# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021104
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## Financing and ordering decisions in a capital-constrained and risk-averse supply chain for the monopolist and non-monopolist supplier

 1 School of Business Administration, Hunan University, Changsha, Hunan Province 410082, China 2 School of Marketing and Logistics Management, Nanjing University of Finance and Economics, Nanjing, Jiangsu Province 210023, China

* Corresponding author: ottoyang@126.com (Honglin Yang)

Received  December 2020 Revised  March 2021 Early access May 2021

Fund Project: This research is supported by the National Natural Science Foundation of China under Grant Nos. 72071072, 71901117 and 71790593 and by the Ministry of Education in China of Humanities and Social Science Project under Grant No. 19YJC630242

The paper focuses on a supply chain consisting of one supplier and one capital-constrained retailer. The retailer can solve the limited working capital problem through a bank, an investor, or its supplier. When facing business risks brought by uncertain demand, the retailer and the supplier are risk-averse behavior. To better explore the financing decision, we consider the supplier has two different cases: monopolist firm and non-monopolist firm. We first use the CVaR criterion to incorporate the members' risk-averse behavior into the objectives function. Then the equilibrium results of the supply chain are derived under three financing schemes, respectively. Our analysis finds when the supplier is a relatively low risk-averse monopolist firm, trade credit financing is the unique financing equilibrium. When the supplier is a relatively high risk-averse monopolist firm and non-monopolist firm, schemes, if the valuation level is relatively low, all members prefer bank credit financing. Otherwise, the members prefer equity financing. By tuning the valuation level, we obtain the conditions in which the supply chain realizes Pareto improvement relative to the other two financing schemes. Finally, we use numerical analysis to verify the above theoretical results.

Citation: Zhiyuan Zhen, Honglin Yang, Wenyan Zhuo. Financing and ordering decisions in a capital-constrained and risk-averse supply chain for the monopolist and non-monopolist supplier. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021104
##### References:
 [1] S. Asian and X. Nie, Coordination in supply chains with uncertain demand and disruption risks: Existence, analysis, and insights, IEEE Transactions on Systems Man & Cybernetics Systems, 44 (2014), 1139-1154. [2] M. Ayyagari, T. Beck and A. Demirguc-Kunt, Small and medium enterprises across the globe, Small Business Economics, (2007), 415–434. [3] P. Bolton and X. Freixas, Equity, bonds, and bank debt: Capital structure and financial market equilibrium under asymmetric information, Journal of Political Economy, 108 (2000), 324-351. [4] H. L. Chang and B.-D. Rhee, Trade credit for supply chain coordination, European Journal of Operational Research, 214 (2011), 136-146.  doi: 10.1016/j.ejor.2011.04.004. [5] X. Chen, A model of trade credit in a capital-constrained distribution channel, International Journal of Production Economics, 159 (2015), 347-357.  doi: 10.1007/s10479-014-1602-x. [6] X. Chen and A. Wang, Trade credit contract with limited liability in the supply chain with budget constraints, Annals of Operations Research, 196 (2012), 153-165.  doi: 10.1007/s10479-012-1119-0. [7] Y. Chen, M. Xu and Z. G. Zhang, Technical note–a risk-averse newsvendor model under the CVaR criterion, Operations Research, 57 (2009), 1040-1044. [8] J. Cong, T. Pang and H. Peng, Optimal strategies for capital constrained low-carbon supply chain under yield uncertainty, Journal of Cleaner Production, 256 (2009), 120339. [9] S. E. Curcuru, C. P. Thomas, F. E. Warnock and J. Wongswan, US international equity investment and past and prospective returns, Operations Research, 101 (2011), 3440-3455. [10] S. Deng and Z. Zheng, Optimal production decision for a risk-averse manufacturer faced with random yield and stochastic demand, International Transactions in Operational Research, 27 (2020), 1622-1637.  doi: 10.1111/itor.12483. [11] B. C. Giri and S. Sharma, Optimal ordering policy for an inventory system with linearly increasing demand and allowable shortages under two levels trade credit financing, Operational Research, 16 (2016), 25-50. [12] B. Jing, X. Chen and G. Cai, Equilibrium financing in a distribution channel with capital constraint, Production and Operations Management, 21 (2012), 1090-1101. [13] T. Koller, D. Lovallo and Z. Williams, Overcoming a bias against risk, McKinsey Quarterly, 4 (2012), 15-17. [14] P. Kouvelis and W. Zhao, The newsvendor problem and price only contract when bankruptcy costs exist, Production and Operations Management, 20 (2011), 921-936.  doi: 10.1287/opre.1120.1040. [15] P. Kouvelis and W. Zhao, Financing the newsvendor: Supplier vs. bank, and the structure of optimal trade credit contracts, Operations Research, 60 (2012), 566-580.  doi: 10.1287/opre.1120.1040. [16] P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Management Science, 62 (2016), 2341-2357. [17] B. Li, P. Chen, Q. Li and W. Wang, Dual-channel supply chain pricing decisions with a risk-averse retailer, International Journal of Production Research, 52 (2014), 7132-7147. [18] Q. Lin, Y. Peng and Y. Hu, Supplier financing service decisions for a capital-constrained supply chain: Trade credit vs. combined credit financing, Journal of Industrial & Management Optimization, 16 (2020), 1731-1752.  doi: 10.3934/jimo.2019026. [19] J. Lerner, M. Sorensen and P. Stromberg, Private equity and long-run investment: The case of innovation, The Journal of Finance, 66 (2011), 445-477. [20] B. Li, S. An and D. Song, Selection of financing strategies with a risk-averse supplier in a capital-constrained supply chain, Transportation Research Part E: Logistics and Transportation Review, 118 (2018), 163-183. [21] S. C. Myers, The capital structure puzzle, Journal of Finance, 39 (1984), 574-592. [22] S. C. Myers and N. S. Majluf, Corporate financing and investment decisions when firms have information that investors do not have, Journal of Finance Economics, 13 (1984), 187-221. [23] A. Nezlobin, M. V. Rajan and S. Reichelstein, Structural properties of the price-to-earnings and price-to-book ratios, Review of Accounting Studies, 21 (2016), 438-472. [24] D. Nissim and S. H. Penman, Financial statement analysis of leverage and how it informs about profitability and price-to-book ratios, Journal of Finance Economics, 8 (2003), 531-560. [25] S. H. Penman, The articulation of price-earnings ratios and market-to-book ratios and the evaluation of growth. Journal of accounting research, Journal of Accounting Research, 34 (1996), 235-259. [26] R. J. Rendleman, Informational asymmetries and optimal project financing, Duke University, (1980). [27] R. Rockafellar and S. Uryasev, Banks versus venture capital: Project evaluation, screening, and expropriation, The Journal of Finance, 59 (2004), 601-621. [28] R. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-42.  doi: 10.1007/978-1-4757-6594-6_17. [29] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, 26 (2002), 1443-1471.  doi: 10.1007/978-1-4757-6594-6_17. [30] V. Roque and M. C. Cortez, The determinants of international equity investment: Do they differ between institutional and noninstitutional investors?, Journal of Banking & Finance, 49 (2014), 469-482. [31] B. Shen, X. Wang, Y. Cao and Q. Li, Financing decisions in supply chains with a capital-constrained manufacturer: Competition and risk, International Transactions in Operational Research, 27 (2020), 2422-2448.  doi: 10.1111/itor.12670. [32] M. Wu, S. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, International Journal of Production Economics, 156 (2014), 13-23. [33] N. Yan, C. Liu, L. Ye and B. Sun, Effects of risk aversion and decision preference on equilibriums in supply chain finance incorporating bank credit with credit guarantee, Applied Stochastic Models in Business & Industry, 33 (2017), 602-625.  doi: 10.1002/asmb.2264. [34] N. Yan, X. He and Y. Liu, Financing the capital-constrained supply chain with loss aversion: Supplier finance vs. supplier investment, Omega, 88 (2019), 162-178. [35] H. Yang, W. Zhuo and L. Shao, Equilibrium evolution in a two-echelon supply chain with financially constrained retailers: The impact of equity financing, International Journal of Production Economics, 185 (2020), 139-149. [36] H. Yang, W. Zhuo and L. Shao, Option contract strategies with risk-aversion and emergency purchase, International Journal of Production Economics, 27 (2020), 3079-3103.  doi: 10.1111/itor.12519. [37] H. Yang, Q. Yan, H. Wan and W. Zhuo, Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer, Journal of Industrial & Management Optimization, 16 (2020), 2723-2741.  doi: 10.3934/jimo.2019077. [38] F. Ye, Q. Lin and Y. Li, Coordination for contract farming supply chain with stochastic yield and demand under CVaR criterion, Operational Research, 20 (2020), 369-397. [39] Z. Zhen and J. Wang, Joint financing and ordering decisions in a capital-constrained supply chain with risk preference, RAIRO-Operations Research, 55 (2021), S2691–S2707. doi: 10.1051/ro/2020094. [40] Y. W. Zhou, J. Li and Y. Zhong, Cooperative advertising and ordering policies in a two-echelon supply chain with risk-averse agents, Omega, 75 (2018), 97-117.

show all references

##### References:
 [1] S. Asian and X. Nie, Coordination in supply chains with uncertain demand and disruption risks: Existence, analysis, and insights, IEEE Transactions on Systems Man & Cybernetics Systems, 44 (2014), 1139-1154. [2] M. Ayyagari, T. Beck and A. Demirguc-Kunt, Small and medium enterprises across the globe, Small Business Economics, (2007), 415–434. [3] P. Bolton and X. Freixas, Equity, bonds, and bank debt: Capital structure and financial market equilibrium under asymmetric information, Journal of Political Economy, 108 (2000), 324-351. [4] H. L. Chang and B.-D. Rhee, Trade credit for supply chain coordination, European Journal of Operational Research, 214 (2011), 136-146.  doi: 10.1016/j.ejor.2011.04.004. [5] X. Chen, A model of trade credit in a capital-constrained distribution channel, International Journal of Production Economics, 159 (2015), 347-357.  doi: 10.1007/s10479-014-1602-x. [6] X. Chen and A. Wang, Trade credit contract with limited liability in the supply chain with budget constraints, Annals of Operations Research, 196 (2012), 153-165.  doi: 10.1007/s10479-012-1119-0. [7] Y. Chen, M. Xu and Z. G. Zhang, Technical note–a risk-averse newsvendor model under the CVaR criterion, Operations Research, 57 (2009), 1040-1044. [8] J. Cong, T. Pang and H. Peng, Optimal strategies for capital constrained low-carbon supply chain under yield uncertainty, Journal of Cleaner Production, 256 (2009), 120339. [9] S. E. Curcuru, C. P. Thomas, F. E. Warnock and J. Wongswan, US international equity investment and past and prospective returns, Operations Research, 101 (2011), 3440-3455. [10] S. Deng and Z. Zheng, Optimal production decision for a risk-averse manufacturer faced with random yield and stochastic demand, International Transactions in Operational Research, 27 (2020), 1622-1637.  doi: 10.1111/itor.12483. [11] B. C. Giri and S. Sharma, Optimal ordering policy for an inventory system with linearly increasing demand and allowable shortages under two levels trade credit financing, Operational Research, 16 (2016), 25-50. [12] B. Jing, X. Chen and G. Cai, Equilibrium financing in a distribution channel with capital constraint, Production and Operations Management, 21 (2012), 1090-1101. [13] T. Koller, D. Lovallo and Z. Williams, Overcoming a bias against risk, McKinsey Quarterly, 4 (2012), 15-17. [14] P. Kouvelis and W. Zhao, The newsvendor problem and price only contract when bankruptcy costs exist, Production and Operations Management, 20 (2011), 921-936.  doi: 10.1287/opre.1120.1040. [15] P. Kouvelis and W. Zhao, Financing the newsvendor: Supplier vs. bank, and the structure of optimal trade credit contracts, Operations Research, 60 (2012), 566-580.  doi: 10.1287/opre.1120.1040. [16] P. Kouvelis and W. Zhao, Supply chain contract design under financial constraints and bankruptcy costs, Management Science, 62 (2016), 2341-2357. [17] B. Li, P. Chen, Q. Li and W. Wang, Dual-channel supply chain pricing decisions with a risk-averse retailer, International Journal of Production Research, 52 (2014), 7132-7147. [18] Q. Lin, Y. Peng and Y. Hu, Supplier financing service decisions for a capital-constrained supply chain: Trade credit vs. combined credit financing, Journal of Industrial & Management Optimization, 16 (2020), 1731-1752.  doi: 10.3934/jimo.2019026. [19] J. Lerner, M. Sorensen and P. Stromberg, Private equity and long-run investment: The case of innovation, The Journal of Finance, 66 (2011), 445-477. [20] B. Li, S. An and D. Song, Selection of financing strategies with a risk-averse supplier in a capital-constrained supply chain, Transportation Research Part E: Logistics and Transportation Review, 118 (2018), 163-183. [21] S. C. Myers, The capital structure puzzle, Journal of Finance, 39 (1984), 574-592. [22] S. C. Myers and N. S. Majluf, Corporate financing and investment decisions when firms have information that investors do not have, Journal of Finance Economics, 13 (1984), 187-221. [23] A. Nezlobin, M. V. Rajan and S. Reichelstein, Structural properties of the price-to-earnings and price-to-book ratios, Review of Accounting Studies, 21 (2016), 438-472. [24] D. Nissim and S. H. Penman, Financial statement analysis of leverage and how it informs about profitability and price-to-book ratios, Journal of Finance Economics, 8 (2003), 531-560. [25] S. H. Penman, The articulation of price-earnings ratios and market-to-book ratios and the evaluation of growth. Journal of accounting research, Journal of Accounting Research, 34 (1996), 235-259. [26] R. J. Rendleman, Informational asymmetries and optimal project financing, Duke University, (1980). [27] R. Rockafellar and S. Uryasev, Banks versus venture capital: Project evaluation, screening, and expropriation, The Journal of Finance, 59 (2004), 601-621. [28] R. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk, Journal of Risk, 2 (2000), 21-42.  doi: 10.1007/978-1-4757-6594-6_17. [29] R. T. Rockafellar and S. Uryasev, Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, 26 (2002), 1443-1471.  doi: 10.1007/978-1-4757-6594-6_17. [30] V. Roque and M. C. Cortez, The determinants of international equity investment: Do they differ between institutional and noninstitutional investors?, Journal of Banking & Finance, 49 (2014), 469-482. [31] B. Shen, X. Wang, Y. Cao and Q. Li, Financing decisions in supply chains with a capital-constrained manufacturer: Competition and risk, International Transactions in Operational Research, 27 (2020), 2422-2448.  doi: 10.1111/itor.12670. [32] M. Wu, S. X. Zhu and R. H. Teunter, A risk-averse competitive newsvendor problem under the CVaR criterion, International Journal of Production Economics, 156 (2014), 13-23. [33] N. Yan, C. Liu, L. Ye and B. Sun, Effects of risk aversion and decision preference on equilibriums in supply chain finance incorporating bank credit with credit guarantee, Applied Stochastic Models in Business & Industry, 33 (2017), 602-625.  doi: 10.1002/asmb.2264. [34] N. Yan, X. He and Y. Liu, Financing the capital-constrained supply chain with loss aversion: Supplier finance vs. supplier investment, Omega, 88 (2019), 162-178. [35] H. Yang, W. Zhuo and L. Shao, Equilibrium evolution in a two-echelon supply chain with financially constrained retailers: The impact of equity financing, International Journal of Production Economics, 185 (2020), 139-149. [36] H. Yang, W. Zhuo and L. Shao, Option contract strategies with risk-aversion and emergency purchase, International Journal of Production Economics, 27 (2020), 3079-3103.  doi: 10.1111/itor.12519. [37] H. Yang, Q. Yan, H. Wan and W. Zhuo, Bargaining equilibrium in a two-echelon supply chain with a capital-constrained retailer, Journal of Industrial & Management Optimization, 16 (2020), 2723-2741.  doi: 10.3934/jimo.2019077. [38] F. Ye, Q. Lin and Y. Li, Coordination for contract farming supply chain with stochastic yield and demand under CVaR criterion, Operational Research, 20 (2020), 369-397. [39] Z. Zhen and J. Wang, Joint financing and ordering decisions in a capital-constrained supply chain with risk preference, RAIRO-Operations Research, 55 (2021), S2691–S2707. doi: 10.1051/ro/2020094. [40] Y. W. Zhou, J. Li and Y. Zhong, Cooperative advertising and ordering policies in a two-echelon supply chain with risk-averse agents, Omega, 75 (2018), 97-117.
The retailer's optimal order quantity with $\eta_r$
The retailer's CVaR with $\eta_r$
The retailer's optimal order quantity with $w$
The retailer's CVaR with $w$
The supplier's CVaR with $w$
Difference of the supplier's CVaR with $\eta_s$
Expected profit and difference of CVaR with $a$
$V(q)$
Notation and Definition
 Notation Definition $p$ Unit retail price. $c$ Unit production cost. $A$ Retailer's fixed asset. $w_{i}$ Unit wholesale price, where $i=I, B, T$ represents three cases under EF, BCF and TCF, respectively. $q_{i}$ Retailer's order quantity. $x$ Uncertain demand. $\eta_{j}$ Risk-averse coefficient, where $j=r, s$ represents the retailer and the supplier, respectively. $a$ The valuation level. $\pi_i$ Retailer's profit function. $\Pi_i$ Supplier's profit function. $\Omega_I$ Investor's expected profit. $r_{f}$ The risk-free interest rate. $r_{B}$ The interest rate charged by the bank to the retailer. $r_{s}$ The interest rate charged by the supplier to the retailer.
 Notation Definition $p$ Unit retail price. $c$ Unit production cost. $A$ Retailer's fixed asset. $w_{i}$ Unit wholesale price, where $i=I, B, T$ represents three cases under EF, BCF and TCF, respectively. $q_{i}$ Retailer's order quantity. $x$ Uncertain demand. $\eta_{j}$ Risk-averse coefficient, where $j=r, s$ represents the retailer and the supplier, respectively. $a$ The valuation level. $\pi_i$ Retailer's profit function. $\Pi_i$ Supplier's profit function. $\Omega_I$ Investor's expected profit. $r_{f}$ The risk-free interest rate. $r_{B}$ The interest rate charged by the bank to the retailer. $r_{s}$ The interest rate charged by the supplier to the retailer.
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