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May  2022, 18(3): 1519-1540. doi: 10.3934/jimo.2021031

## Optimal health insurance with constraints under utility of health, wealth and income

 1 School of Finance, Guangdong University of Foreign Studies, Guangzhou 510006, China 2 Department of Mathematics and Statistics, Curtin University, Western Australia 6845, Australia

* Corresponding author: Haixiang Yao

Received  July 2020 Revised  October 2020 Published  May 2022 Early access  February 2021

Fund Project: This research work is supported by the National Natural Science Foundation of China (Nos. 12001119, 71871071, 72071051), the Natural Science Foundation of Guangdong Province of China (Nos. 2017A030313399, 2018B030311004), and the Featured Innovation Project (Youth Foundation) of Guangdong University of Foreign Studies (19QN21)

We consider an optimal health insurance design problem with constraints under utility of health, wealth and income. The preference framework we establish herein describes the trade-off among health, wealth and income explicitly, which is beneficial to distinguish health insurance design from other nonlife insurance designs. Moreover, the work takes into account the case that if the insured is severely or critically ill, the insured may not fully recover even after necessary medical treatment. By taking these special features into account, the health insurance design problem is formulated as a constrained optimization problem, and the optimal solutions are derived by using the Lagrange multiplier method and optimal control technique. Finally, two numerical examples are given to illustrate our results. Our research work gives new insights into health insurance design.

Citation: Yan Zhang, Yonghong Wu, Haixiang Yao. Optimal health insurance with constraints under utility of health, wealth and income. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1519-1540. doi: 10.3934/jimo.2021031
##### References:
 [1] K. J. Arrow, Uncertainty and the welfare economics of medical care, American Economic Review, 53 (1963), 941-973. [2] K. J. Arrow, Essays in the Theory of Risk-Bearing, North-Holland Publishing Co., Amsterdam-London, 1970. [3] K. J. Arrow, Optimal insurance and generalized deductibles, Scandinavian Actuarial Journal, 1974 (1974), 1-42.  doi: 10.1080/03461238.1974.10408659. [4] C. Bernard, X. He, J.-A. Yan and X. Y. Zhou, Optimal insurance design under rank-dependent expected utility, Mathematical Finance, 25 (2015), 154-186.  doi: 10.1111/mafi.12027. [5] T. J. Besley, Optimal reimbursement health insurance and the theory of Ransey taxation, Journal of Health Economics, 7 (1988), 321-336.  doi: 10.1016/0167-6296(88)90019-7. [6] S. Bhargava, G. Loewenstein and J. Sydnor, Choose to lose: Health plan choices form a menu with dominated option, Quarterly Journal of Economics, 132 (2017), 1319-1372.  doi: 10.1093/qje/qjx011. [7] A. Blomqvist, Optimal non-linear health insurance, Journal of Health Economics, 16 (1997), 303-321.  doi: 10.1016/S0167-6296(96)00529-2. [8] N. A. Doherty and H. Schlesinger, Optimal insurance in incomplete markets, Journal of Political Economy, 91 (1983), 1045-1054.  doi: 10.1086/261199. [9] D. Doiron and N. Kettlewell, Family formation and demand for health insurance, Health Economics, 29 (2020), 523-533.  doi: 10.1002/hec.4000. [10] R. P. Ellis, S. Jiang and W. G. Manning, Optimal health insurance for multiple goods and time periods, Journal of Health Economics, 41 (2015), 89-106.  doi: 10.1016/j.jhealeco.2015.01.007. [11] Y. Fan, The solution of the optimal insurance problem with background risk, Journal of Function Spaces, 2019, Art. ID 2759398, 5 pp. doi: 10.1155/2019/2759398. [12] A. Finkelstein, E. F. P. Luttmer and M. J. Notowidigdo, What good is wealth without health? The effect of health on the marginal utility of consumption, Journal of the European Economic Association, 11 (2013), 221-258.  doi: 10.1111/j.1542-4774.2012.01101.x. [13] A. Finkelstein, N. Hendren and M. Shepard, Subsidizing health insurance for low-income adults: evidence from massachusetts, American Economic Review, 109 (2019), 1530-1567.  doi: 10.1257/aer.20171455. [14] V. R. Fuchs and R. J. Zeckhauser, Valuing health: A priceless commodity, American Economic Review Papers and Proceedings, 77 (1987), 263-268. [15] M. Gerfin, Health insurance and the demand for healthcare, Oxford Research Encyclopedia of Economics and Finance, 1 (2019), 1-22. [16] C. Gollier and H. Schlesinger, Arrow's theorem on the optimality of deductibles: A stochastic dominance approach, Economic Theory, 7 (1996), 359-363. [17] R. E. Hall and C. I. Jones, The value of life and the rise in health spending, Quarterly Journal of Economics, 122 (2007), 39-72.  doi: 10.1162/qjec.122.1.39. [18] T. Y. Ho, P. A. Fishman and Z. B. Zabinsky, Using a game-theoretic approach to design optimal health insurance for chronic disease, IISE Transactions on Healthcare System Engineering, 9 (2019), 26-40. [19] K. Lee, Wealth, income, and optimal insurance, Journal of Risk and Insurance, 74 (2007), 175-184.  doi: 10.1111/j.1539-6975.2007.00206.x. [20] M. Levy and A. R. Nir, The utility of health and wealth, Journal of Health Economics, 31 (2012), 379-392.  doi: 10.1016/j.jhealeco.2012.02.003. [21] T. Li and X. Wang, Connotation and realizing route of Healthy China, Health Economics Research, 1 (2016), 4-10.  doi: 10.1002/rwm3.20392. [22] Z. Lu, S. Meng, L. Liu and Z. Han, Optimal insurance design under background risk with dependence, Insurance: Mathematics and Economics, 80 (2018), 15-28.  doi: 10.1016/j.insmatheco.2018.02.006. [23] H. Markowitz, The utility of wealth, Journal of Political Economy, 60 (1952), 151-156.  doi: 10.1086/257177. [24] K. S. Moore and V. R. Young, Optimal insurance in a continuous-time model, Insurance: Mathematics and Economics, 39 (2006), 47-68.  doi: 10.1016/j.insmatheco.2006.01.009. [25] J. Mossin, Aspects of rational insurance purchasing, Journal of Political Economy, 76 (1968), 552-568.  doi: 10.1086/259427. [26] C. E. Phelps, The Demand for Health Insurance: A Theoretical and Empirical Investigation, Rand Corporation, Santa Monica, Report R-1054-OEO, 1973. [27] J. S. Pliskin, D. S. Shepard and M. C. Weinstein, Utility functions for life years and health status, Operations Research, 28 (1980), 206-224.  doi: 10.1287/opre.28.1.206. [28] S. D. Promislow and V. R. Young, Unifying framework for optimal insurance, Insurance: Mathematics and Economics, 36 (2005), 347-364.  doi: 10.1016/j.insmatheco.2005.04.003. [29] X. Qiao and Y. Hu, Health expectancy of chinese elderly and provincial variabilities, Population & Development, 23 (2017), 2-18. [30] A. Raviv, The design of an optimal insurance policy, American Economic Review, 69 (1979), 84-96. [31] V. L. Smith, Optimal insurance coverage, Journal of Political Economy, 76 (1968), 68-77.  doi: 10.1086/259382. [32] M. Spence and R. Zeckhauser, Insurance, information, and individual action, American Economic Review, 61 (1971), 380-387. [33] S. Wu, Y. Luo, X. Qiu and M. Bao, Building a healthy China by enhancing physical activity: Priorities, challenges, and strategies, Journal of Sport and Health Science, 6 (2017), 125-126.  doi: 10.1016/j.jshs.2016.10.003. [34] W. K. Viscussi and W. N. Evans, Utility functions that depend on health status: Estimates and economic implications, American Economic Review, 80 (1990), 353-374. [35] Z. Q. Xu, X. Y. Zhou and S. C. Zhuang, Optimal insurance under rank-dependent utility and incentive compatibility, Mathematical Finance, 29 (2019), 659-692.  doi: 10.1111/mafi.12185. [36] J. Yan, Why does China's government basic medical insurance system enhance the development of commercial health insurance?, Chinese Review of Financial Studies, 2 (2018), 1-15+221. [37] V. R. Young, Optimal insurance under Wang's premium principle, Insurance: Mathematics and Economics, 25 (1999), 109-122.  doi: 10.1016/S0167-6687(99)00012-8. [38] R. Zeckhauser, Medical insurance: A case study of the tradeoff between risk spreading and appropriate incentives, Journal of Economic Theory, 2 (1970), 10-26. [39] Y. Zhang, Dynamic research on the health level of the elderly in China under the background of Healthy China: Based on CHARLS data empirical analysis, Northwest Population, 40 (2018), 50-59. [40] Y. Zhang and Y. Wu, Optimal health insurance and trade-off between health and wealth, Journal of Applied Mathematics, 2020, Art. ID 2658213, 9 pp. doi: 10.1155/2020/2658213. [41] Y. Zheng, T. Vukina and X. Zheng, Estimating asymmetric information effects in health care with uninsurable costs, International Journal of Health Economics and Management, 19 (2019), 79-98.  doi: 10.1007/s10754-018-9246-z. [42] C. Zhou and C. Wu, Optimal insurance under the insurer's risk constraint, Insurance: Mathematics and Economics, 42 (2008), 992-999.  doi: 10.1016/j.insmatheco.2007.11.005. [43] C. Zhou, W. Wu and C. Wu, Optimal insurance in the presence of insurer's loss limit, Insurance: Mathematics and Economics, 46 (2010), 300-307.  doi: 10.1016/j.insmatheco.2009.11.002.

show all references

##### References:
 [1] K. J. Arrow, Uncertainty and the welfare economics of medical care, American Economic Review, 53 (1963), 941-973. [2] K. J. Arrow, Essays in the Theory of Risk-Bearing, North-Holland Publishing Co., Amsterdam-London, 1970. [3] K. J. Arrow, Optimal insurance and generalized deductibles, Scandinavian Actuarial Journal, 1974 (1974), 1-42.  doi: 10.1080/03461238.1974.10408659. [4] C. Bernard, X. He, J.-A. Yan and X. Y. Zhou, Optimal insurance design under rank-dependent expected utility, Mathematical Finance, 25 (2015), 154-186.  doi: 10.1111/mafi.12027. [5] T. J. Besley, Optimal reimbursement health insurance and the theory of Ransey taxation, Journal of Health Economics, 7 (1988), 321-336.  doi: 10.1016/0167-6296(88)90019-7. [6] S. Bhargava, G. Loewenstein and J. Sydnor, Choose to lose: Health plan choices form a menu with dominated option, Quarterly Journal of Economics, 132 (2017), 1319-1372.  doi: 10.1093/qje/qjx011. [7] A. Blomqvist, Optimal non-linear health insurance, Journal of Health Economics, 16 (1997), 303-321.  doi: 10.1016/S0167-6296(96)00529-2. [8] N. A. Doherty and H. Schlesinger, Optimal insurance in incomplete markets, Journal of Political Economy, 91 (1983), 1045-1054.  doi: 10.1086/261199. [9] D. Doiron and N. Kettlewell, Family formation and demand for health insurance, Health Economics, 29 (2020), 523-533.  doi: 10.1002/hec.4000. [10] R. P. Ellis, S. Jiang and W. G. Manning, Optimal health insurance for multiple goods and time periods, Journal of Health Economics, 41 (2015), 89-106.  doi: 10.1016/j.jhealeco.2015.01.007. [11] Y. Fan, The solution of the optimal insurance problem with background risk, Journal of Function Spaces, 2019, Art. ID 2759398, 5 pp. doi: 10.1155/2019/2759398. [12] A. Finkelstein, E. F. P. Luttmer and M. J. Notowidigdo, What good is wealth without health? The effect of health on the marginal utility of consumption, Journal of the European Economic Association, 11 (2013), 221-258.  doi: 10.1111/j.1542-4774.2012.01101.x. [13] A. Finkelstein, N. Hendren and M. Shepard, Subsidizing health insurance for low-income adults: evidence from massachusetts, American Economic Review, 109 (2019), 1530-1567.  doi: 10.1257/aer.20171455. [14] V. R. Fuchs and R. J. Zeckhauser, Valuing health: A priceless commodity, American Economic Review Papers and Proceedings, 77 (1987), 263-268. [15] M. Gerfin, Health insurance and the demand for healthcare, Oxford Research Encyclopedia of Economics and Finance, 1 (2019), 1-22. [16] C. Gollier and H. Schlesinger, Arrow's theorem on the optimality of deductibles: A stochastic dominance approach, Economic Theory, 7 (1996), 359-363. [17] R. E. Hall and C. I. Jones, The value of life and the rise in health spending, Quarterly Journal of Economics, 122 (2007), 39-72.  doi: 10.1162/qjec.122.1.39. [18] T. Y. Ho, P. A. Fishman and Z. B. Zabinsky, Using a game-theoretic approach to design optimal health insurance for chronic disease, IISE Transactions on Healthcare System Engineering, 9 (2019), 26-40. [19] K. Lee, Wealth, income, and optimal insurance, Journal of Risk and Insurance, 74 (2007), 175-184.  doi: 10.1111/j.1539-6975.2007.00206.x. [20] M. Levy and A. R. Nir, The utility of health and wealth, Journal of Health Economics, 31 (2012), 379-392.  doi: 10.1016/j.jhealeco.2012.02.003. [21] T. Li and X. Wang, Connotation and realizing route of Healthy China, Health Economics Research, 1 (2016), 4-10.  doi: 10.1002/rwm3.20392. [22] Z. Lu, S. Meng, L. Liu and Z. Han, Optimal insurance design under background risk with dependence, Insurance: Mathematics and Economics, 80 (2018), 15-28.  doi: 10.1016/j.insmatheco.2018.02.006. [23] H. Markowitz, The utility of wealth, Journal of Political Economy, 60 (1952), 151-156.  doi: 10.1086/257177. [24] K. S. Moore and V. R. Young, Optimal insurance in a continuous-time model, Insurance: Mathematics and Economics, 39 (2006), 47-68.  doi: 10.1016/j.insmatheco.2006.01.009. [25] J. Mossin, Aspects of rational insurance purchasing, Journal of Political Economy, 76 (1968), 552-568.  doi: 10.1086/259427. [26] C. E. Phelps, The Demand for Health Insurance: A Theoretical and Empirical Investigation, Rand Corporation, Santa Monica, Report R-1054-OEO, 1973. [27] J. S. Pliskin, D. S. Shepard and M. C. Weinstein, Utility functions for life years and health status, Operations Research, 28 (1980), 206-224.  doi: 10.1287/opre.28.1.206. [28] S. D. Promislow and V. R. Young, Unifying framework for optimal insurance, Insurance: Mathematics and Economics, 36 (2005), 347-364.  doi: 10.1016/j.insmatheco.2005.04.003. [29] X. Qiao and Y. Hu, Health expectancy of chinese elderly and provincial variabilities, Population & Development, 23 (2017), 2-18. [30] A. Raviv, The design of an optimal insurance policy, American Economic Review, 69 (1979), 84-96. [31] V. L. Smith, Optimal insurance coverage, Journal of Political Economy, 76 (1968), 68-77.  doi: 10.1086/259382. [32] M. Spence and R. Zeckhauser, Insurance, information, and individual action, American Economic Review, 61 (1971), 380-387. [33] S. Wu, Y. Luo, X. Qiu and M. Bao, Building a healthy China by enhancing physical activity: Priorities, challenges, and strategies, Journal of Sport and Health Science, 6 (2017), 125-126.  doi: 10.1016/j.jshs.2016.10.003. [34] W. K. Viscussi and W. N. Evans, Utility functions that depend on health status: Estimates and economic implications, American Economic Review, 80 (1990), 353-374. [35] Z. Q. Xu, X. Y. Zhou and S. C. Zhuang, Optimal insurance under rank-dependent utility and incentive compatibility, Mathematical Finance, 29 (2019), 659-692.  doi: 10.1111/mafi.12185. [36] J. Yan, Why does China's government basic medical insurance system enhance the development of commercial health insurance?, Chinese Review of Financial Studies, 2 (2018), 1-15+221. [37] V. R. Young, Optimal insurance under Wang's premium principle, Insurance: Mathematics and Economics, 25 (1999), 109-122.  doi: 10.1016/S0167-6687(99)00012-8. [38] R. Zeckhauser, Medical insurance: A case study of the tradeoff between risk spreading and appropriate incentives, Journal of Economic Theory, 2 (1970), 10-26. [39] Y. Zhang, Dynamic research on the health level of the elderly in China under the background of Healthy China: Based on CHARLS data empirical analysis, Northwest Population, 40 (2018), 50-59. [40] Y. Zhang and Y. Wu, Optimal health insurance and trade-off between health and wealth, Journal of Applied Mathematics, 2020, Art. ID 2658213, 9 pp. doi: 10.1155/2020/2658213. [41] Y. Zheng, T. Vukina and X. Zheng, Estimating asymmetric information effects in health care with uninsurable costs, International Journal of Health Economics and Management, 19 (2019), 79-98.  doi: 10.1007/s10754-018-9246-z. [42] C. Zhou and C. Wu, Optimal insurance under the insurer's risk constraint, Insurance: Mathematics and Economics, 42 (2008), 992-999.  doi: 10.1016/j.insmatheco.2007.11.005. [43] C. Zhou, W. Wu and C. Wu, Optimal insurance in the presence of insurer's loss limit, Insurance: Mathematics and Economics, 46 (2010), 300-307.  doi: 10.1016/j.insmatheco.2009.11.002.
Optimal indemnity under Exponential loss distribution
Optimal indemnity under Pareto loss distribution
Sensitivity of $d^{*}$ w.r.t. income $y$ under Exponential loss distribution
Sensitivity of $d^{*}$ w.r.t. safety loading $\rho$ under Exponential loss distribution
Sensitivity of $d^{*}$ w.r.t. degree of health degeneration $\theta$ under Exponential loss distribution
Sensitivity of $d^{*}$ w.r.t. upper limit on coverage $K$ under Exponential loss distribution
Sensitivity of $d^{*}$ w.r.t. current health status $h$ under Exponential loss distribution
Sensitivity of $d^{*}$ w.r.t. current wealth $w$ under Exponential loss distribution
Sensitivity of $d^{*}$ w.r.t. income $y$ under Pareto loss distribution
Sensitivity of $d^{*}$ w.r.t. safety loading $\rho$ under Pareto loss distribution
Sensitivity of $d^{*}$ w.r.t. degree of health degeneration $\theta$ under Pareto loss distribution
Sensitivity of $d^{*}$ w.r.t. upper limit on coverage $K$ under Pareto loss distribution
Sensitivity of $d^{*}$ w.r.t. current health status $h$ under Pareto loss distribution
Sensitivity of $d^{*}$ w.r.t. current wealth $w$ under Pareto loss distribution
Parameter values for the illustration model
 Parameter Symbol Value the probability that the insured gets mild illness and $\Theta = 0$ $p_{1}$ 0.50 the probability that the insured gets severe illness or critical illness and $\Theta = \theta$ $p_{2}$ 0.05 safety loading $\rho$ 0.3 upper limit on coverage $K$ 10 initial wealth $w$ 15 initial income $y$ 4 discount factor $\delta$ 0.9 reduced income level $\alpha$ 0.5 initial health status $h$ 0.9 severity of health degeneration after treatment $\theta$ 0.2
 Parameter Symbol Value the probability that the insured gets mild illness and $\Theta = 0$ $p_{1}$ 0.50 the probability that the insured gets severe illness or critical illness and $\Theta = \theta$ $p_{2}$ 0.05 safety loading $\rho$ 0.3 upper limit on coverage $K$ 10 initial wealth $w$ 15 initial income $y$ 4 discount factor $\delta$ 0.9 reduced income level $\alpha$ 0.5 initial health status $h$ 0.9 severity of health degeneration after treatment $\theta$ 0.2
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