.bar];[[DELTA].sub.12], [[DELTA].sub.23]), [for all] ([q.bar]; [p.bar])[member of][DELTA] x [R.sup.3],

where [[DELTA].sub.i,j] [equivalent to][p.sub.i] - [p.sub.j]. Hence, the probability function depends only on the pairwise differences in firms' price. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

so that Assumption 3 holds.

An example from the literature that conforms to our demand specification is the following m-buyer generalization of the duopoly model of Cabral and Riordan (1994). Let the probability that firm 1 sells to a particular buyer equal F([p.sub.2] - [p.sub.1]), where F : R [right arrow] [0, 1] is continuously differentiable and nondecreasing and F' is symmetric around zero. Assume also that buyers' decisions regarding from whom to buy are i.i.d. That implies that a firm's demand is binomially distributed,

[psi](b, m-b;[p.sub.1], [p.sub.2])=m!/b!(m-b)! F[[p.sub.2]-[p.sub.1].sup.b][(1-F([p.sub.2]-[po.sub.1])).sup.m-b],

so only the price difference matters.