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doi: 10.3934/jimo.2021126
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## Inter-organizational contract control of advertising strategies in the supply chain

 School of Economics and Management, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: Yafei Zu

Received  September 2020 Revised  May 2021 Early access August 2021

Fund Project: The author is supported by the Youth Fund of the MOE (Ministry of Education in China) Project of Humanities and Social Sciences (Grant 20YJC630245), by the Fundamental Research Funds for the Central Universities (Grant 30919013229), by the Youth Fund of the School of Economics and Management, Nanjing University of Science and Technology (Grant JGQN2003), by the National Natural Science Foundation of China (Grant 61901105), and by the Natural Science Foundation of Jiangsu Province (Grant BK20190343)

Advertising has a crucial impact on a product's goodwill. To further improve a product's goodwill and make more profit, member firms in the supply chain use various contracts to coordinate the channel. Considering the dynamic effect of advertising, this paper studies a two-level supply chain consisting of one manufacturer and one retailer. The two members focus on maximizing their profits through advertising and pricing strategies under two types of contracts: the wholesale price contract and the consignment contract. The Stackelberg differential game is introduced, and the optimal advertising effort, wholesale and retail pricing strategies in the two situations are studied. Numerical examples and sensitivity analyses are conducted to explore the models further. The results show that the retailer's revenue proportion and the product's goodwill according to consumers significantly affect the strategies and the contract choice of the partner firms in the supply chain. A proportion of too high or too low revenue may lead to a contract selection conflict between the two partner firms. However, when consumers care more about the product's goodwill, this contract selection conflict can be weakened.

Citation: Yafei Zu. Inter-organizational contract control of advertising strategies in the supply chain. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021126
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##### References:
The overall mechanism
Comparison of situations with a different $\varphi$
Notations and definitions
 Notation Definition Decision variables $A(t)$ Manufacturer's advertising effort for the product, $A(t)>0$. $s(t)$ Retailer's markup (i.e., retail margin) under the wholesale price contract, $s(t)>0$. $w(t)$ Wholesale price of the product under the wholesale price contract, $w(t)>0$. $p(t)$ Retail price of the product under the consignment contract, $p(t)> 0$. Parameters and other variables $\alpha$ The coefficient associated with the product's retail price in the demand function, $\alpha> 0$. $\beta$ The coefficient associated with the product's goodwill in the demand function, $\beta>0$. $\theta$ The coefficient associated with the manufacturer's advertising effort in the product goodwill function, $\theta>0$. $\varphi$ Retailer's proportion of sales revenue under the consignment contract, $0< \varphi< 1$. $\delta$ The decay rate of the product's goodwill, $\delta> 0$. $\mu$ Cost parameter associated with the manufacturer's advertising effort, $\mu>0$. $D(t)$ Demand for the product at time $t$, with initial demand $D_0> 0$. $G(t)$ Goodwill of the product at time $t$, with $G_{0}\geq 0$. $J_R, J_M$ Objective functions (expressed in net profit) of the retailer and the manufacturer, respectively, for $t \in [0, +\infty)$.
 Notation Definition Decision variables $A(t)$ Manufacturer's advertising effort for the product, $A(t)>0$. $s(t)$ Retailer's markup (i.e., retail margin) under the wholesale price contract, $s(t)>0$. $w(t)$ Wholesale price of the product under the wholesale price contract, $w(t)>0$. $p(t)$ Retail price of the product under the consignment contract, $p(t)> 0$. Parameters and other variables $\alpha$ The coefficient associated with the product's retail price in the demand function, $\alpha> 0$. $\beta$ The coefficient associated with the product's goodwill in the demand function, $\beta>0$. $\theta$ The coefficient associated with the manufacturer's advertising effort in the product goodwill function, $\theta>0$. $\varphi$ Retailer's proportion of sales revenue under the consignment contract, $0< \varphi< 1$. $\delta$ The decay rate of the product's goodwill, $\delta> 0$. $\mu$ Cost parameter associated with the manufacturer's advertising effort, $\mu>0$. $D(t)$ Demand for the product at time $t$, with initial demand $D_0> 0$. $G(t)$ Goodwill of the product at time $t$, with $G_{0}\geq 0$. $J_R, J_M$ Objective functions (expressed in net profit) of the retailer and the manufacturer, respectively, for $t \in [0, +\infty)$.
Discussion of the function
 Parameter condition Condition of $\varphi$1 Sign of $y$ $bc>ad$ $\varphi< -b/a$ or $\varphi> - d/c$ $y>0$ $-b/a< \varphi< -d/c$ $y<0$ $\varphi = -b/a$ $y=0$ $bc - b/a$ $y>0$ $-d/c< \varphi< -b/a$ $y<0$ $\varphi = -b/a$ $y=0$ $bc=ad$ Arbitrary $y=a/c>0$ 1 Parameter $\varphi$ needs to meet the condition $\varphi \in (0,1)$ first to ensure the practical implications.
 Parameter condition Condition of $\varphi$1 Sign of $y$ $bc>ad$ $\varphi< -b/a$ or $\varphi> - d/c$ $y>0$ $-b/a< \varphi< -d/c$ $y<0$ $\varphi = -b/a$ $y=0$ $bc - b/a$ $y>0$ $-d/c< \varphi< -b/a$ $y<0$ $\varphi = -b/a$ $y=0$ $bc=ad$ Arbitrary $y=a/c>0$ 1 Parameter $\varphi$ needs to meet the condition $\varphi \in (0,1)$ first to ensure the practical implications.
Sensitivity analysis: wholesale price contract
 $p(0)$ $p(T)$ $w(0)$ $w(T)$ $A(0)$ $A(T)$ $J_R$ $J_M$ $\alpha$ 0.8 9.3750 9.5367 3.1250 3.1789 1.0427 0.0000 16.0238 7.8110 0.7 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 0.6 12.5003 12.7892 4.1667 4.2630 1.3959 0.0000 21.5476 10.4131 $\beta$ 0.6 10.7143 11.0215 3.5714 3.6738 1.4413 0.0000 18.6181 8.9239 0.5 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 0.4 10.7143 10.8489 3.5714 3.6163 0.9502 0.0000 18.1885 8.9277 $\theta$ 0.9 10.7143 10.9834 3.5714 3.6611 1.3479 0.0000 18.5230 8.9250 0.8 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 0.7 10.7143 10.8756 3.5714 3.6252 1.0411 0.0000 18.2548 8.9273 $\delta$ 0.6 10.7143 10.8901 3.5714 3.6300 1.0869 0.0000 18.3113 8.9269 0.5 10.7143 10.9070 3.5714 3.6357 1.1385 0.0000 18.3439 8.9267 0.4 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 $G_0$ 0 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 5 13.3929 12.7621 4.4643 4.2540 1.4428 0.0000 26.7325 12.9890 10 16.0714 14.5983 5.3571 4.8661 1.6918 0.0000 36.6622 17.8197 $T$ 1 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 2 10.7143 11.3267 3.5714 3.7756 2.0493 0.0000 39.0824 17.8136 3 10.7143 11.7422 3.5714 3.9141 2.6798 0.0000 62.9809 26.5708
 $p(0)$ $p(T)$ $w(0)$ $w(T)$ $A(0)$ $A(T)$ $J_R$ $J_M$ $\alpha$ 0.8 9.3750 9.5367 3.1250 3.1789 1.0427 0.0000 16.0238 7.8110 0.7 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 0.6 12.5003 12.7892 4.1667 4.2630 1.3959 0.0000 21.5476 10.4131 $\beta$ 0.6 10.7143 11.0215 3.5714 3.6738 1.4413 0.0000 18.6181 8.9239 0.5 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 0.4 10.7143 10.8489 3.5714 3.6163 0.9502 0.0000 18.1885 8.9277 $\theta$ 0.9 10.7143 10.9834 3.5714 3.6611 1.3479 0.0000 18.5230 8.9250 0.8 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 0.7 10.7143 10.8756 3.5714 3.6252 1.0411 0.0000 18.2548 8.9273 $\delta$ 0.6 10.7143 10.8901 3.5714 3.6300 1.0869 0.0000 18.3113 8.9269 0.5 10.7143 10.9070 3.5714 3.6357 1.1385 0.0000 18.3439 8.9267 0.4 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 $G_0$ 0 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 5 13.3929 12.7621 4.4643 4.2540 1.4428 0.0000 26.7325 12.9890 10 16.0714 14.5983 5.3571 4.8661 1.6918 0.0000 36.6622 17.8197 $T$ 1 10.7143 10.9259 3.5714 3.6420 1.1937 0.0000 18.3797 8.9264 2 10.7143 11.3267 3.5714 3.7756 2.0493 0.0000 39.0824 17.8136 3 10.7143 11.7422 3.5714 3.9141 2.6798 0.0000 62.9809 26.5708
Sensitivity analysis: consignment contract
 $p(0)$ $p(T)$ $w(0)$ $w(T)$ $A(0)$ $A(T)$ $J_R$ $J_M$ $\alpha$ 0.8 6.2500 6.4241 5.0000 5.1393 1.6806 0.0000 6.5087 25.5112 0.7 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.6 8.3333 8.6454 6.6667 6.9164 2.2555 0.0000 8.7988 34.2495 $\beta$ 0.6 7.1429 7.4757 5.7143 5.9805 2.3344 0.0000 7.6410 29.5481 0.5 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.4 7.1429 7.2874 5.7143 5.8299 1.5285 0.0000 7.3570 28.9960 $\theta$ 0.9 7.1429 7.4339 5.7143 5.9471 2.1798 0.0000 7.5775 29.4257 0.8 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.7 7.1429 7.3164 5.7143 5.8531 1.6765 0.0000 7.4004 29.0811 $\delta$ 0.6 7.1429 7.3321 5.7143 5.8657 1.7515 0.0000 7.4374 29.1536 0.5 7.1429 7.3505 5.7143 5.8804 1.8358 0.0000 7.4589 29.1955 0.4 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 $G_0$ 0 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 5 8.9286 8.6119 7.1429 6.8896 2.3277 0.0000 10.8800 42.5342 10 10.7143 9.8528 8.5714 7.8823 2.7293 0.0000 14.9184 58.3372 $T$ 1 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 2 7.1429 7.8186 5.7143 6.2549 3.3651 0.0000 16.5499 61.4808 3 7.1429 8.3091 5.7143 6.6473 4.4927 0.0000 28.0220 97.9091 $\varphi$ 0.2 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.4 7.1429 7.3128 4.2857 4.3877 1.4365 0.0000 14.7899 21.8029 0.6 7.1429 7.2553 2.8571 2.9021 0.9523 0.0000 21.9276 14.4509
 $p(0)$ $p(T)$ $w(0)$ $w(T)$ $A(0)$ $A(T)$ $J_R$ $J_M$ $\alpha$ 0.8 6.2500 6.4241 5.0000 5.1393 1.6806 0.0000 6.5087 25.5112 0.7 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.6 8.3333 8.6454 6.6667 6.9164 2.2555 0.0000 8.7988 34.2495 $\beta$ 0.6 7.1429 7.4757 5.7143 5.9805 2.3344 0.0000 7.6410 29.5481 0.5 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.4 7.1429 7.2874 5.7143 5.8299 1.5285 0.0000 7.3570 28.9960 $\theta$ 0.9 7.1429 7.4339 5.7143 5.9471 2.1798 0.0000 7.5775 29.4257 0.8 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.7 7.1429 7.3164 5.7143 5.8531 1.6765 0.0000 7.4004 29.0811 $\delta$ 0.6 7.1429 7.3321 5.7143 5.8657 1.7515 0.0000 7.4374 29.1536 0.5 7.1429 7.3505 5.7143 5.8804 1.8358 0.0000 7.4589 29.1955 0.4 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 $G_0$ 0 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 5 8.9286 8.6119 7.1429 6.8896 2.3277 0.0000 10.8800 42.5342 10 10.7143 9.8528 8.5714 7.8823 2.7293 0.0000 14.9184 58.3372 $T$ 1 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 2 7.1429 7.8186 5.7143 6.2549 3.3651 0.0000 16.5499 61.4808 3 7.1429 8.3091 5.7143 6.6473 4.4927 0.0000 28.0220 97.9091 $\varphi$ 0.2 7.1429 7.3711 5.7143 5.8969 1.9260 0.0000 7.4825 29.2415 0.4 7.1429 7.3128 4.2857 4.3877 1.4365 0.0000 14.7899 21.8029 0.6 7.1429 7.2553 2.8571 2.9021 0.9523 0.0000 21.9276 14.4509
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