Does health insurance extend life expectancy? Evidence from a structural estimation

Abstract

We apply the simulated method of moments to a lifecycle model to infer the impact of health insurance on life expectancy. The model features endogenous health risks, which are determined jointly by the natural law and one’s own decisions of medical care expenditures. Individual’s optimal decisions of consumption, saving, and medical care expenditure are solved, and life expectancies are calculated in the scenarios with and without health insurance. The comparison result shows that health insurance has a positive impact on life expectancy through its positive impact on medical care expenditures, but this impact is not quantitatively significant.

Appendix: Numerical solution and simulation algorithm

Here, we provide a detailed description of the numerical solution algorithm and simulation method, based on the life-cycle model with annuities.

1.1 Discretization

We first discretize the state spaces for critical illness insurance, annuity, savings, medical care expenditure, and income and restrict optimal choices on these grids. For convenience, we take 10,000 RMB as the unit.

State space for age (${{\varvec{\Omega}}}_{\mathbf{a}}$) and health state (${{\varvec{\Omega}}}_{\mathbf{z}}$) The age ranges from 45 to 90, which we standardize as 1 to 46 in the program, that is, ${\Omega }_{a}=\left\{{1,2},\dots 46\right\}$. The state space for the health state is already discrete, that is, ${\Omega }_{z}=\left\{{1,2},3\right\}$.

State space for critical illness insurance (${\boldsymbol{\Omega }}_{{\varvec{B}}}$) We set 21 optional sum assureds for critical illness insurance, with the minimum sum assured as 0 and the maximum sum assured as 100, that is, $B\in {\Omega }_{B}=\left\{{0,5},\dots ,100\right\}$.

State space for annuity (${\boldsymbol{\Omega }}_{{\varvec{n}}}$) In the model, we set the non-refundable premium ${A}_{t}$ as the control variable. To reduce the number of variables, we set ${A}_{t}=\left({n}_{t}-{n}_{t-1}\right){h}_{t}$ and take the annual annuity payout ${n}_{t}$ after retirement as the state variable as well as the control variable. We assume 21 optional payouts for annuity at each period, ranging from 0 to 10. In addition, the annuity is non-refundable, that is ${n}_{t}\ge {n}_{t-1}$ for $t\in \left[0,T\right]$. Therefore, ${n}_{t}\in {\Omega }_{n,t}=\left\{{n}_{t-1},{n}_{t-1}+0.5,\dots ,10\right\}$ for $t\in \left[0,T\right]$, where ${n}_{-1}$ is appointed to be 0, and ${\Omega }_{n}=\left\{{0,0.5},\dots ,10\right\}$.

State space for saving (${\boldsymbol{\Omega }}_{{\varvec{s}}}$) The relationship between cash wealth ${G}_{t}$ and savings ${s}_{t}$ has been established in the model, that is, ${G}_{t+1}=\left(1+{r}_{{f}}\right){s}_{t}$. Therefore, the state space for ${G}_{t}$ is known if we set the state spaces for ${s}_{t}$. We assume there are 100 optional points for saving at each period, ranging from 0 to 101. Then, ${s}_{t}\in {\Omega }_{s,t}={\Omega }_{s}=\left\{0, {1.01,2.02},\dots ,101\right\}$ for $t\in \left[0,T\right]$.

State space for medical care expenditure (${\boldsymbol{\Omega }}_{{\varvec{e}}}$) We assume 100 optional points for medical care expenditure at each period if the individual is sick, ranging from 0 to 101. Then, ${e}_{t}\in {\Omega }_{e,t}={\Omega }_{e}=\left\{0, {1.01,2.02},\dots ,101\right\}$ for $t\in \left[0,T\right]$.

State space for income process (${\boldsymbol{\Omega }}_{{\varvec{w}}}$) According to Tauchen (1986), we approximate the continuous-valued income process using a three-state Markov chain ${\widetilde{\rho }}_{{a,t}}$ to reduce the numerical burden, that is, ${\widetilde{\rho }}_{x+t,t}\in \left\{-0.5{\sigma }_{\rho },0, 0.5{\sigma }_{\rho }\right\}$, where ${\sigma }_{\rho }={\left({\sigma }_{\upsilon }^{2}/\left(1-{\vartheta }^{2}\right)+{\sigma }_{\varepsilon }^{2}\right)}^{0.5}$. As in Tauchen (1986), we define ${\widetilde{\rho }}^{(1)}=-0.5{\sigma }_{\rho }$, ${\widetilde{\rho }}^{(2)}=0$,${\widetilde{\rho }}^{(3)}=0.5{\sigma }_{\rho }$, and ${w}={\widetilde{\rho }}^{(2)}-{\widetilde{\rho }}^{(1)}=0.5{\sigma }_{\rho }$; then,

$${p}_{i2}=P\left({\widetilde{\rho }}_{x+t,t}=0\left|{\widetilde{\rho }}_{x+t-1,t-1}={\widetilde{\rho }}^{\left(i\right)}\right.\right)\approx P\left(-\frac{{w}}{2}\le {\widetilde{\rho }}_{x+t,t}\le \frac{{w}}{2}\left|{\widetilde{\rho }}_{x+t-1,t-1}={\widetilde{\rho }}^{\left(i\right)}\right.\right)=P\left(-\frac{{w}}{2}\le \vartheta {\widetilde{\rho }}^{\left(i\right)}+{\nu }_{x+t}-\vartheta {\varepsilon }_{t-1}+{\varepsilon }_{t}\le \frac{{w}}{2}\right)=P\left(-\frac{{w}}{2}-\vartheta {\widetilde{\rho }}^{\left(i\right)}\le {\nu }_{x+t}-\vartheta {\varepsilon }_{t-1}+{\varepsilon }_{t}\le \frac{{w}}{2}-\vartheta {\widetilde{\rho }}^{\left(i\right)}\right)={F}\left(\frac{{w}/2-\vartheta {\widetilde{\rho }}^{\left(i\right)}}{\sqrt{{\sigma }_{\upsilon }^{2}+\left(1+{\vartheta }^{2}\right){\sigma }_{\varepsilon }^{2}}}\right)-{F}\left(\frac{-{w}/2-\vartheta {\widetilde{\rho }}^{\left(i\right)}}{\sqrt{{\sigma }_{\upsilon }^{2}+\left(1+{\vartheta }^{2}\right){\sigma }_{\varepsilon }^{2}}}\right),$$

and

$${p}_{i1}={ F}\left(\frac{{\widetilde{\rho }}^{(1)}+{w}/2-\vartheta {\widetilde{\rho }}^{\left(i\right)}}{\sqrt{{\sigma }_{\upsilon }^{2}+\left(1+{\vartheta }^{2}\right){\sigma }_{\varepsilon }^{2}}}\right),$$

$${p}_{i3}=1-{ F}\left(\frac{{\widetilde{\rho }}^{(3)}-{w}/2-\vartheta {\widetilde{\rho }}^{\left(i\right)}}{\sqrt{{\sigma }_{\upsilon }^{2}+\left(1+{\vartheta }^{2}\right){\sigma }_{\varepsilon }^{2}}}\right).$$

Inserting the values of ${\sigma }_{\upsilon }^{2}$, ${\sigma }_{\varepsilon }^{2}$, and $\vartheta$, we have ${\sigma }_{\rho }=0.7168$ and the transition matrix as follows:

$$\left(\begin{array}{ccc}0.5615& 0.1868& 0.2517\\ 0.3985& 0.2030& 0.3985\\ 0.2517& 0.1868& 0.5615\end{array}\right).$$

1.2 Numerical solution

We solve our model using backward induction. The iterations involve the following steps.

Step 1: We first solve the problem in period $T$. For all the grids in state space ${{\Omega }_{s}\times {\Omega }_{n}\times \Omega }_{z}$ and a given sum assured $B$ for critical illness insurance in state space ${\Omega }_{B}$, we solve Eqs. (14) and (20) using the grid search method. For convenience, we transcribe these equations as follows:

$${V}_{{T}}\left({G}_{{T}},{n}_{T-1},{B}_{{T}},{z}_{{T}}=1\left|B\right.\right)=\underset{{c}_{{T}},{s}_{{T}}}{{max}}\left\{{\omega }_{1}U\left({c}_{{T}}\right)+\beta\Psi \left({G}_{T+1}^{\left(1\right)}\right)\right\},$$

$${V}_{{T}}\left({G}_{{T}},{n}_{T-1},{B}_{{T}},{z}_{{T}}=2\left|B\right.\right)=\underset{{c}_{{T}},{s}_{{T}}}{{max}}\left\{{\omega }_{2}U\left({c}_{{T}}\right)+\beta\Psi \left({G}_{T+1}^{\left(1\right)}\right)\right\}.$$

In detail, for any grid in ${{\Omega }_{s}\times {\Omega }_{n}\times \Omega }_{z}$ and the given $B$, we calculate the lifelong utilities in the brace on the right for all possible grids of ${s}_{{T}}$ (${s}_{{T}}\in {\Omega }_{s}$). We find the maximum value ${V}_{{T}}^{*}\left({G}_{{T}},{n}_{T-1},{B}_{{T}},{z}_{{T}}\left|B\right.\right)$ and the corresponding optimal grid ${s}_{{T}}^{*}\left({G}_{{T}},{n}_{T-1},{B}_{{T}},{z}_{{T}}\left|B\right.\right)$.

Step 2: We then solve the problems backing to the first period of retirement ($t=T-1,T-2,\dots ,Tr$). In the same way, we calculate the lifelong utilities in period $t$ for all possible grids of $\left({s}_{t},{e}_{t}\right)\in {\Omega }_{s}\times {\Omega }_{e}$ when sick or ${s}_{t}\in {\Omega }_{s}$ when healthy, based on any grids in ${{\Omega }_{s}\times {\Omega }_{n}\times \Omega }_{z}$, the given $B$, and the solved maximum values ${V}_{t+1}^{*}\left({G}_{t+1},{n}_{t},{B}_{t+1},{z}_{t+1}\left|B\right.\right)$ in period $t+1$. We find the maximum value ${V}_{t}^{*}\left({G}_{t},{n}_{t-1},{B}_{t},{z}_{t}\left|B\right.\right)$ and the optimal grids of ${s}_{t}^{*}\left({G}_{t},{n}_{t-1},{B}_{t},{z}_{t}\left|B\right.\right)$ and ${e}_{t}^{*}\left({G}_{t},{n}_{t-1},{B}_{t},{z}_{t}=2\left|B\right.\right)$ for a given combination of states and the given $B$.

Step 3: We further solve the problems in the working period ($t=Tr-1,Tr-2,\dots ,1$) employing the same method. However, the possible grids are $\left({s}_{t},{n}_{t}\right)\in {\Omega }_{s}\times {\Omega }_{n}$ when healthy and $\left({s}_{t},{n}_{t},{e}_{t}\right)\in {\Omega }_{s}\times {\Omega }_{n}\times {\Omega }_{e}$ when sick. In addition, the state space becomes ${{\Omega }_{s}\times {\Omega }_{n}\times {\Omega }_{w}\times \Omega }_{z}$ in the working period. Then, the maximum value ${V}_{t}^{*}\left({G}_{t},{n}_{t-1},{w}_{t},{B}_{t},{z}_{t}\left|B\right.\right)$ and optimal grids of ${s}_{t}^{*}\left({G}_{t},{n}_{t-1},{w}_{t},{B}_{t},{z}_{t}\left|B\right.\right)$, ${n}_{t}^{*}\left({G}_{t},{n}_{t-1},{w}_{t},{B}_{t},{z}_{t}\left|B\right.\right)$, and ${e}_{t}^{*}\left({G}_{t},{n}_{t-1},{w}_{t},{B}_{t},{z}_{t}=2\left|B\right.\right)$ are found for a given combination of states and the given B. Further, ${V}_{1}^{*}\left({G}_{1},{n}_{0},{w}_{1},{B}_{1},{z}_{1}\left|B\right.\right)$ in the algorithm equals to ${V}_{0}\left({G}_{0},{n}_{-1}=0,{w}_{0},{B}_{0},{z}_{0}\left|B\right.\right)$ in the model.

Step 4: We find the optimal $B$ at last. For every value in ${\Omega }_{B}$, we repeat Steps 1–3 to calculate ${V}_{1}^{*}\left({G}_{1},{n}_{0},{w}_{1},{B}_{1},{z}_{1}\left|B\right.\right)$. We then compare the values of ${V}_{1}^{*}$ with different $B$ and find the optimal ${B}^{*}\left({G}_{1},{n}_{0},{w}_{1},{B}_{1},{z}_{1}\right)$, which maximizes ${V}_{1}^{*}$.

1.3 Simulation

The life cycle decisions are not constant values, depending on the states in the future. Monte Carlo simulations are finally carried out to generate the optimal actions (consumption, savings, medical care expenditure when sick, and critical illness insurance purchase in the initial and annuity purchase before retirement) from the first period forward to the end of life. The simulations involve the following steps.

Step 1: The income path is first drawn using the three-state Markov income process. The process is replicated 10,000 times to produce 10,000 paths, which are said to be the income processes of 10,000 individuals.

Step 2: We calculate the space of transition probabilities. The probabilities from the sick state ${\pi }_{t,t+1}^{2j}$ are endogenous and determined jointly by the natural law and the individual’s own decision of medical care expenditure ${e}_{t}$ (${e}_{t}\ge 0$). This is because we do not know the decision of medical care expenditure before we know the states. Therefore, we calculate all of the possible ${\pi }_{t,t+1}^{2j}$. That is, for any ${e}_{t}\in {\Omega }_{e}$ and $t\in \left[{1,2},\dots ,T\right]$, we calculate $\left\{{\pi }_{t,t+1}^{2j}\left({e}_{t}\right)\right\}$.

Step 3: We conduct simulations from the first period forward to the second period. For individual $i$, the initial states and optimal critical illness insurance are known, and the corresponding optimal decisions in the first period can be determined in ${s}_{1}^{*}\left({G}_{1},{n}_{0},{w}_{1},{B}_{1},{z}_{1}\left|{B}^{*}\right.\right)$ and ${n}_{1}^{*}\left({G}_{1},{n}_{0},{w}_{1},{B}_{1},{z}_{1}\left|{B}^{*}\right.\right)$ by matching the states. Then, the states in the second period can be further produced; that is, ${G}_{2}$ and ${n}_{1}$ are decided by the optimal decisions in the first period, ${w}_{2}$ is cited from the second value on the income path of individual $i$, and ${B}_{1},{z}_{1}$ are generated according to the exogenous probabilities from healthy state ${\pi }_{t,t+1}^{1j}$.

Step 4: We conduct simulations from the second period forward to the end of life. The corresponding optimal decisions in period $t$ ($t={2,3},\dots , T$) can be determined by matching ${G}_{t},{n}_{t-1},{w}_{t},{B}_{t},{z}_{t}$, and the states in period $t+1$ can be further produced. It is worth mentioning that ${B}_{t+1},{z}_{t+1}$ are generated according to the endogenous probabilities from the sick state ${\pi }_{t,t+1}^{2j}\left({e}_{t}\right)$, which is determined in the space of transition probabilities calculated in Step 2, with ${e}_{t}={e}_{t}^{*}\left({G}_{t},{n}_{t-1},{w}_{t},{B}_{t},{z}_{t}=2\left|{B}^{*}\right.\right)$.

Step 5: Steps 3 and 4 are replicated 10,000 times, and then the income path, path of healthy states and life cycle planning of every individual are simulated. The results are calculated by averaging the life cycle planning of all individuals.