Market-Induced Rationalization and Welfare-Enhancing Cartels

Jan Tuinstra; Daan L. in ’t Veld

doi:10.1515/bejeap-2013-0005

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Market-Induced Rationalization and Welfare-Enhancing Cartels

Jan Tuinstra and Daan L. in ’t Veld

Published/Copyright: October 24, 2013

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From the journal The B.E. Journal of Economic Analysis & Policy Volume 14 Issue 1

Abstract

We show that incomplete cartels in quantity-setting oligopolies may increase welfare, without any efficiencies or synergies being internalized by cartel formation. The main intuition is that the cartel has an incentive to contract output and that the firms outside the cartel react to this by expanding output. If the outsiders are more efficient than the cartel firms, average production costs go down. We model collusion in a market structure with imperfect substitute goods. Even for relatively moderate differences in efficiency, total welfare may increase due to this market-induced rationalization, whereas the cartel remains profitable. We discuss why the effect can be relevant for sectors where new, superior products are developed. Because anti-cartel enforcement is costly, it is important for competition authorities to realize that not all cartels lead to a welfare loss.

Keywords: efficient cartels; cournot competition; rationalization of production

JEL Classification: L10; L41; D43

Acknowledgments

The authors are grateful to Maarten Pieter Schinkel, the editor Till Requate, and two anonymous referees for constructive comments. We thank participants of the QED Jamboree 2012 (Copenhagen). Daan L. in ’t Veld would like to thank the Netherlands Organisation for Scientific Research (NWO) for financial support under the Complexity program.

Appendix

Proofs

Proof of Lemma 1. Firm i in industry 1 sets its quantity in order to maximize and firm j in industry 2 sets its quantity to maximize . The first-order conditions are given by

The solution of which is given by for and for . Prices and profits follow straightforwardly from this.

Proof of Proposition 2. Although n is an integer, for mathematical convenience we may treat it as a continuous variable (as Daughety 1990: footnote 1). Differentiating and with respect to n we find that both are decreasing in the number of firms in industry 1:

[5]

Then, the change in aggregate output is

Moreover, both prices decrease:

Since the consumer faces lower prices and consumes positive amounts of both commodities, he will be strictly better off with an increase in n.

Consequently, formation of a cartel in industry 1, which is formally equivalent with a decrease of the number of firms in that industry from n to 1, will increase both prices, decreases production of commodity 1 and increases production of commodity 2. Moreover, the consumer will be strictly worse off, and firms from industry 2, which now sell more and at a higher price, will be better off. The cartel is profitable, if , or when – that is, when the cartel does not restrain production by too much. This is the case for the values of n given by condition [3]. ■

Proof of Lemma 3. Denote by and the Cournot–Nash equilibrium quantities when there are n firms independently producing commodity 1 (where, for notational convenience, we suppress the dependence of and on m, , and ). Consumer surplus is given by

Aggregate profits in industry 1 and industry 2 are given by and , respectively. Total welfare, as a function of n and , is

[6]

Since is continuous in n, we can take the derivative with respect to n. Using eq. [5] from the proof of Proposition 2, we find

Moreover, is decreasing in n. This implies that is unimodal in n and has a unique global maximum which is given by the solution to . Now if for the actual value of n, then in the social optimum at least n firms are active in the production of commodity 1. This gives

On the other hand, if , then in the social optimum no firm should be active in the production of commodity 1. This gives

It is easily checked that for . ■

Proof of Proposition 4. Substituting the expressions for and from eq. [2] into eq. [6] and solving for we obtain, after some straightforward but tedious calculations, .

The fact that is unimodal in n (see the proof of Lemma 3) implies that for all admissible values of n, m, and . Differentiating with respect to n shows that has the same sign as

which is always negative. The derivative with respect to is much more complicated but turns out to be negative for any admissible value of n, m, and . ■

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1
While we borrow this term from Stennek (2003), we use it in a more common setting. The model of Stennek (2003) is special in the sense that the competitive equilibrium is inefficient, because the cost functions are assumed to be private knowledge.
2
Although the effect we describe is very intuitive, we have just found one paper that captures it implicitly. The numerical example of Cheung (1992, 120), which deals with synergies from mergers, includes a welfare-enhancing (but unprofitable) cartel (for the special case of zero synergies).
3
Following Singh and Vives (1984), the linear inverse demand functions can be derived from a utility function that is quadratic in consumption of commodities 1 and 2 and linear in consumption of a numeraire composite commodity. We will depart from Singh and Vives (1984), who consider a (differentiated) duopoly, by allowing for multiple producers of each commodity.
4
In another version of the model, we adopted a constant elasticity of substitution (CES) framework, so that the two goods range from perfect substitutes to perfect complements. We found here as well that welfare-enhancing cartels are possible (and profitable). Because the CES framework does not allow for analytical solutions but requires numerical calculations, we present the results with linear inverse demand functions.
5
Our analysis is centered around the assumption that industry 2 is cost-efficient (i.e. , but it also applies to situations where products from industry 2 are qualitatively superior. In fact, this could be easily incorporated in the model by a reinterpretation of . After generalizing the inverse demand functions to for and , assuming means that industry 2 produces higher quality goods than industry 1, as measured in a vertical sense; see Häckner (2000). Solving the more general model leads us to redefine , a combined measure of the differences in efficiency and quality. The expressions in Lemma 1, Proposition 2, Lemma 3, and Proposition 4 remain unchanged (note that in expression 2 the term ( has to be replaced with ). All our results can, therefore, be interpreted in terms of differences in quality as well as efficiency. We thank an anonymous referee for this suggestion.
6
Note that the profitability of the cartel (relative to the competitive case) does not depend on marginal costs c or d. This is a consequence of assuming linear cost and inverse demand functions. However, the efficiency difference does determine market shares of individual firms. Therefore, the minimal number of firms to join for profitability might represent an arbitrarily small market share. In particular, the market share of industry 1 firms, , equals for which becomes arbitrary small for close enough to .
7
It is straightforward to show that, assuming firms use grim trigger strategies, the critical discount factor for a sustainable cartel in the infinitely repeated game is given by This critical discount factor is increasing in m and , non-monotonic in n and strictly below 1 as long as condition [3] holds.
8
As profitability of a cartel does not depend on marginal costs (see Footnote 5), condition [3] is equally valid for the alternative cartel with m and n reversed. Also, the critical discount factor (see Footnote 6) is identical with m and n reversed. Analysis of the examples in Figure 2 makes clear that the alternative cartel is for most cases either unprofitable or requires a higher discount factor than the cartel of inefficient firms. The only exception is for and , in which case a cartel of efficient firms is sustainable for a wider range of discount factors when or (but not when ).