Entry, collusion, and capacity constraints.

• Author: Mason, Charles F.

1. Introduction

   It is arguable true that no single topic has held more of a fascination for students of Industrial Organization than the issue of entry deterrence. This issue has been analyzed form many perspectives, including: the role of advertising and imperfect information in deterring entry [1; 14; 18; 19]; incentives confronting a monopoly seller of durable goods [3; 4]; incentives in the context of sequential entry [2; 9]; dynamic aspects of entry deterrence [10; 13]; and the role of excess capacity in deterring entry [22; 23].

   Dixit [5; 6] discussed the incentives for an incumbent firm to deter entry in the presence of sunk costs. In his models, a firm must sink some capital prior to entering the industry, and this sunk cost creates an entry barrier. Any potential entrant must pay the sunk costs prior to entering, and the incumbent can deter entry by reducing the stream of profits sufficient so that the entrant's post entry profit flow cannot cover its sunk cost. Ware [24] adapted Dixit's work by allowing for a three-stage game: following the incumbent's decision, the entrant decides upon entry in two, paying the sunk costs at that time; in stage three (if entry occurred the firms play a quantity choosing game.(1)

   Formby and Smith [7] argue that rather than deterring entry, the incumbent firm often has an incentive to allow entry and then collude with the entrant. In this context, the post-entry cartel is asymmetric ,with the incumbent producing the larger amount and realizing the higher profits.(2) (3)Such instability may have characterized the repeated, and unsuccessful, attempts to limit oil production in the 1920s and 1930s. The salient features of these attempted cartels from our perspective were asymmetry of outputs and profit shares, and the observation that smaller firms repeatedly broke ranks when their shares were too small. (4)As the entrant's reaction curve is [q.sub.2] = [R.sub.2]([q.sub.1] = a/2b - [q.sub.1]/2, if the incumbent produced [q.sub.d] in period 1 the entrant would respond by producing a/2b - [q.sub.d]/2. Given these choices, the entrant's period 1 profits would be [(a - [bq.sub.d]).sup.2]/4b and her discounted flor of profits would be [Mathematical Expression Omitted] = [(a - [bq.sub.d]).sup.2]/4b + [[Sigma] a.sup.2]/9b. This implies that [q.sub.d] = a/b - 2[bS - [[[Sigma] a.sup.2]/9].sup.1/2]/b will successfully deter entry. We note for future reference that this gives firm one's period 1 deterrence profits as [Mathematical Expression Omitted]. (5)This presumes that following disagreement, the market reverts to the Cournot/Nash equilibrium. Using the noncooperative Nash equilibrium as the "disagreement point" is common in the bargaining literature [17; 16]. Simith [21] takes a similar approach to modeling the interaction of oil producers extracting from a common pool. (6)This is not simply a hypothetical practice. For example, Wyoming coal producers can, and often do, opt to have their rivals fill part of their deliveries to utilities. An alternative assumption, which would yield equivalent results, is to allow side payments. (7)This is the one aspect of our model which is not robust with respect to the length of the game. If the industry were infinitely-lived, the incumbent would have to commit to [q.sub.d] in each period (so as to always deter entry), and hence he would perennially earn [Mathematical Expression Omitted]. This complication would not alter the fundamental insight of our analysis. (8)Thus, we envision the incumbent offering the entrant a "take-it-or-leave-it" package, with non-cooperative behaviour occurring if the incumbent's offer is declined. This part of our model is included for concreteness, though it can be rationalized as a subgame-perfect equilibrium in a bargaining game of repeated offers if there are very high costs of delaying agreement or if the time horizon is quite short [16]. Alternative constructs, where the collusive pie is shared in accordance with Nash's [15] bargaining model or the risk-dominance concept discussed in Chapter 6 of Harsanyi and Selten [12], could be included in our model. Again this would not radically alter our results, although it would greatly complicate the analysis. (9)The critical value of sunk cost at which the incumbent is just indifferent between deterrence and leadership. S', can be calculated by equating the two profit flows: [Mathematical Expression Omitted]. It is straight forward to verify that period 1 leader profits are [a.sup.2]/8b, Cournot/Nash profits are [a.sup.2]/9b, and monopoly profits are [a.sup.2]/4b; deterrence profits are given in footnote 4. Combining these observation allow us to derive the critical level of sunk costs as [Mathematical Expression Omitted] where x' = a[1 - [(1/2 + 5 [Sigma] /9).sup.1/2]]/4. The value of sunk cost at which the incumbent is just indifferent between collusion and deterrence, S", can be calculated by equating the profit flows: [Mathematical Expression Omitted]. It can be shown that firm one's optimal pre-entry output is [Mathematical Expression Omitted], which gives [Mathematical Expression Omitted]. Then noting that [Mathematical Expression Omitted] and [Mathematical Expression Omitted], we infer that [Mathematical Expression Omitted] when sunk costs equal S". Then using the characterizations in footnote 4, wee obtain: S" = [x.sub."2]/b + [[Sigma] a.sup.2]/9b. where x" = [a[1 - [(1/5 + 4 [Sigma] /9).sup.1/2]].sup.1/2] costs he would prefer deterrence, while with lower sunk costs he would prefer collusion. These observations may be summarized in:

   Proposition 1. There is a range of sunk costs for which entry is effectively impeded and the incumbent prefers to allow entry and then collude.

   It is interesting to note that the entrant is strictly better off when the incumbent chooses the collusion strategy. This follows since her cartel profits equal [Mathematical Expression Omitted], which exceed sunk...