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About the article
Published Online: 2013-10-24
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.
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).
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.
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.
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.
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 .
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  holds.
As profitability of a cartel does not depend on marginal costs (see Footnote 5), condition  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 ).