# American Institute of Mathematical Sciences

October 2017, 13(4): 1815-1840. doi: 10.3934/jimo.2017020

## An uncertain wage contract model for risk-averse worker under bilateral moral hazard

 Institute of Systems Engineering, Tianjin University, Tianjin 300072, China

* Corresponding author: Yanfei Lan, Email: lanyf@tju.edu.cn

Received  February 2016 Revised  October 2016 Published  December 2016

This paper investigates a wage mechanism design problem faced by a risk neutral firm (he) who employs a risk averse worker (she) to sell products for him. The effort levels of both the firm and the worker are unobservable to each other, which results in bilateral moral hazard. The firm offers a wage contract menu to the worker with the objective of maximizing his expected profit. The results show that the firm will provide the same wage contract to the worker when the worker's effort is observable regardless of the market condition being full or private information. The optimal wage contract is related to the worker's risk averse level when the bilateral moral hazard exists. The information values of the worker's effort and the market condition are studied, respectively. The results show that the firm benefits from the worker's observable effort under full information and only when the sales uncertainty is sufficiently low, can the firm profit from that under private information. Moreover, only if the cost coefficient of the firm's effort is sufficiently high, the firm can benefit from full information in the scenario when the worker's effort is unobservable.

Citation: Xiulan Wang, Yanfei Lan, Wansheng Tang. An uncertain wage contract model for risk-averse worker under bilateral moral hazard. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1815-1840. doi: 10.3934/jimo.2017020
##### References:
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Ching, Salesforce contract design, joint pricing and production planning with asymmetric overconfidence sales agent, Journal of Industrial and Management Optimization, (2016). doi: 10.3934/jimo.2016051. [7] X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9 (2010), 69-81. doi: 10.1007/s10700-010-9073-2. [8] L. Cui, R. Zhao and W. Tang, Principal-agent problem in a fuzzy environment, IEEE Transactions on Fuzzy Systems, 15 (2007), 1230-1237. [9] S. Dutta, Managerial expertise, private information, and payperformance sensitivity, Management Science, 54 (2008), 429-442. [10] J. Feng, Y. Lan and R. Zhao, Impact of price cap regulation on supply chain contracting between two monopolists, Journal of Industrial and Management Optimization, (2016). doi: 10.3934/jimo.2016021. [11] M. Goldmanis and K. Ray, Sorting effects of performance pay, Management Science, 61 (2014), 335-353. [12] B. Greenwald, Adverse selection in labour market, The Review of Economic Studies, 53 (1986), 325-347. doi: 10.2307/2297632. [13] S. Grossman and O. Hart, An analysis of the principal-agent problem, Econometrica, 51 (1983), 7-45. doi: 10.2307/1912246. [14] B. Holmstrom, Moral hazard in teams, The Bell Journal of Economics, 13 (1982), 324-340. doi: 10.2307/3003457. [15] S. Kim and S. Wang, Linear contracts and the double moral-hazard, Journal of Economic Theory, 82 (1998), 342-378. doi: 10.1006/jeth.1998.2439. [16] L. Kung and Y. Chen, Monitoring the market or the malesperson? The value of information in a multilayer supply chain, Management Science, 58 (2011), 743-762. doi: 10.1002/nav.20480. [17] Y. Lan, R. Zhao and W. Tang, A yardstick competition approach to a multi-firm regulation problem under asymmetric information, Journal of Computational and Applied Mathematics, 249 (2013), 24-36. doi: 10.1016/j.cam.2013.01.017. [18] B. Liu, Theory and Practice of Uncertain Programming, 2 edition, Springer, Berlin, 2007. [19] B. Liu, Uncertainty Theory, Springer, Berlin, 2009. [20] B. Liu, Uncertain logic for modeling human language, Journal of Uncertain Systems, 5 (2011), 3-20. [21] Y. Liu and M. Ha, Expected value of function of uncertain variables, Journal of Uncertain Systems, 4 (2010), 181-186. [22] G. Manso, Motivating innovation, The Journal of Finance, 66 (2011), 1823-1860. [23] J. Mihm, Incentives in new product development projects and the role of target costing, Management Science, 56 (2010), 1324-1344. [24] R. Mu, Y. Lan and W. Tang, An uncertain contract model for rural migrant worker's employment problems, Fuzzy Optimization and Decision Making, 12 (2013), 29-39. doi: 10.1007/s10700-012-9137-6. [25] R. Myerson, Optimal coordination mechanisms in generalized principal-agent problems, Journal of Mathematical Economics, 10 (1982), 67-81. doi: 10.1016/0304-4068(82)90006-4. [26] Ö. Özalp and G. Raz, Supply chain sourcing under asymmetric information, Production and Operations Management, 20 (2011), 92-115. [27] F. Page, Optimal contract mechanisms for principal-agent problems with moral hazard and adverse selection, Economic Theory, 1 (1991), 323-338. doi: 10.1007/BF01229312. [28] Y. Suzuki, Commitment problem, optimal incentive schemes, and relational contracts in agency with bilateral moral hazard, Journal of International Economic Studies, 21 (2007), 103-124. [29] G. Wang, W. Tang and R. Zhao, An uncertain price discrimination model in labor market, Soft Computing, 17 (2013), 579-585. [30] X. Wang, Y. Lan and J. Wang, An uncertain wage contract model with adverse selection and moral hazard, Journal of Applied Mathematics, 1 (2014), 1-9. [31] X. Wu, R. Zhao and W. Tang, Uncertain agency models with multi-dimensional incomplete information based on confidence level, Fuzzy Optimization and Decision Making, 13 (2013), 231-258. doi: 10.1007/s10700-013-9174-9. [32] W. Xiao and Y. Xu, The impact of royalty contract revision in a multi-stage strategic RD alliance, Management Science, 58 (2005), 2251-2271. [33] K. Yang, Y. Lan and R. Zhao, Monitoring mechanisms in new product development with risk-averse project manager, Journal of Intelligent Manufacturing, (2014), 1-15. [34] Y. Zhu, Uncertain optimal control with aplication to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.

show all references

##### References:
 [1] P. Agrawal, Double moral hazard, monitoring, and the nature of contracts, Journal of Economics, 75 (2002), 33-61. [2] S. Bhattacharyya and F. Lafontaine, Double-sided moral hazard and the nature of share contracts, The Rand Journal of Economics, 26 (1995), 761-781. [3] R. Chao, K. Lichtendahl and Y. Grushka-Cockayne, Incentives in a stage-gate process, Production and Operations Management, 23 (2014), 1286-1298. [4] F. Chen, Salesforce incentives, market information, and production/inventory planning, Management Science, 51 (2005), 60-75. doi: 10.1287/mnsc.1040.0217. [5] F. Chen and M. Deng, Information Sharing in a Manufacturer-Supplier Relationship: Suppliers' Incentive and Production Efficiency, Production & Operations Management, 24 (2015), 619-633. doi: 10.1111/poms.12261. [6] K. Chen, X. Wang, M. Huang and W. Ching, Salesforce contract design, joint pricing and production planning with asymmetric overconfidence sales agent, Journal of Industrial and Management Optimization, (2016). doi: 10.3934/jimo.2016051. [7] X. Chen and B. Liu, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making, 9 (2010), 69-81. doi: 10.1007/s10700-010-9073-2. [8] L. Cui, R. Zhao and W. Tang, Principal-agent problem in a fuzzy environment, IEEE Transactions on Fuzzy Systems, 15 (2007), 1230-1237. [9] S. Dutta, Managerial expertise, private information, and payperformance sensitivity, Management Science, 54 (2008), 429-442. [10] J. Feng, Y. Lan and R. Zhao, Impact of price cap regulation on supply chain contracting between two monopolists, Journal of Industrial and Management Optimization, (2016). doi: 10.3934/jimo.2016021. [11] M. Goldmanis and K. Ray, Sorting effects of performance pay, Management Science, 61 (2014), 335-353. [12] B. Greenwald, Adverse selection in labour market, The Review of Economic Studies, 53 (1986), 325-347. doi: 10.2307/2297632. [13] S. Grossman and O. Hart, An analysis of the principal-agent problem, Econometrica, 51 (1983), 7-45. doi: 10.2307/1912246. [14] B. Holmstrom, Moral hazard in teams, The Bell Journal of Economics, 13 (1982), 324-340. doi: 10.2307/3003457. [15] S. Kim and S. Wang, Linear contracts and the double moral-hazard, Journal of Economic Theory, 82 (1998), 342-378. doi: 10.1006/jeth.1998.2439. [16] L. Kung and Y. Chen, Monitoring the market or the malesperson? The value of information in a multilayer supply chain, Management Science, 58 (2011), 743-762. doi: 10.1002/nav.20480. [17] Y. Lan, R. Zhao and W. Tang, A yardstick competition approach to a multi-firm regulation problem under asymmetric information, Journal of Computational and Applied Mathematics, 249 (2013), 24-36. doi: 10.1016/j.cam.2013.01.017. [18] B. Liu, Theory and Practice of Uncertain Programming, 2 edition, Springer, Berlin, 2007. [19] B. Liu, Uncertainty Theory, Springer, Berlin, 2009. [20] B. Liu, Uncertain logic for modeling human language, Journal of Uncertain Systems, 5 (2011), 3-20. [21] Y. Liu and M. Ha, Expected value of function of uncertain variables, Journal of Uncertain Systems, 4 (2010), 181-186. [22] G. Manso, Motivating innovation, The Journal of Finance, 66 (2011), 1823-1860. [23] J. Mihm, Incentives in new product development projects and the role of target costing, Management Science, 56 (2010), 1324-1344. [24] R. Mu, Y. Lan and W. Tang, An uncertain contract model for rural migrant worker's employment problems, Fuzzy Optimization and Decision Making, 12 (2013), 29-39. doi: 10.1007/s10700-012-9137-6. [25] R. Myerson, Optimal coordination mechanisms in generalized principal-agent problems, Journal of Mathematical Economics, 10 (1982), 67-81. doi: 10.1016/0304-4068(82)90006-4. [26] Ö. Özalp and G. Raz, Supply chain sourcing under asymmetric information, Production and Operations Management, 20 (2011), 92-115. [27] F. Page, Optimal contract mechanisms for principal-agent problems with moral hazard and adverse selection, Economic Theory, 1 (1991), 323-338. doi: 10.1007/BF01229312. [28] Y. Suzuki, Commitment problem, optimal incentive schemes, and relational contracts in agency with bilateral moral hazard, Journal of International Economic Studies, 21 (2007), 103-124. [29] G. Wang, W. Tang and R. Zhao, An uncertain price discrimination model in labor market, Soft Computing, 17 (2013), 579-585. [30] X. Wang, Y. Lan and J. Wang, An uncertain wage contract model with adverse selection and moral hazard, Journal of Applied Mathematics, 1 (2014), 1-9. [31] X. Wu, R. Zhao and W. Tang, Uncertain agency models with multi-dimensional incomplete information based on confidence level, Fuzzy Optimization and Decision Making, 13 (2013), 231-258. doi: 10.1007/s10700-013-9174-9. [32] W. Xiao and Y. Xu, The impact of royalty contract revision in a multi-stage strategic RD alliance, Management Science, 58 (2005), 2251-2271. [33] K. Yang, Y. Lan and R. Zhao, Monitoring mechanisms in new product development with risk-averse project manager, Journal of Intelligent Manufacturing, (2014), 1-15. [34] Y. Zhu, Uncertain optimal control with aplication to a portfolio selection model, Cybernetics and Systems, 41 (2010), 535-547.
Optimal bonus coefficient of the wage
Optimal effort of the firm
Optimal effort of the worker
Impact of the worker's risk adverse level on VE1 and VE2
Impact of the worker's risk adverse level on VM1 and VM2
The four information cases
 $X$ known $X$ unknown The worker's observable effort Case $\rm{OF}$ Case $\rm{OP}$ The worker's unobservable effort Case $\rm{UF}$ Case $\rm{UP}$
 $X$ known $X$ unknown The worker's observable effort Case $\rm{OF}$ Case $\rm{OP}$ The worker's unobservable effort Case $\rm{UF}$ Case $\rm{UP}$
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