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doi: 10.3934/jimo.2021031
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## 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 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 & Management Optimization, doi: 10.3934/jimo.2021031
##### References:

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##### References:
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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