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November  2021, 17(6): 3559-3580. doi: 10.3934/jimo.2020133

Incentive contract design for supplier switching with considering learning effect

Qian Wei 1, , Jianxiong Zhang 2, and  Xiaojie Sun 2,,

1. 

School of Management, Tianjin University of Technology, Tianjin, 300384, China
2. 

College of Management and Economics, Tianjin University, Tianjin, 300072, China

* Corresponding author: Xiaojie Sun

Received  April 2019 Revised  July 2020 Published  November 2021 Early access  August 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant No. 71971152)

For minimizing purchase cost, a buying firm would switch to suppliers with providing more favorable prices. This paper investigates the optimal switching decision of a buyer that may switch to an entrant supplier with production learning ability (which is regarded as a private information) under a principal-agent framework. The results obtained show that the switching cost and the learning effect have significant impacts on the buyer's switching decision. Only when the fixed component of the switching cost is relatively low, the buyer can be better off from a partial switching strategy; otherwise, the buyer should take an all-or-nothing switching strategy or no switching strategy. As the learning ability of the entrant supplier increases, the buyer prefers to make more switching. Finally, a benefit-sharing contract is proposed to evaluate the performance of the principal-agent contract, and we demonstrate that the principal-agent contract almost completely dominates the benefit-sharing contract.

Keywords: Supplier switching, principal-agent, asymmetric information, learning effect, benefit-sharing.
Mathematics Subject Classification: Primary: 90B05.
Citation: Qian Wei, Jianxiong Zhang, Xiaojie Sun. Incentive contract design for supplier switching with considering learning effect. Journal of Industrial & Management Optimization, 2021, 17 (6) : 3559-3580. doi: 10.3934/jimo.2020133
References:
[1]

P. S. Adler and K. B. Clark, Behind the learning curve: A sketch of the learning process, Management Science, 37 (1991), 267-281.  doi: 10.1287/mnsc.37.3.267.  Google Scholar

[2]

K. J. Arrow, The economic implications of learning by doing, Review of Economic Studies, 29 (1962), 155-173.  doi: 10.2307/2295952.  Google Scholar

[3]

M. Ben-daya and M. Hariga, Lead-time reduction in a stochastic inventory system with learning consideration, International Journal of Production Research, 41 (2003), 571-579.  doi: 10.1080/00207540210158807.  Google Scholar

[4]

L. E. Bygballe, Toward a conceptualization of supplier-switching processes in business relationships, Journal of Purchasing and Supply Management, 23 (2017), 40-53.  doi: 10.1016/j.pursup.2016.04.007.  Google Scholar

[5]

C. K. Chen, C. C. Lo and Y. X. Liao, Optimal lot size with learning consideration on an imperfect production system with allowable shortages, International Journal of Production Economics, 113 (2008), 459-469.  doi: 10.1016/j.ijpe.2007.03.025.  Google Scholar

[6]

M. Cheng, S. Xiao and G. Liu, Single-machine rescheduling problems with learning effect under disruptions, Journal of Industrial & Management Optimization, 14 (2018), 967-980.  doi: 10.3934/jimo.2017085.  Google Scholar

[7]

P. Dasgupta and J. Stiglitz, Learning-by-doing, market structure and industrial and trade policies, Oxford Economic Papers, 40 (1988), 246-268.  doi: 10.7916/D8183HJM.  Google Scholar

[8]

J. S. Demski, D. E. M. Sappington and P. T. Spiller, Managing Supplier Switching, The RAND Journal of Economics, 18 (1987), 77-97.  doi: 10.2307/2555536.  Google Scholar

[9]

L. Feng and Y. L. Chan, Joint pricing and production decisions for new products with learning curve effects under upstream and downstream trade credits, European Journal of Operational Research, 272 (2019), 905-913.  doi: 10.1016/j.ejor.2018.07.003.  Google Scholar

[10]

G. Friedl and S. M. Wagner, Supplier development or supplier switching?, International Journal of Production Research, 50 (2012), 3066-3079.  doi: 10.1080/00207543.2011.588804.  Google Scholar

[11]

D. Fudenberg and J. Tirole, Capital as a commitment: Strategic investment to deter mobility, Journal of Economic Theory, 31 (1983), 227-250.  doi: 10.1016/0022-0531(83)90075-3.  Google Scholar

[12]

A. Gamble, E. A. Juliusson and T. Gärling, Consumer attitudes towards switching supplier in three deregulated markets, The Journal of Socio-Economics, 38 (2009), 814-819.  doi: 10.1016/j.socec.2009.05.002.  Google Scholar

[13]

B. C. Giri and C. H. Glock, A closed-loop supply chain with stochastic product returns and worker experience under learning and forgetting, International Journal of Production Research, 55 (2017), 6760-6778.  doi: 10.1080/00207543.2017.1347301.  Google Scholar

[14]

J. B. Heide and A. M. Weiss, Vendor consideration and switching behavior for buyers in high-technology markets, Journal of Marketing, 59 (1995), 30-43.  doi: 10.1177/002224299505900303.  Google Scholar

[15]

T. W. Hung and P. T. Chen, On the optimal replenishment in a finite planning horizon with learning effect of setup costs, Journal of Industrial & Management Optimization, 6 (2010), 425-433.  doi: 10.3934/jimo.2010.6.425.  Google Scholar

[16]

M. Y. Jaber and M. Bonney, The economic manufacture/order quantity (EMQ/EOQ) and the learning curve: Past, present, and future, International Journal of Production Economics, 59 (1999), 93-102.  doi: 10.1016/S0925-5273(98)00027-9.  Google Scholar

[17]

B. Kamrad and A. Siddique, Supply contracts, profit sharing, switching, reaction options, Management Science, 50 (2004), 64-82.  doi: 10.1287/mnsc.1030.0157.  Google Scholar

[18]

S. L. Li, A. Madhok, G. Plaschka and R. Verma, Supplier-switching inertia and competitive asymmetry: A demand-side perspective, Decision Sciences, 37 (2006), 547-576.  doi: 10.1111/j.1540-5414.2006.00138.x.  Google Scholar

[19]

T. Li, S. P. Sethi and X. L. He, Dynamic pricing, production, and channel coordination with stochastic learning, Production and Operations Management, 24 (2015), 857-882.  doi: 10.1111/poms.12320.  Google Scholar

[20]

C. Löffler, T. Pfeiffer and G. Schneider, Controlling for supplier switching in the presence of real options and asymmetric information, European Journal of Operational Research, 223 (2012), 690-700.  doi: 10.1016/j.ejor.2012.07.018.  Google Scholar

[21]

M. March, S. Zanoni and M. Y. Jaber, Economic production quantity model with learning in production, quality, reliability and energy efficiency, Computers & Industrial Engineering, 129 (2019), 502-511.  doi: 10.1016/j.cie.2019.02.009.  Google Scholar

[22]

G. Mosheiov, Scheduling problems with a learning effect, European Journal of Operational Research, 132 (2012), 687-693.  doi: 10.1016/S0377-2217(00)00175-2.  Google Scholar

[23]

R. B. Myerson, Incentive compatibility and the bargaining problem, Econometrica: Journal of the Econometric Society, 47 (1979), 61-73.  doi: 10.2307/1912346.  Google Scholar

[24]

T. Pfeiffer, A dynamic model of supplier switching, European Journal of Operational Research, 207 (2010), 697-710.  doi: 10.1016/j.ejor.2010.05.030.  Google Scholar

[25]

M. Plaza, O. K. Ngwenyama and K. Rohlf, A comparative analysis of learning curves: Implications for new technology implementation management, European Journal of Operational Research, 200 (2010), 518-528.  doi: 10.1016/j.ejor.2009.01.010.  Google Scholar

[26]

S. Shum, S. L. Tong and T. T. Xiao, On the impact of uncertain cost reduction when selling to strategic consumers, Management Science, 63 (2016), 843-860.  doi: 10.1287/mnsc.2015.2355.  Google Scholar

[27]

L. Silbermayr and S. Minner, Dual sourcing under disruption risk and cost improvement through learning, European Journal of Operational Research, 250 (2016), 226-238.  doi: 10.1016/j.ejor.2015.09.017.  Google Scholar

[28]

A. M. Spence, The learning curve and competition, The Bell Journal of Economics, 12 (1981), 49-70.  doi: 10.2307/3003508.  Google Scholar

[29]

A. T. Tsekrekos and A. N. Yannacopoulos, Optimal switching decisions under stochastic volatility with fast mean reversion, European Journal of Operational Research, 251 (2016), 148-157.  doi: 10.1016/j.ejor.2015.12.011.  Google Scholar

[30]

M. Uluskan, A. B. Godfrey and J. A. Joines, Impact of competitive strategy and cost-focus on global supplier switching (reshore and relocation) decisions, The Journal of The Textile Institute, 108 (2017), 1308-1318.  doi: 10.1080/00405000.2016.1245596.  Google Scholar

[31]

S. M. Wagner and G. Friedl, Supplier switching decisions, European Journal of Operational Research, 183 (2007), 700-717.  doi: 10.1016/j.ejor.2006.10.036.  Google Scholar

[32]

T. P. Wright, Factors affecting the cost of airplanes, Journal of Aeronautical Sciences, 3 (1936), 122-128.  doi: 10.2514/8.155.  Google Scholar

[33]

K. Xu, W. Y. Chiang and L. Liang, Dynamic pricing and channel efficiency in the presence of the cost learning effect, International Transactions in Operational Research, 18 (2011), 579-604.  doi: 10.1111/j.1475-3995.2011.00816.x.  Google Scholar

[34]

L. E. Yelle, The learning curve: Historical review and comprehensive survey, Decision Sciences, 10 (1979), 302-328.  doi: 10.1111/j.1540-5915.1979.tb00026.x.  Google Scholar

[35]

W. I. Zangwill and P. B. Kantor, The learning curve: A new perspective, International Transactions in Operational Research, 7 (2000), 595-607.  doi: 10.1016/S0969-6016(00)00022-8.  Google Scholar

[36]

J. X. Zhang, W. S. Tang and M. M. Hu, Optimal supplier switching with volume-dependent switching costs, International Journal of Production Economics, 161 (2015), 96-164.  doi: 10.1016/j.ijpe.2014.11.021.  Google Scholar

[37]

J. X. Zhang, Q. Wei, G. W. Liu and W. S. Tang, A supplier switching model with the competitive reactions and economies of scale effects, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2831-2843.  doi: 10.1109/TSMC.2016.2523926.  Google Scholar

[38]

Q. Zhang, W. S. Tang and J. X. Zhang, Green supply chain performance with cost learning and operational inefficiency effects, Journal of Cleaner Production, 112 (2016), 3267-3284.  doi: 10.1016/j.jclepro.2015.10.069.  Google Scholar

[39]

X. Zhao, Z. Pang and K. E. Stecke, When does a retailer's advance selling capability benefit manufacturer, retailer, or both?, Production and Operations Management, 25 (2016), 1073-1087.  doi: 10.1111/poms.12535.  Google Scholar

show all references

References:
[1]

P. S. Adler and K. B. Clark, Behind the learning curve: A sketch of the learning process, Management Science, 37 (1991), 267-281.  doi: 10.1287/mnsc.37.3.267.  Google Scholar

[2]

K. J. Arrow, The economic implications of learning by doing, Review of Economic Studies, 29 (1962), 155-173.  doi: 10.2307/2295952.  Google Scholar

[3]

M. Ben-daya and M. Hariga, Lead-time reduction in a stochastic inventory system with learning consideration, International Journal of Production Research, 41 (2003), 571-579.  doi: 10.1080/00207540210158807.  Google Scholar

[4]

L. E. Bygballe, Toward a conceptualization of supplier-switching processes in business relationships, Journal of Purchasing and Supply Management, 23 (2017), 40-53.  doi: 10.1016/j.pursup.2016.04.007.  Google Scholar

[5]

C. K. Chen, C. C. Lo and Y. X. Liao, Optimal lot size with learning consideration on an imperfect production system with allowable shortages, International Journal of Production Economics, 113 (2008), 459-469.  doi: 10.1016/j.ijpe.2007.03.025.  Google Scholar

[6]

M. Cheng, S. Xiao and G. Liu, Single-machine rescheduling problems with learning effect under disruptions, Journal of Industrial & Management Optimization, 14 (2018), 967-980.  doi: 10.3934/jimo.2017085.  Google Scholar

[7]

P. Dasgupta and J. Stiglitz, Learning-by-doing, market structure and industrial and trade policies, Oxford Economic Papers, 40 (1988), 246-268.  doi: 10.7916/D8183HJM.  Google Scholar

[8]

J. S. Demski, D. E. M. Sappington and P. T. Spiller, Managing Supplier Switching, The RAND Journal of Economics, 18 (1987), 77-97.  doi: 10.2307/2555536.  Google Scholar

[9]

L. Feng and Y. L. Chan, Joint pricing and production decisions for new products with learning curve effects under upstream and downstream trade credits, European Journal of Operational Research, 272 (2019), 905-913.  doi: 10.1016/j.ejor.2018.07.003.  Google Scholar

[10]

G. Friedl and S. M. Wagner, Supplier development or supplier switching?, International Journal of Production Research, 50 (2012), 3066-3079.  doi: 10.1080/00207543.2011.588804.  Google Scholar

[11]

D. Fudenberg and J. Tirole, Capital as a commitment: Strategic investment to deter mobility, Journal of Economic Theory, 31 (1983), 227-250.  doi: 10.1016/0022-0531(83)90075-3.  Google Scholar

[12]

A. Gamble, E. A. Juliusson and T. Gärling, Consumer attitudes towards switching supplier in three deregulated markets, The Journal of Socio-Economics, 38 (2009), 814-819.  doi: 10.1016/j.socec.2009.05.002.  Google Scholar

[13]

B. C. Giri and C. H. Glock, A closed-loop supply chain with stochastic product returns and worker experience under learning and forgetting, International Journal of Production Research, 55 (2017), 6760-6778.  doi: 10.1080/00207543.2017.1347301.  Google Scholar

[14]

J. B. Heide and A. M. Weiss, Vendor consideration and switching behavior for buyers in high-technology markets, Journal of Marketing, 59 (1995), 30-43.  doi: 10.1177/002224299505900303.  Google Scholar

[15]

T. W. Hung and P. T. Chen, On the optimal replenishment in a finite planning horizon with learning effect of setup costs, Journal of Industrial & Management Optimization, 6 (2010), 425-433.  doi: 10.3934/jimo.2010.6.425.  Google Scholar

[16]

M. Y. Jaber and M. Bonney, The economic manufacture/order quantity (EMQ/EOQ) and the learning curve: Past, present, and future, International Journal of Production Economics, 59 (1999), 93-102.  doi: 10.1016/S0925-5273(98)00027-9.  Google Scholar

[17]

B. Kamrad and A. Siddique, Supply contracts, profit sharing, switching, reaction options, Management Science, 50 (2004), 64-82.  doi: 10.1287/mnsc.1030.0157.  Google Scholar

[18]

S. L. Li, A. Madhok, G. Plaschka and R. Verma, Supplier-switching inertia and competitive asymmetry: A demand-side perspective, Decision Sciences, 37 (2006), 547-576.  doi: 10.1111/j.1540-5414.2006.00138.x.  Google Scholar

[19]

T. Li, S. P. Sethi and X. L. He, Dynamic pricing, production, and channel coordination with stochastic learning, Production and Operations Management, 24 (2015), 857-882.  doi: 10.1111/poms.12320.  Google Scholar

[20]

C. Löffler, T. Pfeiffer and G. Schneider, Controlling for supplier switching in the presence of real options and asymmetric information, European Journal of Operational Research, 223 (2012), 690-700.  doi: 10.1016/j.ejor.2012.07.018.  Google Scholar

[21]

M. March, S. Zanoni and M. Y. Jaber, Economic production quantity model with learning in production, quality, reliability and energy efficiency, Computers & Industrial Engineering, 129 (2019), 502-511.  doi: 10.1016/j.cie.2019.02.009.  Google Scholar

[22]

G. Mosheiov, Scheduling problems with a learning effect, European Journal of Operational Research, 132 (2012), 687-693.  doi: 10.1016/S0377-2217(00)00175-2.  Google Scholar

[23]

R. B. Myerson, Incentive compatibility and the bargaining problem, Econometrica: Journal of the Econometric Society, 47 (1979), 61-73.  doi: 10.2307/1912346.  Google Scholar

[24]

T. Pfeiffer, A dynamic model of supplier switching, European Journal of Operational Research, 207 (2010), 697-710.  doi: 10.1016/j.ejor.2010.05.030.  Google Scholar

[25]

M. Plaza, O. K. Ngwenyama and K. Rohlf, A comparative analysis of learning curves: Implications for new technology implementation management, European Journal of Operational Research, 200 (2010), 518-528.  doi: 10.1016/j.ejor.2009.01.010.  Google Scholar

[26]

S. Shum, S. L. Tong and T. T. Xiao, On the impact of uncertain cost reduction when selling to strategic consumers, Management Science, 63 (2016), 843-860.  doi: 10.1287/mnsc.2015.2355.  Google Scholar

[27]

L. Silbermayr and S. Minner, Dual sourcing under disruption risk and cost improvement through learning, European Journal of Operational Research, 250 (2016), 226-238.  doi: 10.1016/j.ejor.2015.09.017.  Google Scholar

[28]

A. M. Spence, The learning curve and competition, The Bell Journal of Economics, 12 (1981), 49-70.  doi: 10.2307/3003508.  Google Scholar

[29]

A. T. Tsekrekos and A. N. Yannacopoulos, Optimal switching decisions under stochastic volatility with fast mean reversion, European Journal of Operational Research, 251 (2016), 148-157.  doi: 10.1016/j.ejor.2015.12.011.  Google Scholar

[30]

M. Uluskan, A. B. Godfrey and J. A. Joines, Impact of competitive strategy and cost-focus on global supplier switching (reshore and relocation) decisions, The Journal of The Textile Institute, 108 (2017), 1308-1318.  doi: 10.1080/00405000.2016.1245596.  Google Scholar

[31]

S. M. Wagner and G. Friedl, Supplier switching decisions, European Journal of Operational Research, 183 (2007), 700-717.  doi: 10.1016/j.ejor.2006.10.036.  Google Scholar

[32]

T. P. Wright, Factors affecting the cost of airplanes, Journal of Aeronautical Sciences, 3 (1936), 122-128.  doi: 10.2514/8.155.  Google Scholar

[33]

K. Xu, W. Y. Chiang and L. Liang, Dynamic pricing and channel efficiency in the presence of the cost learning effect, International Transactions in Operational Research, 18 (2011), 579-604.  doi: 10.1111/j.1475-3995.2011.00816.x.  Google Scholar

[34]

L. E. Yelle, The learning curve: Historical review and comprehensive survey, Decision Sciences, 10 (1979), 302-328.  doi: 10.1111/j.1540-5915.1979.tb00026.x.  Google Scholar

[35]

W. I. Zangwill and P. B. Kantor, The learning curve: A new perspective, International Transactions in Operational Research, 7 (2000), 595-607.  doi: 10.1016/S0969-6016(00)00022-8.  Google Scholar

[36]

J. X. Zhang, W. S. Tang and M. M. Hu, Optimal supplier switching with volume-dependent switching costs, International Journal of Production Economics, 161 (2015), 96-164.  doi: 10.1016/j.ijpe.2014.11.021.  Google Scholar

[37]

J. X. Zhang, Q. Wei, G. W. Liu and W. S. Tang, A supplier switching model with the competitive reactions and economies of scale effects, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 47 (2017), 2831-2843.  doi: 10.1109/TSMC.2016.2523926.  Google Scholar

[38]

Q. Zhang, W. S. Tang and J. X. Zhang, Green supply chain performance with cost learning and operational inefficiency effects, Journal of Cleaner Production, 112 (2016), 3267-3284.  doi: 10.1016/j.jclepro.2015.10.069.  Google Scholar

[39]

X. Zhao, Z. Pang and K. E. Stecke, When does a retailer's advance selling capability benefit manufacturer, retailer, or both?, Production and Operations Management, 25 (2016), 1073-1087.  doi: 10.1111/poms.12535.  Google Scholar

Figure 1.  The optimal switching ratio $ r^{*}(x) $ and transfer payment $ t^*(x) $
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Figure 2.  The expected total cost reduction $ \hat{R}(\phi) $ with respect to the sharing proportion $ \phi $
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Figure 3.  $ \rm{S_2} $'s profit and the total cost reduction difference under the two contracts via $ x $
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Table 1.  Optimal switching strategies with $ I\geq m(0) $
$ \Psi\!=c_0\!-\gamma\!+k\!+I\!-p $ $ \Psi\geq 0 $ $ -m(0)-\gamma\leq\Psi <0 $ $ \Psi <-m(0)-\gamma $
$ r^*(x) $ $ \begin{array}{ll}r^*(x)=0,\\ x\in [0,\gamma].\end{array} $ $ r^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\1,x\in [x_1^*,\gamma]. \end{array}\right. $ $ \begin{array}{ll}r^*(x)=1,\\x\in [0,\gamma].\end{array} $
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$ \Psi\!=c_0\!-\gamma\!+k\!+I\!-p $ $ \Psi\geq 0 $ $ -m(0)-\gamma\leq\Psi <0 $ $ \Psi <-m(0)-\gamma $
$ r^*(x) $ $ \begin{array}{ll}r^*(x)=0,\\ x\in [0,\gamma].\end{array} $ $ r^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\1,x\in [x_1^*,\gamma]. \end{array}\right. $ $ \begin{array}{ll}r^*(x)=1,\\x\in [0,\gamma].\end{array} $
Table 2.  Optimal switching strategies with $ I<m(0) $
$ \Psi\!=c_0\!-\gamma\!+k\!+I\!-p $ $ \Psi\geq 0 $ $ -I-\gamma\leq\Psi<0 $
$ r^*(x) $ $ r^*(x)\!=\!\!0,\ x\in [0,\gamma]. $ $ {r^*}(x) = \left\{ {\begin{array}{*{20}{l}} {0,\;x\; \in \;[0,x_1^*),}\\ {1,\;x\; \in \;[x_1^*,\gamma ].} \end{array}} \right.$
$ \!\!I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma <\!\!\Psi\!\! <-\!\!I\!\!-\!\!\gamma $ $ \quad \!\!I\!\!-\!\!2m(0)\!\!-\gamma\!\! <\!\!\Psi\!\!\leq I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma $ $ \Psi\leq I-2m(0)-\gamma $
$ r^*(x)\!\!=\!\!\left\{ \begin{array}{lll}0,\quad x\in [0,x_3^*),\\ \!\!-\frac{c_0-p+k}{2[m(x)-x]},\\ \quad x\in [x_3^*,x_2^*),\\ 1, \quad x\in [x_2^*,\gamma]. \end{array}\right. $ $ r^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{ll} \!\!-\frac{c_0-p+k}{2[m(x)-x]}, \\ \quad \quad \quad x\in [0,x_2^*), \\1,\quad \quad x\in [x_2^*,\gamma]. \end{array}\right. $ $ \begin{array}{l} r^*(x)=1,\\ x\in [0,\gamma]. \end{array} $
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$ \Psi\!=c_0\!-\gamma\!+k\!+I\!-p $ $ \Psi\geq 0 $ $ -I-\gamma\leq\Psi<0 $
$ r^*(x) $ $ r^*(x)\!=\!\!0,\ x\in [0,\gamma]. $ $ {r^*}(x) = \left\{ {\begin{array}{*{20}{l}} {0,\;x\; \in \;[0,x_1^*),}\\ {1,\;x\; \in \;[x_1^*,\gamma ].} \end{array}} \right.$
$ \!\!I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma <\!\!\Psi\!\! <-\!\!I\!\!-\!\!\gamma $ $ \quad \!\!I\!\!-\!\!2m(0)\!\!-\gamma\!\! <\!\!\Psi\!\!\leq I\!\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma $ $ \Psi\leq I-2m(0)-\gamma $
$ r^*(x)\!\!=\!\!\left\{ \begin{array}{lll}0,\quad x\in [0,x_3^*),\\ \!\!-\frac{c_0-p+k}{2[m(x)-x]},\\ \quad x\in [x_3^*,x_2^*),\\ 1, \quad x\in [x_2^*,\gamma]. \end{array}\right. $ $ r^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{ll} \!\!-\frac{c_0-p+k}{2[m(x)-x]}, \\ \quad \quad \quad x\in [0,x_2^*), \\1,\quad \quad x\in [x_2^*,\gamma]. \end{array}\right. $ $ \begin{array}{l} r^*(x)=1,\\ x\in [0,\gamma]. \end{array} $
Table 3.  Optimal transfer payments with $ I\geq m(0) $
$ \Psi=c_0-\gamma+k+I-p $ $ \Psi\geq 0 $ $ -m(0)-\gamma\leq\Psi <0 $ $ \Psi <-m(0)-\gamma $
$ t^*(x) $ $ \begin{array}{lll} t^*(x)=0,\\ x\in [0,\gamma].\end{array} $ $ t^*(x)=\left\{ \begin{array}{ll}0, \\ \quad x\in [0,x_1^*),\\ c_0-x_1^*,\\ \quad x\in [x_1^*,\gamma]. \end{array}\right. $ $ \begin{array}{lll} t^*(x)\!=\!c_0\!-\!x_1^*,\\ x\in [0,\gamma].\end{array} $
Table Options

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$ \Psi=c_0-\gamma+k+I-p $ $ \Psi\geq 0 $ $ -m(0)-\gamma\leq\Psi <0 $ $ \Psi <-m(0)-\gamma $
$ t^*(x) $ $ \begin{array}{lll} t^*(x)=0,\\ x\in [0,\gamma].\end{array} $ $ t^*(x)=\left\{ \begin{array}{ll}0, \\ \quad x\in [0,x_1^*),\\ c_0-x_1^*,\\ \quad x\in [x_1^*,\gamma]. \end{array}\right. $ $ \begin{array}{lll} t^*(x)\!=\!c_0\!-\!x_1^*,\\ x\in [0,\gamma].\end{array} $
Table 4.  Optimal transfer payments with $ I<m(0) $
$ \Psi\!=\!c_0\!-\!\gamma\!+\!k\!+\!I\!-\!p $ $ \Psi\!\geq\!0 $
$ t^*(x) $ $ t^*(x)=0,\ x\in [0,\gamma]. $
$ -I\!-\!\gamma\!\leq\!\Psi\! <\!0 $ $ I\!-\!2\sqrt{Im(0)}\!-\!\gamma\! <\!\Psi\! <\!-I\!-\!\gamma $
$ t^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\c_0-x_1^*, x\in [x_1^*,\gamma]. \end{array}\right. $ $ \!\!\!t^*(x)\!\!=\!\!\left\{ \begin{array}{lll}\!\!0, \\ \quad\quad\quad\quad \quad\quad x\in [0,x_3^*),\\ \!\!c_0 r^*(x)-x{r^2}^*(x)+ \int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad\quad \quad\quad \quad\quad x\in [x_3^*,x_2^*), \\\!\!c_0-x_2^*+\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\\quad\quad \quad\quad \quad\quad x\in [x_2^*,\gamma]. \end{array}\right. $
$ \!\!I\!-\!\!2m(0)\!\!-\!\!\gamma\! <\!\Psi\!\leq \!I\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma $ $ \Psi\leq I-2m(0)-\gamma $
$ \!\!\!t^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{lll} \!\!c_0r^*(x)\!\!-\!\!x{r^2}^*(x)\!\!+\!\!\int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad x\in [0,x_2^*), \\\!\!c_0\!-\!x_2^*\!+\!\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\ \quad x\in [x_2^*,\gamma]. \end{array}\right. $ $ t^*(x)=c_0-x_1^*,\ x\in [0,\gamma]. $
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$ \Psi\!=\!c_0\!-\!\gamma\!+\!k\!+\!I\!-\!p $ $ \Psi\!\geq\!0 $
$ t^*(x) $ $ t^*(x)=0,\ x\in [0,\gamma]. $
$ -I\!-\!\gamma\!\leq\!\Psi\! <\!0 $ $ I\!-\!2\sqrt{Im(0)}\!-\!\gamma\! <\!\Psi\! <\!-I\!-\!\gamma $
$ t^*(x)=\left\{ \begin{array}{ll}0, x\in [0,x_1^*), \\c_0-x_1^*, x\in [x_1^*,\gamma]. \end{array}\right. $ $ \!\!\!t^*(x)\!\!=\!\!\left\{ \begin{array}{lll}\!\!0, \\ \quad\quad\quad\quad \quad\quad x\in [0,x_3^*),\\ \!\!c_0 r^*(x)-x{r^2}^*(x)+ \int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad\quad \quad\quad \quad\quad x\in [x_3^*,x_2^*), \\\!\!c_0-x_2^*+\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\\quad\quad \quad\quad \quad\quad x\in [x_2^*,\gamma]. \end{array}\right. $
$ \!\!I\!-\!\!2m(0)\!\!-\!\!\gamma\! <\!\Psi\!\leq \!I\!-\!\!2\sqrt{Im(0)}\!\!-\!\!\gamma $ $ \Psi\leq I-2m(0)-\gamma $
$ \!\!\!t^*(x)\!\!\!=\!\!\!\left\{ \begin{array}{lll} \!\!c_0r^*(x)\!\!-\!\!x{r^2}^*(x)\!\!+\!\!\int_{x_3^*}^{x}r^2(v){\rm d}v, \\ \quad x\in [0,x_2^*), \\\!\!c_0\!-\!x_2^*\!+\!\int_{x_3^*}^{x_2^*}r^2(v){\rm d}v, \\ \quad x\in [x_2^*,\gamma]. \end{array}\right. $ $ t^*(x)=c_0-x_1^*,\ x\in [0,\gamma]. $
Table 5.  Sensitivity analysis with respect to system parameters $ c_0, p, k, I. $
-80% -60% $ -40\% $ $ -20\% $ $ 0 $ $ 20\% $ $ 40\% $ $ 60\% $ $ 80\% $
$ \rm{ \rm{E}}(C) $ 1.0200 1.3400 1.6982 1.9775 2.2772 2.5239 2.7191 2.8631 2.9559
$ c_0 $$ \hat{ \rm{E}}(C) $ 1.3609 1.6088 1.8492 2.0805 2.3026 2.5406 2.7265 2.8656 2.9559
$ \Delta \rm{E} $ 0.3409 0.2688 0.1510 0.1030 0.0254 0.0167 0.0074 0.0025 0
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 33% 20% 8.9% 5.2% 1.1% 0.7% 0.3% 0.09% 0%
$ \rm{E}(C) $ 0.6000 1.2000 1.7375 2.0975 2.2772 2.3000 2.3000 2.3000 2.3000
$ p $$ \hat{ \rm{E}}(C) $ 0.6000 1.2000 1.7375 2.1060 2.3026 2.4786 2.6235 2.7449 2.8488
$ \Delta \rm{E} $ 0 0 0 0.0085 0.0254 0.1786 0.3235 0.4449 0.5488
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 0% 0% 0% 0.4% 1.1% 7.7% 14% 19% 24%
$ \rm{E}(C) $ 2.1340 2.1728 2.2114 2.2498 2.2772 2.3261 2.3639 2.3759 2.4071
$ k $$ \hat{ \rm{E}}(C) $ 2.1922 2.2196 2.2469 2.2739 2.3026 2.3340 2.3661 2.3974 2.4278
$ \Delta \rm{E} $ 0.0582 0.0468 0.0355 0.0241 0.0254 0.0079 0.0022 0.0215 0.0207
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 2.7% 2.2% 1.6% 1.1% 1.1% 0.34% 0.1% 0.9% 0.86%
$ \rm{E}(C) $ 1.8881 1.9881 2.0880 2.1870 2.2772 2.3600 2.4375 2.5100 2.5775
$ I $$ \hat{ \rm{E}}(C) $ 2.0793 2.1365 2.1925 2.2473 2.3026 2.3789 2.4507 2.5178 2.5810
$ \Delta \rm{E} $ 0.1912 0.1484 0.1045 0.0603 0.0254 0.0189 0.0132 0.0078 0.0035
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 10.1% 7.5% 5.0% 2.8% 1.1% 0.8% 0.5% 0.3% 0.1%
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-80% -60% $ -40\% $ $ -20\% $ $ 0 $ $ 20\% $ $ 40\% $ $ 60\% $ $ 80\% $
$ \rm{ \rm{E}}(C) $ 1.0200 1.3400 1.6982 1.9775 2.2772 2.5239 2.7191 2.8631 2.9559
$ c_0 $$ \hat{ \rm{E}}(C) $ 1.3609 1.6088 1.8492 2.0805 2.3026 2.5406 2.7265 2.8656 2.9559
$ \Delta \rm{E} $ 0.3409 0.2688 0.1510 0.1030 0.0254 0.0167 0.0074 0.0025 0
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 33% 20% 8.9% 5.2% 1.1% 0.7% 0.3% 0.09% 0%
$ \rm{E}(C) $ 0.6000 1.2000 1.7375 2.0975 2.2772 2.3000 2.3000 2.3000 2.3000
$ p $$ \hat{ \rm{E}}(C) $ 0.6000 1.2000 1.7375 2.1060 2.3026 2.4786 2.6235 2.7449 2.8488
$ \Delta \rm{E} $ 0 0 0 0.0085 0.0254 0.1786 0.3235 0.4449 0.5488
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 0% 0% 0% 0.4% 1.1% 7.7% 14% 19% 24%
$ \rm{E}(C) $ 2.1340 2.1728 2.2114 2.2498 2.2772 2.3261 2.3639 2.3759 2.4071
$ k $$ \hat{ \rm{E}}(C) $ 2.1922 2.2196 2.2469 2.2739 2.3026 2.3340 2.3661 2.3974 2.4278
$ \Delta \rm{E} $ 0.0582 0.0468 0.0355 0.0241 0.0254 0.0079 0.0022 0.0215 0.0207
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 2.7% 2.2% 1.6% 1.1% 1.1% 0.34% 0.1% 0.9% 0.86%
$ \rm{E}(C) $ 1.8881 1.9881 2.0880 2.1870 2.2772 2.3600 2.4375 2.5100 2.5775
$ I $$ \hat{ \rm{E}}(C) $ 2.0793 2.1365 2.1925 2.2473 2.3026 2.3789 2.4507 2.5178 2.5810
$ \Delta \rm{E} $ 0.1912 0.1484 0.1045 0.0603 0.0254 0.0189 0.0132 0.0078 0.0035
$ \frac{\Delta \rm{E}}{ \rm{E}(C)} $ 10.1% 7.5% 5.0% 2.8% 1.1% 0.8% 0.5% 0.3% 0.1%
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