Review ofIndustrialOrganization 11: 551-561,1996. @ 1996 Kluwer Academic Publishers. Printed in the Netherlands.

551

Optimal Cartel Trigger Strategies and the Number of Firms

HUGH BRIGGS, III Department of Economics, Miami University, Oxford, Ohio 45056, U.S.A.

Abstract, Recent empirical papers have analyzed collusion in the Joint Executive Committee in an attempt to determine which of several theories of cartel behavior is supported by the behavior of this 19th century railroad cartel. Non-parametric tests of whether high and low profit regimes followed a first-order Markov process when one controls for the number of firms support the theory of optimal collusion given by Abreu, Pearce, and Stachetti. Results on whether transition probabilities depend on the number of firms are inconclusive.

Key words: Cartels, collusion, oligopoly theory, Markov processes,

Modern game-theoretic theories of non-cooperative collusion yield different pre- dictions about the causes of “price wars.” For example, Rotemberg and Saloner (1986) show that if a cartel observes demand shocks as they occur, then it will have to change the collusive price (or quantity) countercyclically. Other authors suppose demand shocks are unobservable, and for that reason, changes in the cartel’s sales of a homogeneous product that are due to demand shocks cannot be disentangled from cheating on the cartel. For example, Green and Porter (1984) and Porter (1983a) presume that fhms choose quantities collusively, but can neither observe each other’s actual sales nor perfectly infer industry sales from the market price which, given a homogeneous product, is determined by a random draw from an identical distribution conditional on the industry quantity. To deter defections from a low quantity, high profit regime, the industry punishes itself by switching to a regime of Cournot-Nash quantities and their associated lower profits if the market price falls below some trigger level which would provide an imperfect signal that a fhm or fhms deviated from the prescribed low quantities. Under less restrictive assumptions that those employed by Green and Porter, Abreu, Pearce, and Stachetti (1986) show that the optimal collusive scheme till have the property that the high and low profit regimes will follow a first-order Markov process.

The present paper extends the investigation undertaken by Berry and Briggs (1988), who examine whether the regimes of the EC follow a first-order Markov process. The method they employ is used to examine whether the results they obtain survive when one controls for the number of firms in the cartel. Since traditional wisdom suggests the prospects for successful collusion decrease with the number of firms and narrative accounts of the JEC’s behavior suggest that

552 HUGH BRIGGS, III

the cartel was less successful after entry by new firms, the conclusion that the JEC’s regimes followed a first-order Markov process is strengthened by the finding reported below that it continues to hold when one controls for the number of firms. Whether transition probabilities from between high and low profit regimes depends on the number of f?rms turns out to be sensitive to the classification of regimes. The paper is organized as follows. Section I briefly reviews the operations of the JEC and previous empirical literature. Section II outlines the non-parametric test proposed by Berry and Briggs and applies a similar test while controlling for the number of f?rms. Section III concludes.

I. Non-Cooperative Collusion with Imperfect Monitoring and The JEC

The Joint Executive Committee was formed in 1879 as a formal cartel to control railroad rates on various types of traffic after the failure of the trunk-line railroads to maintain collusive rates in the late 1870’s.’ The original three members of the JEC were the New York Central railroad, the Penn, and the Baltimore and Ohio. Here, as in most of the previous empirical literature, the focus is on these firms’ control of grain rates for grain bound for an eastern seaport, and ultimately, export.

The primary means by which collusive rates were to be maintained by the cartel was through the pooling of freight. Each member of the cartel was allocated a market share, and plans were made for roads that ran deficits below their allocated shares to receive transfers of freight from roads that ran surpluses.2 In theory, the pooling of freight would of course remove the incentive for firms to undercut the collusive rate set by the cartel, since the increase in market share realized by a cut would have to be compensated for later. In practice, transfers of freight were sporadic, as members of the cartel squabbled over the appropriate market shares and the appropriate size of price differentials on rates to ports of different distance from Chicago.

A second problem faced by the cartel was entry. Here, one should be careful to distinguish among entrants who joined the cartel and entrants who competed with the cartel for traffic. In 1880 a Canadian road, the Grand Trunk railroad, completed an extension to Chicago, joined the cartel, and was allocated a share of the eastbound traffic in grain. In 1883, the Erie entered the Chicago all-rail market through an agreement with the Chicago and Atlantic Railway, which linked up with the Erie’s previous western terminus at Marion, Ohio; it was also granted a share of the eastbound traffic.

Throughout the operation of the cartel, transportation services by steamship and sail through the Great Lakes to ports along Lake Erie, and from there by barge

’ Excellent historical accounts of the trunk-line railroad cartels are given by MacAvoy (1965) and Ulen (1979), which the brief summary here relies on.

’ Ulen (1979, p. 192) notes that firms made “good faith depositis” towards a money pool for transfers of revenues to deficit lines in 1882, but apparently never subsequently continued to pool revenues.

OPTIMAL CARTEL TRIGGER STRATEGIES AND THE NUMBER OF FIRMS 553

or rail to a seaport were an alternative to all-rail transport by the JEC. The firms involved in these operations were never allocated market shares by the cartel, and may have competed with the firms in the JEC.3 Additionally, other lines were developed between St. Louis and the east coast in the 1880’s, and additional lines were developed between Buffalo and New York city. Changes in the share of the market served by non-cartel members should change the collusive price, but would seem to be less likely to affect the stability of the cartel than change in the number of members in the cartel.

Given that the JEC pooled the market and fixed prices, the theories of quantity- setting collusion referred to in the introduction would not seem to be directly applicable. However, a variety of authors have used switching regression models and a variety of explanatory variables to estimate price setting equations that indicate that the prices set by the cartel come from different behavioral regimes, as in the theory models of quantity-setting collusion.4 In the most recent such model, Ellison (1994) also shows that the probability of entering a price war regime from a high profit regime is decreasing in the unobserved component of a serially correlated demand shock, a second prediction of the models of quantity setting collusion.5 The uncertainty faced by the firms in the pool was not only over whether rivals were shading the collusive price, but also presumably over whether deficits from allocated shares would be made up. Thus periods of unanticipated low demand may have appeared to each firm to be either the result of cheating by a single firm or may have made the transfers of freight required to make up the deficits seem less likely to occur.

II. Testing Regimes for a First-Order Markov Process

Define the binary variable 1, such that 1t = 1 if the cartel is in the high profit state in period t and 1t = 0 if the cartel is in the low profit state at time t. The theory of optimal collusion under imperfect information of Abreu, Pearce, and Stachetti (1986) implies that the regime series {It}gl will follow a first-order Markov process. That is, the probability that the cartel is in the high profit state at time t depends only on the state of the cartel at time t - 1.

Berry and Briggs (1988) show how to test the null hypothesis that a series of binary variables follows a K-th order Markov process against an alternative hypothesis of a Markov process of any order h4 > K. A brief description follows. Let {lt}~z=o be a binary series that represents the state of the cartel at time t, with It = 1 if the cartel is in the high profit state in period t and 1, = 0 if the cartel is in the low profit state in period t. If {It}T=c follows an M-th order

3 Ulen (1979, pp. 231-2) notes that some of the steamship lines were at times owned by the railroads. Over time, the trend was for all-rail transport to capture a larger share of the market.

4 See for example, Porter (1983b), Lee and Porter (1984), Coslett and Lee (1985), Hajivassiliou (1989X and Ellison (1994).

5 He also finds that larger the deviation from its allocated share recieved by the firm with the largest deviation from its allocated share, the more likely is the start of a price war.

554 HUGHBRlGGS,III

Markov process, then the probability that It = 1 depends on the M periods of history immediately predating period t; let H,v denote the i-th A4 period history, i = l,..., 2”. Similarly, if {It}~=c follows a K-th order Markov process, then the probability that 1, = 1 depends on only the K period history immediately preceding period t; let HK denote the i-th K period history, i = 1, . . .2K. Let P(It = 1 /HM) be the probability that It = 1 given the M period history HM. Under the null hypothesis that I follows a K-th order stationary Markov process, P(It = l/H%M) = P(It = l/H,M) for all M period histories H$F and HJv that include the same K period histories HK . Thus one can test the null hypothesis that { It}~=c follows a K-th order Markov process by testing for the equality of means across A4 period histories that contain the same K period histories.

To implement the test given a sample of observations on a indicator variable I, {lt}Tz=l, partition the sample into the sets SF, i = 1, . . ., 2”, such that It E Sty if the M period history prior to It is given by HM. Now let p; = CltEs+4 It/N; be the sample proportion of It = 1, given 1t E S%w, where N; is the number of observations in Sty; if there are observations on It for each A4 period history, then there will be 2M such sample proportions. Further, observe that since each It E SF is an independent draw from an identical binomial distribution, the sample means ,~t; are consistent estimators of the population means &, o; = pu;( 1 - ,~i) is a consistent estimator of the population variance 00, and a[(~; - pp)/m converges in distribution to a standard normal distribution. Under the null hypothesis that {I} follows a K-th order Markov process there will be 2” different groups of means (with 2”-K means in each group) which have the same K period history and therefore the same mean. Therefore, under the null hypothesis, within each of the 2” different groups of means we can impose 2”-K - 1 restrictions that means are equal. Let ~1 be the 2M by 1 vector of sample means and let R be a 2” (2”-K - 1) by 2M matrix of full row rank that imposes the restrictions that sample proportions with the same K period histories have the same means; that is, let Rp” = 0, where p” is the vector of population means. Then under the null hypothesis that {l~}~zo follows a K-th order Markov process, Rp is distributed normally with mean 0 and variance RVR*, where V = diag{wl/Nt , D~/Nz, . . ., UZM/N~M} is the variance matrix of p. Therefore, under the null hypothesis, ( RP)~( RVR’)-’ (Rp) follows a x2 distribution with parameter given by the number of restrictions imposed by the null hypothesis, 2K (2”-K - 1). One can choose a critical value from this chi-square distribution to reject the null hypothesis that {I} follows a K-th order Markov process.

It is assumed that the Markov process is stationary for all 1, observed while the cartel had the same number of firms, but that transition probabilities differ when there are different numbers of firms in the cartel. Only the number of firms is used as a control, since controlling for additional variables causes the problem of no or scant observations in many cells. Thus to apply the hypothesis testing methodology

OPTIMAL CARTEL TRIGGER STRATEGIES AND THE NUMBER OF FIRMS

TABLE I. Descriptive statistics of Series 1 and 2

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Number of weeks Number of low profit episodes Mean length of a low profit episode, weeks Median length of a low profit episode, weeks Minimum length of a low profit episode, weeks Maximum length of a low profit episode, weeks Total number of low profit weeks Proportion of weeks in high profit regime

Series la Series 2b

328 328 11 8 11.4 11.7 8 7 1 3

37 37 125 94

0.62 0.71

’ Series 1 is derived from contemporaneous accounts of the cartel’s behavior. b Series 2 is a regime classification estimated by Lee and Porter (1984).

described above, the data was partitioned by the number of firms in the cartel prior to partitioning by M period histories.

To obtain some robustness against misclassification of regimes, two sets of observations on the regime series {I} are used. The series referred to as Series 1 is a regime classification developed by Thomas Ulen, who derived the series by looking contemporaneous accounts of the cartel’s performance for reports of price-cutting below the cartel’s list prices. Series 2 is an estimated classification of the regime series into high and low profit periods provided by the study of Lee and Porter (1984). They use a switching regression model to estimate separate high and low profit supply regimes for the JEC. Note that their estimation procedure does not constrain the regime series to follow a Markov process of any order. For each series a period is one week, which is the length of time used by the JEC to gather and disseminate reports on the volume of shipments and an index of prices charged by member fhms.

Table I provides some simple descriptive statistics of the two series. In both series, regimes tend to occur in fairly long runs of weeks of high profit and week of low profit; each such run is labeled an episode in Table I. Note that the two series are quite similar; they agree on the classification of 271 of the 328 weeks.

Table II lists the proportion of weeks that each series deems high profit when we control for the number of firms. For the first 27 weeks, starting from the first week in January of 1880, there were three firms in the cartel: the New York Central, the Penn, and the Baltimore and Ohio. A fourth firm, the Grand Trunk Railway, entered the Chicago market in April, 1880 and the cartel in the summer of 1880. A fifth firm, the Erie Railroad, entered the cartel in 1883. Table II reveals that the cartel became less successful over time with the entry of additional firms. Both Series 1 and Series 2 indicate that the cartel was able to sustain collusion for all of the twenty-seven weeks in which there were only three firms in the cartel. When four firms were in the cartel, Series 1 shows that the cartel was successful in sustaining

556 HUGH BRIGGS, III

TABLE II. The proportion of high profit weeks by number of firms in the cartel

Series 1 Series 2 Number firms Number of weeks Proportion high profit Proportion high profit

3 27 1.0 1.0 4 154 0.714 0.760 5 147 0.449 0.612

collusion 7 1% of the weeks, while Series 2 indicates it was successful 76% of the weeks. With five firms in the cartel, collusive prices were maintained 44% of the weeks according to Series 1, and 61% of the weeks according to Series 2.

Table III applies the hypothesis test for a fist-order Markov process against the alternative of a second-order Markov process. I omit the histories during which the cartel had three firms since all weeks were high profit weeks then. The first row of the table indicates that when the cartel had four firms, there were forty weeks preceded by the two period history of low profits in both periods, and that 7.5% of these forty weeks were high profit weeks. The variance .0694 is an estimate of the population variance, .0750 (l-.0750). Observe that descriptively, the means appear to satisfy the hypothesis of a first-order Markov process; the four means with single period histories of low profits are all .25 or less while the four means with single period histories of high profits are all .8852 or higher. Note that for two of the listed means, the parameter estimate is at the boundary of the parameter space [0, 11; when this occurs, one cannot apply the central limit theorem necessary to obtain normality and therefore cannot use these observations in a formal hypothesis test. Under the null hypothesis that regimes follow a fist-order Markov process, Table II(A) gives estimates on the interior of the parameter space for two groups of two means. There are two degrees of freedom in the chi-square distribution used for the test, and a test statistic of .9313. The test statistic is associated with a marginal significance level of .6277; thus one should accept the null hypothesis that the regimes follow a first-order Markov process.

There is one interior estimate in Table III(A) for when there were four firms in the industry and the previous period was high profit and one interior estimate for when there were five fhms in the industry and the previous period was high profit. If these relatively precise estimates are used as the probability that the cartel would remain in the high profit state given that it was in the high profit state in the previous week, then the probability of observing the estimates that are at the boundary of the parameter space can be calculated. Thus, for example, when there are four firms in the industry and the preceding period was high profit, the probability of observing a mean of 1 .O for the 4 periods which had a history of (1,0) is (.9623)4 or .8575. Thus, the mean of 1.0 for the history (0, 1) when there are four firms in the industry is not implausible. Similarly, when the cartel had five ohms, the

OPTIMAL CARTEL TRIGGER STRATEGIES AND THE NUMBER OF FIRMS 557

TABLE III. Tests for a first-order Markov process against a second-order alternative

History (t - 1,1 - 2) Mean Variance Number of observations

(A) Series 1 * (Number of firms = 4) w9 0.0750 0.0694 W) 0.2500 0.1875 (1,O) 1.0 0.0 (1J) 0.9623 0.0363

Number of firms = 5 (OS9 0.0685 0.0638 al) 0.1429 0.1225 (l?O) 1.0 0.0 (191) 0.8852 0.1016

(B) Series 2 (Number of firms = 4) (w9 0.0278 0.0270 (031) 0.0 0.0 (190) 1.0 0.0 (191) 0.9914 0.0085

Number of firms = 5 W) 0.1400 0.1204 641) 0.0 0.0 (1~0) 1.0 0.0 (Ll) 0.0772 0.0772

40 4 4

106

73 7 6

61

36 1 1

116

50 7 7

83

* Degrees of freedom: 2 Test statistic: 0.93 13 Marginal signifcance of test: 0.6277.

probability of obtaining 6 high profit weeks with the history (1, 0) is (.8852)6 or .48 11, once again not an unlikely event.

Means for different two period histories for Series 2 are listed in Table III(B). Series 2 does not permit a formal test of the hypothesis that regimes follow a first-order Markov process because for each of the four groups of two means, one of the two means is at the boundary of the parameter space. However, it is once again the case that given the interior estimates based on one period histories, the fact that estimates at the boundary of the parameter space were obtained is not an unusual event. For example, the probability of obtaining seven l’s for the history (1,O) with five firms is (.9157)7, or .5948, and the probability of obtaining seven O’s for the history (0, 1) with five firms is .3479.

Table TV tests the null hypothesis that the regimes follow a first-order Markov process against the alternative of a second or third-order Markov process. To save space, histories which have no observations have been omitted. For Series 1, there

558 HUGH BRIGGS, III

are four degrees of freedom and a test statistic of 1.45 13 which is associated with a marginal significance level of .8352. Therefore, we should once again accept the null hypothesis that the regimes follow a first-order Markov process. For Series 2 there is one degree of freedom and test statistic of .2214 which is associated with a marginal significance level of .6380. Thus, Series 2 also provides support for the null hypothesis.

Observe that if the probability of any period being high profit is independent of history, then the regimes follow a zeroeth-order Markov process. The null hypoth- esis of a zeroeth-order Markov process can be tested by imposing the restriction that all means are equal. One can overwhelmingly reject the null hypothesis of a zeroeth-order Markov process against an alternative of a Markov process of up to any order we have considered. For example, for the null hypothesis that the regimes follow a zeroeth-order Markov process against the alternative of up to a second-order Markov process, Table III (A) would yield a test statistic of 85 with 2 degrees of freedom.

Table V provides a test of the null hypothesis that the number of firms does not influence the transition probabilities of the first-order Markov process. Consider Series 1 depicted in Table V(A). Under the null hypothesis that the number of firms does not influence the transition probabilities there are two restrictions on the means in Table 5. Therefore, the test statistic of 2.8 183 has two degrees of freedom and a marginal significance level of .2444. Thus, Series 1 indicates that there is no difference in regime transition probabilities when we control for the number of firms. Series 2, depicted in Table V(B), provides conflicting evidence in this regard. For Series 2, the test statistic of 8.8660 implies we could reject the null hypothesis that regime transition probabilities do not differ when the number of firms goes from four to five with a significance level of a little more than .O 1.

The result that the number of firms does not affect the transition probabilities may seem surprising, especially in light of the data presented in Table II, which shows that Series 2 finds less difference between the probability of success when there were four and five firms in the cartel. However, the explanation for these results is clear. First, for interpreting Table II, note that the probability of a high profit week is very sensitive to the number of times the cartel enters the low profit regime, since it is difficult to leave the low profit regime once it has been entered, that is, the low profit regimes occur in long runs. Series 2 indicates one such episode of thirty-seven weeks duration when there were four firms in the cartel. Since all of the runs of non-cooperative weeks were much smaller than this when there were five firms in the cartel, Series 2 finds a much lower probability of switching from the low profit regime to the high profit regime when there were four firms than when there were five firms. In contrast, Series 1 indicates four episodes of low profit regimes when there were four firms, with these episodes averaging eleven weeks in duration. When the cartel had five firms, there were seven episodes of low profits that lasted a total of eighty weeks, which is again an average of about eleven weeks duration. Therefore, for Series 1 the probability of leaving the low

OPTIMAL CARTEL TRIGGER STRATEGIES AND THE NUMBER OF FIRMS

TABLE IV. Tests for a first-order Markov process against a second-order or third- order alternative

History (t - 1, t - 2, t - 3) Mean Variance Number of Observations

(A) Series 1 l (Number of firms = 4) @,O,O) 0.0540 Kml) 0.3333 (0,191) 0.2500 U,O,O) 1.0 U,l,O) 1.0 U,O,l) 1.0 (l,l,l) 0.9608

Number of Firms = 5 w,o) aos) (0,191) W,O) U,l,O) (l,O,l) U,l,l)

0.0597 0.0561 67 0.1667 0.1389 6 0.1429 0.1225 7 1.0 0.0 5 1.0 0.0 6 1.0 0.0 1 0.8727 0.1111 55

(B) Series 2 * * (Number of firms = 4) K40s9 0.0286 @,W) 0.0 (WJ) 0.0 (1,0,0) 1.0 (l,l,O) 1.0 (1,1,1) 0.9913

Number of Firms = 5 wu9 @,W) (O,l,l) (1 ,w U,l,O) (1,1,1)

0.1628 0.1363 43 0.0 0.0 7 0.0 0.0 7 1.0 0.0 7 0.8571 0.1225 7 0.9211 0.0733 76

0.0511 37 0.2222 3 0.1875 4 0.0 3 0.0 4 0.0 1 0.0377 102

0.0278 35 0.0 1 0.0 1 0.0 1 0.0 1 0.0086 115

*Degrees of freedom: 4 Test statistic: 1.45 13. Marginal significance of test: 0.8352. **Degrees of freedom: 1 Test staistic: .2214. Marginal significance of test: 0.6380.

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profit state once it has been entered was about the same regardless of the number of firms.

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TABLE V. Hypothesis tests on the effects of the number of firms

History@ - 1) N p V

(A) Series l* (Number of Firms = 4) 0 44 0.0909 0.0826 1 110 0.9636 0.0350

Number of Firms = 5 0 80 0.0750 0.0694 1 67 0.8955 0.0936

(B) Series 2” (Number of Firms = 4) 0 37 0.0270 0.0263 1 117 0.9915 0.0084

Number of Firms = 5 0 57 0.1228 0.1077 1 90 0.9222 0.0717

*Degrees of freedom: 2 Test statistic: 2.8183. Marginal significance of test: 0.2444. **Degrees of freedom: 2 Test statistic: 8.860.

‘Marginal significance of test: 0.0 119.

III. Conclusion

The results of the present paper are consistent with earlier empirical research that concludes that price wars occurred during unanticipated slumps in demand. Further, they extend that conclusion by suggesting that the regimes of the cartel followed a first-order Markov process, as predicted by the model of Abreu, Pearce, and Stachetti.

Because no functional form or distributional assumptions were made to obtain results, they provide an appropriate starting point for constructing fully parametric models based on the Abreu, Pearce and Stachetti model. For example, one should consider modeling two sets of transition probabilities: the probability of entering the high profit state from the low profit state, and the probability of entering the low profit state from the high profit state. The Abreu, Pearce and Stachetti model suggests that these transition probabilities will have differently signed correlations with unobserved components of demand.

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OPTIMAL CARTEL TRIGGER STRATEGIES AND THE NUMBER OF FIRMS 561

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