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doi: 10.3934/jimo.2021194
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## 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

Received  April 2021 Revised  August 2021 Early access November 2021

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.

Citation: Mingxia Li, Kebing Chen, Shengbin Wang. Retail outsourcing strategy in Cournot & Bertrand retail competitions with economies of scale. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021194
##### References:

show all references

##### References:
Possible regions for equilibrium structures in the pure Cournot competition
Possible regions for the more profitable competition mode
Possible regions for equilibrium structure in pure Bertrand competition
Possible regions for equilibrium structures in the Cournot-Bertrand competition
Possible regions for the equilibrium competition mode
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses direct sales
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses retail outsourcing
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses direct sales
The effect of the retail outsourcing strategy on $M_{1}$'s profit when $M_{2}$ uses retail outsourcing
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 Bertrand Cournot 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 √ √ √ √ √ √ √
 Article Model Market competition mode Economies of scale Product complementation Outsourcing decision Structure equilbrium Competition mode equilbrium Bertrand Cournot 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 √ √ √ √ √ √ √
Equilibrium solutions for the four different structures
 X= CC CD(DC) DD Chain i Chain 1(2) Chain 2(1) Chain i $q_{{X_i}}^{*QQ}$ $\frac{{a - {c_0}}}{{A + b}}$ $\frac{{({A^2} + 2A - {b^2} - Ab)(a - {c_0})}}{{A({A^2} + 2A - 2{b^2})}}$ $\frac{{(A - b)(a - {c_0})}}{{{A^2} + 2A - 2{b^2}}}$ $\frac{{2({c_0} - a)}}{{(b - 4)(b + 2) + 4\theta }}$ $\pi _{{X_{Mi}}}^{*QQ}$ $(1 - \theta ){(q_{C{C_i}}^{QQ})^2}$ $(1 - \theta ){(q_{C{D_1}}^{QQ})^2}$ $\frac{{(2 - \theta )A - {b^2}}}{A}{(q_{C{D_2}}^{QQ})^2}$ $\frac{{(4 - 2\theta - {b^2})}}{2}{(q_{D{D_i}}^{QQ})^2}$
 X= CC CD(DC) DD Chain i Chain 1(2) Chain 2(1) Chain i $q_{{X_i}}^{*QQ}$ $\frac{{a - {c_0}}}{{A + b}}$ $\frac{{({A^2} + 2A - {b^2} - Ab)(a - {c_0})}}{{A({A^2} + 2A - 2{b^2})}}$ $\frac{{(A - b)(a - {c_0})}}{{{A^2} + 2A - 2{b^2}}}$ $\frac{{2({c_0} - a)}}{{(b - 4)(b + 2) + 4\theta }}$ $\pi _{{X_{Mi}}}^{*QQ}$ $(1 - \theta ){(q_{C{C_i}}^{QQ})^2}$ $(1 - \theta ){(q_{C{D_1}}^{QQ})^2}$ $\frac{{(2 - \theta )A - {b^2}}}{A}{(q_{C{D_2}}^{QQ})^2}$ $\frac{{(4 - 2\theta - {b^2})}}{2}{(q_{D{D_i}}^{QQ})^2}$
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 $\frac{{Sa - (1 + b)(2 - b)({b^2} - 2){c_0}}}{{S - (1 + b)(2 - b)({b^2} - 2)}}$ $\frac{{a - w_{D{D_i}}^{*PP}}}{{(1 + b)(2 - b)}}$ $[\frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}} - \theta ]{(q_{D{D_i}}^{*PP})^2}$
 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 $\frac{{Sa - (1 + b)(2 - b)({b^2} - 2){c_0}}}{{S - (1 + b)(2 - b)({b^2} - 2)}}$ $\frac{{a - w_{D{D_i}}^{*PP}}}{{(1 + b)(2 - b)}}$ $[\frac{{(4 - {b^2})(1 - {b^2})}}{{2 - {b^2}}} - \theta ]{(q_{D{D_i}}^{*PP})^2}$
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 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 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 $\frac{{(2 - b)a + (2 + b){c_0}}}{4}$ $\frac{{(1 - {b^2})(2 - b)a + b(1 - {b^2}){c_0} + (2 - {b^2})w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - b)a + b{c_0} - 2w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{C_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{C_1}}^{*QP})^2}$ 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 $\frac{{({b^3} - 5{b^2} - 2b + 8)a - ({b^3} + 4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - {b^2})(2 - b)a + (2 - {b^2})w_{D{D_1}}^{*QP} + b(1 - {b^2})w_{D{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{2({b^3} - 5{b^2} - 2b + 8)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{D_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{D_1}}^{*QP})^2}$ 2 $\frac{{(8 - 5{b^2} - 2b)a - (4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - b)(2 + b)a + 2(1 - {b^2})w_{D{D_2}}^{*QP} + bw_{D{D_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(5{b^2} + 2b - 8)({b^2} - 2)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{{2 - {b^2}}}{(q_{D{D_2}}^{*QP})^2}$ ${(q_{D{D_2}}^{*QP})^2}$
 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 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 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 $\frac{{(2 - b)a + (2 + b){c_0}}}{4}$ $\frac{{(1 - {b^2})(2 - b)a + b(1 - {b^2}){c_0} + (2 - {b^2})w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(2 - b)a + b{c_0} - 2w_{D{C_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{C_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{C_1}}^{*QP})^2}$ 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 $\frac{{({b^3} - 5{b^2} - 2b + 8)a - ({b^3} + 4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - {b^2})(2 - b)a + (2 - {b^2})w_{D{D_1}}^{*QP} + b(1 - {b^2})w_{D{D_2}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{2({b^3} - 5{b^2} - 2b + 8)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{2}{(q_{D{D_1}}^{*QP})^2}$ $(1 - {b^2}){(q_{D{D_1}}^{*QP})^2}$ 2 $\frac{{(8 - 5{b^2} - 2b)a - (4{b^2} - 2b - 8){c_0}}}{{16 - 9{b^2}}}$ $\frac{{(1 - b)(2 + b)a + 2(1 - {b^2})w_{D{D_2}}^{*QP} + bw_{D{D_1}}^{*QP}}}{{4 - 3{b^2}}}$ $\frac{{(5{b^2} + 2b - 8)({b^2} - 2)(a - {c_0})}}{{(4 - 3{b^2})(16 - 9{b^2})}}$ $\frac{{4 - 3{b^2}}}{{2 - {b^2}}}{(q_{D{D_2}}^{*QP})^2}$ ${(q_{D{D_2}}^{*QP})^2}$
Payoff table for the strategic competition choice of the X supply chain structure
 Supply chain 2 Cournot Bertrand Supply chain 1 Cournot $(\pi _{{X_{M1}}}^{*QQ},\pi _{{X_{M2}}}^{*QQ})$ $(\pi _{{X_{M1}}}^{*QP},\pi _{{X_{M2}}}^{*QP})$ Bertrand $(\pi _{{X_{M1}}}^{*PQ},\pi _{{X_{M2}}}^{*PQ})$ $(\pi _{{X_{M1}}}^{*PP},\pi _{{X_{M2}}}^{*PP})$
 Supply chain 2 Cournot Bertrand Supply chain 1 Cournot $(\pi _{{X_{M1}}}^{*QQ},\pi _{{X_{M2}}}^{*QQ})$ $(\pi _{{X_{M1}}}^{*QP},\pi _{{X_{M2}}}^{*QP})$ Bertrand $(\pi _{{X_{M1}}}^{*PQ},\pi _{{X_{M2}}}^{*PQ})$ $(\pi _{{X_{M1}}}^{*PP},\pi _{{X_{M2}}}^{*PP})$
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 $\frac{{(T + b - {b^2})(a - {c_0})}}{{AT + {b^2}}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{C_1}}^{*QP})^2}$ 2 NA $\frac{{(T + b)(a - {c_0})}}{{AT + {b^2}}}$ $(1 - \theta ){(\tilde q_{C{C_2}}^{*QP})^2}$ CD 1 NA $\frac{{(2 - b)a - 2{c_0} + b\tilde w_{C{D_2}}^{*QP}}}{{ - U}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{D_1}}^{*QP})^2}$ 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 $\frac{{(U - 2\theta T)(A - b)a + [U(b + A) - 2\theta b(T + Ab)]{c_0}}}{{2A[U + 2\theta (1 - {b^2} - \theta )]}}$ $\frac{{(A - b)a + b{c_0} - A\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}}$ $\frac{{U + 2\theta (1 - {b^2} - \theta )}}{{ - A}}{(\tilde q_{D{C_1}}^{*QP})^2}$ 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 $\frac{{U[ - {b^3} + 5{b^2} + 2b - 8 + 2\theta (2 - {b^2})]a - (2 - {b^2})[5U + 4\theta (b - {b^2} - 3)]{c_0}}}{{(3{b^2} - 4)(9{b^2} - 16) + 2\theta (4 - {b^2})(5{b^2} - 8) + 8{\theta ^2}(2 - {b^2})}}$ $\frac{{2(\tilde w_{D{D_1}}^{*QP} - {c_0})}}{{ - U}}$ $(\frac{{4 - 3{b^2}}}{2} - \theta ){(\tilde q_{D{D_1}}^{*QP})^2}$ 2 $\frac{{U(b - 2)a - 2(U - 4\theta ){c_0} + 4(U - 2\theta )\tilde w_{D{D_1}}^{*QP}}}{{Ub}}$ $\frac{{(2 - {b^2})(\tilde w_{D{D_2}}^{*QP} - {c_0})}}{{2\theta {b^2} - U}}$ $(\frac{{4 - 3{b^2}}}{{2 - {b^2}}} - \theta ){(\tilde q_{D{D_2}}^{*QP})^2}$
 Chain i $\tilde w_{{X_i}}^{*QP}$ $\tilde q_{{X_i}}^{*QP}$ $\tilde \pi _{{X_{Mi}}}^{*QP}$ CC 1 NA $\frac{{(T + b - {b^2})(a - {c_0})}}{{AT + {b^2}}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{C_1}}^{*QP})^2}$ 2 NA $\frac{{(T + b)(a - {c_0})}}{{AT + {b^2}}}$ $(1 - \theta ){(\tilde q_{C{C_2}}^{*QP})^2}$ CD 1 NA $\frac{{(2 - b)a - 2{c_0} + b\tilde w_{C{D_2}}^{*QP}}}{{ - U}}$ $(1 - {b^2} - \theta ){(\tilde q_{C{D_1}}^{*QP})^2}$ 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 $\frac{{(U - 2\theta T)(A - b)a + [U(b + A) - 2\theta b(T + Ab)]{c_0}}}{{2A[U + 2\theta (1 - {b^2} - \theta )]}}$ $\frac{{(A - b)a + b{c_0} - A\tilde w_{D{C_1}}^{*QP}}}{{2\theta {b^2} - U}}$ $\frac{{U + 2\theta (1 - {b^2} - \theta )}}{{ - A}}{(\tilde q_{D{C_1}}^{*QP})^2}$ 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 $\frac{{U[ - {b^3} + 5{b^2} + 2b - 8 + 2\theta (2 - {b^2})]a - (2 - {b^2})[5U + 4\theta (b - {b^2} - 3)]{c_0}}}{{(3{b^2} - 4)(9{b^2} - 16) + 2\theta (4 - {b^2})(5{b^2} - 8) + 8{\theta ^2}(2 - {b^2})}}$ $\frac{{2(\tilde w_{D{D_1}}^{*QP} - {c_0})}}{{ - U}}$ $(\frac{{4 - 3{b^2}}}{2} - \theta ){(\tilde q_{D{D_1}}^{*QP})^2}$ 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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