Bankruptcy & Life-Cycle Consumption

The paper addresses the limitations of traditional models assuming full loan repayment by incorporating the possibility of Chapter 7 bankruptcy into a four-period life-cycle model of consumer consumption behavior. This innovative approach uses stochastic dynamic programming to solve the model, accounting for the complexities of individual financial decision-making under uncertainty. A significant aspect of this model is the introduction of an endogenous borrowing interest rate, which is dynamically adjusted to reflect the default risk inherent in bankruptcy. This rate is not fixed but rather adjusts to equate loan demand and supply, ensuring that creditors expect zero profit in aggregate. Consequently, the borrowing rate tends to increase with the loan amount because higher borrowing increases the expected loan losses both directly and indirectly by raising the likelihood of default. The model aims to analyze the effects of relaxing the assumption of certain loan repayment on consumer behavior and economic outcomes.

The model incorporates a realistic depiction of Chapter 7 bankruptcy, considering the associated costs. These costs comprise legal fees, asset losses paid to creditors, reputational damage, and potential social stigma. Conversely, bankruptcy offers benefits such as debt reduction, cessation of wage garnishments and other legal actions, and a 'fresh start' for consumers with the ability to retain some assets. Insolvency, the inability to fully repay debts on time, is a central element, driven by factors like unforeseen income losses, increased spending needs, rising interest rates, or simple errors in debt evaluation. The model simplifies by focusing on exogenous uncertainty regarding future labor income, recognizing that this is a primary driver of insolvency risk in consumer financial decision-making, while ignoring other sources of uncertainty for the sake of analysis. The model simplifies lender behavior to immediately forcing insolvent borrowers into bankruptcy, acknowledging this might not always be optimal in reality but simplifying the analysis to isolate the impact of limited liability.

To maintain analytical tractability, the model incorporates several simplifying assumptions. Only one bankruptcy filing per lifetime is allowed, consistent with legal constraints on successive Chapter 7 discharges within a six-year period. Loan maturities are assumed to be the longest possible, aligning with typical consumer loan structures. Non-human assets are perfectly liquid and earn the risk-free rate of interest; durable goods are excluded, and bequests are absent. All debt is unsecured. The credit supply constraint is based on the zero-profit condition for risk-neutral creditors in a perfectly competitive market; there are no transaction costs. The model utilizes a four-period solution tree to track the consumer's financial state, with distinct branches representing the post-bankruptcy scenario, after which the consumer faces a different borrowing constraint reflecting limited liability. This unique framework is designed to analyze the impact of bankruptcy risk on optimal consumer behavior and to illustrate the complex interactions between consumption, borrowing, and insolvency risk.

The model's key variables include consumption, borrowing, and financial asset holdings. The primary constraints are the standard cash flow constraint and a zero-profit credit supply constraint, which endogenously determines the borrowing interest rate. A bankruptcy indicator variable helps track bankruptcy events across time and various states. The model tracks 'discretionary funds', defined as the sum of current income and previous period's financial assets less total loan payments due. This model departs from the standard Yaari (1964) model with perfect capital markets; the primary difference lies in the introduction of bankruptcy risk and the consequent impact on the consumption constraint. The model explicitly considers the conditional probabilities of bankruptcy throughout the time horizon, which informs the decision making process. The interaction between these variables is crucial in understanding the model's results, especially regarding the effects of bankruptcy on optimal consumer behavior.

The model focuses on Chapter 7 bankruptcy, simplifying its representation for analytical clarity. In this simplified version, the consumer receives a discharge of all current debts in exchange for surrendering all financial assets and any current-period labor income exceeding a minimum subsistence level. This minimum level is defined as the minimum value of the probability density function for labor income in the bankruptcy period. This simplified Chapter 7 model captures the essence of limited liability, setting it apart from Chapter 13, which is characterized as a court-sponsored debt repayment plan rather than true bankruptcy. The model's simplification is further emphasized by the constraint that creditors will not lend to individuals who have previously filed for bankruptcy; this restriction forces former bankrupts back into the Yaari (1964) model's framework of certain debt repayment. This assumption, while a simplification, allows for a focused analysis of the effects of bankruptcy in the model. The consequences of Chapter 7 bankruptcy are thus the loss of assets and income exceeding the minimum, coupled with a complete prohibition on future bankruptcy filings.

To streamline the analysis, the model assumes that consumers can only file for bankruptcy once during the four-period time horizon. This is justified by the fact that the model only considers four periods, and US bankruptcy laws prevent successive Chapter 7 discharges within six years. The assumption of a single bankruptcy aligns with real-world constraints. Furthermore, the model incorporates loan maturities reflective of real-world consumer lending practices. These maturities are not single-period loans but rather span the remaining periods of the consumer's life cycle when the loan is initiated, ensuring the loan maturities are aligned with standard consumer lending structures. Loans taken in the first period are repaid over the following three periods, those in the second period are repaid over two periods, and third-period loans are repaid in the final period. Borrowing is disallowed in the final period to avoid the complications of modeling bequests, simplifying the model and making the analysis more focused.

The model incorporates several key assumptions regarding assets and debts. Non-human assets are perfectly liquid and earn a risk-free rate of return. Consumers cannot own durable goods, only rent their services, and there are no bequests. All debt is unsecured. This simplification focuses attention on the direct effects of bankruptcy and avoids complexities associated with collateral or asset-specific risks. Insolvency is defined as the failure to repay all debts from current income and non-human wealth. Importantly, the loan maturity specification is crucial in determining the strictness of the bankruptcy rule. Single-period loans would imply a stricter rule and tighter borrowing constraint. In this case, the model assumes a multiperiod loan structure, making the bankruptcy rule less stringent but also more realistic. The model also considers only one borrowing rate charged before bankruptcy for simplicity, highlighting the intended focus on other crucial elements of the model.

The credit market is modeled as perfectly competitive with risk-neutral creditors who aim to maximize their discounted expected profits. The credit supply constraint is derived from the first-order condition for profit maximization, equating the discounted expected cost of funds with the discounted expected revenues from loans. The borrowing rate is the equilibrating mechanism. The model assumes a perfectly elastic supply of funds at the risk-free rate of interest, and the absence of any transaction costs. Creditors are assumed to make contingent contracts, specifying a single borrowing interest rate and borrowing amounts for all periods before bankruptcy. This simplifies the analysis by eliminating the complexities of varying interest rates or moral hazard, thereby allowing for a concentrated analysis on the fundamental impact of bankruptcy on consumer behavior and the interaction with the credit market. The model’s assumption of perfect competition and risk-neutrality facilitates the derivation of the credit supply constraint, which plays a critical role in determining the endogenous borrowing rate.

The simulation results highlight a key effect of incorporating bankruptcy: a shift in consumption patterns across the life cycle. Compared to the standard Yaari model (which assumes certain repayment), the introduction of bankruptcy causes a notable shift of consumption from later periods to earlier periods. Specifically, the simulation shows a 39% increase in first-period consumption and a more than 14% increase in average second-period consumption in the bankruptcy model compared to the Yaari model. This demonstrates how the possibility of bankruptcy significantly influences consumer spending decisions, prioritizing consumption in the early stages of the life cycle. This finding suggests that the potential for bankruptcy alters the intertemporal allocation of resources, reflecting consumers' risk aversion and preference for immediate gratification in the face of potential financial distress. The results underscore the importance of considering bankruptcy risk when modeling consumer behavior.

A crucial finding is the creation of a wedge between the borrowing rate and the risk-free rate of interest due to default risk. The model's optimal borrowing levels imply a non-zero probability of bankruptcy in all future periods, although these probabilities decline over time. The simulations show a probability of bankruptcy of 0.02 in the second period and 0.0192 in the third period under specific scenarios. This demonstrates that the introduction of bankruptcy risk into the model leads to a higher borrowing rate than the risk-free rate, reflecting the additional risk lenders assume when facing the possibility of borrower default. The magnitude of this wedge highlights the cost of default risk and its influence on credit markets. The results show that the potential for bankruptcy significantly affects the cost of borrowing, and understanding this cost is crucial for a complete understanding of consumer financial decision-making.

The simulation also analyzes the marginal propensity to consume (MPC), revealing notable differences between the bankruptcy model, the Yaari model, and a certainty-equivalence model. While first- and second-period consumption levels are similar in the bankruptcy and certainty-equivalence models, their MPCs differ significantly. The bankruptcy model's MPC generally falls between those of the other two models, showing an irregular pattern as income rises, sometimes exhibiting negative values. This irregularity highlights the non-linear relationship between income and consumption when bankruptcy risk is considered. In contrast, the certainty-equivalence model exhibits constant MPCs, while the Yaari model's MPCs decrease monotonically as income rises. These differences highlight how the incorporation of bankruptcy fundamentally alters the relationship between income and consumption, offering a more nuanced understanding of consumer behavior under uncertainty.

Further simulations explore the impact of changes in initial wealth, first-period income, and bankruptcy costs on borrowing, consumption, and the probability of bankruptcy. Increases in first-period income lead to complex adjustments in borrowing and consumption across periods. Changes in initial wealth influence the MPC differently in the bankruptcy, Yaari, and certainty-equivalence models. The model demonstrates a decrease in borrowing and the borrowing rate of interest as the cost of bankruptcy increases. Specifically, simulations altering the fraction of minimum labor income retained post-bankruptcy show that higher costs reduce the demand for borrowing, indicating the significant influence of bankruptcy costs on consumer financial decisions. The results are consistent with the increased consumer bankruptcy filings observed after the Bankruptcy Reform Act of 1978, which lowered bankruptcy costs. This sensitivity analysis provides valuable insights into how various factors interact to shape consumer financial choices under the possibility of bankruptcy.

The model presented incorporates several simplifying assumptions to enhance analytical tractability. The limitation of allowing only one bankruptcy filing per lifetime simplifies the analysis but might not fully capture the complexities of repeated bankruptcies. The assumption of symmetric information between borrowers and lenders simplifies the interaction in the credit market, neglecting potential complexities like adverse selection or moral hazard. The model also simplifies the bankruptcy process itself, focusing solely on Chapter 7 bankruptcy and neglecting other possible avenues for resolving insolvency, such as debt refinancing, consumer credit counseling services, or wage-earner trusteeships. Ignoring these alternative mechanisms may affect the model's ability to capture the full range of consumer responses to financial distress. The model also utilizes a discrete three-point probability distribution for labor income, which, while simplifying calculations, can lead to differences in results compared to a continuous income distribution. These simplifications should be considered when interpreting the model's findings.

Several extensions to the model are suggested to address its limitations and enhance its realism. Incorporating asymmetric information would introduce complexities such as credit rationing and time-varying interest rates, reflecting information asymmetry in the credit market. This would more accurately reflect the challenges lenders face in assessing borrower risk. Allowing for multiple bankruptcies within the consumer's lifetime would provide a more complete representation of consumer behavior in the face of repeated financial crises. Considering stochastic interest rates or consumption needs would introduce additional sources of uncertainty, leading to a more comprehensive analysis of financial decisions under variable external factors. These extensions could yield additional insights into the factors driving consumer bankruptcy and its consequences. The authors also note that the model's results could vary if it incorporated other options consumers have when facing insolvency, such as debt restructuring or consumer credit advisory services.

The relatively small risk premia observed in the model result from the assumption of symmetric information, neglecting the uncertainty about borrower behavior and creditworthiness. Adding asymmetric information would likely increase these premia and offer a more realistic assessment of the value of information to creditors. While the model produces noteworthy findings, it is crucial to acknowledge that the magnitudes of the results should not necessarily be viewed as accurate real-world estimates. The observed effects can be influenced by the specific parameters used, such as the discrete nature of the income distribution. The model's results can be seen as a first step in understanding the effects of limited liability on consumer borrowing and consumption. Further research incorporating the suggested extensions would significantly improve the realism and comprehensive understanding of consumer behavior under the threat of bankruptcy.