Endogenous Insurance Purchase Model (Health/Life/LTC)

[alanlujan91]

Summary

Out-of-pocket medical spending for retirees averages roughly $4,300/year but has a long right tail: the 95th percentile exceeds $20,000 (MEPS, 2019). HARK's MedShockConsumerType models exogenous medical expenditure shocks, and BasicHealthConsumerType adds health capital investment, but neither model includes endogenous insurance purchase, the agent's choice of whether and how much coverage to buy. Because insurance smooths consumption across health states, the purchase decision interacts with precautionary saving, portfolio allocation, and labor supply. A model with insurance choice would address questions about health insurance market design (ACA, Medicare, Medicaid), long-term care policy, and the welfare effects of mandates and subsidies.

Connection to Financial Services Practice

The wealth-management plugin financial-plan skill (wealth-management/skills/financial-plan/SKILL.md) includes a full risk management module: life insurance needs analysis (income replacement, debt payoff, education funding), disability insurance adequacy, long-term care planning, and umbrella liability coverage. The skill notes that insurance gaps are one of the top 6 prioritized action items in every financial plan. A HARK model would answer the questions these heuristics leave open: why households are over- or under-insured, how insurance demand varies with wealth and health, and how Medicaid and Medicare crowd out private coverage.

Mathematical Model

Environment

Time is discrete, $t = 0, 1, \ldots, T$. Each period the agent faces a stochastic medical expenditure need $\tilde{x}_t$ drawn from a health- and age-dependent distribution. Before the medical need is realized, the agent chooses an insurance contract from a menu. The insurance contract determines the out-of-pocket cost function. The agent also chooses consumption and saving.

State Variables

The individual state is $(m_t, h_t, t)$, where we denote by $m_t$ cash-on-hand (financial wealth plus current income), and by $h_t \in {1, 2, \ldots, H}$ a discrete health state (e.g., $H = 3$: good, fair, poor).

Insurance Contract Menu

The agent chooses from a finite set of insurance plans $\iota_t \in {0, 1, \ldots, J}$, where $\iota = 0$ denotes no insurance. Each plan $j$ is characterized by:

• A premium $p_j(h_t, t)$ (possibly health- and age-rated, or community-rated depending on the policy regime)
• A deductible $d_j$
• A coinsurance rate $\gamma_j \in [0, 1]$
• An out-of-pocket maximum $\bar{x}_j$

The out-of-pocket cost function for plan $j$ given medical need $x$ is:

$$\text{oop}_j(x) = \min\Bigl(\bar{x}_j, ;; \min(x, d_j) + \gamma_j \cdot \max(x - d_j, 0)\Bigr)$$

The agent pays the full cost up to the deductible $d_j$, then the coinsurance share $\gamma_j$ of costs above the deductible, capped at the out-of-pocket maximum $\bar{x}_j$. For the uninsured ($j = 0$), we have $\text{oop}_0(x) = x$. For plan $j = 0$, we set $p_0 = 0$, $d_0 = 0$, $\gamma_0 = 1$, and $\bar{x}_0 = +\infty$, so that $\text{oop}_0(x) = x$.

Health Dynamics

Health transitions follow a first-order Markov chain with age-dependent transition probabilities:

$$\Pr(h_{t+1} = h' \mid h_t = h) = \Pi_t(h, h')$$

Mortality depends on health: the survival probability is $\varsigma_t(h) = \Pr(\text{alive at } t+1 \mid h_t = h)$. The transition matrix $\Pi_t(h, h')$ gives the probability of transitioning to health state $h'$ conditional on surviving to period $t+1$, so that $\sum_{h'} \Pi_t(h,h') = 1$. The overall probability of being alive in state $h'$ at $t+1$ is $\varsigma_t(h) \cdot \Pi_t(h, h')$.

Medical Need Distribution

The medical expenditure need $\tilde{x}_t$ is drawn from a health- and age-dependent distribution:

$$\tilde{x}_t \sim F_{h_t, t}(x)$$

A standard specification uses a mixture: with probability $p_0(h, t)$ the agent has zero medical need, and with probability $1 - p_0(h, t)$ the need is drawn from a lognormal:

$$\log \tilde{x}_t \mid \tilde{x}_t > 0 ;\sim; \mathcal{N}\bigl(\mu_x(h, t),; \sigma_x^2(h, t)\bigr)$$

Preferences

$$u(c) = \frac{c^{1-\rho}}{1-\rho}$$

with warm-glow bequest motive $\Phi(a) = \phi_b (a + \underline{a})^{1-\rho}/(1-\rho)$ upon death.

Timing Within a Period

1. The agent enters period $t$ with cash-on-hand $m_t$ and health $h_t$.
2. The agent chooses insurance plan $\iota_t$ and pays premium $p_{\iota_t}(h_t, t)$.
3. The medical need $\tilde{x}_t$ is realized.
4. The agent pays out-of-pocket costs $\text{oop}_{\iota_t}(\tilde{x}_t)$.
5. The agent chooses consumption $c_t$ and saves the remainder.

Recursive Formulation

Stage 1: Insurance choice (before medical need realization):

$$V_t(m, h) = \max_{\iota \in {0, 1, \ldots, J}} ; W_t(m - p_\iota(h, t),; h,; \iota)$$

where we require $m - p_\iota(h, t) \geq 0$ (the agent must be able to afford the premium).

Stage 2: Post-insurance, pre-medical-need value:

$$W_t(\tilde{m}, h, \iota) = \mathbb{E}_{\tilde{x} \sim F_{h,t}} \bigl[\hat{V}_t\bigl(\max(\tilde{m} - \text{oop}_\iota(\tilde{x}),; \underline{c}),; h\bigr)\bigr]$$

where $\tilde{m} = m - p_\iota(h,t)$ is post-premium resources.

Stage 3: Consumption-saving (after all costs are paid):

$$\hat{V}_t(\hat{m}, h) = \max_{c} ; u(c) + \beta, \varsigma_t(h) \sum_{h'} \Pi_t(h, h'), V_{t+1}(m', h') + \beta\bigl(1-\varsigma_t(h)\bigr)\Phi(\hat{m} - c)$$

subject to

$$m' = R(\hat{m} - c) + Y_{t+1}$$

$$c \leq \hat{m}, \qquad c > 0$$

where we define end-of-period assets $a_t = \hat{m} - c$ and next-period income $Y_{t+1}$ follows the standard permanent-transitory process used in HARK's IndShockConsumerType.

Medicaid as a Consumption Floor

Following De Nardi, French, and Jones (2010), Medicaid provides a consumption floor $\underline{c}$ for agents whose resources fall below subsistence after medical costs. We incorporate this by modifying the Stage 2 integration:

$$W_t(\tilde{m}, h, \iota) = \mathbb{E}_{\tilde{x} \sim F_{h,t}} \bigl[\hat{V}_t\bigl(\max(\tilde{m} - \text{oop}_\iota(\tilde{x}),; \underline{c}),; h\bigr)\bigr]$$

where $\tilde{m} = m - p_\iota(h,t)$ is post-premium resources.

Medicaid effectively provides free insurance to the poorest agents, which crowds out private insurance demand among the near-poor (the Medicaid notch).

Insurance Pricing

In equilibrium, premiums satisfy the insurer's zero-profit condition:

$$p_j(h, t) = (1 + \mu) \cdot \mathbb{E}\bigl[x - \text{oop}_j(x) ;\big|; h, t, \text{ agent chooses plan } j\bigr]$$

where the loading factor $\mu > 0$ covers administrative costs, and where the distribution of $x$ conditional on plan choice is endogenous, creating a fixed-point condition solved jointly with the agent's optimal plan selection. Adverse selection arises because sicker agents select more generous plans, driving up premiums for those plans, the classic Rothschild-Stiglitz (1976) problem. Community rating (requiring $p_j$ to be independent of $h$) exacerbates this.

Extension: Life Insurance and Long-Term Care Insurance

Life insurance: The agent chooses a face value $F_t \geq 0$ at actuarially fair (plus loading) premium $p^{\text{life}} = (1+\mu^{\text{life}})(1-\varsigma_t(h)) \cdot F_t / R$. Upon death, the bequest becomes $a_t + F_t$ rather than $a_t$. This connects to the bequest motive.

Long-term care insurance: Similar structure, but the insurance pays out upon entering a "disabled" health state $h = H$ that requires nursing home care. The premium is $p^{\text{LTC}} = (1+\mu^{\text{LTC}}) \Pr(h_{t+1} = H \mid h_t) \cdot \text{benefit}$.

Key References

• De Nardi, M., French, E., and Jones, J. B. (2010). "Why Do the Elderly Save? The Role of Medical Expenses." Journal of Political Economy, 118(1), pp. 39-75.
• Ameriks, J., Briggs, J., Caplin, A., Shapiro, M. D., and Tonetti, C. (2020). "Long-Term-Care Utility and Late-in-Life Saving." Journal of Political Economy, 128(6), pp. 2375-2451.
• Handel, B. R. (2013). "Adverse Selection and Inertia in Health Insurance Markets: When Nudging Hurts." American Economic Review, 103(7), pp. 2643-2682.
• Rothschild, M. and Stiglitz, J. (1976). "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information." Quarterly Journal of Economics, 90(4), pp. 629-649.
• Pashchenko, S. and Porapakkarm, P. (2013). "Quantitative Analysis of Health Insurance Reform: Separating Regulation from Redistribution." Review of Economic Dynamics, 16(3), pp. 383-404.
• Braun, R. A., Kopecky, K. A., and Koreshkova, T. (2017). "Old, Sick, Alone, and Poor: A Welfare Analysis of Old-Age Social Insurance Programmes." Review of Economic Studies, 84(2), pp. 580-612.

Implementation Notes

• The three-stage structure (insurance choice, medical realization, consumption-saving) maps to HARK's multi-stage-within-period architecture, as in RiskyContribConsumerType.
• Insurance choice is a discrete optimization over $J+1$ plans, computed by evaluating the continuation value for each plan and selecting the maximum; no continuous first-order condition is needed.
• Medical expenditure integration (expectation over $\tilde{x}$) can use HARK's existing calc_expectation machinery with the discrete approximation to $F_{h,t}$.
• Health dynamics connect directly to MarkovConsumerType and BasicHealthConsumerType.
• Natural calibration sources: Medical Expenditure Panel Survey (MEPS) for medical cost distributions, Healthcare.gov for plan parameters.
• Medicaid consumption floor is a max operation on post-cost resources, already a feature of the De Nardi et al. specification.
• Shares health transition structure with the Social Security claiming model (Issue Social Security Claiming Decision Model #1732). Builds on MedShockConsumerType by adding the insurance purchase margin.