# Entry, collusion, and capacity constraints.

I. 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 costs from remarks above. Correspondingly, for values of S only slightly larger than S", firm two's potential gain from collusion outweighs the profits firm one would sacrifice by not deterring. In such a scenario the entrant would like to forge a cartel with terms less attractive to her. But since such an offer would not be credible, in that entrant's best action would be to cheat on the cartel once she has entered, the rational incumbent would not agree to such terms.

The foregoing discussion suggests that one constraint on the successful implementation of a cartel is the inability of the entrant to credibly limit her period 2 output. One is then lead to ask: are there structural conditions under which the entrant can credibly limit the level of profits she would obtain as part of the cartel? For example, if production were subject to capacity constraints, as in Spence [22] or Dixit [6], the entrant's ability to expand output following entry would be limited, and she might be able to credibly limit her share of the cartel. Since this would insure that the incumbent received a proportionately larger share of the cartel, it might make the collusion strategy more attractive.

III. Capacity Constraints and Collusion

In this section, we investigate the effect of capacity constraints on the incumbent's choice between the deterrence strategy and the collusion strategy. The model we utilize is a two-period variant of the model of entry deterrence in the presence of capacity in Dixit [6]. The essential feature is cost asymmetry: as the incumbent's capacity is already in place, his marginal costs are nil for outputs at or below capacity. On the other hand, the entrant must build from scratch, and so her marginal costs explicitly include the marginal cost of an additional unit of capacity. This asymmetry affects the nature of both cooperative and non-cooperative equilibria following entry, and the entry-barring equilibrium.

In the discussion which follows, we will denote the marginal cost of an additional unit of capacity by r and firm j's capacity by kj, where again j = 1 incumbent and j = 2 for the entrant. As is common in the literature, we choose units of capacity to equal units of output. Thus, our capacity constraint may be written as [q.sub.j] [is less than or equal to] [k.sub.j].

It is straightforward to show that the asymmetric static Cournot-Nash equilibrium entails production of (a + r)/3b by the incumbent and (a - 2r)/3b by the entrant, and the Cournot-Nash profits for the entrant are equal to(10) (8) [Mathematical Expression Omitted]

As in section II, if the incumbent is to deter entry, he must select some output [Mathematical Expression Omitted] so large that the entrant's best response earns her zero profits ex ante. In this event, the incumbent earns deterrence profits [Mathematical Expression Omitted]; as in section II these profits are increasing in S within the relevant range.(11)

Also as in section II, there exists a range of sunk costs for which entry is effectively impeded and the incumbent prefers to allow entry and then collude. Because of the cost asymmetry, the combination of outputs corresponding to a collusive regime will be more complex than the corresponding collusive frontier developed in section II. In particular, the collusive frontier in this section's model will be non-linear, moreover, if one rules out side payments any post-entry cartel cannot maximize industry profits.(12)

The main result of this section is that increase in r reduce the attractiveness of the collusion strategy. Equivalently, when entry is effectively impeded, allowing entry and colluding is more likely to be adopted for smaller values of r, and correspondingly cost asymmetries.

Proposition 2. [Mathematical Expression Omitted]

The proof of this proposition is neither enlightening nor straightforward, and so we relegate it to an appendix.

At first blush, this result is somewhat surprising. By inviting entry and then colluding, the incumbent's optimal capacity level is significantly reduced, so that this capacity costs are much lower if he colludes. Based on this incentive, one would expect the likelihood of adoption to be positively related to the relative costs of capacity. There is, however, a competing incentive which the incumbent must consider. This second incentive is that the discounted flow of revenues earned by choosing deterrence will rise more rapidly than the discounted flow of collusive revenues as r increase.(13) Based on this second incentive, one would expect the likelihood of adoption to be negatively related to the relative cost of capacity. The proposition suggests that this second incentive is more important than the first: increases in the unit cost of capacity raise deterrence revenues more than they raise the incumbent's revenues as part of a cartel; this difference in added revenues is larger than the cost savings he would realize from colluding rather than deterring entry.

The above result is particularly striking when related to Spulber's Proposition 4 [23, 511]. He showed that deterring entry by holding excess capacity was unlikely unless capacity costs are low. Taken together with our results, this suggests that when holding excess capacity to deter entry is credible, it is frequently preferable to adopt the collusion strategy.

Including capacity constraints complicates the analysis, but not the essential intuition. The decision as to collude or deter entry follows a comparison of the associated profit flows. As before, there exists a value S" of sunk cost at which the incumbent is indifferent between the two strategies; collusion is preferred for smaller sunk costs. Unlike the variant in section 2, and because of the computational difficulty in deriving the incumbent's period 1 collusive output, it is not possible to characterize S" in closed form.(14) Instead, we conduct a numerical analysis to demonstrate the effect of capital costs on the desirability of the collusion strategy.

IV. Numerical Analysis

In this section we use a numerical example to demonstrate the impact of an asymmetric cost structure on the willingness of the incumbent to adopt the collusion strategy. For various values of r, one can numerically determine [Mathematical Expression Omitted], the incumbent's period 1 output associated with the collusion strategy, [Mathematical Expression Omitted], the incumbent's period 2 output under collusion, and [Mathematical Expression Omitted], the period 1 entry deterring output. Based on these numbers, a comparison of the profit flows can be made and a strategy selected. As noted above, the optimal strategy depends on the level of sunk costs, and so we seek the critical value at which the two strategies are equally attractive. Results of the simulation analysis are provided in Table 1. The illustrates the impact of changes in the entrant's marginal cost of capital on output, profit, and entry deterrence. The demand parameters used in the simulation are a = b = 1 (which are those used in Schmalansee [20]). Finally, we used a discount factor of 90% ([Delta] = .9).

The first column in Table I gives the unit cost of capacity, ranging from 0 to .20. The preentry output selected by the incumbent when he chooses the collusive strategy, [Mathematical Expression Omitted], is given in column two. The incumbent's period 2 collusive output, [Mathematical Expression Omitted], is given in column three. The entrant's collusive outputs, which are [Mathematical Expression Omitted] in period 1 and [Mathematical Expression Omitted] in period 2, are listed in columns four and five. The incumbent's discounted collusive profit flow is given in column six, and the entrant's flow is column seven. Column eight gives the value of sunk cost at which the incumbent is indifferent between collusion and deterrence. The sunk cost at which entry is just effectively impeded is listed in column nine. Finally, the ratio of column eight to column nine is reported in column ten.

Examination of Table I reveals that as the entrant's unit cost of capital increases, the second period cartel output of the dominant firm increases while at the same time the cartel output of the entrant declines. Note also that when r > 0, total output of the two firms in period 2 is less than the zero-cost monopoly output, due to the convexity of the collusive frontier. Period 1 output for both firms falls as r rises. As one would expect, collusive profits for the entrant decrease with positive changes in r. The incumbent's collusive profit flow also declines as the unit cost of capacity rises. For while larger values of r increase his advantage, allowing for increases in his discounted flow of revenues, they also increase his capacity costs. [Tabular Data I Omitted]

Note too that higher values of r reduce S". the crucial level of sunk costs at which the incumbent is indifferent between deterrence and collusion. This is the essence of our proposition: For fixed S, increases in r lower collusive profits more rapidly than they lower deterrence profits; is follows that the level of sunk costs for which deterrence profits just equal collusive profits must shrink. But increases in the unit capacity costs also facilitate deterrence; this is manifested by the decrease in S', the value of sunk costs at which entry is just effectively impeded, as r rises. It is interesting to note that S' falls more rapidly than S". The interpretation is that the range of sunk costs for which entry is effectively impeded and the collusive strategy is preferred, S' [is less than or equal to] < S", is increasing in the unit cost of capacity. That is, while increase in r render the collusion strategy less attractive in an absolute sense, such increases make it more attractive in a relative sense.

V. Concluding Remarks

This paper has investigated an alternative strategy to entry deterrence: inviting entry and then colluding with the new entrant. We showed that this strategy will be preferred to deterring entry for sufficiently small levels of sunk cost. This holds true even if the incumbent has a cost advantage because of capacity costs, although higher capacity cost reduce the attractiveness of colluding. We should point out that these results differ in an important respect from Gallini's [9] result that an incumbent may invite entry by one firm so as to deter entry by additional firms. In our analysis, the incumbent faces only one potential entrant, so that allowing entry has no impact on future entrants. Obviously, this limits its applications to observed industrial behavior; it does have the benefit that it makes the analysis considerably less opaque. Here, entry is allowed when an acceptably asymmetric cartel can be formed because relative to deterrence, this allows the incumbent to eliminate costly excess capacity without greatly reducing, and indeed perhaps increasing, his revenues.

It has been argued that the circumstances under which an incumbent might credibly deter entry by holding excess capacity may not be very general [23, 511]. In particular, holding excess capacity to deter entry may only be credible if the cost of capacity is small. To the extent that this is true, our results have added significance: when excess capacity can credibly deter entry, the collusion strategy might commonly be adopted. This suggests that the strategy of holding excess capacity to deter entry has limited theoretical support. (1)Ware argues that this should be a three stage problem because there should be a lag between the time that the entrant decides to enter and the time she first produces. Note that this approach abstracts from an important possibility. By design, the incumbent is not allowed to alter his capital in the second stage, even as the entrant is choosing her capital. It would seem that the incumbent could choose the monopoly level of output and capital prior to entry, and then expand upon entry, much as the incumbent might accelerate the pace of its research and development efforts if another firm enters the patent race as in Gilbert and Newberry [11]. By contrast, in Dixit's models [5;6], there is no time lag between the entry decision and the time of production. In the model below, we adopt Dixit's interpretation of the entry deterrence game. (2)Such behavior may be consistent with the experience in one of the most famous Sherman Act anti-trust cases. During the early part of this century, U.S. Steel continually allowed entry, and then invited the entrant to participate in the infamous "Gary Dinners," where steel prices were collusively fixed. The possibility that an incumbent may prefer to present an accommodating appearance to the entrant is considered in a more general context in Fudenberg and Tirole [8]. (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] (10)Since the incumbent's marginal costs are zero, his static reaction function is [q.sub.1] = [R.sub.1] ([q.sub.2]) = a/2b - [q.sub.2] /2. On the other hand, the entrant has marginal costs of r, so her static reaction function is [q.sub.2] = [R.sub.2] ([q.sub.1]) = (a - r)/2b - [q.sub.1] /2. Combining these results produces the values given in the text. Further this yields market price of [P.sup.N] = (a + r) /3; noting that [[Pi].sub.2]] = ([P.sup.N] - r) [q.sub.2] gives eq. (8). (11)Using the information in footnote 10,th entrant's period 1 follow profits can be calculated as (a -- r -- [bq.sub.1])[.sup.2] /4b. [Mathematical Expression Omitted] It follows that the incumbent can deter entry by producing. in period 1. For any S such that entry is not blockaged, [Mathematical Expression Omitted] exceeds the monopoly output since [Mathematical Expression Omitted] is decreasing in S, increases in S raise deterrrence profits. (12)The collusive frontier in our model corresponds to the notion of "market sharing"" in Schmalansee [20]. The collusive frontier under market sharing is a quadratic relation, with x-intercept at the low cost monopoly output (which is a / 2b in our framework) and y-intercept at the high cost monopoly output (which is [a - r]/2b in our framework). Because the incumbent's marginal costs are lower, movements along the collusive frontier which raise his output while reducing the entrant's output yield increasing in industry profits. In the limit, industry profits are maximized when the incumbent is the only producer. Of course, for the entrant to accept such an allocation, the incumbent would necessarily have to bride her to keep her output at zero. (13)On the one hand, increases in r enable the incumbent to deter entry with smaller outputs, so that his revenues rise markedly. On the other hand, while increases in r raise the discounted flow of cartel profits, the incumbent does not capture all these gains. (14)Owing to the added complexity in this section's model, calculation of the critical value of sunk cost at which the incumbent is different between deterrence and leadership, S', and the value at which he is just indifferent between deterrence and collusion, S", is complicated considerably. Following the same logic as in footnote 9, and in Figure 2, one first determines the disounted flow of profits earned by colluding, call it [[Pi].sup.c], and by leading, call in [[Pi].sup.L]. Next, subtract off period 2 monopoly profits net of capacity costs discounted back [Mathematical Expression Omitted] The value of sunk costs which makes period 1 deterrence equal [Mathematical Expression Omitted] The value of sunk costs which gives deterrence profits equal to [Mathematical Expression Omitted]. One can express thse critical values of sunk costs as [Mathematical Expression Omitted] and [Mathematical Expression Omitted] where q* is that value of [Mathematical Expression Omitted] for which the incumbent is just indifferent between leadership and deterrence, and q** is that value of [Mathematical Expression Omitted] for which he is just indifferent between collusion and deterrence. Unfortunately, closed form solutions for q* and q** are difficult to obtain when r [is to equal to] 0. Correspondingly, we rely on numerical methods in the analysis below.

References

[1]Baldani, Jeffrey and Robert Masson, "Economies of Scale, Strategic Advertising and Fully Credible Entry Deterrence." Review of Industrial Organization, Spring 1984, 190-205. [2]Bernheim, B. Douglas, "Strategic Deterrence of Sequential Entry into an Industry." Rand Journal of Economics, Spring 1984, 1-11. [3]Bucovetsky, Sam and John Chilton, "Concurrent Renting and Selling in a Durable-Goods Monopoly under Threat of Entry." Rand Journal of Economics, Summer 1996, 261-75. [4]Bulow, Jeremy, "Durable Goods Monopolists." Journal of Political Economy, June 1982, 314-32. [5]Dixit, Avinash, "A Model of Duopoly Suggesting a Theory of Entry Barriers." Bell Journal of Economics, Spring 1979, 20-32. [6]--, "The Role of Investment in Entry-Deterrence." Economic Journal, March 1980, 95-106. [7]Formby, John and W. James Smith, "Collusion, Entry, and Market Shares." Review of Industrial Organization, Spring 1984, 15-25. [8]Fudenberg, Drew and Jean Tirole, "The Fat-Cat Effect, the Puppy Dog Ploy, and the Lean and Hungry Look." American Economic Review, May 1984, 361-66. [9]Gallini, Nancy, "Determine by Market Sharing." American Economic Review, December 1984, 931-41. [10]Gaskins, Darius, "Dynamic Limit Pricing: Optimal Pricing Under Threat of Entry." Journal of Economic Theory, September 1971, 306-22. [11]Gilbert, Richard J. and David Newbery, "Pre-emptive Patenting and the Persistence of Monopoly." American Economic Review, June 1982, 514-26. [12]Harsanyi, John C. and Reinhard Selten. A General Theory of Equilibrium Selection in Games Cambridge, Mass: MIT Press, 1988. [13]Judd, Ken and Bruce Petersen, "Dynamic Limit Pricing: A Reformulation." Review of Industrial Organization, Summer 1985, 160-77. [14]Mason, Charles F., "Predation by Noisy Advertising." Review of Industrial Organization v. 3, Spring 1987, 78-93. [15]Nash, John F., "The Bargaining Problem." Econometrica, January 1950, 155-62. [16]Osborne, Martin J. and Ariel Rubinstein. Bargaining and Markets. San Diego, Cal.: Academic Press, 1990. [17]Roth, Alvin E. Axiomatic Models of Bargaining. Berlin: Springer-Verlag. 1979. [18]Schmalansee, Richard, "Entry Deterrence in the Ready-to-Eat Breakfast Cereal Industry." Bell Journal of Economics, Autumn 1978, 305-27. [19]--, "Advertising and Entry Deterrence: An Exploratory Model." Journal of Political Economy, August 1983, 636-53. [20]--, "Competitive Advantage and Collusive Optima." International Journal of Industrial Organization, December 1987, 351-68. [21]Smith, James L., "The Common Pool, Bargaining, and the Rule of Capture." Economic Inquiry, October 1987, 631-44. [22]Spence, A. Michael, "Entry, Capacity, Investment and Oligopolistic Pricing." Bell Journal of Economics, Autumn 1977, 534-44. [23]Spulber, Daniel, "Capacity, Output, and Sequential Entry." American Economic Review, June 1981, 503-14. [24]Ware, Roger, "Sunk and Costs and Strategic Commitment: A proposed Three-Stage Equilibrium." Economic Journal, June 1984, 370-78.

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 costs from remarks above. Correspondingly, for values of S only slightly larger than S", firm two's potential gain from collusion outweighs the profits firm one would sacrifice by not deterring. In such a scenario the entrant would like to forge a cartel with terms less attractive to her. But since such an offer would not be credible, in that entrant's best action would be to cheat on the cartel once she has entered, the rational incumbent would not agree to such terms.

The foregoing discussion suggests that one constraint on the successful implementation of a cartel is the inability of the entrant to credibly limit her period 2 output. One is then lead to ask: are there structural conditions under which the entrant can credibly limit the level of profits she would obtain as part of the cartel? For example, if production were subject to capacity constraints, as in Spence [22] or Dixit [6], the entrant's ability to expand output following entry would be limited, and she might be able to credibly limit her share of the cartel. Since this would insure that the incumbent received a proportionately larger share of the cartel, it might make the collusion strategy more attractive.

III. Capacity Constraints and Collusion

In this section, we investigate the effect of capacity constraints on the incumbent's choice between the deterrence strategy and the collusion strategy. The model we utilize is a two-period variant of the model of entry deterrence in the presence of capacity in Dixit [6]. The essential feature is cost asymmetry: as the incumbent's capacity is already in place, his marginal costs are nil for outputs at or below capacity. On the other hand, the entrant must build from scratch, and so her marginal costs explicitly include the marginal cost of an additional unit of capacity. This asymmetry affects the nature of both cooperative and non-cooperative equilibria following entry, and the entry-barring equilibrium.

In the discussion which follows, we will denote the marginal cost of an additional unit of capacity by r and firm j's capacity by kj, where again j = 1 incumbent and j = 2 for the entrant. As is common in the literature, we choose units of capacity to equal units of output. Thus, our capacity constraint may be written as [q.sub.j] [is less than or equal to] [k.sub.j].

It is straightforward to show that the asymmetric static Cournot-Nash equilibrium entails production of (a + r)/3b by the incumbent and (a - 2r)/3b by the entrant, and the Cournot-Nash profits for the entrant are equal to(10) (8) [Mathematical Expression Omitted]

As in section II, if the incumbent is to deter entry, he must select some output [Mathematical Expression Omitted] so large that the entrant's best response earns her zero profits ex ante. In this event, the incumbent earns deterrence profits [Mathematical Expression Omitted]; as in section II these profits are increasing in S within the relevant range.(11)

Also as in section II, there exists a range of sunk costs for which entry is effectively impeded and the incumbent prefers to allow entry and then collude. Because of the cost asymmetry, the combination of outputs corresponding to a collusive regime will be more complex than the corresponding collusive frontier developed in section II. In particular, the collusive frontier in this section's model will be non-linear, moreover, if one rules out side payments any post-entry cartel cannot maximize industry profits.(12)

The main result of this section is that increase in r reduce the attractiveness of the collusion strategy. Equivalently, when entry is effectively impeded, allowing entry and colluding is more likely to be adopted for smaller values of r, and correspondingly cost asymmetries.

Proposition 2. [Mathematical Expression Omitted]

The proof of this proposition is neither enlightening nor straightforward, and so we relegate it to an appendix.

At first blush, this result is somewhat surprising. By inviting entry and then colluding, the incumbent's optimal capacity level is significantly reduced, so that this capacity costs are much lower if he colludes. Based on this incentive, one would expect the likelihood of adoption to be positively related to the relative costs of capacity. There is, however, a competing incentive which the incumbent must consider. This second incentive is that the discounted flow of revenues earned by choosing deterrence will rise more rapidly than the discounted flow of collusive revenues as r increase.(13) Based on this second incentive, one would expect the likelihood of adoption to be negatively related to the relative cost of capacity. The proposition suggests that this second incentive is more important than the first: increases in the unit cost of capacity raise deterrence revenues more than they raise the incumbent's revenues as part of a cartel; this difference in added revenues is larger than the cost savings he would realize from colluding rather than deterring entry.

The above result is particularly striking when related to Spulber's Proposition 4 [23, 511]. He showed that deterring entry by holding excess capacity was unlikely unless capacity costs are low. Taken together with our results, this suggests that when holding excess capacity to deter entry is credible, it is frequently preferable to adopt the collusion strategy.

Including capacity constraints complicates the analysis, but not the essential intuition. The decision as to collude or deter entry follows a comparison of the associated profit flows. As before, there exists a value S" of sunk cost at which the incumbent is indifferent between the two strategies; collusion is preferred for smaller sunk costs. Unlike the variant in section 2, and because of the computational difficulty in deriving the incumbent's period 1 collusive output, it is not possible to characterize S" in closed form.(14) Instead, we conduct a numerical analysis to demonstrate the effect of capital costs on the desirability of the collusion strategy.

IV. Numerical Analysis

In this section we use a numerical example to demonstrate the impact of an asymmetric cost structure on the willingness of the incumbent to adopt the collusion strategy. For various values of r, one can numerically determine [Mathematical Expression Omitted], the incumbent's period 1 output associated with the collusion strategy, [Mathematical Expression Omitted], the incumbent's period 2 output under collusion, and [Mathematical Expression Omitted], the period 1 entry deterring output. Based on these numbers, a comparison of the profit flows can be made and a strategy selected. As noted above, the optimal strategy depends on the level of sunk costs, and so we seek the critical value at which the two strategies are equally attractive. Results of the simulation analysis are provided in Table 1. The illustrates the impact of changes in the entrant's marginal cost of capital on output, profit, and entry deterrence. The demand parameters used in the simulation are a = b = 1 (which are those used in Schmalansee [20]). Finally, we used a discount factor of 90% ([Delta] = .9).

The first column in Table I gives the unit cost of capacity, ranging from 0 to .20. The preentry output selected by the incumbent when he chooses the collusive strategy, [Mathematical Expression Omitted], is given in column two. The incumbent's period 2 collusive output, [Mathematical Expression Omitted], is given in column three. The entrant's collusive outputs, which are [Mathematical Expression Omitted] in period 1 and [Mathematical Expression Omitted] in period 2, are listed in columns four and five. The incumbent's discounted collusive profit flow is given in column six, and the entrant's flow is column seven. Column eight gives the value of sunk cost at which the incumbent is indifferent between collusion and deterrence. The sunk cost at which entry is just effectively impeded is listed in column nine. Finally, the ratio of column eight to column nine is reported in column ten.

Examination of Table I reveals that as the entrant's unit cost of capital increases, the second period cartel output of the dominant firm increases while at the same time the cartel output of the entrant declines. Note also that when r > 0, total output of the two firms in period 2 is less than the zero-cost monopoly output, due to the convexity of the collusive frontier. Period 1 output for both firms falls as r rises. As one would expect, collusive profits for the entrant decrease with positive changes in r. The incumbent's collusive profit flow also declines as the unit cost of capacity rises. For while larger values of r increase his advantage, allowing for increases in his discounted flow of revenues, they also increase his capacity costs. [Tabular Data I Omitted]

Note too that higher values of r reduce S". the crucial level of sunk costs at which the incumbent is indifferent between deterrence and collusion. This is the essence of our proposition: For fixed S, increases in r lower collusive profits more rapidly than they lower deterrence profits; is follows that the level of sunk costs for which deterrence profits just equal collusive profits must shrink. But increases in the unit capacity costs also facilitate deterrence; this is manifested by the decrease in S', the value of sunk costs at which entry is just effectively impeded, as r rises. It is interesting to note that S' falls more rapidly than S". The interpretation is that the range of sunk costs for which entry is effectively impeded and the collusive strategy is preferred, S' [is less than or equal to] < S", is increasing in the unit cost of capacity. That is, while increase in r render the collusion strategy less attractive in an absolute sense, such increases make it more attractive in a relative sense.

V. Concluding Remarks

This paper has investigated an alternative strategy to entry deterrence: inviting entry and then colluding with the new entrant. We showed that this strategy will be preferred to deterring entry for sufficiently small levels of sunk cost. This holds true even if the incumbent has a cost advantage because of capacity costs, although higher capacity cost reduce the attractiveness of colluding. We should point out that these results differ in an important respect from Gallini's [9] result that an incumbent may invite entry by one firm so as to deter entry by additional firms. In our analysis, the incumbent faces only one potential entrant, so that allowing entry has no impact on future entrants. Obviously, this limits its applications to observed industrial behavior; it does have the benefit that it makes the analysis considerably less opaque. Here, entry is allowed when an acceptably asymmetric cartel can be formed because relative to deterrence, this allows the incumbent to eliminate costly excess capacity without greatly reducing, and indeed perhaps increasing, his revenues.

It has been argued that the circumstances under which an incumbent might credibly deter entry by holding excess capacity may not be very general [23, 511]. In particular, holding excess capacity to deter entry may only be credible if the cost of capacity is small. To the extent that this is true, our results have added significance: when excess capacity can credibly deter entry, the collusion strategy might commonly be adopted. This suggests that the strategy of holding excess capacity to deter entry has limited theoretical support. (1)Ware argues that this should be a three stage problem because there should be a lag between the time that the entrant decides to enter and the time she first produces. Note that this approach abstracts from an important possibility. By design, the incumbent is not allowed to alter his capital in the second stage, even as the entrant is choosing her capital. It would seem that the incumbent could choose the monopoly level of output and capital prior to entry, and then expand upon entry, much as the incumbent might accelerate the pace of its research and development efforts if another firm enters the patent race as in Gilbert and Newberry [11]. By contrast, in Dixit's models [5;6], there is no time lag between the entry decision and the time of production. In the model below, we adopt Dixit's interpretation of the entry deterrence game. (2)Such behavior may be consistent with the experience in one of the most famous Sherman Act anti-trust cases. During the early part of this century, U.S. Steel continually allowed entry, and then invited the entrant to participate in the infamous "Gary Dinners," where steel prices were collusively fixed. The possibility that an incumbent may prefer to present an accommodating appearance to the entrant is considered in a more general context in Fudenberg and Tirole [8]. (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] (10)Since the incumbent's marginal costs are zero, his static reaction function is [q.sub.1] = [R.sub.1] ([q.sub.2]) = a/2b - [q.sub.2] /2. On the other hand, the entrant has marginal costs of r, so her static reaction function is [q.sub.2] = [R.sub.2] ([q.sub.1]) = (a - r)/2b - [q.sub.1] /2. Combining these results produces the values given in the text. Further this yields market price of [P.sup.N] = (a + r) /3; noting that [[Pi].sub.2]] = ([P.sup.N] - r) [q.sub.2] gives eq. (8). (11)Using the information in footnote 10,th entrant's period 1 follow profits can be calculated as (a -- r -- [bq.sub.1])[.sup.2] /4b. [Mathematical Expression Omitted] It follows that the incumbent can deter entry by producing. in period 1. For any S such that entry is not blockaged, [Mathematical Expression Omitted] exceeds the monopoly output since [Mathematical Expression Omitted] is decreasing in S, increases in S raise deterrrence profits. (12)The collusive frontier in our model corresponds to the notion of "market sharing"" in Schmalansee [20]. The collusive frontier under market sharing is a quadratic relation, with x-intercept at the low cost monopoly output (which is a / 2b in our framework) and y-intercept at the high cost monopoly output (which is [a - r]/2b in our framework). Because the incumbent's marginal costs are lower, movements along the collusive frontier which raise his output while reducing the entrant's output yield increasing in industry profits. In the limit, industry profits are maximized when the incumbent is the only producer. Of course, for the entrant to accept such an allocation, the incumbent would necessarily have to bride her to keep her output at zero. (13)On the one hand, increases in r enable the incumbent to deter entry with smaller outputs, so that his revenues rise markedly. On the other hand, while increases in r raise the discounted flow of cartel profits, the incumbent does not capture all these gains. (14)Owing to the added complexity in this section's model, calculation of the critical value of sunk cost at which the incumbent is different between deterrence and leadership, S', and the value at which he is just indifferent between deterrence and collusion, S", is complicated considerably. Following the same logic as in footnote 9, and in Figure 2, one first determines the disounted flow of profits earned by colluding, call it [[Pi].sup.c], and by leading, call in [[Pi].sup.L]. Next, subtract off period 2 monopoly profits net of capacity costs discounted back [Mathematical Expression Omitted] The value of sunk costs which makes period 1 deterrence equal [Mathematical Expression Omitted] The value of sunk costs which gives deterrence profits equal to [Mathematical Expression Omitted]. One can express thse critical values of sunk costs as [Mathematical Expression Omitted] and [Mathematical Expression Omitted] where q* is that value of [Mathematical Expression Omitted] for which the incumbent is just indifferent between leadership and deterrence, and q** is that value of [Mathematical Expression Omitted] for which he is just indifferent between collusion and deterrence. Unfortunately, closed form solutions for q* and q** are difficult to obtain when r [is to equal to] 0. Correspondingly, we rely on numerical methods in the analysis below.

References

[1]Baldani, Jeffrey and Robert Masson, "Economies of Scale, Strategic Advertising and Fully Credible Entry Deterrence." Review of Industrial Organization, Spring 1984, 190-205. [2]Bernheim, B. Douglas, "Strategic Deterrence of Sequential Entry into an Industry." Rand Journal of Economics, Spring 1984, 1-11. [3]Bucovetsky, Sam and John Chilton, "Concurrent Renting and Selling in a Durable-Goods Monopoly under Threat of Entry." Rand Journal of Economics, Summer 1996, 261-75. [4]Bulow, Jeremy, "Durable Goods Monopolists." Journal of Political Economy, June 1982, 314-32. [5]Dixit, Avinash, "A Model of Duopoly Suggesting a Theory of Entry Barriers." Bell Journal of Economics, Spring 1979, 20-32. [6]--, "The Role of Investment in Entry-Deterrence." Economic Journal, March 1980, 95-106. [7]Formby, John and W. James Smith, "Collusion, Entry, and Market Shares." Review of Industrial Organization, Spring 1984, 15-25. [8]Fudenberg, Drew and Jean Tirole, "The Fat-Cat Effect, the Puppy Dog Ploy, and the Lean and Hungry Look." American Economic Review, May 1984, 361-66. [9]Gallini, Nancy, "Determine by Market Sharing." American Economic Review, December 1984, 931-41. [10]Gaskins, Darius, "Dynamic Limit Pricing: Optimal Pricing Under Threat of Entry." Journal of Economic Theory, September 1971, 306-22. [11]Gilbert, Richard J. and David Newbery, "Pre-emptive Patenting and the Persistence of Monopoly." American Economic Review, June 1982, 514-26. [12]Harsanyi, John C. and Reinhard Selten. A General Theory of Equilibrium Selection in Games Cambridge, Mass: MIT Press, 1988. [13]Judd, Ken and Bruce Petersen, "Dynamic Limit Pricing: A Reformulation." Review of Industrial Organization, Summer 1985, 160-77. [14]Mason, Charles F., "Predation by Noisy Advertising." Review of Industrial Organization v. 3, Spring 1987, 78-93. [15]Nash, John F., "The Bargaining Problem." Econometrica, January 1950, 155-62. [16]Osborne, Martin J. and Ariel Rubinstein. Bargaining and Markets. San Diego, Cal.: Academic Press, 1990. [17]Roth, Alvin E. Axiomatic Models of Bargaining. Berlin: Springer-Verlag. 1979. [18]Schmalansee, Richard, "Entry Deterrence in the Ready-to-Eat Breakfast Cereal Industry." Bell Journal of Economics, Autumn 1978, 305-27. [19]--, "Advertising and Entry Deterrence: An Exploratory Model." Journal of Political Economy, August 1983, 636-53. [20]--, "Competitive Advantage and Collusive Optima." International Journal of Industrial Organization, December 1987, 351-68. [21]Smith, James L., "The Common Pool, Bargaining, and the Rule of Capture." Economic Inquiry, October 1987, 631-44. [22]Spence, A. Michael, "Entry, Capacity, Investment and Oligopolistic Pricing." Bell Journal of Economics, Autumn 1977, 534-44. [23]Spulber, Daniel, "Capacity, Output, and Sequential Entry." American Economic Review, June 1981, 503-14. [24]Ware, Roger, "Sunk and Costs and Strategic Commitment: A proposed Three-Stage Equilibrium." Economic Journal, June 1984, 370-78.

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Author: | Nowell, Clifford |
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Publication: | Southern Economic Journal |

Date: | Apr 1, 1992 |

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