Tacit collusion under interest rate fluctuations.

Previous literature has shown that demand fluctuations affect the scope for tacit collusion. I study whether discount factor fluctuations can have similar effects. I find that collusion depends not only on the level of the discount factor but also, and more surprisingly, on its volatility. Collusive prices and profits increase with a higher discount factor level, but decrease with its volatility. These results have important implications for empirical studies of collusive pricing, the role that collusive pricing may play in economic cycles and the study of cooperation in repeated games.

1. Introduction

* It is well known that oligopolies can use the threat of future price wars to sustain prices above competitive levels if firms care enough about the future (Friedman, 1971). The extent to which firms care about the future depends primarily on the interest rate if the firms' objective is to maximize the present value of profits. The firms' discount factor may also depend on other forces, such as the probability that the product may become obsolete and the time needed for cheating to be detected. Given that the interest rate and other variables that affect the discount factor are constantly changing, it is important to study tacit collusion under discount factor fluctuations.

I characterize collusive prices and profits when the discount factor changes over time and show that collusive prices and profits increase with both present and future levels of the discount factor but decrease with its volatility. These results have important implications not only for the study of collusion but also for repeated game theory in general.

Oligopoly games are one example among many of an environment in which it is natural to assume that the discount factor changes over time. Another example would be that of a partnership, where the probability that the partnership might end varies over time. Thus, the volatility of the discount factor may be an important determinant of cooperation for many kinds of repeated games, not just oligopoly.

The environments I study and the specific results I find are as follows. I consider the case in which the discount factor, identical for all firms, is randomly and independently drawn every period. I characterize the maximum symmetric tacit collusion prices and profits that can be supported in an environment in which firms are identical and compete repeatedly on price. The three main results derived from this characterization are as follows. (1)

First and more interestingly, I show that the higher the volatility of the discount factor, the lower the collusive prices and profits that can be supported in equilibrium. The reason for this is twofold. First, given that the combination of the incentive compatibility and feasibility constraints results in a concave collusive profit function (as a function of the discount factor), an increase in volatility leads to a decrease in expected profits. Second, this decrease in expected profits reduces the size of future punishment and hence results in a decrease in equilibrium profits and prices.

Second, the higher the discount factor in a given period, the higher the collusive prices and profits that can be supported in equilibrium in that period. The intuition behind this is straightforward: the higher the discount factor, the stronger the threat of future price wars and the higher prices and profits can be without firms deviating.

Third, a shift in the distribution function toward higher discount factors would result in an increase in the profits and prices that can be supported in equilibrium for each discount factor. Again, the intuition is straightforward. From the previous result, we know that the higher the realization of the discount factor, the higher collusive prices and profits will be. Hence, a shift in the distribution function toward higher discount factors would result in an increase in the expected value of collusive profits and an increase in the threat of future punishment, allowing higher equilibrium prices and profits.

The rest of the article is organized as follows. In Section 2, I relate this article to the previous literature. In Section 3, I study the optimal tacit collusion solution and provide comparative statics results. In Section 4, I conclude.

2. Related literature

* To my knowledge, there is only one article that considers a fluctuating discount factor. This is Baye and Jansen (1996), which provides folk theorem results for repeated games with stochastic discount factors.

The well-known article by Rotemberg and Saloner (1986) offers interesting results with respect to tacit collusion that follow, as do the results in this article, from changes in the relative importance of present and future profits. In their article, however, those changes are driven by changes in demand, not the discount factor. This difference is not trivial and leads to significantly different results.

First, in this article, an increase in the discount factor always has a nonnegative effect on the equilibrium price, while in Rotemberg and Saloner, an increase in demand may result in either a decrease or an increase in price (depending on whether the incentive compatibility restriction is binding or not). In addition, the effect of an increase of demand on prices may not be robust to assuming quantity competition instead of price competition, as Rotemberg and Saloner note, or to the existence of capacity constraints, as Staiger and Wolak (1992) note. The effect of an increase in the discount factor is robust. Second, while in this article an increase in the volatility of the discount factor always results in a decrease in profits and prices, in Rotemberg and Saloner's model, an increase in the volatility of demand is again ambiguous. Third, under discount factor fluctuations, the issue of serial positive correlation of the shocks is less important than under demand fluctuations. The fact that high demand today makes it difficult to support collusion, while high demand in the future makes it easy has led a number of authors to study the consequences of demand correlation on collusion (see Kandori, 1991; Haltiwanger and Harrington, 1991; Bagwell and Staiger, 1997). In contrast, both high discount factors today and in the future facilitate collusion. Therefore, positive correlation per se does not affect the positive effect of an increase of the discount factor on collusive prices.

Finally, the literature on customer markets also relates oligopoly prices with the discount factor (see Phelps and Winter, 1970; Gottfries, 1991 ; Klemperer, 1995; Chevalier and Scharfstein, 1996). In those models, an increase in the discount factor increases the incentives to invest in new customers and results in lower prices, contrary to the results of this article.

3. Model and results

* The model is as follows. Consider a market with N identical firms with a constant marginal cost of c and facing a demand function D(p) for p [member of] R. Assume that D(p) is bounded, continuous and decreasing in p and that there exists a price [bar.p] > c such that D(p) = 0 for any p > [bar.p]. Firms compete repeatedly on price and, in each period, demand is divided equally among those firms charging the lowest price. Firms care only about profits and are risk neutral and, hence, their objective is to maximize the discounted stream of profits. The distinctive feature of this model is that the discount factor, [[delta].sub.t], which discounts earnings from t + 1 to t, is a continuous, independent and identically distributed random variable between a and b, with p.d.f, f([[delta].sub.t]) and c.d.f. F([[delta].sub.t]).

The timing of the game in a given period t is as follows: the firms observe the realization of the discount factor, [[delta].sub.t], then they choose the price for that period and finally they observe all the chosen prices, quantities and payoffs. All characteristics of the environment are common knowledge.

Given that firms cannot commit to charge a given price, or sign contracts among themselves or with third parties regarding prices, any equilibrium of the model must be a subgame perfect equilibrium of the infinitely repeated oligopoly game. I restrict my attention to equilibria in which all the firms charge the same price, p. In this symmetric case, I can write the profits of each firm as [pi](p) = (p - c)D(p)/N. Given the assumptions regarding the demand function, it is straightforward to show that [pi](p) is continuous and attains a maximum. Denote as [[pi].sup.m] the maximum (monopoly or perfect collusion) profit per firm. Assume that there is a unique price, [p.sup.m], that results in profits [[pi].sup.m] (a sufficient condition is for the demand function not to be too convex relative to its slope: [d.sup.2]D(p)/[dp.sup.2] < -[2/(p - c)](dD(p)/dp) for every p).

[] Optimal tacit collusion with a random discount factor. In this section, I characterize the optimal symmetric tacit collusion solution. First, I characterize the profits that can be supported by subgame perfect equilibria. Second, I define and prove existence and uniqueness of the optimal tacit collusion solution. Finally, I characterize this solution.

I start by calculating the expected value of profit per firm if firms agree to set prices to achieve profits of [pi]([delta]) when the discount factor is [delta]. Using the recursiveness of the problem, the present value at t of a stream of profits to a firm can be written as

V([[delta].sub.t]) = [pi]([[delta].sub.t]) + [[delta].sub.t] [[integral].sup.b.sub.a] V([[delta].sub.t + 1])f([[delta].sub.t + 1])d[[delta].sub.t + 1], (1)