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doi: 10.3934/jimo.2021167
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Equity-based incentive to coordinate shareholder-manager interests under information asymmetry

1. 

School of Management, Hefei University of Technology, Hefei 230009, China

2. 

Key Laboratory of Process Optimization and Intelligent Decision-Making of Ministry of Education, Hefei 230009, China

3. 

Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL 32611, USA

4. 

LANTA, High School of Economics, Moscow, Russia

*Corresponding author: Zhiping Zhou

Received  January 2021 Revised  July 2021 Early access September 2021

Fund Project: This research is supported by National Natural Science Foundation of China (Grant No. 72101077), and Anhui Provincial Natural Science Foundation (Grant No. 2008085QG341)

The shareholder's interest oriented from business operation relies on opportunism regulation of the manager under asymmetry. Effective motivation incentives should be exploited to facilitate the manager's effort devotion enthusiasms. This paper establishes a theoretic model in which the shareholder offers equity-based incentive to a fairness-preferred manager to coordinate their interest conflicts and maximize her expected revenue. The manager exerts unverifiable levels of efforts toward both decision and coordination tasks making the most of his private information about fairness preference. Two interrelated performance measures on different hierarchical levels are considered for contracting purposes. In each situation, we derive the equilibrium effort choices and incentive coefficients of both participants, and investigate how these decisions are affected by fairness preference. Research findings suggest that the incorporation of firm equity dominates pure profit incentive in eliciting high effort levels toward two distinctive managerial tasks. Besides, the equity-based incentive weakens the perceived unfairness and facilitates the participants' expected revenue. Comparative statics and numerical analysis are conducted to demonstrate our results and the effectiveness of the proposed equity-based incentive. Finally, we summarize the contributions of this paper and put forward directions for further study.

Citation: Zhiping Zhou, Yao Yin, Mi Zhou, Hao Cheng, Panos M. Pardalos. Equity-based incentive to coordinate shareholder-manager interests under information asymmetry. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021167
References:
[1]

H. Ashbaugh-SkaifeD. W. CollinsW. R. Kinney and R. LaFond, The effect of SOX internal control deficiencies and their remediation on accrual quality, The Accounting Review, 83 (2008), 217-250.  doi: 10.2308/accr.2008.83.1.217.  Google Scholar

[2]

F. Ballestín and R. Leus, Resource-constrained project scheduling for timely project completion with stochastic activity durations, Production and Operations Management, 18 (2009), 459-474.  doi: 10.1111/j.1937-5956.2009.01023.x.  Google Scholar

[3]

F. Balmaceda, Optimal task assignments with loss-averse agents, European Economic Review, 105 (2018), 1-26.  doi: 10.1016/j.euroecorev.2018.03.006.  Google Scholar

[4]

R. Bergmann and G. Friedl, Controlling innovative projects with moral hazard and asymmetric information, Research Policy, 37 (2008), 1504-1514.  doi: 10.1016/j.respol.2008.05.004.  Google Scholar

[5]

F. Bova and L. Yang, Employee bargaining power, inter-firm competition, and equity-based compensation, Journal of Financial Economics, 126 (2017), 342-363.  doi: 10.1016/j.jfineco.2017.07.006.  Google Scholar

[6]

S. Cato, The first-order approach to the principal-agent problems under inequality aversion, Operations Research Letters, 41 (2013), 526-529.  doi: 10.1016/j.orl.2013.06.012.  Google Scholar

[7]

A. ChandrasekaranK. Linderman and R. Schroeder, The role of project and organizational context in managing high-tech R & D projects, Production and Operations Management, 24 (2014), 560-586.  doi: 10.1111/poms.12253.  Google Scholar

[8]

Z. ChenY. Lan and R. Zhao, Impacts of risk attitude and outside option on compensation contracts under different information structures, Fuzzy Optimization and Decision Making, 17 (2018), 13-47.  doi: 10.1007/s10700-016-9263-7.  Google Scholar

[9]

W. ChiT. X. LiuX. Qian and Q. Ye, An experimental study of incentive contracts for short- and long-term employees, Journal of Economic Behavior and Organization, 159 (2019), 366-383.  doi: 10.1016/j.jebo.2019.02.006.  Google Scholar

[10]

B. CorgnetJ. Gómez-Miñambres and R. Hernán-González, Goal setting in the principal-agent model: Weak incentives for strong performance, Games and Economic Behavior, 109 (2018), 311-326.  doi: 10.1016/j.geb.2017.12.017.  Google Scholar

[11]

T. DaiZ. Li and D. Sun, Equity-based incentives and supply chain buy-back contracts, Decision Sciences, 43 (2012), 661-685.  doi: 10.1111/j.1540-5915.2012.00363.x.  Google Scholar

[12]

C. DingK. Wang and S. Lai, Channel coordination mechanism with retailers having fairness preference-An improved quantity discount mechanism, Journal of Industrial and Management Optimization, 9 (2013), 967-982.  doi: 10.3934/jimo.2013.9.967.  Google Scholar

[13]

A. El-Tannir, Optimal project deadlines for mean-variance incentive contract designs, Computers & Industrial Engineering, 137 (2019), 106018.  doi: 10.1016/j.cie.2019.106018.  Google Scholar

[14]

F. Englmaier and A. Wambach, Optimal incentive contracts under inequity aversion, Games and Economic Behavior, 69 (2010), 312-328.  doi: 10.1016/j.geb.2009.12.007.  Google Scholar

[15]

H. FuM. Liu and B. Chen, Supplier's investment in manufacturers's quality improvement with equity holding, Journal of Industrial and Management Optimization, 17 (2021), 649-668.  doi: 10.3934/jimo.2019127.  Google Scholar

[16]

J. FuX. Chen and Q. Hu, Subsidizing strategies in a sustainable supply chain, Journal of the Operational Research Society, 69 (2017), 283-295.  doi: 10.1057/s41274-017-0199-2.  Google Scholar

[17]

H. Gan and M. S. Park, Are more able CEOs getting more compensated? Evidence from the pay-for-performance sensitivity of equity-based incentives, Advances in Accounting, 34 (2016), 64-76.  doi: 10.1016/j.adiac.2016.07.008.  Google Scholar

[18]

H. Gao, Optimal compensation contracts when managers can hedge, Journal of Financial Economics, 97 (2010), 218-238.  doi: 10.1016/j.jfineco.2010.03.015.  Google Scholar

[19]

X.-N. Gao and J. Tian, Multi-period incentive contract design in the agent emergency supplies reservation strategy with asymmetric information, Computers & Industrial Engineering, 120 (2018), 94-102.  doi: 10.1016/j.cie.2018.04.030.  Google Scholar

[20]

J. Han and A. Rapoport, Intention-based fairness preferences in multi-partner project teams, Journal of Behavioral and Experimental Economics, 81 (2019), 84-90.  doi: 10.1016/j.socec.2019.06.003.  Google Scholar

[21]

M. Hoffmann and M. Kolmar, Intention-based fairness preferences in two-player contests, Economics Letters, 120 (2013), 276-279.  doi: 10.1016/j.econlet.2013.04.038.  Google Scholar

[22]

A. Intezari and D. J. Pauleen, Conceptualizing wise management decision-making: A grounded theory approach, Decision Sciences, 49 (2018), 335-400.  doi: 10.1111/deci.12267.  Google Scholar

[23]

J.-P. KallunkiE. Pyykkö and T. Laamanen, Stock market valuation, profitability and R & D spending of the firm: The effect of technology mergers and acquisitions, Journal of Business Finance & Accounting, 36 (2009), 838-862.  doi: 10.1111/j.1468-5957.2009.02161.x.  Google Scholar

[24]

E. Katok and V. Pavlov, Fairness in supply chain contracts: A laboratory study, Journal of Operations Management, 31 (2013), 129-137.  doi: 10.1016/j.jom.2013.01.001.  Google Scholar

[25]

L. P. Kerkhove and M. Vanhoucke, Incentive contract design for projects: The owner's perspective, Omega, 62 (2016), 93-114.  doi: 10.1016/j.omega.2015.09.002.  Google Scholar

[26]

Y. Khoroshilov and M. P. Narayanan, The role of profit-based and stock-based components in incentive compensation, Journal of Financial Intermediation, 17 (2008), 357-378.  doi: 10.1016/j.jfi.2008.02.005.  Google Scholar

[27]

M. KongJ. PeiJ. XuX. LiuX. Yu and P. M. Pardalos, A robust optimization approach for integrated steel production and batch delivery scheduling with uncertain rolling times and deterioration effect, International Journal of Production Research, 58 (2020), 5132-5154.  doi: 10.1080/00207543.2019.1693659.  Google Scholar

[28]

H. D. KwonS. A. Lippman and C. S. Tang, Optimal time-based and cost-based coordinated project contracts with unobservable work rates, International Journal of Production Economics, 126 (2010), 247-254.  doi: 10.1016/j.ijpe.2010.03.013.  Google Scholar

[29]

H.-H. Lee and C. Li, Supplier quality management: Investment, inspection, and incentives, Production and Operations Management, 27 (2017), 304-322.  doi: 10.1111/poms.12802.  Google Scholar

[30]

J. LiuR. GaoC. Y. J. Cheah and J. Luo, Incentive mechanism for inhibiting investors'opportunistic behavior in PPP projects, International Journal of Project Management, 34 (2016), 1102-1111.  doi: 10.1016/j.ijproman.2016.05.013.  Google Scholar

[31]

T. Nohel and S. Todd, Compensation for managers with career concerns: The role of stock options in optimal contracts, Journal of Corporate Finance, 11 (2005), 229-251.  doi: 10.1016/S0929-1199(03)00047-6.  Google Scholar

[32]

V. O'ConnellJ.-H. Lee and D. O'Sullivan, The influence of CEO equity incentives on licensing, European Management Journal, 36 (2018), 266-277.  doi: 10.1016/j.emj.2018.01.005.  Google Scholar

[33]

J. Oxley and G. Pandher, Equity-based incentives and collaboration in the modern multibusiness firm, Strategic Management Journal, 37 (2016), 1379-1394.  doi: 10.1002/smj.2392.  Google Scholar

[34]

E. SiemsenS. Balasubramanian and A. V. Roth, Incentives that induce task-related effort, helping, and knowledge sharing in workgroups, Management Science, 53 (2007), 1533-1550.  doi: 10.1287/mnsc.1070.0714.  Google Scholar

[35]

N. Siggelkow, Misperceiving interactions among complements and substitutes: Organizational consequences, Management Science, 48 (2002), 900-916.  doi: 10.1287/mnsc.48.7.900.2820.  Google Scholar

[36]

R. Silvers, The value of information in a principal-agent model with moral hazard: The ex ante contracting case, Games & Economic Behavior, 74 (2012), 352-365.  doi: 10.1016/j.geb.2011.07.002.  Google Scholar

[37]

G. Strobl, Stock-based managerial compensation, price informativeness, and the incentive to overinvest, Journal of Corporate Finance, 29 (2014), 594-606.  doi: 10.1016/j.jcorpfin.2013.12.003.  Google Scholar

[38]

M. SupraptoH. L. M. BakkerH. G. Mooi and M. J. C. M. Hertogh, Hertogh, How do contract types and incentives matter to project performance?, International Journal of Project Management, 34 (2015), 1071-1087.  doi: 10.1016/j.ijproman.2015.08.003.  Google Scholar

[39]

X. YanH.-Y. ChongJ. ZhouZ. Sheng and F. Xu, Fairness preference based decision-making model for concession period in PPP projects, Journal of Industrial and Management Optimization, 16 (2020), 11-23.  doi: 10.3934/jimo.2018137.  Google Scholar

[40]

K. YangR. Zhao and Y. Lan, Impacts of uncertain project duration and asymmetric risk sensitivity information in project management, International Transactions in Operational Research, 23 (2016), 749-774.  doi: 10.1111/itor.12156.  Google Scholar

[41]

C. ZhouJ. PengZ. Liu and B. Dong, Optimal incentive contracts under loss aversion and inequity aversion, Fuzzy Optimization and Decision Making, 18 (2019), 85-102.  doi: 10.1007/s10700-018-9288-1.  Google Scholar

show all references

References:
[1]

H. Ashbaugh-SkaifeD. W. CollinsW. R. Kinney and R. LaFond, The effect of SOX internal control deficiencies and their remediation on accrual quality, The Accounting Review, 83 (2008), 217-250.  doi: 10.2308/accr.2008.83.1.217.  Google Scholar

[2]

F. Ballestín and R. Leus, Resource-constrained project scheduling for timely project completion with stochastic activity durations, Production and Operations Management, 18 (2009), 459-474.  doi: 10.1111/j.1937-5956.2009.01023.x.  Google Scholar

[3]

F. Balmaceda, Optimal task assignments with loss-averse agents, European Economic Review, 105 (2018), 1-26.  doi: 10.1016/j.euroecorev.2018.03.006.  Google Scholar

[4]

R. Bergmann and G. Friedl, Controlling innovative projects with moral hazard and asymmetric information, Research Policy, 37 (2008), 1504-1514.  doi: 10.1016/j.respol.2008.05.004.  Google Scholar

[5]

F. Bova and L. Yang, Employee bargaining power, inter-firm competition, and equity-based compensation, Journal of Financial Economics, 126 (2017), 342-363.  doi: 10.1016/j.jfineco.2017.07.006.  Google Scholar

[6]

S. Cato, The first-order approach to the principal-agent problems under inequality aversion, Operations Research Letters, 41 (2013), 526-529.  doi: 10.1016/j.orl.2013.06.012.  Google Scholar

[7]

A. ChandrasekaranK. Linderman and R. Schroeder, The role of project and organizational context in managing high-tech R & D projects, Production and Operations Management, 24 (2014), 560-586.  doi: 10.1111/poms.12253.  Google Scholar

[8]

Z. ChenY. Lan and R. Zhao, Impacts of risk attitude and outside option on compensation contracts under different information structures, Fuzzy Optimization and Decision Making, 17 (2018), 13-47.  doi: 10.1007/s10700-016-9263-7.  Google Scholar

[9]

W. ChiT. X. LiuX. Qian and Q. Ye, An experimental study of incentive contracts for short- and long-term employees, Journal of Economic Behavior and Organization, 159 (2019), 366-383.  doi: 10.1016/j.jebo.2019.02.006.  Google Scholar

[10]

B. CorgnetJ. Gómez-Miñambres and R. Hernán-González, Goal setting in the principal-agent model: Weak incentives for strong performance, Games and Economic Behavior, 109 (2018), 311-326.  doi: 10.1016/j.geb.2017.12.017.  Google Scholar

[11]

T. DaiZ. Li and D. Sun, Equity-based incentives and supply chain buy-back contracts, Decision Sciences, 43 (2012), 661-685.  doi: 10.1111/j.1540-5915.2012.00363.x.  Google Scholar

[12]

C. DingK. Wang and S. Lai, Channel coordination mechanism with retailers having fairness preference-An improved quantity discount mechanism, Journal of Industrial and Management Optimization, 9 (2013), 967-982.  doi: 10.3934/jimo.2013.9.967.  Google Scholar

[13]

A. El-Tannir, Optimal project deadlines for mean-variance incentive contract designs, Computers & Industrial Engineering, 137 (2019), 106018.  doi: 10.1016/j.cie.2019.106018.  Google Scholar

[14]

F. Englmaier and A. Wambach, Optimal incentive contracts under inequity aversion, Games and Economic Behavior, 69 (2010), 312-328.  doi: 10.1016/j.geb.2009.12.007.  Google Scholar

[15]

H. FuM. Liu and B. Chen, Supplier's investment in manufacturers's quality improvement with equity holding, Journal of Industrial and Management Optimization, 17 (2021), 649-668.  doi: 10.3934/jimo.2019127.  Google Scholar

[16]

J. FuX. Chen and Q. Hu, Subsidizing strategies in a sustainable supply chain, Journal of the Operational Research Society, 69 (2017), 283-295.  doi: 10.1057/s41274-017-0199-2.  Google Scholar

[17]

H. Gan and M. S. Park, Are more able CEOs getting more compensated? Evidence from the pay-for-performance sensitivity of equity-based incentives, Advances in Accounting, 34 (2016), 64-76.  doi: 10.1016/j.adiac.2016.07.008.  Google Scholar

[18]

H. Gao, Optimal compensation contracts when managers can hedge, Journal of Financial Economics, 97 (2010), 218-238.  doi: 10.1016/j.jfineco.2010.03.015.  Google Scholar

[19]

X.-N. Gao and J. Tian, Multi-period incentive contract design in the agent emergency supplies reservation strategy with asymmetric information, Computers & Industrial Engineering, 120 (2018), 94-102.  doi: 10.1016/j.cie.2018.04.030.  Google Scholar

[20]

J. Han and A. Rapoport, Intention-based fairness preferences in multi-partner project teams, Journal of Behavioral and Experimental Economics, 81 (2019), 84-90.  doi: 10.1016/j.socec.2019.06.003.  Google Scholar

[21]

M. Hoffmann and M. Kolmar, Intention-based fairness preferences in two-player contests, Economics Letters, 120 (2013), 276-279.  doi: 10.1016/j.econlet.2013.04.038.  Google Scholar

[22]

A. Intezari and D. J. Pauleen, Conceptualizing wise management decision-making: A grounded theory approach, Decision Sciences, 49 (2018), 335-400.  doi: 10.1111/deci.12267.  Google Scholar

[23]

J.-P. KallunkiE. Pyykkö and T. Laamanen, Stock market valuation, profitability and R & D spending of the firm: The effect of technology mergers and acquisitions, Journal of Business Finance & Accounting, 36 (2009), 838-862.  doi: 10.1111/j.1468-5957.2009.02161.x.  Google Scholar

[24]

E. Katok and V. Pavlov, Fairness in supply chain contracts: A laboratory study, Journal of Operations Management, 31 (2013), 129-137.  doi: 10.1016/j.jom.2013.01.001.  Google Scholar

[25]

L. P. Kerkhove and M. Vanhoucke, Incentive contract design for projects: The owner's perspective, Omega, 62 (2016), 93-114.  doi: 10.1016/j.omega.2015.09.002.  Google Scholar

[26]

Y. Khoroshilov and M. P. Narayanan, The role of profit-based and stock-based components in incentive compensation, Journal of Financial Intermediation, 17 (2008), 357-378.  doi: 10.1016/j.jfi.2008.02.005.  Google Scholar

[27]

M. KongJ. PeiJ. XuX. LiuX. Yu and P. M. Pardalos, A robust optimization approach for integrated steel production and batch delivery scheduling with uncertain rolling times and deterioration effect, International Journal of Production Research, 58 (2020), 5132-5154.  doi: 10.1080/00207543.2019.1693659.  Google Scholar

[28]

H. D. KwonS. A. Lippman and C. S. Tang, Optimal time-based and cost-based coordinated project contracts with unobservable work rates, International Journal of Production Economics, 126 (2010), 247-254.  doi: 10.1016/j.ijpe.2010.03.013.  Google Scholar

[29]

H.-H. Lee and C. Li, Supplier quality management: Investment, inspection, and incentives, Production and Operations Management, 27 (2017), 304-322.  doi: 10.1111/poms.12802.  Google Scholar

[30]

J. LiuR. GaoC. Y. J. Cheah and J. Luo, Incentive mechanism for inhibiting investors'opportunistic behavior in PPP projects, International Journal of Project Management, 34 (2016), 1102-1111.  doi: 10.1016/j.ijproman.2016.05.013.  Google Scholar

[31]

T. Nohel and S. Todd, Compensation for managers with career concerns: The role of stock options in optimal contracts, Journal of Corporate Finance, 11 (2005), 229-251.  doi: 10.1016/S0929-1199(03)00047-6.  Google Scholar

[32]

V. O'ConnellJ.-H. Lee and D. O'Sullivan, The influence of CEO equity incentives on licensing, European Management Journal, 36 (2018), 266-277.  doi: 10.1016/j.emj.2018.01.005.  Google Scholar

[33]

J. Oxley and G. Pandher, Equity-based incentives and collaboration in the modern multibusiness firm, Strategic Management Journal, 37 (2016), 1379-1394.  doi: 10.1002/smj.2392.  Google Scholar

[34]

E. SiemsenS. Balasubramanian and A. V. Roth, Incentives that induce task-related effort, helping, and knowledge sharing in workgroups, Management Science, 53 (2007), 1533-1550.  doi: 10.1287/mnsc.1070.0714.  Google Scholar

[35]

N. Siggelkow, Misperceiving interactions among complements and substitutes: Organizational consequences, Management Science, 48 (2002), 900-916.  doi: 10.1287/mnsc.48.7.900.2820.  Google Scholar

[36]

R. Silvers, The value of information in a principal-agent model with moral hazard: The ex ante contracting case, Games & Economic Behavior, 74 (2012), 352-365.  doi: 10.1016/j.geb.2011.07.002.  Google Scholar

[37]

G. Strobl, Stock-based managerial compensation, price informativeness, and the incentive to overinvest, Journal of Corporate Finance, 29 (2014), 594-606.  doi: 10.1016/j.jcorpfin.2013.12.003.  Google Scholar

[38]

M. SupraptoH. L. M. BakkerH. G. Mooi and M. J. C. M. Hertogh, Hertogh, How do contract types and incentives matter to project performance?, International Journal of Project Management, 34 (2015), 1071-1087.  doi: 10.1016/j.ijproman.2015.08.003.  Google Scholar

[39]

X. YanH.-Y. ChongJ. ZhouZ. Sheng and F. Xu, Fairness preference based decision-making model for concession period in PPP projects, Journal of Industrial and Management Optimization, 16 (2020), 11-23.  doi: 10.3934/jimo.2018137.  Google Scholar

[40]

K. YangR. Zhao and Y. Lan, Impacts of uncertain project duration and asymmetric risk sensitivity information in project management, International Transactions in Operational Research, 23 (2016), 749-774.  doi: 10.1111/itor.12156.  Google Scholar

[41]

C. ZhouJ. PengZ. Liu and B. Dong, Optimal incentive contracts under loss aversion and inequity aversion, Fuzzy Optimization and Decision Making, 18 (2019), 85-102.  doi: 10.1007/s10700-018-9288-1.  Google Scholar

Figure 1.  The sequence of events and actions
Figure 2.  The impact of fairness preference on managerial effort levels
Figure 3.  The impact of fairness preference on bonus coefficients
Figure 4.  The impact of fairness preference on participants' benefits
Figure 5.  The impacts of effort contributions and performance variances on shareholder's revenue
Table 1.  NOTATIONS AND PARAMETERS
Symbol Meaning
$ a_d,a_c $ The managerial effort levels toward decision and coordination tasks, $ a_d,a_c\ge 0 $;
$ k(a_d,a_c) $ The manager's total effort cost, in which the parameter $ \delta $ captures the task synergy effect;
$ p $ The firm's stochastic accounting profit with noise term $ {\varepsilon _p} \sim N(0,\sigma_p^2) $;
$ \mu_d $ The contribution of manager's decision task effort on firm's accounting profit, $ \mu_d >0 $;
$ v $ The firm's stochastic stock value with noise term $ {\varepsilon _v} \sim N(0,\sigma_v^2) $;
$ \mu_c,\lambda_p $ The contribution of coordination task effort and firm's accounting profit on its stock value respectively, $ \mu_c >0,\lambda_p >0 $;
$ \alpha_0,\alpha_p,\alpha_v $ The fixed compensation and incentive coefficients respectively, $ \alpha_p,\alpha_v\in(0,1) $;
$ \beta,\hat\beta $ The manager's actual and reported fairness preference coefficients, $ \beta,\hat\beta\in[0,\bar \beta] $;
$ \gamma $ The manager's comparative fair ratio between the participants'net incomes, $ \gamma\in(0,1) $;
$ \rho $ The risk-averse coefficient of the manager, $ \rho >0 $;
$ \pi_m,\pi_s $ The net incomes of the manager and the shareholder respectively;
$ CE_m $ The manager's certainty equivalent based on his perceived utility $ U_m $.
Symbol Meaning
$ a_d,a_c $ The managerial effort levels toward decision and coordination tasks, $ a_d,a_c\ge 0 $;
$ k(a_d,a_c) $ The manager's total effort cost, in which the parameter $ \delta $ captures the task synergy effect;
$ p $ The firm's stochastic accounting profit with noise term $ {\varepsilon _p} \sim N(0,\sigma_p^2) $;
$ \mu_d $ The contribution of manager's decision task effort on firm's accounting profit, $ \mu_d >0 $;
$ v $ The firm's stochastic stock value with noise term $ {\varepsilon _v} \sim N(0,\sigma_v^2) $;
$ \mu_c,\lambda_p $ The contribution of coordination task effort and firm's accounting profit on its stock value respectively, $ \mu_c >0,\lambda_p >0 $;
$ \alpha_0,\alpha_p,\alpha_v $ The fixed compensation and incentive coefficients respectively, $ \alpha_p,\alpha_v\in(0,1) $;
$ \beta,\hat\beta $ The manager's actual and reported fairness preference coefficients, $ \beta,\hat\beta\in[0,\bar \beta] $;
$ \gamma $ The manager's comparative fair ratio between the participants'net incomes, $ \gamma\in(0,1) $;
$ \rho $ The risk-averse coefficient of the manager, $ \rho >0 $;
$ \pi_m,\pi_s $ The net incomes of the manager and the shareholder respectively;
$ CE_m $ The manager's certainty equivalent based on his perceived utility $ U_m $.
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