Retail outsourcing strategy in Cournot &amp; Bertrand retail competitions with economies of scale

Mingxia Li; Kebing Chen; Shengbin Wang

doi:10.3934/jimo.2021194

Article Contents

2023, Volume 19, Issue 1: 491-528. Doi: 10.3934/jimo.2021194

This issue Previous Article Zinc ore supplier evaluation and recommendation method based on nonlinear adaptive online transfer learning Next Article An accelerated differential equation system for generalized equations

Retail outsourcing strategy in Cournot & Bertrand retail competitions with economies of scale

1.
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211100, China
2.
College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu 211100, China
3.
School of Business, The College of New Jersey, Ewing, NJ 08618, USA

^* Corresponding author: Kebing Chen
^* Corresponding author: Kebing Chen

Received: April 2021

Revised: August 2021

Early access: November 2021

Published: January 2023

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Abstract

This paper investigates a manufacturer's retail outsourcing strategies under different competition modes with economies of scale. We focus on the effects of market competition modes, economies of scale and competitor's behavior on manufacturer's retail outsourcing decisions, and then we develop four game models under three competition modes. Firstly, we find the channel structure where both manufacturers choose retail outsourcing cannot be an equilibrium structure under the Cournot competition. The Cournot competition mode is less profitable to the firm than the Bertrand competition when the products are complements. Secondly, under the hybrid Cournot-Bertrand competition mode, there is only one equilibrium supply chain structure where neither manufacturer chooses retail outsourcing, when the substitutability and complementarity levels are not sufficiently high. In addition, setting price (quantity) contracts as the strategic variables is the dominant strategy for the direct-sale manufacturer who provides complementary (substitutable) products. Thirdly, both competitive firms will benefit from the situation where they choose the same competition mode. When the products are substitutes (complements), both of them choose the Cournot (Bertrand) competition mode. Finally, we show that the economies of scale have little impact on the equilibrium of the outsourcing structure but a great impact on the competition mode equilibrium.

Keywords:

Game theory,
retail outsourcing,
supply chain management and competition,
economies of scale.

Mathematics Subject Classification: Primary: 91A40, 90B50; Secondary: 91B06.

Citation:

Full Text(HTML)

Figure 1. Possible regions for equilibrium structures in the pure Cournot competition

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Figure 2. Possible regions for the more profitable competition mode

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Figure 3. Possible regions for equilibrium structure in pure Bertrand competition

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Figure 4. Possible regions for equilibrium structures in the Cournot-Bertrand competition

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Figure 5. Possible regions for the equilibrium competition mode

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Figure 6. The effect of the retail outsourcing strategy on $ M_{1} $'s profit when $ M_{2} $ uses direct sales

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Figure 7. The effect of the retail outsourcing strategy on $ M_{1} $'s profit when $ M_{2} $ uses retail outsourcing

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Figure 8. The effect of the retail outsourcing strategy on $ M_{1} $'s profit when $ M_{2} $ uses direct sales

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Figure 9. The effect of the retail outsourcing strategy on $ M_{1} $'s profit when $ M_{2} $ uses retail outsourcing

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Table 1. Compare and contrast our model with the extant literature

Article	Model	Market competition mode		Economies of scale	Product complementation	Outsourcing decision	Structure equilbrium	Competition mode equilbrium
Article	Model	Bertrand	Cournot	Economies of scale	Product complementation	Outsourcing decision	Structure equilbrium	Competition mode equilbrium
Chen & Lee [10]	Price vs. quantity under R&D competition	√	√
Nariu et al [33]	Product-differentiated Cournot competition		√				√
Arya et al [2]	Price vs. quantity competition	√	√			√
Farahat & Perakis [15]	Equilibrium profits of price & quantity competition	√	√
Matsumura & Ogawa [30]	Choice of a price or a quantity contract in a mixed duopoly		√			√	√
Fang & Shou [13]	Cournot competition between two supply chains	√	√
Zhao et al [44]	Pricing of complementary products in dual channel chain	√			√		√
Atkins & Liang [4]	Competitive supply chains with generalised supply costs	√		√			√
Haraguchi & Matsumura [22]	Comparing price and quantity competition	√	√					√
Din & Sun [12]	Choice of prices versusquantities with patent licensing	√	√					√
Wang et al [39]	pricing problem of complementary products	√			√	√
Our paper	Equilibrium analysis under the supply chain competition	√	√	√	√	√	√	√

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Table 2. Equilibrium solutions for the four different structures

X=	CC	CD(DC)		DD
X=	Chain i	Chain 1(2)	Chain 2(1)	Chain i
$ q_{{X_i}}^{*QQ} $	$ \frac{{a - {c_0}}}{{A + b}} $	$ \frac{{({A^2} + 2A - {b^2} - Ab)(a - {c_0})}}{{A({A^2} + 2A - 2{b^2})}} $	$ \frac{{(A - b)(a - {c_0})}}{{{A^2} + 2A - 2{b^2}}} $	$ \frac{{2({c_0} - a)}}{{(b - 4)(b + 2) + 4\theta }} $
$ \pi _{{X_{Mi}}}^{*QQ} $	$ (1 - \theta ){(q_{C{C_i}}^{QQ})^2} $	$ (1 - \theta ){(q_{C{D_1}}^{QQ})^2} $	$ \frac{{(2 - \theta )A - {b^2}}}{A}{(q_{C{D_2}}^{QQ})^2} $	$ \frac{{(4 - 2\theta - {b^2})}}{2}{(q_{D{D_i}}^{QQ})^2} $

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Table 3. Equilibrium solutions for the four different structures

X	chain	$ w_{{X_i}}^{*PP} $	$ q_{{X_i}}^{*PP} $	$ \pi _{{X_{Mi}}}^{*PP} $
CC	i	NA	$ \frac{{a - {c_0}}}{{(2 - b)(1 + b) - 2\theta }} $	$ (1 - {b^2} - \theta ){(q_{C{C_i}}^{*PP})^2} $
CD	1(2)	NA	$ \frac{{(1 - b)(2 + b)a + ({b^2} - 2){c_0} + bw_{C{D_2}}^{*PP}}}{S} $	$ (1 - {b^2} - \theta ){(q_{C{D_1}}^{*PP})^2} $
(DC)	2(1)	$ \frac{{(S + 2\theta T)(T + b)a + (ST - Sb - 2\theta Tb){c_0}}}{{2T(S + \theta T)}} $	$ \frac{{ - (T + b)a + b{c_0} + Tw_{C{D_2}}^{*PP}}}{S} $	$ \frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}}{(q_{C{D_2}}^{*PP})^2} $
DD	i	$ \frac{{Sa - (1 + b)(2 - b)({b^2} - 2){c_0}}}{{S - (1 + b)(2 - b)({b^2} - 2)}} $	$ \frac{{a - w_{D{D_i}}^{*PP}}}{{(1 + b)(2 - b)}} $	$ [\frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}} - \theta ]{(q_{D{D_i}}^{*PP})^2} $

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Table 4. Equilibrium solutions for the four supply chain structures

	Chain i	$ w_{{X_i}}^{*QP} $	$ p_{{X_i}}^{*QP} $	$ q_{{X_i}}^{*QP} $	$ \pi _{{X_{Mi}}}^{*QP} $	$ \pi _{{X_{Ri}}}^{*QP} $
CC	1	NA	$ \frac{{({b^2} - b - 2)(1 - b)(a - {c_0}) + (3{b^2} - 4){c_0}}}{{3{b^2} - 4}} $	$ \frac{{(b - 2)(a - {c_0})}}{{3{b^2} - 4}} $	$ (1 - {b^2}){(q_{C{C_1}}^{*QP})^2} $	NA
CC	2	NA	$ \frac{{({b^2} + b - 2)(a - {c_0}) + (3{b^2} - 4){c_0}}}{{3{b^2} - 4}} $	$ \frac{{({b^2} + b - 2)(a - {c_0})}}{{3{b^2} - 4}} $	$ {(q_{C{C_2}}^{*QP})^2} $	NA
CD	1	NA	$ \frac{{(1 - {b^2})(2 - b)a + (2 - {b^2}){c_0} + b(1 - {b^2})w_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}} $	$ \frac{{(2 - b)a - 2{c_0} + bw_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}} $	$ (1 - {b^2}){(q_{C{D_1}}^{*QP})^2} $	NA
CD	2	$ \frac{{(1 - b)(2 + b)a - (1 + b)(b - 2){c_0}}}{{2(2 - {b^2})}} $	$ \frac{{(1 - b)(2 + b)a + b{c_0} + 2(1 - {b^2})w_ {C{D_2}}^ {*QP} }}{{4 - 3{b^2}}} $	$ \frac{{(2 - {b^2} - b)a + b{c_0} + ({b^2} - 2)w_{C{D_2}}^{*QP}}}{{4 - 3{b^2}}} $	$ \frac{{4 - 3{b^2}}}{{2 - {b^2} }} {(q_{C{D_2}}^{*QP})^2} $	$ {(q_{C{D_2}}^{*QP})^2} $
DC	1	$ \frac{{(2 - b)a + (2 + b){c_0}}}{4} $	$ \frac{{(1 - {b^2})(2 - b)a + b(1 - {b^2}){c_0} + (2 - {b^2})w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}} $	$ \frac{{(2 - b)a + b{c_0} - 2w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}} $	$ \frac{{4 - 3{b^2}}}{2}{(q_{D{C_1}}^{*QP})^2} $	$ (1 - {b^2}){(q_{D{C_1}}^{*QP})^2} $
DC	2	NA	$ \frac{{(1 - b)(2 + b)a + 2(1 - {b^2}){c_0} + bw_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}} $	$ \frac{{(2 - {b^2} - b)a + ({b^2} - 2){c_0} + bw_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}} $	$ {(q_{D{C_2}}^{*QP})^2} $	NA
DD	1	$ \frac{{({b^3} - 5{b^2} - 2b + 8)a - ({b^3} + 4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}} $	$ \frac{{(1 - {b^2})(2 - b)a + (2 - {b^2})w_{D{D_1}}^{QP} + b(1 - {b^2})w_{D{D_2}}^{QP}}}{{4 - 3{b^2}}} $	$ \frac{{2({b^3} - 5{b^2} - 2b + 8)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}} $	$ \frac{{4 - 3{b^2}}}{2}{(q_{D{D_1}}^{*QP})^2} $	$ (1 - {b^2}){(q_{D{D_1}}^{*QP})^2} $
DD	2	$ \frac{{(8 - 5{b^2} - 2b)a - (4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}} $	$ \frac{{(1 - b)(2 + b)a + 2(1 - {b^2})w_{D{D_2}}^{QP} + bw_{D{D_1}}^{QP}}}{{4 - 3{b^2}}} $	$ \frac{{(5{b^2} + 2b - 8)({b^2} - 2)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}} $	$ \frac{{4 - 3{b^2}}}{{2 - {b^2}}}{(q_{D{D_2}}^{*QP})^2} $	$ {(q_{D{D_2}}^{*QP})^2} $

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Table 5. Payoff table for the strategic competition choice of the X supply chain structure

		Supply chain 2
		Cournot	Bertrand
Supply chain 1	Cournot	$ (\pi _{{X_{M1}}}^{QQ},\pi _{{X_{M2}}}^{QQ}) $	$ (\pi _{{X_{M1}}}^{QP},\pi _{{X_{M2}}}^{QP}) $
Supply chain 1	Bertrand	$ (\pi _{{X_{M1}}}^{PQ},\pi _{{X_{M2}}}^{PQ}) $	$ (\pi _{{X_{M1}}}^{PP},\pi _{{X_{M2}}}^{PP}) $

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Table 6. Equilibrium solutions for the four structures with economies of scale

	Chain i	$ \tilde w_{{X_i}}^{*QP} $	$ \tilde q_{{X_i}}^{*QP} $	$ \tilde \pi _{{X_{Mi}}}^{*QP} $
CC	1	NA	$ \frac{{(T + b - {b^2})(a - {c_0})}}{{AT + {b^2}}} $	$ (1 - {b^2} - \theta ){(\tilde q_{C{C_1}}^{*QP})^2} $
CC	2	NA	$ \frac{{(T + b)(a - {c_0})}}{{AT + {b^2}}} $	$ (1 - \theta ){(\tilde q_{C{C_2}}^{*QP})^2} $
CD	1	NA	$ \frac{{(2 - b)a - 2{c_0} + b\tilde w_{C{D_2}}^{*QP}}}{{ - U}} $	$ (1 - {b^2} - \theta ){(\tilde q_{C{D_1}}^{*QP})^2} $
CD	2	$ \frac{{(U - 2\theta T)(T + b)a + [U(T - b) + 2\theta bT]{c_0}}}{{2T(U - \theta T)}} $	$ \frac{{( - T - b)a + b{c_0} + T\tilde w_{C{D_2}}^{*QP}}}{{ - U}} $	$ (\frac{U}{T} - \theta ){(\tilde q_{C{D_2}}^{*QP})^2} $
DC	1	$ \frac{{(U - 2\theta T)(A - b)a + [U(b + A) - 2\theta b(T + Ab)]{c_0}}}{{2A[U + 2\theta (1 - {b^2} - \theta )]}} $	$ \frac{{(A - b)a + b{c_0} - A\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}} $	$ \frac{{U + 2\theta (1 - {b^2} - \theta )}}{{ - A}}{(\tilde q_{D{C_1}}^{*QP})^2} $
DC	2	NA	$ \frac{{(2 + b)(1 - b)a + ({b^2} - 2){c_0} + b\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}} $	$ (1 - \theta ){(\tilde q_{D{C_2}}^{*QP})^2} $
DD	1	$ \frac{{U[ - {b^3} + 5{b^2} + 2b - 8 + 2\theta (2 - {b^2})]a - (2 - {b^2})[5U + 4\theta (b - {b^2} - 3)]{c_0}}}{{(3{b^2} - 4)(9{b^2} - 16) + 2\theta (4 - {b^2})(5{b^2} - 8) + 8{\theta ^2}(2 - {b^2})}} $	$ \frac{{2(\tilde w_{D{D_1}}^{*QP} - {c_0})}}{{ - U}} $	$ (\frac{{4 - 3{b^2}}}{2} - \theta ){(\tilde q_{D{D_1}}^{*QP})^2} $
DD	2	$ \frac{{U(b - 2)a - 2(U - 4\theta ){c_0} + 4(U - 2\theta )\tilde w_{D{D_1}}^{*QP}}}{{Ub}} $	$ \frac{{(2 - {b^2})(\tilde w_{D{D_2}}^{*QP} - {c_0})}}{{2\theta {b^2} - U}} $	$ (\frac{{4 - 3{b^2}}}{{2 - {b^2}}} - \theta ){(\tilde q_{D{D_2}}^{*QP})^2} $

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