Abstract
A leitmotiv of the analysis of marketing channels’ behaviour is the possibility of designing contractual relations so as to replicate the performance of vertically integrated firms, whenever this is efficient for firms. This is particularly relevant when the vertical externality provokes distortions in the firms’ incentives to invest in R&D or advertising. The present model illustrates the possibility of using two-part tariffs endogenously defined as linear functions of firms’ efforts to sterilize the vertical externality altogether in a duopoly where firms’ invest in advertising to increase brand equity. This is done first in a static model and then replicated in the differential game based upon the same building blocks.
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Notes
- 1.
- 2.
The same quadratic functional form is also adopted in several other models discussing advertising campaigns (see Chu and Desai 1995; Jørgensen et al. 2001; Karray 2015, among many others). Of course, a linear advertising cost coupled with a concave impact (usually, the square root of the effort) also ensures concavity while at the same time accounting for decreasing returns (see Kim and Staelin 1999; Karray and Zaccour 2006).
- 3.
I have intentionally assumed quantity to be the relevant market variable, for two unrelated reasons. The first is that the resulting expressions are slightly more compact, while the second is that the formal structure and properties of the model, in particular, the emergence of a prisoner’s dilemma, do not depend on whether retailers (or downstream divisions) are price or quantity setters.
- 4.
Profit functions (3) are additively separable in the vector of advertising efforts. Therefore, the simultaneous and sequential choice of k iU and k iD yield the same equilibrium outcome.
- 5.
Additionally, a few words suffice to stress an aspect which is usually left aside: the presence of decreasing returns to advertising, embodied in the convex cost functions, implies that it is surely efficient to smooth investments onto divisions (or separated firms) by Jensen’s inequality.
- 6.
The lower bound to b has been chosen so as to satisfy all concavity and non-negativity requirements. It is also worth noting that the asymptotic value of \(\widetilde {s}\) as b tends to infinity coincides with that identified in Sect. 2.2 in absence of advertising. This is due to the fact that all equilibrium advertising efforts are monotonically decreasing in b , with \(\lim _{b\rightarrow \infty }k^{II},k_{U}^{SS},k_{D}^{SS}=0.\)
- 7.
It is worth stressing that, should divisions along the same supply chain cooperate, i.e., maximize joint profits, this would obviously replicate the outcome of the vertically integrated case.
- 8.
Henceforth, the time argument will be omitted for the sake of brevity.
References
Bonanno, G., & Vickers, J. (1988). Vertical separation. Journal of Industrial Economics, 36, 257–265.
Cachon, G. P. (2003). Supply chain coordination with contracts. In S. Graves & T. de Kok (Eds.), Handbook in operation research and management science: Supply chain management. Amsterdam: North-Holland.
Cachon, G. P., & Lariviere, M. A. (2005). Supply chain coordination with revenue-sharing contracts: Strengths and limitations. Management Science, 51, 30–44.
Chu, W., & Desai, P. S. (1995). Channel coordination mechanisms for customer satisfaction. Marketing Science, 14, 343–359.
Coughlan, A. (1985). Competition and cooperation in marketing channel choice: Theory and application. Marketing Science, 4, 110–129.
Dragone, D., Lambertini, L., Leitmann, G., & Palestini, A. (2015). Hamiltonian potential functions for differential games. Automatica, 62, 134–138.
Giannoccaro, I., & Pontrandolfo, P. (2004). Supply chain coordination by revenue sharing contracts. International Journal of Production Economics, 89, 131–139.
Grossman, S., & Hart, O. (1986). The costs and benefits of ownership: A theory of vertical and lateral integration. Journal of Political Economy, 94, 691–719.
Grout, P. (1984). Investment and wages in the absence of binding contracts: A Nash bargaining approach. Econometrica, 52, 449–460.
Ingene, C. A., & Parry, M. E. (1995). Channel coordination when retailers compete. Marketing Science, 14, 360–377.
Ingene, C. A., & Parry, M. E. (2004). Mathematical models of distribution channels. Boston, MA: Kluwer.
Ingene, C. A., Taboubi, S., & Zaccour, G. (2012). Game-theoretic coordination mechanisms in distribution channels: Integration and extensions for models without competition. Journal of Retailing, 88, 476–496.
Jeuland, A. P., & Shugan, S. M. (1983). Managing channel profits. Marketing Science, 2, 239–272.
Jeuland, A. P., & Shugan, S. M. (1988). Reply to: Managing channel profits: Comment. Marketing Science, 7, 103–106.
Jørgensen, S., Taboubi, S., & Zaccour, G. (2001). Cooperative advertising in a marketing channel. Journal of Optimization Theory and Applications, 110, 145–158.
Karray, S. (2015). Modeling brand advertising with heterogeneous consumer response: Channel implications. Annals of Operations Research, 233, 181–199.
Karray, S., & Zaccour, G. (2006). Could co-op advertising be a manufacturer’s counterstrategy to store brands? Journal of Business Research, 59, 1008–1015.
Kim, S. Y., & Staelin, R. (1999). Manufacturer allowances and retailer pass-through rates in a competitive environment. Marketing Science, 18, 59–76.
Lambertini, L. (2010). Oligopoly with hyperbolic demand: A differential game approach. Journal of Optimization Theory and Applications, 145, 108–119.
Lambertini, L. (2014). Coordinating static and dynamic supply chains with advertising through two-part tariffs. Automatica, 50, 565–569.
Lambertini, L. (2018). Coordinating research and development efforts for quality improvement along a supply chain. European Journal of Operational Research, 270, 599–605.
Lambertini, L., & Zaccour, G. (2015). Inverted-U aggregate investment curves in a dynamic game of advertising. Economics Letters, 132, 34–38.
Lee, E., & Staelin, R. (1997). Toward the general theory of channel management strategy. Marketing Science, 1b, 185–207.
Leng, M., & Parlar, M. (2010). Game-theoretic analyses of decentralized assembly supply chains: Non-cooperative equilibria vs coordination with cost-sharing contracts. European Journal of Operational Research, 204, 96–104.
MacLeod, W. B., & Malcomson, J. M. (1993). Investments, holdup, and the form of market contracts. American Economic Review, 83, 811–837.
McGuire, T., & Staelin, R. (1983). An industry equilibrium analysis of downstream vertical integration. Marketing Science, 2, 161–191.
Monderer, D., & Shapley, L. S. (1996). Potential games. Games and Economic Behavior, 14, 124–143.
Moorthy, K. S. (1987). Managing channel profits: Comment. Marketing Science, 6, 375–379.
Nagarajan, M., & Sošić, G. (2008). Game-theoretic analysis of cooperation among supply chain agents: Review and extensions. European Journal of Operational Research, 187, 719–745.
Reynolds, S. (1987). Preemption and commitment in an infinite horizon model. International Economic Review, 28, 69–88.
Rogerson, W. P. (1992). Contractual solutions to the hold-up problem. Review of Economic Studies, 59, 777–793.
Singh, N., & Vives, X. (1984). Price and quantity competition in a differentiated duopoly. RAND Journal of Economics, 15, 546–554.
Wang, C.-J., Chen, Y.-J., & Wu, C.-C. (2011). Advertising competition and industry channel structure. Marketing Letters, 22, 79–99.
Williamson, O. E. (1971). The vertical integration of production: Market failure considerations. American Economic Review, 61, 112–123.
Wirl, F. (2010). Dynamic demand and noncompetitive intertemporal output adjustments. International Journal of Industrial Organization, 28, 220–229.
Zaccour, G. (2008). On the coordination of dynamic marketing channels and two-part tariffs. Automatica, 44, 1233–1239.
Zusman, P., & Etgar, M. (1981). The marketing channel as an equilibrium set of contracts. Management Science, 27, 284–302.
Acknowledgements
The author would like to thank Davide Dragone, Arsen Palestini and two anonymous referees for helpful comments and suggestions. The usual disclaimer applies.
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Lambertini, L. (2020). On the Coordination of Static and Dynamic Marketing Channels in a Duopoly with Advertising. In: Pineau, PO., Sigué, S., Taboubi, S. (eds) Games in Management Science. International Series in Operations Research & Management Science, vol 280. Springer, Cham. https://doi.org/10.1007/978-3-030-19107-8_4
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