Strategic Default and Bankruptcy Model

[alanlujan91]

Summary

In HARK, borrowing is governed by hard constraints: the agent simply cannot borrow beyond a limit. Real-world borrowing involves default risk, because agents who borrow may choose not to repay if the cost of repayment exceeds the cost of default (bankruptcy, credit exclusion, garnishment). The default option changes the pricing of debt, the distribution of credit, and the response of consumption to income shocks. A HARK model with endogenous default would address questions about consumer bankruptcy policy (Chapter 7 vs. Chapter 13), credit card regulation, mortgage default (strategic and distress-driven), and the macroeconomic effects of credit market frictions.

Connection to Financial Services Practice

The financial-analysis plugin integrates Moody's credit data (financial-analysis/.mcp.json), and the LSEG partner plugin (partner-built/lseg/skills/bond-relative-value/SKILL.md) performs credit spread decomposition, separating government yield from credit curve spreads and residual (liquidity + technicals). The equity-research plugin models credit metrics such as Net Debt/EBITDA and Interest Coverage ratios (financial-analysis/skills/3-statements/SKILL.md). These practitioner tools assess default risk from the lender's perspective. A HARK model provides the borrower's micro-foundation: why agents default, how default probabilities depend on the business cycle, and how credit pricing responds endogenously to policy changes.

Mathematical Model

Environment

Time is discrete and infinite. A continuum of agents indexed by $i$ receive stochastic endowment income $Y_t^i$ and can borrow or save using one-period non-contingent bonds. The key feature is that borrowers can default on their debt, wiping out their obligations at the cost of temporary exclusion from credit markets and a direct utility penalty. Lenders (a competitive financial sector) price bonds to break even given rational expectations about default.

State Variables

The individual state is $(a_t, z_t, \eta_t)$, where we denote by $a_t \in \mathbb{R}$ the agent's asset position (positive = savings, negative = debt), by $z_t$ the persistent income component following

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

and by $\eta_t \in {\text{good standing}, \text{excluded}}$ the credit-market access flag.

Preferences

$$\mathbb{E}_0 \sum_{t=0}^{\infty} \beta^t u(c_t), \quad u(c) = \frac{c^{1-\rho}}{1-\rho}$$

with discount factor $\beta$ and relative risk aversion $\rho$.

Bond Pricing

A competitive risk-neutral financial sector prices one-period bonds. An agent in good standing who borrows $a' < 0$ (i.e., issues a bond with face value $|a'|$ due next period) receives $q(a', z) \cdot |a'|$ today, where the bond price $q(a', z)$ reflects the lender's rational expectation of repayment:

$$q(a', z) = \frac{1}{1 + r + \iota} ; \mathbb{E}\bigl[1 - \mathbb{1}{(a', z') \in \mathcal{D}} ;\big|; z\bigr]$$

where $r$ is the risk-free rate, $\iota$ is an intermediation spread $\iota \geq 0$ representing the lender's administrative cost per unit of lending, so that even a riskless borrower pays rate $r + \iota$ rather than $r$, and $\mathcal{D}$ is the default set (the region of the state space where the agent chooses to default). The default set $\mathcal{D} \subseteq \mathbb{R} \times Z$ is defined over beginning-of-period asset-income pairs. In the bond pricing equation, the condition $\mathbb{1}{(a', z') \in \mathcal{D}}$ evaluates whether the borrower will default next period when the debt $a'$ matures and the income state is $z'$. For savers ($a' \geq 0$), the bond price is simply $q = 1/(1+r)$. The dependence of $q$ on $(a', z)$ means that borrowing terms reflect the agent's individual default risk: agents with worse income prospects or more debt face higher borrowing costs.

Recursive Formulation

We suppress $\eta$ from the notation: $V(a,z)$ denotes the value in good standing ($\eta = \text{good}$), and $V^X(a,z)$ denotes the value during exclusion ($\eta = \text{excluded}$).

Agent in good standing:

$$V(a, z) = \max\bigl{V^R(a, z),; V^D(a, z)\bigr}$$

The agent compares the value of repaying against the value of defaulting.

Value of repaying:

$$V^R(a, z) = \max_{c,, a'} ; u(c) + \beta, \mathbb{E}\bigl[V(a', z')\bigr]$$

subject to the budget constraint

$$c + q(a', z), a' = Y(z) + a \quad \text{if } a' < 0$$ $$c + \frac{a'}{1+r} = Y(z) + a \quad \text{if } a' \geq 0$$

and a natural borrowing limit $a' \geq \underline{a}$.

Value of defaulting:

$$V^D(a, z) = u\bigl(Y^D(z)\bigr) + \beta, \mathbb{E}\bigl[(1 - \lambda), V^X(0, z') + \lambda, V(0, z')\bigr]$$

Default is relevant only when $a &lt; 0$ (the agent owes debt). Upon default, the agent's liabilities are discharged and they emerge with $a = 0$. The income penalty $Y^D(z) = \min(Y(z), \hat{y})$ applies in the period of default only. With probability $\lambda$ each period, the credit record clears and the agent re-enters with $a = 0$.

Value during exclusion:

$$V^X(a, z) = \max_{c,, a' \geq 0} ; u(c) + \beta, \mathbb{E}\bigl[(1 - \lambda), V^X(a', z') + \lambda, V(a', z')\bigr]$$

subject to $c + a'/(1+r) = Y(z) + a$, $a' \geq 0$

During exclusion, the agent can save but not borrow, and earns full income $Y(z)$.

Default Set and Equilibrium

The default set is the collection of states where the agent prefers default:

$$\mathcal{D} = \bigl{(a, z) : V^D(a, z) > V^R(a, z)\bigr}$$

An equilibrium consists of a value function $V$, a bond price function $q$, and a default set $\mathcal{D}$ such that:

1. Given $q(a', z)$, the agent optimizes and $\mathcal{D}$ is the resulting default set.
2. Given $\mathcal{D}$, the bond price satisfies the zero-profit condition for lenders.

This is a fixed-point problem in $q$ (or equivalently in $\mathcal{D}$): default incentives determine bond prices, and bond prices determine borrowing costs, which in turn affect default incentives.

Euler Equation and Bond Price Schedule

For an interior solution in the repayment region, the first-order condition is

$$u'(c) \left[q(a', z) + a' \frac{\partial q}{\partial a'}(a', z)\right] = \beta, \mathbb{E}\bigl[u'(c') \cdot \mathbb{1}{(a', z') \notin \mathcal{D}}\bigr]$$

The term $a' (\partial q / \partial a')$ captures the credit spread effect: additional borrowing worsens the terms on all borrowing, creating an endogenous borrowing cost that depends on the agent's financial position.

Extension: Mortgage Default

For a secured (mortgage) version, replace unsecured debt with collateralized borrowing against housing. Here $d \geq 0$ denotes outstanding mortgage debt (face value) and $H$ denotes the housing stock (using uppercase $H$ to avoid collision with the health state $h$ in other models):

$$V^R_{\text{mortgage}}(a, H, d, z) = \max_{c, a'} ; u(c, H) + \beta, \mathbb{E}[V(a', H', d', z')]$$

$$V^D_{\text{mortgage}}(a, z) = V^{\text{rent}}(a, z) - \kappa$$

where the agent retains liquid assets $a$ but loses the house and mortgage (both equity $(1-\phi^s)p_t^H H - d$ and the collateral obligation are surrendered). Default is more attractive when the house is underwater ($d &gt; p^H H$). This connects directly to a housing model.

Key References

• Chatterjee, S., Corbae, D., Nakajima, M., and Rios-Rull, J.-V. (2007). "A Quantitative Theory of Unsecured Consumer Credit with Risk of Default." Econometrica, 75(6), pp. 1525-1589.
• Livshits, I., MacGee, J., and Tertilt, M. (2007). "Consumer Bankruptcy: A Fresh Start." American Economic Review, 97(1), pp. 402-418.
• Athreya, K. B. (2002). "Welfare Implications of the Bankruptcy Reform Act of 1999." Journal of Monetary Economics, 49(8), pp. 1567-1595.
• Arellano, C. (2008). "Default Risk and Income Fluctuations in Emerging Economies." American Economic Review, 98(3), pp. 690-712. (Sovereign default, but same mathematical structure.)
• Campbell, J. Y. and Cocco, J. F. (2015). "A Model of Mortgage Default." Journal of Finance, 70(4), pp. 1495-1554.

Implementation Notes

• Solving for the equilibrium bond price schedule $q(a', z)$ is the main computational challenge: it requires iterating on the value function and bond price simultaneously until convergence, a fixed-point iteration not currently in HARK's solver toolkit.
• This fixed-point structure goes beyond EGM. While the agent's problem resembles IndShockConsumerType, the standard approach is value function iteration with an inner loop updating $q(a', z)$ at each step.
• Credit-access flag $\eta \in {\text{good}, \text{excluded}}$ fits naturally in MarkovConsumerType.
• For the mortgage default extension, shared state variables and solution methods connect to a housing model (see related issue).
• Calibration targets include the cross-section of debt, default rates, and credit spreads from consumer credit market data.
• HARK's Market class could close the model in general equilibrium (endogenous $r$ clearing the bond market), analogous to the existing Krusell-Smith implementation.
• Connects to the housing model (Issue Housing and Mortgage Choice Model #1730) via mortgage default, and to the entrepreneurship model (Issue Entrepreneurship and Occupational Choice Model #1736) where the endogenous bond price schedule pins down the borrowing rate.