Housing and Mortgage Choice Model

[alanlujan91]

Summary

Housing is the largest asset on most household balance sheets and the dominant source of leverage for the non-wealthy, yet HARK has no housing model. Issue #825 (durable goods) has been open and stalled. A dedicated model with tenure choice (rent vs. own), mortgage origination, and optional refinancing/default would connect HARK to the quantitative macro literature on housing booms, mortgage policy, and wealth inequality.

Connection to Financial Services Practice

The wealth-management plugin in the Claude Financial Services Plugins marketplace models mortgage payoff timing, real estate in estate planning, and home equity as a component of net worth in comprehensive financial plans (wealth-management/skills/financial-plan/SKILL.md). The private-equity plugin models leveraged asset acquisition more generally (private-equity/skills/returns-analysis/SKILL.md). A HARK housing model would ground these practitioner tools in dynamic programming, connecting mortgage contract design, down-payment requirements, and interest rate policy to their effects on household consumption, saving, and default.

Mathematical Model

Environment

Time is discrete, indexed by $t = 0, 1, \ldots, T$. Each period, the agent chooses consumption of a non-durable good $c_t$, a tenure mode $\iota_t \in {\text{rent}, \text{own}}$, and, conditional on tenure, either a rental quantity $s_t$ or a house size $H_t$ with associated mortgage debt $d_t$. The agent faces idiosyncratic labor income risk $Y_t$ and aggregate house price risk $p_t^H$.

Preferences

The agent has non-separable CRRA preferences over non-durable consumption $c_t$ and housing services $s_t$, where we define the housing service flow as the rental equivalent:

$$U = \mathbb{E}_0 \sum_{t=0}^{T} \beta^t \varsigma_t ; u(c_t, s_t)$$

with the period utility function

$$u(c, s) = \frac{\bigl(c^{,\alpha}, s^{,1-\alpha}\bigr)^{1-\rho}}{1-\rho}$$

with the convention that $u(c,s) = \alpha \log c + (1-\alpha)\log s$ when $\rho = 1$, where the discount factor is $\beta \in (0,1)$, the one-period conditional survival probability from age $t$ to $t+1$ is $\varsigma_t$, the consumption share parameter is $\alpha \in (0,1)$, and the coefficient of relative risk aversion is $\rho > 0$. For homeowners, the housing service flow equals the owned stock: $s_t = H_t$.

State Variables

The individual state is $(a_t, H_t, d_t, z_t, t)$, where we denote by $a_t$ liquid financial assets, by $H_t$ the housing stock (zero for renters), by $d_t$ the outstanding mortgage principal, and by $z_t$ the log persistent component of labor income following an AR(1) process:

$$z_{t+1} = \varrho z_t + \epsilon_{t+1}, \quad \epsilon_{t+1} \sim \mathcal{N}(0, \sigma_\epsilon^2)$$

Recursive Formulation

The value function involves a discrete choice over tenure mode. We write the overall value as

$$V_t(a, H, d, z) = \max\bigl{V_t^{\text{stay}}(a, H, d, z),; V_t^{\text{sell}}(a, H, d, z),; V_t^{\text{rent}}(a, z),; V_t^{\text{buy}}(a, z)\bigr}$$

where each option is defined as follows.

Homeowner who stays (no housing transaction):

$$V_t^{\text{stay}}(a, H, d, z) = \max_{c,, a'} ; u(c, H) + \beta, \varsigma_t ;\mathbb{E}\bigl[V_{t+1}(a', (1-\delta^H)H, d', z')\bigr]$$

subject to the budget constraint

$$c + a' = Y(z, t) + Ra - \pi(d, r^m) - \tau^p, p_t^H, H - \delta^H, p_t^H, H$$

where the mortgage payment function is $\pi(d, r^m)$, the property tax rate is $\tau^p$, the housing depreciation rate is $\delta^H$, the gross risk-free rate is $R = 1 + r_f$, and the mortgage balance amortizes according to a standard fixed-rate schedule $d' = (1 + r^m)d - \pi(d, r^m)$. The maintenance cost $\delta^H p_t^H H$ represents the expenditure required to offset physical depreciation.

Homeowner who sells (liquidates housing, becomes renter or buyer):

$$V_t^{\text{sell}}(a, H, d, z) = \max\bigl{V_t^{\text{rent}}(\tilde{a}, z),; V_t^{\text{buy}}(\tilde{a}, z)\bigr}$$

where $\tilde{a} = a + (1-\phi^s) p_t^H H - d$ is the liquid wealth after selling, and the selling cost is $\phi^s$ as a fraction of house value (typically 6-8%).

Renter:

$$V_t^{\text{rent}}(a, z) = \max_{c,, s,, a'} ; u(c, s) + \beta, \varsigma_t; \mathbb{E}\bigl[V_{t+1}(a', 0, 0, z')\bigr]$$

subject to

$$c + q, s + a' = Y(z, t) + Ra$$

where the rental price per unit of housing services is $q$.

Renter who buys:

$$V_t^{\text{buy}}(a, z) = \max_{c,, H',, d',, a'} ; u(c, H') + \beta, \varsigma_t; \mathbb{E}\bigl[V_{t+1}(a', (1-\delta^H)H', d'', z')\bigr]$$

subject to

$$c + (1 + \phi^b),p_t^H, H' - d' + a' = Y(z, t) + Ra$$

and the loan-to-value (LTV) constraint

$$d' \leq \theta, p_t^H, H'$$

where the maximum LTV ratio is $\theta$ (e.g., 0.80 for a 20% down payment), the buying transaction cost is $\phi^b$, and $d'' = (1+r^m)d' - \pi(d', r^m)$ is the mortgage balance after one period of amortization on the newly originated mortgage $d'$.

Collateral and Borrowing Constraints

The liquid asset is subject to a standard no-borrowing constraint $a' \geq 0$ (all borrowing is collateralized through the mortgage). The mortgage collateral constraint $d' \leq \theta, p_t^H, H'$ binds at origination and may also be relevant for refinancing decisions.

Optional Extension: Refinancing and Default

If the agent can refinance, we add a refinancing option within the "stay" branch that replaces the existing mortgage with a new one at the current rate $r_t^m$, subject to paying a refinancing cost $\phi^{refi}$ and meeting the current LTV constraint.

If the agent can default on the mortgage (strategic default), we add

$$V_t^{\text{default}}(a, z) = V_t^{\text{rent}}(a, z) - \kappa$$

where the utility cost of default is $\kappa$, representing stigma and credit-record damage. The agent keeps liquid assets $a$ but loses the house. After default, the agent is excluded from mortgage markets for a stochastic number of periods.

Key References

• Kaplan, G., Mitman, K., and Violante, G. L. (2020). "The Housing Boom and Bust: Model Meets Evidence." Journal of Political Economy, 128(9), pp. 3285-3345.
• Davis, M. A. and Van Nieuwerburgh, S. (2015). "Housing, Finance, and the Macroeconomy." Handbook of Regional and Urban Economics, Vol. 5, pp. 753-811.
• Berger, D., Guerrieri, V., Lorenzoni, G., and Vavra, J. (2018). "House Prices and Consumer Spending." Review of Economic Studies, 85(3), pp. 1502-1542.
• Iacoviello, M. (2005). "House Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle." American Economic Review, 95(3), pp. 739-764.
• Campbell, J. Y. and Cocco, J. F. (2015). "A Model of Mortgage Default." Journal of Finance, 70(4), pp. 1495-1554.

Implementation Notes

• Because the tenure choice (rent/own/sell/buy) is endogenous rather than exogenous as in MarkovConsumerType, the DBlock architecture with a discrete Control variable for tenure mode is the appropriate framework, combined with DC-EGM (Iskhakov et al. 2017) for the mixed discrete-continuous problem.
• Within each tenure branch, the continuous choices (consumption, savings, house size, mortgage) can use EGM, similar to IndShockConsumerType.
• House prices $p_t^H$ can be exogenous (partial equilibrium) or endogenous in a Market equilibrium with a housing supply function.
• With 4 continuous state variables $(a, H, d, z)$, computation is demanding. A common simplification restricts housing to a discrete grid of sizes, reducing the problem to $(a, d, z)$ continuous with discrete $H$.
• Connects to the strategic default model (Issue Strategic Default and Bankruptcy Model #1733) and the inter-vivos transfer model (Issue Inter-Vivos Transfers and Dynastic Bequest Model #1737, where parental gifts relax the down-payment constraint), and to BequestWarmGlowConsumerType for housing as a bequeathable asset.