Coordination in a supply chain with two manufacturers, two substitute products, and one retailer

ABSTRACT

This paper analyses a model of coordination in a supply chain consisting of two manufacturers, two products, and a single retailer under full information. Market demand for each of the manufacturer's products allows for both price and cross-price elasticities. We consider a Stackelberg game between the retailer and the two manufacturers and solve for the subgame perfect equilibrium wholesale price chosen by each of the manufacturers, the retail price charged by the retailer for each of two products, as well as the equilibrium demands for the two products. Unlike a classical dyadic supply chain, we show that only under certain allocations of the total profit between the manufacturers and the retailer is it the case that the vertically integrated chain is the preferred supply chain structure, even though it provides the highest total profit. An important result is that vertical integration is less advantageous when products are closer substitutes. We also show that a revenue sharing contract can coordinate this chain, but only when the manufacturers set their wholesale prices below their marginal costs of production. Finally, we show that the retailer can choose to integrate partially with one manufacturer to achieve a Pareto improving profit outcome.

KEYWORDS:

• Supply chain coordination
• vertical integration
• revenue sharing
• Stackelberg game

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 While our work focuses on two-echelon supply chains, there has also been research on three-echelon dual supply chains focussing on pricing, return policy, replenishment, and coordination issues (see Modak, Panda, & Sana, 2015; Modak et al., 2016; Taleizadeh & Noori-daryan, 2015 for details).

2 The second-order conditions are satisfied because ${\mathrm{\partial }}^2{\mathrm{\Pi }}^R/\mathrm{\partial }{d}_1^2=-2{\beta }_1<0$, ${\mathrm{\partial }}^2{\mathrm{\Pi }}^R/\mathrm{\partial }{d}_2^2=-2{\beta }_2<0$, and $\left({\mathrm{\partial }}^2{\mathrm{\Pi }}^R/\mathrm{\partial }{d}_1^2\right)\left({\mathrm{\partial }}^2{\mathrm{\Pi }}^R/\mathrm{\partial }{d}_2^2\right)-({\mathrm{\partial }}^2{\mathrm{\Pi }}^R/\mathrm{\partial }{d}_1\mathrm{\partial }{d}_2{)}^2=4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2>0$.

3 The second-order conditions are satisfied because ${\mathrm{\partial }}^2{\pi }_1^M/\mathrm{\partial }{w}_1^2=-4{\beta }_2/(4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2)<0$, ${\mathrm{\partial }}^2{\pi }_2^M/\mathrm{\partial }{w}_2^2=-4{\beta }_1/(4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2)<0$.

4 The second-order conditions are satisfied because ${\mathrm{\partial }}^2{\pi }^{SC}/\mathrm{\partial }{d}_1^2=-2{\beta }_1<0$, ${\mathrm{\partial }}^2{\pi }^{SC}/\mathrm{\partial }{d}_2^2=-2{\beta }_2<0$, and $\left({\mathrm{\partial }}^2{\pi }^{SC}/\mathrm{\partial }{d}_1^2\right)\left({\mathrm{\partial }}^2{\pi }^{SC}/\mathrm{\partial }{d}_2^2\right)-({\mathrm{\partial }}^2{\pi }^{SC}/\mathrm{\partial }{d}_1\mathrm{\partial }{d}_2{)}^2=4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2>0$.

5 The analysis is (of course) completely symmetric if the retailer integrates only with manufacturer 2.

6 The second-order conditions are satisfied because ${\mathrm{\partial }}^2{\mathrm{\Pi }}_1^R/\mathrm{\partial }{d}_1^2=-2{\beta }_1<0$, ${\mathrm{\partial }}^2{\mathrm{\Pi }}_1^R/\mathrm{\partial }{d}_2^2=-2{\beta }_2<0$, and $\left({\mathrm{\partial }}^2{\mathrm{\Pi }}_1^R/\mathrm{\partial }{d}_1^2\right)\left({\mathrm{\partial }}^2{\mathrm{\Pi }}_1^R/\mathrm{\partial }{d}_2^2\right)-({\mathrm{\partial }}^2{\mathrm{\Pi }}_1^R/\mathrm{\partial }{d}_1\mathrm{\partial }{d}_2{)}^2=4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2>0$.

7 Note that this is equivalent to a single manufacturer or supplier producing two goods. However, we are concerned with comparing the coalition profits with the decentralised scenario. This comparison yields some interesting insights regarding the incentives for the manufacturers to form a coalition or not.

8 The stage 2 problem for the retailer is the same as that for the decentralised case.

9 The second-order conditions are satisfied because ${\mathrm{\partial }}^2{\pi }^{M_1+M_2}/\mathrm{\partial }{w}_1^2=-4{\beta }_2/(4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2)<0$, ${\mathrm{\partial }}^2{\pi }^{M_1+M_2}/\mathrm{\partial }{w}_2^2=-4{\beta }_1/(4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2)<0$, and $\left({\mathrm{\partial }}^2{\pi }^{M_1+M_2}/\mathrm{\partial }{w}_1^2\right)\left({\mathrm{\partial }}^2{\pi }^{M_1+M_2}/\mathrm{\partial }{w}_2^2\right)-({\mathrm{\partial }}^2{\pi }^{M_1+M_2}/\mathrm{\partial }{w}_1\mathrm{\partial }{w}_2{)}^2=4/(4{\beta }_1{\beta }_2-({\gamma }_1+{\gamma }_2{)}^2)>0$.

Additional information

Notes on contributors

Anders Thorstenson

Anders Thorstenson is a Professor in the Section Logistics, Department of Economics and Business Economics at Aarhus University, Denmark. He received his M.Sc. and Ph.D. from the Institute of Technology, Linkoping University, Sweden. His research interests are in Supply Chain and Inventory Management, and Logistics.

Vinay Ramani

Vinay Ramani is is an Associate Professor in the Economics area at the Indian Institute of Management Udaipur. He received his M.A. and Ph.D. from the University at Buffalo, The State University of New York. His main research interests are in Industrial Organisation, Supply Chain Contracts, and Distribution Channels.