Abstract
Game-theoretic analyses of distribution channels have generated six widely held beliefs (we call them Channel Hypotheses) whose universal soundness has not been examined. To assess the validity of these Hypotheses, we develop a general, linear-demand model in which distributors face heterogeneity in demand, heterogeneity in costs, and any degree of intensity of inter-distributor competition. For ease of comparison, we nest the bilateral-monopoly model and the identical-distributors model within our general model. Our analysis reveals that the Channel Hypotheses do not generalize beyond the specific game-theoretic models from which they were derived. This lack of generality is critical, because these beliefs have led to intuitively appealing (but inadvertently misleading) strategic advice for managers and modeling advice for game theorists. From our general, linear-demand model, we derive six Channel Propositions that correct these accumulated errors of conceptualization and that generate a richer, more broadly applicable set of managerial and modeling implications. We also present a Channel-Modeling Proposition that we believe will help modelers avoid the errors of conceptualization described in this paper.
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Notes
There are three generating models in the game-theoretic literature on distribution: inter-channel competition (e.g., McGuire & Staelin, 1983), inter-manufacturer competition (e.g., Choi, 1991), and inter-retailer competition (e.g., Ingene & Parry, 1995b, 2004). Choi (1996) combined the latter two models for the case of identical manufacturers and retailers. The bilateral-monopoly model (e.g., Edgeworth, 1881; Jeuland & Shugan, 1983) is embedded within each of these models. The seminal articles were by Jeuland and Shugan (1983) and McGuire and Staelin (1983).
A channel dyad comprises firms upstream (a “manufacturer”) and downstream (a “retailer”). In a decentralized channel, the manufacturer and the retailer are independent; in a vertically integrated system they are jointly owned.
A bilateral-monopoly channel comprises one manufacturer selling to one retailer; in turn, the retailer buys exclusively from that manufacturer (Edgeworth, 1881). The marketing science literature has concentrated on two-level channels.
Profit sharing between retailer and manufacturer is necessary for a dyadic member to accept a zero margin; negotiation between channel members is one method of profit sharing (Jeuland & Shugan, 1983).
Because marketing scientists are concerned with number of distributors, not differences between them, they typically assume that distributors are identical; that is, distributors are modeled with equal demands and equal costs.
Gerstner and Hess (1995) provide a clear statement of the belief that a vertically integrated system maximizes total channel profit.
This unique mapping would break down if demand was a rectangular hyperbola—which generates the same total revenue at all price levels.
We stress that all our examples are derived from rigorous game-theoretic analyses.
Section 2(a) of the Robinson–Patman Act “...prohibits sellers from charging different prices to different buyers for similar products where the effect might be to injure, destroy, or prevent competition, in either the buyers’ or sellers’ markets” (Monroe, 1990, p. 394).
In practice, a manufacturer may have a small set of wholesale-price schedules; what is important for our analysis is that multiple distributors face a common schedule.
Choi (1991) has shown that price, quantity and the distribution of channel profit are invariant with respect to who occupies the leader’s role in a bilateral monopoly. We distinguish between net revenue and profit (net revenue minus fixed cost); Choi did not make this distinction because he assumed zero fixed cost.
Desiraju and Moorthy (1997) have addressed informational asymmetry.
The symbol “∈” is defined as “an element of.” Thus i,j∈(1,N) is read as “the values of i and j are elements of the integers from 1 to N.” In simple English, i is an integer between 1 and N, and j is an integer between 1 and N.
This demand system was introduced to marketing by McGuire and Staelin (1983).
The symbol ∀ is defined as “for all.” Thus Eq. 6 is merely a statement that “for all values of p j greater than or equal to \( {{\left( {A_{j} + \theta p_{i} } \right)}} \mathord{\left/ {\vphantom {{{\left( {A_{j} + \theta p_{i} } \right)}} b}} \right. \kern-\nulldelimiterspace} b \), the value of \( Q^{{{\text{One}}}}_{i} \,is\,\alpha _{i} - \beta p_{1} \) and the value of Q j is 0.”
We believe that this definition of logically consistent demand first appeared in Ingene and Parry (2004).
Profit is \( \Pi = {\left( {p_{1} - c_{1} - C} \right)}q_{1} - f_{1} - F \); it is maximized by solving for the optimal value of p 1.
The symbol “≡” means “is defined as;” a trivial example is the high school formula for area (Area≡WL).
The model is solved by backward induction: first the π1-equation is optimized over p1 to obtain the retailer’s quantity-reaction function, then the Π1-equation is optimized over W 1, given the retailer’s quantity-reaction function. This approach is called Stackelberg maximization in honor of its originator (Stackelberg, 1934); we refer to it as naïve Stackelberg maximization for reasons that will become clear in sub-Section Heterogeneous Demand and Costs: a Manufacturer’s Profit-Maximizing Tariff.
An alternative that is rarely discussed in the literature is for the manufacturer to set its margin at \( \widehat{M}^{ * } = \mu ^{ * }_{1} \) and pay the distributor a fixed fee (say φ) not to markup its merchandise (i.e., \( \widehat{m}^{ * }_{1} = 0 \)); this will also lead to coordination. This alternative approach reinforces our central point that double marginalization is incompatible with coordination.
A two-part tariff consists of a per-unit fee and a lump-sum payment (a fixed fee).
Fixed costs affect the participation constraint that determines the allowable set of fixed fees: \( {\left( {R^{{VI^{ * } }}_{1} - f_{1} } \right)} \geqslant \widehat{\phi }_{1} \geqslant F \).
There are a minimal number of identical retailers (\( \underline{N} \)) needed to generate sufficient revenue to cover the manufacturer’s fixed costs. The channel cannot exist if \( N < \underline{N} \).
We address the third popular model, the identical-competitors model, in “Competitive substitutability” below.
Formula (14), and equivalence of output with marginal-cost pricing, generalizes to any number of outlets.
Expanding a VIS to a third outlet would generate the same conclusion as shown in Eq. 18.
Similar results can be obtained by varying f j .
A negative fixed fee (a payment from manufacturer to retailers) occurs for all f i ≥ $1,633.33.
Although the non-coordinated channel has more outlets than the coordinated channel in this example, this is not a general principle. It is a function of the specific demand curve that has been chosen.
\( U \equiv {\sum\limits_{k = 1}^N {{\left( {A_{k} Q_{k} - {BQ^{2}_{k} } \mathord{\left/ {\vphantom {{BQ^{2}_{k} } 2}} \right. \kern-\nulldelimiterspace} 2} \right)} - T{\sum\limits_{k = 1}^N {Q_{k} } }{\sum\limits_{^{{m = 1}}_{{m > k}} }^N {Q_{m} } }} } \); we develop utility-based demand in the Technical Appendix.
In the special case of identical competitors we have (A i = A j ≡ A) implies \( {{\text{d}}A} \mathord{\left/ {\vphantom {{{\text{d}}A} {{\text{d}}T}}} \right. \kern-\nulldelimiterspace} {{\text{d}}T} = - {\left( {b - \theta } \right)}A < 0 \).
We use the nomenclature that identical, non-competing distributors are “identical retailers” and that identical, competing distributors are “identical competitors;” the term “identical distributors” encompasses both identical retailers and identical competitors.
When a channel can be characterized by VSS, an increase in one channel member’s margin leads to an optimal decrease in the other channel member’s margin. In contrast, with VSC a rise in one margin leads to an increase in the other member’s margin. Taken to its logical conclusion, under VSC both channel members can set their margins as high as possible. No real-world situation can be accurately characterized by VSC on a continuing basis.
References
Battacharyya, S., & Lafontaine, F. (1995). Double-sided moral hazard and the nature of share contracts. Rand Journal of Economics, 26(4), 761–781.
Betancourt, R. (2004). The economics of retailing and distribution. Cheltenham, UK: Edward Elgar.
Choi, S. C. (1991). Price competition in a channel structure with a common retailer. Marketing Science, 10, 271–296 (Fall).
Choi, S. C. (1996). Price competition in a duopoly common retailer channel. Journal of Retailing, 72, 117–137 (Summer).
Desiraju, R., & Moorthy, S. (1997). Managing a distribution channel under asymmetric information with performance requirements. Management Science, 43, 1628–1644 (December).
Edgeworth, F. (1881). Mathematical psychics: An essay on the application of mathematics to the moral science. London: Kegan Paul (Reprinted 1967. New York: Augustus M. Kelley).
Edgeworth, F. (1897). Teoria pura del monopolio. Giornale degli Economisti, 15:13–31 (English translation by F. Edgeworth Theory of Monopoly. In F. Edgeworth 1925 (Eds.), Papers relating to political economy. v. 1. London: Macmillan).
Gerstner, E., & Hess, J. (1995). Pull promotions and channel coordination. Marketing Science, 14, 43–60 (Winter).
Horowitz, A. (2000). Pricing in a vertical channel with fixed costs: A sticky price equilibrium with e-commerce implications. Mimeo: Department of Economics, University of Arkansas.
Ingene, C., & Parry, M. (1995a). Coordination and manufacturer profit maximization: The multiple retailer channel. Journal of Retailing, 71, 129–151 (Summer).
Ingene, C., & Parry, M. (1995b). Channel coordination when retailers compete. Marketing Science, 14, 360–377 (Fall).
Ingene, C., & Parry, M. (1998). Manufacturer-optimal wholesale pricing when retailers compete. Marketing Letters, 9, 61–73 (February).
Ingene, C., & Parry, M. (2000). Is channel coordination all it is cracked up to be? Journal of Retailing, 76, 511–548 (Winter).
Ingene, C., & Parry, M. (2004). Mathematical models of distribution channels. New York: Kluwer.
Jeuland, A., & Shugan, S. (1983). Managing channel profits. Marketing Science, 2, 239–272 (Summer).
Lafontaine, F. (1990). An empirical look at franchise contracts as signaling devices. Graduate school of industrial administration, Mimeo. Pittsburgh, PA: Carnegie-Mellon University.
Lafontaine, F. (1992). Agency theory and franchising: Some empirical results. The Rand Journal of Economics, 23, 263–283 (Summer).
Lafontaine, F., & Shaw, K. (1999). The dynamics of franchising: Evidence from panel data. The Journal of Political Economy, 107, 1041–1080 (October).
Lee, E., & Staelin, R. (1997). Vertical strategic interaction: Implications for channel pricing strategy. Marketing Science, 16, 185–207 (Fall).
McGuire, T., & Staelin, R. (1983). An industry equilibrium analysis of downstream vertical integration. Marketing Science, 2, 161–192 (Spring).
McGuire, T., & Staelin, R. (1986). Channel efficiency, incentive compatibility, transfer pricing and market structure: An equilibrium analysis of channel relationships. In L. Bucklin & J. Carman (Eds.), Research in marketing: Distribution channels and institutions, 8, 181–223.
Monroe, K. (1990). Pricing: Making profitable decisions. New York: McGraw-Hill.
Moorthy, S. (1987). Managing channel profits: Comment. Marketing Science, 6, 375–379, (Fall).
Moorthy, S. (1993). Theoretical modeling in marketing. Journal of Marketing, 57, 92–106 (April).
Moorthy, S., & Fader, P. (1988). Strategic interaction within a channel. In L. Pellegrini & S. Reddy (Eds.), Retail and marketing channels. London: Routledge.
Spengler, J. (1950). Vertical integration and antitrust policy. Journal of Political Economy, 58 (4), 347–352.
Stackelberg, H. (1934). Marktform und gleichgewicht. Wein und Berlin. (Translated by A. T. Peacock 1952. The theory of the market economy. London: William Hodge).
Acknowledgements
We thank George Zinkhan, Dave Stewart, and the anonymous reviewers for their insightful comments on earlier drafts. Any remaining errors are our own responsibility.
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Bilateral Monopoly, Identical Distributors, and Game-Theoretic Analyses of Distribution Channels (PDF 798 kb)
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Ingene, C.A., Parry, M.E. Bilateral monopoly, identical distributors, and game-theoretic analyses of distribution channels. J. of the Acad. Mark. Sci. 35, 586–602 (2007). https://doi.org/10.1007/s11747-006-0006-0
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DOI: https://doi.org/10.1007/s11747-006-0006-0