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Supply chain coordination based on revenue-sharing contract with a loss-averse retailer and capital constraint

  • Optimization
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Abstract

The paper aims to provide a theoretical basis for the application of revenue-sharing contract under bounded rationality and capital constraints. We consider an uncooperative ordering model in a supplier-Stackelberg game and coordination strategy with revenue-sharing contract for a loss-averse and capital-constrained retailer. We drive the existence and uniqueness conditions of the optimal solutions under bank financing and revenue-sharing contract. We also develop a series of propositions and corollaries to determine the optimal solutions and offer some managerial insights. The key contribution of the paper is to deepen and expand the revenue-sharing contract under the risk-neutral assumption, and to provide a theoretical basis for the application of revenue-sharing contract under bounded rationality and capital constraints. We find that the revenue-sharing ratio of loss-averse and capital-constrained retailer is larger than that of neutral retailer and the expected utility of loss-averse and capital-constrained retailer is larger than that of neutral retailer under coordination strategy with revenue-sharing contract.

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Funding

This work is supported by the National Natural Science Foundation of China (72021001, 72174019), Shandong Provincial Natural Science Foundation of China (ZR2020MG008), Shandong Humanities and Social Science Fund of China (2022-JCGL-03), Qingdao City Social Science Planning Research Project (QDSKL2201225), Humanities and Social Science Fund of Qingdao University of Science and Technology (22XA01).

Author information

Authors and Affiliations

  1. School of Economics and Management, Qingdao University of Science and Technology, 99 Songling Rd., Laoshan Dist., Qingdao, 266061, People’s Republic of China

    Chengfeng Wu, Shuaicheng Lin & Chunfeng Xu

  2. School of Economics and Management, Beihang University, 37 Xueyuan Rd., Haidian Dist., Beijing, 100191, People’s Republic of China

    Qiuhong Zhao

Authors
  1. Chengfeng Wu
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  2. Qiuhong Zhao
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  4. Chunfeng Xu
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Contributions

Investigation, CW; methodology, CW; numerical calculation, CX; formal analysis, CX; supervision, QZ; introduction writing, SL; graphics processing, SL. All authors read and approved the revised manuscript.

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Correspondence to Chengfeng Wu.

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Appendices

Appendix A

Proof of Corollary 1

According to Eq. (6), we know \({q}_{1}\left({w}_{1}\right)\) is an implicit function of \({w}_{1}\). Taking the derivative of the right and left sides of Eq. (6) with respect to \({w}_{1}\), we have the following expression:

$$f\left({q}_{1}\left({w}_{1}\right)\right)\frac{d{q}_{1}\left({w}_{1}\right)}{d{w}_{1}}=-\frac{\left(1+\theta \right)\left(1+\left(\lambda -1\right)F\left({A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\right)\right)+\left(\lambda -1\right)\left({w}_{1}+{w}_{1}\theta -s\right)f\left({A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\right)\frac{d{A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)}{d{w}_{1}}}{p+{c}_{u}-s},$$
(25)

where \(\frac{d{A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)}{d{w}_{1}}=\frac{\left(1+\theta \right){q}_{1}\left({w}_{1}\right)+\left({w}_{1}+{w}_{1}\theta -s\right)\frac{d{q}_{1}\left({w}_{1}\right)}{d{w}_{1}}}{p-s}\).

By applying simplification and merging similar items, the first derivative function \({q}_{1}\left({w}_{1}\right)\) with respect to \({w}_{1}\) will be given by

$$\frac{d{q}_{1}\left({w}_{1}\right)}{d{w}_{1}}=-\frac{\left(1+\theta \right)\left(p-s\right)\left(1+\left(\lambda -1\right)F\left({A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\right)\right)+\left(\lambda -1\right)\left(1+\theta \right)\left({w}_{1}+{w}_{1}\theta -s\right){q}_{1}\left({w}_{1}\right)f\left({A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\right)}{f\left({q}_{1}\left({w}_{1}\right)\right)\left(p+{c}_{u}-s\right)\left(p-s\right)+\left(\lambda -1\right){\left({w}_{1}+{w}_{1}\theta -s\right)}^{2}f\left({A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\right)}<0.$$
(26)

Corollary 1 can be proved.

Appendix B

Proof of Proposition 1

The supplier’s goal is to get the optimal wholesale price such that its total profit is maximized. According to Eq. (7), we can obtain the first derivative function and second derivative function \({\pi }_{s1}\) with respect to \({w}_{1}\), which are given by, respectively,

$$\frac{d{\pi }_{s1}}{d{w}_{1}}={q}_{1}\left({w}_{1}\right)+\left({w}_{1}-c\right)\frac{d{q}_{1}\left({w}_{1}\right)}{d{w}_{1}},$$
(27)
$$\frac{{d}^{2}{\pi }_{s1}}{d{w}_{1}^{2}}=2\frac{d{q}_{1}\left({w}_{1}\right)}{d{w}_{1}}+\left({w}_{1}-c\right)\frac{{d}^{2}{q}_{1}\left({w}_{1}\right)}{d{w}_{1}^{2}}.$$
(28)

To know the positive and negative values of \(\frac{{d}^{2}{\pi }_{s1}}{d{w}_{1}^{2}}\), we need to know the expression of \(\frac{{d}^{2}{q}_{1}\left({w}_{1}\right)}{d{w}_{1}^{2}}\). Therefore, taking the derivative of the right and left sides of Eq. (26) with respect to \({w}_{1}\), the second derivative function \({q}_{1}\left({w}_{1}\right)\) with respect to \({w}_{1}\) will be given by

$$\frac{{d}^{2}{q}_{1}\left({w}_{1}\right)}{d{w}_{1}^{2}}=\frac{2\left(1-\lambda \right)\left(1+\theta \right)f\left({A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\right)\left(\left(1+\theta \right){q}_{1}\left({w}_{1}\right)+2\left({w}_{1}+{w}_{1}\theta -s\right)\frac{d{q}_{1}\left({w}_{1}\right)}{d{w}_{1}}\right)}{f\left({q}_{1}\left({w}_{1}\right)\right)\left(p+{c}_{u}-s\right)\left(p-s\right)+\left(\lambda -1\right){\left({w}_{1}+{w}_{1}\theta -s\right)}^{2}f\left({A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\right)}.$$
(29)

Next, substituting the expressions of \(\frac{{d}^{2}{q}_{1}\left({w}_{1}\right)}{d{w}_{1}^{2}}\) and \(\frac{d{q}_{1}\left({w}_{1}\right)}{d{w}_{1}}\) into Eq. (29), we can obtain the new function expression of \(\frac{{d}^{2}{\pi }_{s1}}{d{w}_{1}^{2}}\), which is given by

$$\frac{{d}^{2}{\pi }_{s1}}{d{w}_{1}^{2}}=2\frac{\begin{array}{c}\left(1+\theta \right){\left(\lambda -1\right)}^{2}{\left({w}_{1}+{w}_{1}\theta -s\right)}^{2}{q}_{1}f{\left({A}_{1}\right)}^{2}\left(s-c\left(1+\theta \right)\right)\\ -\left(\lambda -1\right)\left(1+\theta \right){q}_{1}f\left({A}_{1}\right)f\left({q}_{1}\right)\left(p+{c}_{u}-s\right)\left(p-s\right)\left(\left(2{w}_{1}-c\right)\left(1+\theta \right)-s\right)\\ -\left(1+\theta \right)\left(p-s\right)\left(1+\left(\lambda -1\right)F\left({A}_{1}\right)\right)\left(f\left({q}_{1}\right)\left(p+{c}_{u}-s\right)\left(p-s\right)-f\left({A}_{1}\right)\left({w}_{1}+{w}_{1}\theta -s\right)\left(\left({w}_{1}-2c\right)\left(1+\theta \right)+s\right)\right)\end{array}}{{\left(f\left({q}_{1}\right)\left(p+{c}_{u}-s\right)\left(p-s\right)+\left(\lambda -1\right){\left({w}_{1}+{w}_{1}\theta -s\right)}^{2}f\left({A}_{1}\right)\right)}^{2}},$$
(30)

where \({q}_{1}={q}_{1}\left({w}_{1}\right)\), \({A}_{1}={A}_{1}\left({q}_{1}\left({w}_{1}\right)\right)\). According to Table 1, we know that \(p+{c}_{u}-s>p-s>{w}_{1}\left(1+\theta \right)-s>{w}_{1}\left(1+\theta \right)-c\left(1+\theta \right)>{w}_{1}+s-2c\left(1+\theta \right)>0\). Therefore, in general, the sum of the three terms in the numerator of Eq. (30) are negative, i.e., \(\frac{{d}^{2}{\pi }_{s1}}{d{w}_{1}^{2}}<0\). Consequently, when the first-order condition \(\frac{d{\pi }_{s1}}{d{w}_{1}}=0\), we obtain the optimal wholesale price \({w}_{1}^{*}\). Similarly, \({w}_{1}^{*}\) does not have a closed-form expression. Proposition 1 can be proved.

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Wu, C., Zhao, Q., Lin, S. et al. Supply chain coordination based on revenue-sharing contract with a loss-averse retailer and capital constraint. Soft Comput 28, 3015–3028 (2024). https://doi.org/10.1007/s00500-023-09229-y

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