Inter-organizational contract control of advertising strategies in the supply chain

Yafei Zu

doi:10.3934/jimo.2021126

Article Contents

2022, Volume 18, Issue 5: 3561-3585. Doi: 10.3934/jimo.2021126

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Inter-organizational contract control of advertising strategies in the supply chain

Yafei Zu^,

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

^* Corresponding author: Yafei Zu
^* Corresponding author: Yafei Zu

Received: August 31, 2020

Revised: April 30, 2021

Early access: August 2021

Published: November 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. The overall mechanism

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Figure 2. Comparison of situations with a different $ \varphi $

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Table 1. 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
$ s(t) $	price contract, $ s(t)>0 $.
$ w(t) $	Wholesale price of the product under the wholesale price contract,
$ w(t) $	$ 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
$ \alpha $	demand function, $ \alpha> 0 $.
$ \beta $	The coefficient associated with the product's goodwill in the
$ \beta $	demand function, $ \beta>0 $.
$ \theta $	The coefficient associated with the manufacturer's advertising effort
$ \theta $	in the product goodwill function, $ \theta>0 $.
$ \varphi $	Retailer's proportion of sales revenue under the consignment contract,
$ \varphi $	$ 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 $	$ \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
$ J_R, J_M $	the manufacturer, respectively, for $ t \in [0, +\infty) $.

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Table 2. 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<ad $	$ \varphi< -d/c $ or $ \varphi> - 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.

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Table 3. 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

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Table 4. 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

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