Costas Meghir
email: c.meghir@yale.edu

Florian Oswald
email: florian.oswald@sciencespo.fr

Wenli Li
email: wenli.li@phil.frb.org

Consumer Bankruptcy, Mortgage Default and Labor Supply *

Keywords: JEL Classification: G33, K35, J22, J31, D14, D18, D52, D53, E21 Lifecycle, bankruptcy, Mortgage Default, housing, Labor Supply, Consumption, Education, Insurance, moral hazard Lifecycle, Bankruptcy, Housing, Mortgage Default, Labor Supply, Consumption, Education, Insurance, Moral hazard. JEL Codes: G33, K35, J22, J31, D14, D18, D52, D53, E21

We specify and estimate a lifecycle model of consumption, housing demand and labor supply in an environment where individuals may file for bankruptcy or default on their mortgage. Uncertainty in the model is driven by house price shocks, {education specific} productivity shocks, and catastrophic consumption events, while bankruptcy is governed by the basic institutional framework in the US as implied by Chapter 7 and Chapter 13. The model is estimated using micro data on credit reports and mortgages combined with data from the American Community Survey. We use the model to understand the relative importance of the two chapters (7 and 13) for each of our two education groups that differ in both preferences and wage profiles. We also provide an evaluation of the BACPCA reform. Our paper demonstrates importance of distributional effects of Bankruptcy policy.

Introduction

A number of countries, including the US and the UK, have bankruptcy legislation allowing a fresh start for people who are unable to repay debts. Such legislation is an attempt to balance the legitimate rights of creditors with the need to offer a degree of insurance for adverse events in a world with incomplete markets.

The legislation is complex and treats secured and unsecured debt differently. The US even offers alternative choices for filing for bankruptcy. For example, under certain conditions one can choose to file under Chapter 7, which completely clears all outstanding debt, in return for the individual foregoing their assets, such as housing. Alternatively, one can file under Chapter 13, keeping all assets but agreeing to a gradual repaying scheme for at least part of the debt, clearing the rest. Chapters 7 and 13 offer different levels of protection and target the insurance provided by the system to different needs and populations. The tradeoff between Chapter 7 and Chapter 13 is crucial and varies depending on other institutional features, such as the level of homestead exemption in the particular state or the level of recourse. The homestead extension is the extent to which housing equity has to be used to repay outstanding unsecured debts following bankruptcy and varies widely across US states from 0% to the entire level of housing equity. Recourse legislation defines the extent to which non-housing assets can be used to repay debts following residential mortgage default. The design of the system implies a trade off between insurance and moral hazard and will tend to have important disrtibutional implications, not just between creditors and borrowers but also importantly, between different types of borrowers, in terms of their risk profile and level of (permanent) income. Thus, an important focus of this paper is the way the bankruptcy system affects different education groups, that are characterized by very different income profiles and preferences for housing and work.

Understanding the effects of the institutional framework requires a model of the household that encompasses the complex interactions of various decisions such as labor supply, savings, house purchase, and, of course, the decision to file for bankruptcy and/or to default on a mortgage. Together these decisions help capture the many margins of household adjustment as well as moral hazard and thus the costs and benefits of bankruptcy legislation. Furthermore, the effects of such legislation need to be considered in an equilibrium framework to account for the feedback effect of its implications on the price of credit.

In this paper we investigate the impact of bankruptcy legislation on different education groups by specifying and estimating a microeconomic life-cycle model that generalizes the [START_REF] Aiyagari | Uninsured idiosyncratic risk and aggregate saving[END_REF] economy.

Our model is closest to that of [START_REF] Mitman | Macroeconomic Effects of Bankruptcy & Foreclosure Policies[END_REF], but with a number of differences that we discuss below.

Specifically, in the model individuals choose to buy a house of a certain size or to rent, save in a separate liquid asset, and supply labor. Their labor productivity is subject to shocks, which, together with house price volatility and the random arrival of infrequent but catastrophic spending shocks (such as medical expenses), can lead to financial distress and an inability to service existing loans or mortgages. Thus, at each period individuals may file for bankruptcy and/or default on their mortgage. Lenders set interest rates for unsecured debt and for mortgages conditional on borrowers' probability of filing for bankruptcy and/or default.

To estimate the model we use the simulated method of moments (SMM, see [START_REF] Mcfadden | A method of simulated moments for estimation of discrete response models without numerical integration[END_REF][START_REF] Pakes | Simulation and the asymptotics of optimization estimators[END_REF]) combined with the Markov Chain Monte Carlo method (MCMC) adapted for SMM by [START_REF] Chernozhukov | An MCMC approach to classical estimation[END_REF]. This approach makes estimation tractable and at the same time avoids problems with moments that may not be differentiable with respect to parameters, as is often the case when simulation methods are used.

Our model is estimated using US data including those drawn from the Federal Reserve Bank of New York/Equifax Consumer Credit Panel (FRBNY CCP/Equifax) and based on the period before the implementation of the Bankruptcy Abuse and Prevention Consumer Protection Act (BAPCPA reform) in October 2005. We use anonymized microeconomic data that records all loan and mortgage activity as well as bankruptcies. We combine US county level information on house price variability, bankruptcy and mortgage default rates with information from the census. The final data allow us to estimate a rich model of individual consumption and labor supply with differences across education groups. The latter is important because individuals with lower education have lower income and lower accumulated housing wealth and are more likely to find themselves in financial difficulty. Moreover, they are more likely to benefit from filing under Chapter 7 both before and after BAPCPA because of their low assets.

We use the estimated model to understand the extent and value of insurance offered by the system. Specifically, we examine the impact of shutting down either Chapter 13 or Chapter 7 filings, consider the impact of changing the probability of a deficiency judgment for those who default on their mortgage, and examine the impact of the BAPCPA reform which introduced a means test for for Chapter 7 bankruptcy filing, among other changes.

Our paper builds on the thesis by Oswald (2015), and has a number of common features with Mitman See [START_REF] Sullivan | As we forgive our debtors: Bankruptcy and consumer credit in America[END_REF] pp. 128 for an account of the importance of housing shocks as drivers of bankruptcy. BAPCA: The Bankruptcy Abuse and Prevention Consumer Protection Act on the Department of Justice website. (2016). Both papers allow for Chapter 13 and Chapter 7 filing as well as mortgage default and both use an equilibrium setting, which is of particular importance for this type of problem. Both also model explicitly a rental market. However, our model differs in several substantive ways. First, we allow for labor supply; this is important because by varying labor supply an individual can change the probability of bankruptcy (by increasing labor income) and because it allows us to capture the labor market implications of post-bankruptcy repayment plan; the anticipation of such an event can itself change effort levels, possibly limiting bankruptcy; and ex post the type of filing and the actual settlement can affect work incentives. Second, our model features a finite-horizon life-cycle setup, which implies a life-cycle structure on assets and borrowing. For instance, a mortgage contract has a finite maturity and the loan to value ratio declines as borrowers age. This is important, since mortgage vintage, which is highly correlated with borrower age, is a strong predictor of default and bankruptcy. Third, we allow households to differ by education, an important source of heterogeneity for income risk as well as preferences. Lower education individuals have lower income and are thus more likely to file under Chapter 7. The explicit modeling of differences by education enables us to examine the distributional effects of policy changes.

In addition to [START_REF] Mitman | Macroeconomic Effects of Bankruptcy & Foreclosure Policies[END_REF], a number of other papers have addressed the effects of institutional arrangements on bankruptcy. Important examples include [START_REF] Pavan | Consumer durables and risky borrowing: The effects of bankruptcy protection[END_REF], who investigates the effect of exemption levels on bankruptcy and durable purchases and finds that welfare gains from greater insurance are canceled out by losses due to tighter credit constraints, making the impact of homestead exemptions small. [START_REF] Hintermaier | Debt Portfolios and Homestead Exemptions[END_REF] arrive at a similar conclusion but approaching the problem differently.

They use a calibrated model of housing, collateralized and uncollateralized debt, and show that variations in homestead exemptions have little impact on the price and amount of unsecured debt because in equilibrium individuals who hold such debt invest little in housing. Based on reduced form analysis [START_REF] Gropp | Personal Bankruptcy and Credit Supply and Demand[END_REF] find that homestead exemptions tend to favor high asset households and make credit more expensive harming lower income groups. [START_REF] Fay | The household bankruptcy decision[END_REF] provide evidence of strategic bankruptcy, which is central for our approach, since households in our model can choose to file and indeed the availability of filling will affect their borrowing and housing decisions. Since bankruptcy laws affect available insurance they can be expected to crowd out other insurance mechanisms available. Thus [START_REF] Traczynski | Divorce rates and bankruptcy exemption levels in the US[END_REF] notes that more generous bankruptcy laws reduce the risk sharing advantages of marriage and based on within state variation of exemption levels, finds substantial effects of generous exemption levels on divorce rates.

Our respective projects started independently at about the same time.

More broadly, our paper extends the [START_REF] Aiyagari | Uninsured idiosyncratic risk and aggregate saving[END_REF]-type economy where borrowing is allowed only up to an amount the consumer can repay with probability one, to one where bankruptcy and mortgage default laws effectively permit the violation of the life-cycle budget constraint. The theoretical foundation of this is laid out in [START_REF] Chatterjee | A Quantitative Theory of Unsecured Consumer Credit with Risk of Default[END_REF]; examples of applications to different aspects of risk-sharing and welfare implications are [START_REF] Athreya | Default, insurance, and debt over the life-cycle[END_REF], who examines the interaction of bankruptcy with social insurance, Li and Sarte (2006), who model bankruptcy chapter choices explicitly, and [START_REF] Livshits | Consumer bankruptcy: A fresh start[END_REF], who calibrate a life-cycle model to investigate welfare differences of alternative bankruptcy schemes.

In a recent paper [START_REF] Chen | The impact of personal bankruptcy on labor supply decisions[END_REF] analyze labor supply and bankruptcy choices in a partial equilibrium search and matching model. They use the model to infer the value of the existence of Chapter 7 bankruptcy. Their model does not consider housing, however, and predicts that higher wage earners would prefer Chapter 7 over Chapter 13. Given that high wage earners are more likely to be homeowners, who are in turn more likely to file for Chapter 13, we reach different conclusions. [START_REF] Han | Fresh start or head start? the effects of filing for personal bankruptcy on work effort[END_REF] estimate the effects of bankruptcy on labor supply using an IV approach. They find no impact, which they interpret as reflecting the opposing impacts of wealth effects of the debt write-off and the incentive provided from the fresh start aspect of bankruptcy. [START_REF] Dobbie | Debt Relief and Debtor Outcomes: Measuring the Effects of Consumer Bankruptcy Protection[END_REF] consider the ex post effects of being granted Chapter 13 bankruptcy protection conditional on filing; when judges (assumed to be randomly assigned) deny protection many do not take this further and others file under Chapter 7. With respect to such a comparison group, they find important effects on earnings, foreclosure and even mortality. Our model abstracts from the possibility of being denied protection, given formal eligibility but does allow randomness in the terms offered under Chapter 13. Li et al. (2011) study the effects of the BAPCPA reform on mortgage default and find that the reform caused prime and subprime mortgage default rates to rise. Finally, [START_REF] Albanesi | Insolvency After the 2005 Bankruptcy Reform[END_REF] also study the effects BAPCPA the reform. They find that the reform reduced Chapter 7 fillings and increased insolvency, but did not increase Chapter 13 fillings; they attribute this to liquidity constraints and the increased costs of filling. Our model does allow for such costs and for liquidity constraints, shown to be important by this work and others.

The paper proceeds as follows. In Section 2 we present the data and some descriptive facts about bankruptcy, default and the institutional context. We then present our model in Section 3, followed by Section 4 on estimation. Section 5 discusses the implications of the model and Section 6 discusses counterfactuals.

Finally Section 7 concludes.

Data and Descriptive facts

Our data is drawn from several sources. We use the proprietary Federal Reserve Bank of New York Consumer Credit Panel/Equifax data ( FRBNY CCP/Equifax ) data merged with the Black Knight McDash data also known as LPS Mortgage Loan Level Data or the "McDash" data (Black Knight McDash) data to compute bankruptcy and default rates at county level. We supplement this with county level house prices obtained from Zillow Research, as well as county level demographic and economic characteristics from the American Community Survey (ACS).

The FRBNY CCP/Equifax is a 5% random sample representative of those with a credit history and includes details of various types of debt as well as loan-level detailed anonymized information on mortgages they have taken out. It also contains anonymized information on various demographic characteristics, such as age, zip code, census tract and Equifax credit risk score. The Black Knight McDash data includes detailed anonymized information at the time of origination, such as the amount of the loan, estimated house prices, interest rate and, importantly, termination date and reason for termination. We match the FRBNY CCP/Equifax with the Black Knight data thus creating a data set that allows for a detailed study of mortgage defaults.

We then merge county level information on education and employment using the American Community Survey (ACS) anonymized Public Use Micro Data Sample. Finally, we use the Panel Study of Income Dynamics (PSID) data to estimate the life-cycle profiles and associated stochastic processes for income, as well as hours worked and homeownership rates for each education group. We present summary statistics for data from PSID in Table 1 by educational attainment.

We use the Zillow House Value Index for All Homes at County level, see http://www.zillow.com/research/data/. Such information includes loan origination date, amount, current balance, requested payment amount or term of the loan, credit limit (on HELOCs), individual/joint account and payment status, whether GSE guaranteed, whether for a mobile home, whether second mortgage, and whether the account was closed in bankruptcy or foreclosure.

A detailed description of the panel can be found at http://www.newyorkfed.org/research/staff_reports/sr479.pdf.

The termination types include paid off, foreclosure and other negative termination events such as REO sale. The data includes among other variables, the loan amount, house price, documentation status, source of the loan (e.g., whether it was broker-originated), property location (zip code), type of loan (fixed-rate, ARM, prime, subprime, etc.), the prepayment penalty period (if any).

The match is based on mortgage loan origination date, origination amount, the zipcode of the property, purpose of the mortgage (purchase versus refinance), lien status (first lien versus second lien or home equity), type of mortgage (agency loans or not)) and occupancy type (primary residence, second homes or investment properties).

This data have been used extensively over the past few years to study mortgage defaults. See, among others, [START_REF] Demyanyk | Understanding the Subprime Mortgage Crisis[END_REF], Li and[START_REF] White | Mortgage Default, Foreclosure, and Bankruptcy[END_REF][START_REF] Brown | The Impact of Housing Markets on Consumer Debt: Credit Report Evidence from 1999 to 2012[END_REF][START_REF] Brown | The Impact of Housing Markets on Consumer Debt: Credit Report Evidence from 1999 to 2012[END_REF].

Institutional Features

Individuals can file for bankruptcy either under Chapter 13, or if eligible, under Chapter 7. Those filing under Chapter 13 get to keep all their assets but have to agree to a repayment plan typically over five years and as a function of their usual earnings. After that period all debt will be discharged. By contrast those filing under Chapter 7 have all their debt immediately discharged but have to give up their assets, including their housing equity. However, the way this is done varies by state: each state defines a homestead exemption, which is an amount of protected housing equity that creditors cannot access. So if housing equity is below the exemption the individual keeps the house. If it is above, then the house is sold and the amount of equity remaining after any mortgage has been repaid and after the exemption is accounted for is handed over to the creditors as part repayment of the unsecured debt. Homestead exemptions vary across the states of the US from nothing to an unlimited amount (as in Florida, for example).

BAPCPA limited the eligibility to Chapter 7 to those with incomes below the state median income.

Finally, the other key feature of the institutional set up is the extent to which mortgage lenders can access other financial assets to repay any equity not covered by the sale of a house under repossession by obtaining a so called Deficiency Judgment or mortgage lender Recourse (see [START_REF] Ghent | Recourse and Residential Mortgage Default: Theory and Evidence from US States[END_REF][START_REF] Li | Recourse and residential mortgages: the case of Nevada[END_REF], for instance). These features make mortgage default and bankruptcy interrelated and their design provides alternative tradeoffs between the insurance value of bankruptcy and the moral hazard they induce.

The institutional details across the US are given in Appendix Table D.2.

Descriptive Facts on Bankruptcy

Table 2 presents the annual non-business bankruptcy and mortgage default rates in our sample. About 1% of individuals file for bankruptcy, and in our sample period three quarters of these file under Chapter 7. About 2.1% mortgage borrowers default on their mortgage. Time trends in non-business bankruptcy filings and mortgage defaults are depicted in Figure 1. The spike in Chapter 7 bankruptcies in 2005, followed by a speedy decline, is associated with the introduction of the BAPCPA reform, which introduced a means test for filling under Chapter 7 and led to a rush to file before the rules changed. Our model will be fit to the period before the implementation of this reform; amongst other counterfactuals we will also offer our version of the impact of this reform. The following years are marked by the Great Recession and the associated upheaval in the housing and mortgage market, which is beyond the scope of our paper. We now describe how the bankruptcy and mortgage default relate to individual characteristics and the institutional framework by a regression. Since we do not observe individual level education we run the regression at the county level using the log-odds ratio as a dependent variable:

log y jt 1 -y jt = γ 0 + γ 1 College j,2010 + βX jt + ε j,t , (1)
where y j,t represents in turn overall bankruptcy rate, Chapter 7, Chapter 13 and mortgage default in county j and period t. We define two education groups, a high level which includes all those with a college degree and more and a lower level, comprising the rest. Hence, College j,2010 is the percentage of the population of j in year 2010 with a college degree or more. X jt is a set of control variables. At county-year level we include the Zillow house price index, the default rate, and when explaining default behaviour, the bankruptcy rate.

At county level but fixed to 2010, we include the homeownership rate. Finally, the state legal environment is captured by whether the state offers lender recourse, whether homestead exemption is unlimited, and if not, the size of the homestead exemption. We report the regression results in Table 3. More generous bankruptcy provisions increase the demand for credit but the amount of credit supplied declines pushing up interest rates, reducing the equilibrium amount of unsecured debt. This interplay between insurance offered by generous provisions and moral hazard is central to the motivation of our paper and lies at the heart of discussions in [START_REF] Gropp | Personal Bankruptcy and Credit Supply and Demand[END_REF] and [START_REF] Mitman | Macroeconomic Effects of Bankruptcy & Foreclosure Policies[END_REF]. The interplay between these two forces is demonstrated in the table : The size of the homestead exemption is not associated with higher or lower rates of bankruptcy, implying increased moral hazard is fully counteracted by reduced supply of credit. When we consider the states with unlimited exemption, which effectively allow people to borrow and shelter the assets into housing we find reduced levels of bankruptcy reflecting an even stronger reaction in the supply of credit.

The next two columns show opposing associations for Chapters 7 and 13. Interestingly, the difference across chapters carries over to the unlimited exemption, which is positively associated with the rate of Chapter 13 bankruptcy but negatively with the rate of chapter 7 filings. Whether a state allows for recourse is not significantly associated with bankruptcy, but as we expect it reduces the default rate (last column), because it becomes more costly to the individual since their other assets can be seized to pay for the uncovered loan.

Education may be a particularly important factor in driving both behavior towards bankruptcy and the risks that people face, which are reflected in differences in the income process. Higher education individuals earn more but their income is subject to higher variance shocks (for example, [START_REF] Meghir | Income variance dynamics and heterogeneity[END_REF]. These different processes will imply different borrowing and savings behavior and different resilience to shocks.

In understanding the value of alternative bankruptcy arrangements and how well they are targeted it is thus important to allow for this heterogeneity, which is one of the motivations of the paper.

Table 3 demonstrates a strong association between education bankruptcy filing rates and mortgage default rates. To quantify the effect, Table 4 shows how the coefficients in the regression translate to different predicted rates of bankruptcy and default, which we use for estimation: the rates of bankruptcy and default are more than fifty percent higher for the low education group. Note: Prediction of bankruptcy filing rates and mortgage default rates by educational achievement in countylevel data. We take the group of counties with % of college below the 25-th percentile as reference level for low education and compute the predicted value for a high educated county (above the 75-th percentile) using the estimate for γ 1 in (1), as shown in Table 3.

Model

Overview

Individuals are life cycle expected utility maximizers. The active life period starts at age 25 and lasts until age T = 60, the last period we explicitly model. Individuals differ by their completed level of schooling, but are identical in all other respects ex ante. They face persistent wage rate and house price uncertainty. Given this, in each period they decide how much to consume, work and save. They can either rent a house or buy one of a range of different sizes. Selling a house is subject to transaction costs. Houses are financed by a long term mortgage contract of a finite number of periods and secured against the value of the house. The mortgage interest rate depends on the probability of defaulting on the mortgage. Individuals can also obtain unsecured loans at an interest rate that is a function of the probability of bankruptcy and is set so that the expected profits of making the loans are zero.

Finally, indebted individuals can file for bankruptcy under Chapter 7 or Chapter 13. Depending on the institutional arrangements in place they may have to give up their house to repay some of the delinquent debt, if they file under Chapter 7. Under Chapter 13 they will have to agree on a repayment scheme, deducted at source from their wages for a random number of periods of fixed expected duration. In addition to filing for bankruptcy they can separately decide to default on their mortgage. In this case, the recourse legislation in place governs the extent to which they have to use their other assets to cover the remaining mortgage.

We next describe the essential elements of the model and provide examples of some of the value functions and choices. A detailed description of the model is provided in Appendix A.

Preferences

Households derive utility from consumption of a composite non-durable consumption good c, leisure, and housing good h.

Labor supply decisions are modeled as choices from a set of five hours values l ∈ {l 1 , l 2 , . . . , l 5 } where l 1 = 0 hours and P ≡ 1 [l > 0] and L is maximal leisure time. Houses come in

H sizes, h ∈ {h, h 1 , . . . , h H } and H = 1 [h > h].
Only the smallest type h is also available for rent. The This simplification is made for computational purposes and focuses the model on periods where housing decisions and mortgage repayment are central.

We use "individuals", "households" and "agents" interchangeably.

per-period utility function for individuals of education level e is

u e (c, l, h) = c ω e (L -l -Pθ Pe ) 1-ω e 1-γ 1 -γ exp (θ He h) + µ e Hh, (2)
where θ Pe denotes the fixed cost of working. The additive term µ e Hh makes preferences nonhomothetic, and the sign of µ e governs whether housing is a necessity or a luxury as in [START_REF] Attanasio | Modelling the demand for housing over the life cycle[END_REF].

House Prices

We normalize the unit price of housing to be 1 at the time of purchase and thereafter we assume house prices follow an autoregressive process with idiosyncratic shocks it :

p it = ρ p p it-1 + it-1 with p i0 = 1 and it ∼ N(0, σ 2 p ). (3)
We abstract from aggregate shocks, which lead to a number of complications that are beyond the scope of this paper.

Labor productivity

Log hourly wages of an individual with education level e grow deterministically with age t and are subject to idiosyncratic shocks as follows,

ln y it = e i + f e (t) + w e it , (4)
where f e (t) is an education specific polynomial in age (t) and w e it follows an age-t-dependent Markov chain of order one, where -importantlyboth state space and transition matrix of the Markov chain depend on age and education level e. We interpret w e it as the wage residual, and include it in the consumer's state space.

We take this representation of the wage process from De [START_REF] Nardi | Nonlinear household earnings dynamics, self-insurance, and welfare[END_REF] and estimate it on PSID data following their procedure (see Appendix B). We suppress the index for education e henceforth but note that all preference parameters are separately estimated for each group. In case of a labor supply choice of l = 0, the individual receives benefits b instead of labor income y × l.

Catastrophic Spending Shocks

In addition to wage and house price shocks, the individual is also subject to the arrival of catastrophic spending shocks. As pointed out in [START_REF] Sullivan | As we forgive our debtors: Bankruptcy and consumer credit in America[END_REF], the main reasons cited for consumer bankruptcy are job loss (67.5% of respondents), family issues like divorce (22.2%), and medical expenses (19.3%). We introduce an independently distributed spending shock to capture the latter two categories, over and above our income and price shocks. We adopt the setup and calibration in [START_REF] Livshits | Consumer bankruptcy: A fresh start[END_REF], which estimate dollar values for spending shocks from individual level medical spending data. We denote the spending shock by a ∈ {0, 32, 102} (in $1000), and the probability of spending shock a arising by π a (a) (see Table 5).

Institutions Governing Bankruptcy and Mortgage Default

Under Chapter 7, all non-exempt assets are seized to help repay the debt and the remaining debt is discharged. The extent to which housing equity has to be used for debt repayment is regulated by the state's homestead exemption (ξ), which varies from zero to the entire value of the house. Under Chapter 13, by contrast, filers keep their house regardless of the exemption level but they sign up to a repayment plan stipulating debt repayments for as long as they are in the bankruptcy punishment state. In our model exit from such a state is random with a probability of 20% each year leading to an average duration of 5 years.

Note that Chapter 13 is only an option if the creditor can expect to recoup at least as much as under a Chapter 7 liquidation. Associated with filing for bankruptcy are certain costs: First, the individual is excluded from financial markets for the duration of the punishment. Second, bankruptcy incurs a chapter specific psychic costs, λ je , j = 7, 13, associated with the stigma of a bad credit record.

Homestead exemption is a legal clause which exempts certain amount of home equity from liquidation.

As a result, if an owner finds himself with unsecured debt and at the same time has equity in the home below the exemption level, he could file for bankruptcy without risking losing his home in a forced sale, since the unsecured lender is prevented from claiming the exempt equity. If an owner in excess of the exemption limit files for Chapter 7, he loses the house, which is sold at market price, but he gets to keep the exemption level from the proceeds of the sale.

The second institution concerns default on mortgage debt. An owner may default if the mortgage debt is higher than the equity value of the house, although even then he may decide not to because defaulting is

We set the exit to be random so as to avoid introducing an extra state variable in the model.

costly. Additionally, after default the owner becomes a renter and the house is sold to recover the debt.

In all but seven US Statesa lender may lay claim to a fraction of other assets via recourse (we denote the probability of success of a resulting deficiency judgment as ψ ∈ [0, 1]) to cover remaining outstanding debt after default [START_REF] Ghent | Recourse and Residential Mortgage Default: Theory and Evidence from US States[END_REF].

Financial Market

In our model financial institutions borrow at a fixed interest rate r from the world market and issue unsecured debt and mortgages ensuring zero profits on each type of debt. We now describe the terms under which this debt is supplied.

Mortgage Market

A mortgage is a loan of a fixed initial amount m 0 = (1χ)p 0 h, depending on the value of the house on which it is secured (p 0 h), and bearing a fixed interest rate r m , leading to constant payments κ(m 0 ), computed using the standard amortization formula, over its exogenously given term T m < T years. The term χ represents the mandatory downpayment. Using the prime to denote the next period, the loan amount evolves according to

m = (1 + r m )m -κ(m 0 ) (5)
To adjust the loan for the individual default risk over the term we define a price that will ensure zero profits on each individual mortgage contract (see [START_REF] Kaplan | The housing boom and bust: Model meets evidence[END_REF]. Thus, the individual will be making repayments based on the nominal value of the loan m 0 = (1χ)p 0 h but will only receive (1

-χ × q m 0 )p 0 h,
where q m 0 is the price of the loan. To see how this is set, define as d(x) the default indicator at state of the world x, which includes wages, the house value, the outstanding mortgage as well as net financial assets. The mortgage price is then determined according to the recursive formulation

q m (m|S) = 1 m(1 + r m ) E w |w,p |p,a d δ d p h + (1 -d ) κ + q m (m |S )m , (6)
where E w |w,p |p,a is the expectation operator taken over all future states, included in the full state space S, see [START_REF] Bajari | An empirical model of subprime mortgage default from 2000 to 2007[END_REF] and [START_REF] Guiso | Moral and social constraints to strategic default on mortgages[END_REF] for discussions of these issues

The standard amortization formula is

κ(m) = m r m (1+r m ) Tm -t (1+r m ) Tm -t -1 .
introduced below, m is the outstanding loan at the beginning of the period before the default decision. The curly bracket in (6) describes the repayment to the mortgage lender in case of default (d = 1), when the house is sold in foreclosure at a discount δ d < 1, and in the case of continuing mortgage payments: The value to the lender in that case is the next payment κ plus the next period's pricing function times the remaining mortgage debt m = (1 + r m )mκ.

Unsecured Debt Market

Individuals can take out one year unsecured loans, which they can choose to repay or file for bankruptcy as in [START_REF] Chatterjee | A Quantitative Theory of Unsecured Consumer Credit with Risk of Default[END_REF] or [START_REF] Athreya | Default, insurance, and debt over the life-cycle[END_REF]. The equilibrium cost of the loan needs to take into account the possibility of bankruptcy filing under Chapter 7 or 13. Under Chapter 13, a 5-year repayment plan is set, where the amount repaid each year is equal to a fifth of the amount owed. In the model we place a cap of 15% of annual earnings, reflecting the practice in courts. Alternatively, if the individual files under Chapter 7 (subject to eligibility) no repayments are made from future income but any available assets above an exemption level (housing in our case) is seized to repay as much of the debt as possible with the rest discharged. We define this as non-exempt equity, given by ν = max((1φ) (phm)ξ, 0), with φ a proportional transaction cost when selling the house and ξ the homestead exemption. In either case, the amount an individual can borrow depends on the expected repayment for the lender next period. Thus, the implicit bond price for a one-period loan is given by

q a (a |S) = 1 (1 + r)(-a ) -a (1 -π 7 (a , S) -π 13 (a , S)) + π 7 (a , S)νζ + π 13 (a , S) ȳT bk , (7)
where π j (a , S) is the probability of filing for Chapter j, given a savings choice a . The squared bracket denotes the expected repayment amount for the lender, where ζ < 1 parametrizes inefficiency in the bankruptcy technology of the lender, that is, the lender recovers only ζν from a Chapter 7 bankruptcy. Hence, expression (7) for a renter (or an owner with only exempted equity, such that ν = 0) implies a smaller unsecured loan than for an owner with ν > 0, all else equal. Notice that decreasing homestead exemption ξ will increase ν, and therefore ceteris paribus will lead to larger unsecured borrowing. Finally, note that equation ( 7) reduces to the familiar q a (a |S) = 1 1+r if the probability of bankruptcy is zero.

Consumer Choice

The household maximizes lifetime utility given by

U e = E 0 T-1 ∑ t=1 β t u e (c t , l t , h t ) + β T V Te (a, h, p, m, τ) (8)
with respect to sequences {c t , h t , l t , d t } T t=1 of consumption, housing, labor supply, and a set of discrete choices d relating to bankruptcy and default decisions. The term β < 1 is the discount factor and V Te is the terminal value function, which depends on the amount of home equity at the end of the active life cycle and the start of retirement. The expectation is taken with respect to contingent paths of wages, house prices, and spending shocks.

The optimization is subject to a budget constraint, whose detailed form is given with the value functions depending on the circumstances. The flow of labor income is endogenous and depends on labor supply. However, we also introduce a basic insurance b, set at 23% of median income, so that even when the individual is unemployed a low level of consumption is guaranteed. In addition, we allow for basic consumption support c equal to 10% of median income, which is applicable following a catastrophic consumption shock once the individual has filed for bankruptcy. Neither of these amounts vary by education.

The timing in each period is as follows: First, wage, house price and spending shocks are resolved.

Then, optimal savings, housing, labor and discrete choices are made. In each period t < T -1, in addition to the consumption decision, the renter's problem in the non-bankruptcy state is to choose the maximal value In what follows we drop the education subscript e without loss of clarity, to simplify somewhat the notation.

The Choice of Renters

The state space for a renter includes their liquid assets a, wage w, and age t. The corresponding optimal value function is W(a, w, t). If she has filed for bankruptcy before then her value function is denoted by Wj (a, w, t), for j = 7, 13. Defining med(y) as state median income, we write the problem of a renter in the post BAPCPA period as:

W(a, w, t) =                max W rent , W buy , W file.7 , W file.13 if a < 0, yl < med(y); max W rent , W buy , W file.13 if a < 0, yl ≥ med(y); max W rent , W buy if a ≥ 0. (9)
The restriction on the discrete choice set of the renter in ( 9) makes explicit that one only file for bankruptcy if there exists unsecured debt. Moreover, when simulating the BAPCPA version of the model, we implement the means test by preventing individuals with labor income above med(y) to file for Chapter 7. The pre-BAPCPA model corresponds to setting med(y) = ∞. We define the conditional value functions next.

Value of Buying

As an illustration, consider the value of buying (W buy ) for an individual that is currently renting and not in bankruptcy. A prime denotes the next period and V denotes the owner's value:

W buy (a, w, t) = max a ,h,l u(c, l, h) + βE w |w,p |p,a V(S ) (10) subject to S = (a -a, w , p , n , h, t + 1), c + a q a (a |w, p, n, h) = yl + a -κ((1 -χ)p i0 h) -Γ(p 0 , h, w, t), (11)
π 7 (a |w, h) = E w |w,p |p,a 1 V file.7 S > V -file.7 S , ( 12
)
π 13 (a |w, h) = E w |w,p |p,a 1 V file.13 S > V -file.13 S , ( 13
) n = n -1, w t+1 |w t ∼ Π w (t), (14)
p |p ∼ Π p . ( 15
)
The state space of the owner S contains liquid assets, wage shock, house price, mortgage vintage, house size and age. For simplicity, we use vintage of mortgage n ∈ { n, n -1, . . . , 0}, where n = 0 denotes a paid-off mortgage, instead of the actual mortgage amount. The budget constraint of the buyer (11) includes the first mortgage payment κ and the downpayment Γ ≡ (1q m 0 (1χ))p 0 h, where q m 0 is the price of the loan given by ( 6). The function q a , reflecting the cost of unsecured debt, depends on mortgage debt and the house size, (m, h), and is defined in (7). The respective probabilities of bankruptcy, that drive the cost of unsecured debt, are defined in ( 12) and ( 13). Finally, equations ( 14) and ( 15) denote the transition matrices associated to the discretizations of wage and house price processes in ( 4) and (3). We refer the reader to Appendix A for details on the value functions listed in equation ( 9), and on computation.

The Problem of the Owner

The problem of an owner not in a bankruptcy state is complex due to the large number of subcases, depending both on the asset position, the house value and the level of income. It is given by

V(S) =                                      max V stay , V sell if a ≥ 0, hp t -m t ≥ 0; max V stay , V sell , V def if a ≥ 0, hp t -m t < 0; max V stay , V sell , V file.7 , V file.13 if a < 0, hp t -m t ≥ 0, yl < med(y); max V stay , V sell , V file.13 if a < 0, hp t -m t < 0, yl ≥ med(y); max V stay , V sell , V def , V file.7 , V file.13 , V file.def if a < 0, hp t -m t < 0, yl < med(y); max V stay , V sell , V def , V file.13 , V file.def if a < 0, hp t -m t < 0, yl ≥ med(y), (16)
where def denotes default, a ≥ 0 indicates non-negative financial assets, and hp tm t is equity in the house.

Again, not all discrete choices are available everywhere on the state space, as can be seen from the restrictions for each case. For example, the admissible chapter of bankruptcy depends on labor income lying below the threshold med(y), as before (whenever BAPCPA is active). The default choice is only an option if home equity is negative. In other words we assume that a person who has difficulty repaying a mortgage (say because of a negative income shock) but has positive equity in the house will always choose to sell. Owners with home equity in excess of the exemption level face eviction should they file for bankruptcy under Chapter 7. The level of homestead exemption determines whether an owner filing under Chapter 7 stays on in the Given (h, p), there is a one-to-one mapping between n and m in the model Note the i.i.d. spending shock a appears only in the future state of assets, hence is part of the expectations operator.

house or is evicted. We define the sub-problems in Appendix A, but we illustrate the value of default.

Value of Default

The default value is similar to the value of selling with the exception that for a defaulter unsecured borrowing is not allowed, and a one-time utility penalty is incurred. Regarding recourse legislation, we introduce parameter ψ ∈ [0, 1] here that relates to the probability with which negative equity ((1φ)(phm)) will be rolled over into post default life. Notice that the future value is that of a renter, but the asset state takes into account any remaining mortgage debt d ≤ 0 brought forward.

V def (S) = max a >0,l∈L u (c, l, h) + βE w |w,a W( d + a -a, w , t + 1) (17)
subject to

c + 1 1 + r a = yl + a, d = ψ((1 -φ)ph -m).

Owner in Bankruptcy State

An owner who has declared bankruptcy but has kept his home can either remain a home owner (stay), sell or default on the mortgage. Thus his valuation is Ṽj (S) = max Ṽstay j , Ṽsell j , Ṽdef j , j = 7, 13.

In case of default, the amount of assets that the person carries over to the next period depends both on the extent of recourse and on the amount of mortgage debt. In any case financial assets cannot be negative since the person has already filed for bankruptcy and cannot borrow. We need to provide a social safety net to individuals in bankrutpcy state, in case a negative spending shocks or a deficiency judgement arrives. In such a case, the individual receives u(c, l 1 , h), i.e. the flow utility of consuming the safety-net assignment, full leisure and the smallest house in the concerned period.

Leaving the rest to Appendix A we show here the value for the case where the owner keeps his home

Ṽstay j (S) = max a >0,l∈L u λ j c, l, h + βE w |w,p |p,a (1 -δ) Ṽj S + δV(S ) (18)
subject to S = (max(aa, 0), w , p , n , h, t + 1), S = (aa, w , p , n , h, t + 1),

c + 1 1 + r a = a + yl -1 [j = 13] ȳ -κ(p 0 h, r m , T m ), j = 7, 13,
where δ is the probability of exiting the state of bankruptcy, ȳ is the loan payment agreed under Chapter 13 and λ j is the psychic cost of a bad credit record, which depends on chapter (j).

Terminal Value function

We model the active part of the life cycle until a mandatory retirement age. At that point we specific a terminal value function of the form

V T (a, h, p, m) = θ W log(a + ph -m) -θ BK 1 [in bankruptcy state] , (19)
where θ BK is a penalty if one enters the last period in bankruptcy (BK) state.

Parameterization and Estimation

We limit parameter estimation to the states in group 5 in Table D.2, which we observe annually from years 2000 to 2006. The states in group 5 have similar institutions governing bankruptcy and mortgage default, in particular, they are in the same quartile of homestead exemption relative to state median income, and they allow for mortgage deficiencies. A number of parameters are set based on earlier results from the literature and are shown in Table 5 with the exception of safety net and unemployment benefit which we chose to be very small numbers and our model simulations are not sensitive to our choices.

In principle we could estimate the model for all groups, but it would add computational burden for little extra insight. [START_REF] Chernozhukov | An MCMC approach to classical estimation[END_REF]. This approach makes estimation tractable and at the same time avoids problems with moments that may not be differentiable with respect to parameters, as is often the case when simulation methods are used.

Estimation Results and Model fit

The estimates together with their asymptotic standard errors are presented in Table 6. The education groups display very different preferences for housing and for work (fixed costs). The stigma effects associated with bankruptcy λ 7,13 and with default λ de f are high. For example, the stigma of Chapter 7 bankruptcy for college graduates reduces period consumption by 51% (1 -0.49). The other preference parameters reflect the patterns in the data, and in combination with the budget constraint determine the fit of the model, to which we now turn. Model Fit Table 7 displays model and data moments on bankruptcy filing rates by chapter, mortgage default rates, homeownership rates, and annual hours worked for each education group and for households on average and near retirement, respectively. Overall, the model does a good job at matching bankruptcy filing rates as well as mortgage default rates for both education groups and for the near retirement age group except that we overpredict Chapter 13 bankruptcy filing rates for the low education group. In terms of homeownership rates, our model slightly underpredicts relative to the data for both education groups. The average number of hours worked is a bit higher in the model than in the data for both education groups, likely a consequence of not modeling disability that pushes hours down from about the age of 45 (see [START_REF] Meghir | Chapter 9 -Earnings, Consumption and Life Cycle Choices[END_REF][START_REF] Low | Employment Risk and Wage Risk over the Life Cycle[END_REF].

Model Implications

Defining Consumption Equivalent Welfare

In what follows, we use ι to denote the willingness to pay as a proportion of consumption c for a consumer to be indifferent between the baseline and an alternative scenario. We define the equalizing willingness to pay for the group of individuals indexed by initial simulation condition Ω as

ι(Ω) = 1 -       E 0 U 0 (Ω) -E 0 Σ t β t µ Ĥit (Ω) ĥit (Ω) + β T-1 ViT (Ω) E 0 Σ t β t ĉit (Ω) ω (L-lit (Ω)-Pit (Ω)θ P ) 1-ω 1-γ 1-γ exp θ Ĥit ĥit (Ω)       1 ω(1-γ) , ( 20
)
where E 0 is the expectation with respect to information in the initial period. Thus E 0 U 0 (Ω) is the ex-ante baseline lifetime utility, given initial condition Ω. Choice variables under the counterfactual policy are shown with a hat. We compute the weighted average over all ι(Ω), where the weights are the empirical frequencies of the initial conditions, drawn for asset position, ownership status, and initial wage shock.

Policy Functions

As an example of the trade offs involved in households' decisions, consider Figure 2, where we illustrate a set of discrete choice policy functions from the baseline model. In the figure, we depict owners in the top row and renters in the bottom row. We project the high-dimensional policy function into wage-unsecured debt space, shading areas of certain discrete choices with different colors. Starting thus with an owner at young age in Figure 2a, we observe that at moderate levels of unsecured debt and relative high hourly wage, the owners optimally stays in the house (north east in the plot). As we traverse the picture going south west, hence increasing unsecured debt and decreasing wage, there is first a region with optimal sale and last a

For a detailed derivation see Appendix ??.

We report the measure in terms of percent, such that ι = 1.1 means that the average consumer is willing to pay 1.1% of consumption (hence prefers the policy scenario), whereas ι = -2.1 means the consumer has a negative willingness to pay for the policy, i.e. would demand an increase of 2.1% of consumption in order to be indifferent. region where filing for Chapter 7 is the optimal choice.

In Figure 2b we examine what happens to the owner at the same stage in the life cyle, but upon receiving a negative house price shock. The young age of the owner suggests that he has little equity in the house, which turns quickly negative after the price shock -hence mortgage default becomes the optimal action in the north eastern region. As accompanying unsecured debt increases and as wages change, there are different regions where the owner will either file for Chapter 7, Chapter 13 or a combination of default and Chapter 7 in the south west corner. The choice between Chapter 7 and Chapter 13 involves whether the house would be lost in a Chapter 7 bankruptcy (which is not the case here, as there is no excess equity, hence nothing to gain for the mortgage lender), and the wage rate, which determines the required Chapter 13 repayments in the repayment plan. Here, too high a wage rate deters the owner from choosing Chapter 13, because Chapter 7 offers the same benefits (keep the house), without the associated costs (future wage garnishments).

Figure 2c presents decisions made by a young renter after receiving a negative spending shock in the first period of life. Note that receiving a spending shock is the only way for an individual to have unsecured debt in the first period of life as we assume individuals start life with non-negative assets. At high hourly wage levels, the renter will buy. However, as unsecured debt grows larger and wages smaller, the renter first decides to keep renting, and finally file for bankruptcy under Chapter 13. In the adjacent Figure 2d, which depicts the renter at period 20 (age 45), hence with a substantially higher wage rate, we see that buying a house is an option that is optimal on a larger subset of the state space, and that the choice of bankruptcy chapter has changed to Chapter 7 -a result of the higher wage as he would incur larger Chapter 13 repayments.

The Role of Price and Wage Risk

Our model includes many complex interactions. To better understand them, in this section we derive elasticities in several economic outcomes as well as welfare effects with respect to various risks, emphasizing differences across education groups. Specifically, in Table 8 we show the elasticities in bankruptcy filing rates, interest rates, mortgage default rates, homeownership rates, and hours worked with respect to changes in the standard deviation of the house price shock and of the wage shock. It is important to note that when we perturb risk, we arrive at a new equilibrium with different interest rates. Moreover, changes in risk cause wealth effects, which we do not compensate for. This is particularly the case here, because the bankruptcy system, together with the safety net provisions we have allowed for insure against the worst outcomes. The elasticities are calculated as the percent change in an outcome as a result of a one percent change in the standard deviation of the house price shock or the wage shock. There are no low-educated owners filing for Chapter 13 in the high wage variance model.

The elasticities are reported in Table 8. Chapters 7 and 13 bankruptcies are sensitive to increases in house price risk, especially for the low education group. Interestingly, the mortgage default rates for high education households are only weakly negatively related to house price risk, while, as expected, the low education households' mortgage default rates are highly positively related to house price risk. Related to the increased bankruptcy rates, the interest rate elasticity with respect to house price risk is small and positive and a bit higher for the lower education group. Homeownership rates rise for the high education group but decline for the low. Finally, hours worked are not responsive to changes in house price risk, with two exceptions: low education owners in Chapter 13 work more to compensate for their income loss, while high education renters who have filed under chapter 7 work less as house price risk increases.

Turning to wage risk, the importance of distinguishing between education groups becomes evident.

For higher education households an increase in wage risk leads to a decline in overall bankruptcies, while the opposite is true for low education households. Moreover, within the high education group there is a large shift away from Chapter 7 towards Chapter 13, mortgage default rates decline and homeownership rates rise.

Initial mortgage rates, on the other hand, change little. For the low education group both Chapter filings rise and particularly for Chapter 7. Consistent with these changes, equilibrium interest rates for unsecured debt decline for the high education group but increase for the low education group. Finally, hours worked for the low education households decline with increased wage risk, reflecting in part the resulting wealth effects. This is driven by owners while renters increase their hours worked. For high education individuals, homeowners also work somewhat less particularly those in Chapter 13 bankruptcy.

To provide further understanding of the underlying equilibrium, Figure 3 demonstrates how the cost of a mortgage changes with assets (measured as a fraction of median income) as well as with the amount of house price risk. The cost is forward looking and accounts for the possibility of unsecured debt being accumulated following the issue of the mortgage. Overall, mortgage rates do not vary very much, reflecting the importance of collateral. The little variation that exists is more driven by the amount of unsecured debt at origination than by house price risk. The rate quickly converges to the low risk rate as unsecured debt declines. In addition the interest rate faced by lower education individuals is higher at all levels of assets than that faced by high education individuals, reflecting the higher probability of an income shock leading to bankruptcy for this group of people. Note: House price risk is set at 80% of baseline, at baseline (100%), and at 120% of baseline.

Finally, Figure 4 shows the close relationship between unsecured borrowing and the level of the mortgage loan. All else equal, a higher mortgage loan implies an increased risk of default, which can then trigger bankruptcy when there is recourse. Moreover, with some of the home equity protected by homestead exemption, a large mortgage loan may also lead to bankruptcy so that the households can have their unsecured debt discharged and thus have more resources to service the mortgages.

Labor Supply, Assets and Consumption Before and After Bankruptcy Filing

Labor supply offers a mechanism for individuals to mitigate adverse income shocks and potentially to stave off bankruptcy. Shocks to wages, however, can lead to bankruptcy or even mortgage default and discourage work effort. The implicit earnings tax induced by the repayment scheme under Chapter 13 can add to these disincentives. In the Figures that follow we present an event study of labor supply and wage rates before and after a bankruptcy event. In interpreting these figures it is important to note that the vast majority of bankruptcies in the model are caused be a large negative wage shock, only occasionally accompanied by spending shocks: 93% for the higher education people and 95% for the low. The main role of the catastrophic spending shocks is to force unsecured borrowing, which then makes people vulnerable to such wage/employment shocks.

In Figure 5a we show the proportion of individuals receiving adverse wage shocks (w = 1, 2, 3) versus positive ones (w = 4, 5) in the run up to bankruptcy. Figure 5b shows an overview with respect to assets, consumption, and labor income. As is evident the adverse shocks dominate and (not shown) there is a shift towards the worst shocks within the adverse group as we approach the date of filing. Labor income, which is the product of these shocks and endogenous labor supply, remains relatively stable. This is due to the increase in labor supply seen in Figure 5c for those who will eventually choose to file under Chapter 7 and Figure 5d for Chapter 13 filers. On the one hand, those who eventually file for bankruptcy work much fewer hours than average in ensuring five years of bankruptcy filing. On the other hand, they did increase labor supply in their attempt to avoid bankruptcy filing 2-3 years before the event, despite the decline in wages. Strikingly, consumption remains remarkably stable, demonstrating the amount of consumption insurance provided by bankruptcy and the safety net. First, bankruptcy filers work much fewer hours than their high education counterparts, a result of the lower wages they face despite the fact their hours increase rapidly in the year immediately before filing for Chapter 7 bankruptcy. Second, the recovery in income and hours is protracted.

Policy Counterfactuals

We now examine the impacts and insurance value of alternative structures of bankruptcy law. We start with an analysis of the value of bankruptcy protection. After that, we conduct our evaluation of BAPCPA reform as in [START_REF] Mitman | Macroeconomic Effects of Bankruptcy & Foreclosure Policies[END_REF], and [START_REF] Albanesi | Insolvency After the 2005 Bankruptcy Reform[END_REF] amongst others, with added discussion on the heterogeneity based on education and labor market outcome.

In Appendix D.1 we report the impact of different levels of recourse, relative to our baseline, providing some comparability with [START_REF] Mitman | Macroeconomic Effects of Bankruptcy & Foreclosure Policies[END_REF]. We find that increasing recourse, tends to increase bankruptcy filings and interest rates for unsecured debt, but decreases mortgage default. Hours worked are not sensitive to this change, except for those with low education filing under chapter 7. Welfare decreases as we move from no recourse to 20% recourse, reflecting the fall in insurance protection.

The Value of Bankruptcy Protection

Chapter 7 and Chapter 13 differ substantially in the insurance they offer and in their moral hazard implications. Their value is likely to be very different depending on the level and volatility of income and on preferences for work and housing. In this section we quantify these differences across education groups, which differ in these dimensions. In doing this we demonstrate important distributional implications of such policies. To achieve this we shut down each of the chapters in turn and assess the implications. A complete abolition of all bankruptcy provisions in the presence of catastrophic consumption shocks would require introducing other forms of consumption support and consequently entails a degree of arbitrariness.

The results are reported in Table 9. For the higher educated group the complete abolition of Chapter 7 leads to a dramatic decline in bankruptcies and as a result a fall in interest rates for unsecured debt. In addition mortgage defaults decline significantly. While hours worked do not change on aggregate, the hours of those filing under Chapter 13 increase by about five weeks a year. The net effect is an increase of welfare equivalent to 10% of consumption. This exceptionally high effect is driven by the almost complete elimination of moral hazard, making unsecured borrowing much cheaper, while still offering some necessary protection via Chapter 13.

By contrast if we consider the lower education group, while bankruptcy declines substantially, interest rates actually increase a bit for those who borrow because the composition of borrowers shifts towards individuals with a higher risk of filing. On the other hand mortgage default declines substantially because of the reduction in unsecured debt, the decline in bankruptcies overall and the elimination of Chapter 7. Hours worked decline overall and particularly for renters. While there are many complex interactions underlying the fall, the reduction in hours worked in part reflects the decline in ownership and the reduced need to accumulate for a down payment. By contrast, hours worked by those filing under Chapter 13 increase by about three weeks a year, reflecting an income effect. There is also a substantial decline in homeownership.

The net effect of all these changes is a large decline in welfare by 12%, reflecting a large decline in insurance for this group. However, the overall aggregate welfare effect, accounting for the relative size of the groups is a loss of -0.06%. In other words this policy has strong distributional effects, but little aggregate effect on

In some counties such as UK, bankruptcy laws are much less generous than those in the US; catastrophic consumption shocks, however, are insured. For example, such shocks often relate to uninsured healthcare expenses, which are insured by public health systems such as the NHS. An interesting research question is to examine the relative merits of insuring the shocks themselves in this way rather than having to resort to bankruptcy, but this is beyond the scope of this paper.

The percentage of those with high education is 54%.

the economy.

When Chapter 13 is abolished and only Chapter 7 is available, for the higher education group this reduces bankruptcy only slightly. But since creditors lose more we observe a large increase in interest rates.

Hours worked change little except for those in bankruptcy where we see a decline. The result is a fall in welfare for this group of about 1.7%. For the lower education group, shutting down Chapter 13 also reduces bankruptcy by a small amount and has a large impact on the interest rate, because all those who now switch to Chapter 7 impose larger losses on creditors. The net effect is a small increase in welfare of about 0.5%, implying an aggregate impact of -0.73% in consumption equivalent.

These exercises demonstrate that the way bankruptcy law is organized has important distributional consequences masked by the aggregate effect. Moreover, these distributional effects are not a straightforward implication of the differences in the income processes because preferences also differ in a substantial way.

The Impact of the BAPCPA reform

The Bankruptcy Abuse Prevention and Consumer Protection Act (BAPCPA) was signed in to law in April 2005 and took effect after October 17 of the same year. The bill was specifically designed to make it more difficult to file for bankruptcy, and in particular, to file under Chapter 7. For instance, under the reform, to be eligible for filing under Chapter 7, individual earnings have to be lower than the state median. The cost of bankruptcy filing has also gone up significantly after the reform. According to a report by the US Government Accountability Office (2008), average total of filing and attorney costs of Chapter 7 increased from $900 to $1500 after the reform, and those for Chapter 13 from $3700 to $5700. This reform has been evaluated extensively in the literature as we have reviewed in the introduction. We revisit it here from the perspective of our model, which allows for labor supply adjustments and distinguishes between education groups. to BAPCPA, with a negative willingness to pay of about 0.05%.

For the lower education group, bankruptcy rates are similar to the baseline, but there is some substitution towards Chapter 7 bankruptcy. This results in an increase in interest rates for unsecured debt from 25.3% to 28.2%. In terms of labor supply, those filing under Chapter 13 work slightly more. On balance, the BAPCPA reform implies a 0.6% reduction in consumption equivalent welfare for the low education group. The nature of income and spending shock processes implies a more frequent need for bankruptcy protection for this group and BACPCA made them substantially worse off.

In terms of aggregate headline numbers, our estimation indicates a smaller welfare loss associated with BAPCPA than that in [START_REF] Mitman | Macroeconomic Effects of Bankruptcy & Foreclosure Policies[END_REF]. The comparison is not straightforward, because Mitman's calculations includes the transition period. Moreover, there are important differences in the two models, including the fact ours is a lifecycle model with labor supply and the presence of catastrophic consumption shocks.

Conclusions

In this paper we evaluate central aspects of bankruptcy legislation and consider how these effects differ across education groups with different exposure to risk and different preferences. We do so by specifying and estimating a lifecycle model of consumption, housing demand and labor supply in an environment where individuals may file for bankruptcy or default on their mortgage. Uncertainty in the model is driven by house price shocks, catastrophic consumption shocks (such as health expenditures) and wage/productivity shocks, while bankruptcy is governed by the basic institutional framework in the US as implied by Chapter 7 and Chapter 13. Importantly, the supply of unsecured and secured credit is endogenous leading to interest rates that depend on the probability of an individual filing for bankruptcy protection or defaulting on their mortgage. Allowing for such endogenous interest rates is central to understanding the role of bankruptcy legislation because there is a very stark tradeoff between the effects of insurance protection and moral hazard.

There are a number of key insights from our paper. First, labor supply adjusts before bankruptcy to mitigate the effects of negative shocks and to avoid filing, implying that a number of bankruptcies are avoided by such preventive action. Labor supply drops at the moment of filing, because of the large negative productivity shocks that induce it. It only recovers gradually. Though the overall population labor supply is not sensitive to changes in bankruptcy legislation, it is sensitive to changes in risk and associated wealth.

Second, the availability of Chapter 7 has starkly different implications for low education and high education groups, increasing substantially the welfare of the former and decreasing it for the latter, despite the presence of Chapter 13. The aggregate effect on welfare is however, zero. For the high education group the moral hazard aspect of Chapter 7 is the dominant factor, while for the lower education group the insurance component is central. Third, the availability of Chapter 13 also has opposing effects for the two education groups, improving the welfare of the higher group but decreasing slightly the one for the lower (given the presence of Chapter 7). However, the impacts are much smaller. A key implication of this result is that in evaluating bankruptcy legislation the distributional component across demographic groups is important to consider. We emphasize that this is not just due to differences in the income process and level but also due to important differences in preferences for housing and work.

Finally, we reevaluate BAPCPA and find it has very small effects on bankruptcy and only on the high education group. Overall the welfare effects were negative and mostly due to the lower education group, which is consistent with the value of chapter 7 for this group. Future research should also consider how family labor supply interacts with bankruptcy legislation and with filing and what role does the added worker effect have in mitigating extreme shocks. While this is an important issue, it is left for future work.

The future value in the bankruptcy state 7 is denoted by W7 . Upon filing for Chapter j, a fee of f j is collected, and no savings are possible.

W file.7 (R) = max l∈L u(cλ 7,e , l, h) + βE w |w W7 (R ) (A.7) subject to c = yl -f 7 .
The individual suffers the utility (stigma) cost λ 7,e and cannot borrow until she exits this state. This happens with probability δ in each period. Thus the expected duration of the punishment state is 1 δ . The value W7 in the bankruptcy state is

W7 (a, w, t) = max a ∈R + ,l∈L u(cλ 7,e , l, h) + βE w |w δW(R ) + (1 -δ) W7 (R ) subject to c + 1 1 + r a = yl + a.

Renter Bankruptcy Chapter 13

Individuals may not be eligible for Chapter 7, or indeed may choose Chapter 13. This problem is very similar to the previous one except that a repayment ȳ needs to be made. Hence moving into the bankruptcy state we have W file.13 (a, w, t) = max l∈L u(cλ 13,e , l, h) + βE w |w W13 ( ȳ(a, w), 0, w, t + 1) (A.8) subject to c = ylf 13 > 0,

where ȳ(a, w) is defined in (A.9).

We denote by ȳ the amount to be repaid in each period of chapter 13, and by Ȳ total expected income over the next T bk years. Federal law imposes a maximum debt to income ratio ŷ = 0.15 that the repayment ȳ must respect, hence we define ȳ = ι(a, Ȳ, ŷ) Ȳ

T bk ι(a, Ȳ, ŷ) = -a Ȳ if -a Ȳ < ŷ ŷ
else. (A.9)

Then the corresponding punishment state following filing for chapter 13 is given by W13 ( ȳ, a, w, t) = max a ∈R + ,l∈L u(cλ 13,e , l, h) + βE w |w δW(R ) + (1δ) W13 (R )

subject to c + 1 1 + r a = yl -ȳ + a > 0.

B Wage Process

We employ the same procedure as in De [START_REF] Nardi | Nonlinear household earnings dynamics, self-insurance, and welfare[END_REF] to estimate wage support points by age as well as age-specific transition matrices Π w (t). The method in De [START_REF] Nardi | Nonlinear household earnings dynamics, self-insurance, and welfare[END_REF] allows for a generalization of the traditional income process used in macroeconomics. For further details and thorough comparison to the traditional approach, please refer to De [START_REF] Nardi | Nonlinear household earnings dynamics, self-insurance, and welfare[END_REF]. We replicate their results using the outline in section 4 of their paper, but splitting the data into two education groups. We use PSID data from 1969-1997, because thereafter we only have bi-annual observations on wage. We subset the data to household heads between age 25 and 60, whom we observe in at least two consecutive periods, and we drop observations with annual earnings below $900. We use 2003 as the base year for CPI adjustment. Finally, we only keep observations with a valid education entry, and real hourly wage in [4,999]. We classify all observations with less than 13 years of education as "low" education, and the remainder as "high".

We regress log real earnings on a year dummy and a fourth order polynomial in age (term f e (t) in equation 4), by education group. From the residual of the regressions, we recover the support of the wage shock w it at each age t for 5 bins, and for each education group, as well as their respective transition across time. The binning classifies the residuals at each age into 5 regions delimited by the quantiles of the corresponding residual distribution. We illustrate the outcome in figure B.1. Note: We cut the empirical distribution of wage residuals into 5 quantiles, and take the midpoints to be the points of support for the corresponding bin. The transition matrix at each age (right panel of plot) is estimated by counting proportion of bin i in period t which end up in bin j in period t + 1. The right panels illustrate the probability associated with each transition from state i to state j.

D Tables

In this appendix, we prresent two additional tables that provide information on our computation and state legislation.

Figure

  Figure 1: Bankruptcies by Chapter and Mortgage Default over Time Trends in Bankruptcy Filings Mortgage Defaults

  among three discrete choices, rent, buy, and file for bankruptcy either under Chapter 7 or under Chapter 13. The owner's problem in the non-bankruptcy state is to choose between staying in the existing house, selling it, defaulting on the mortgage, and filing for bankruptcy under one of the two chapters. If they are in the punishment state due to an earlier bankruptcy they choose between staying, selling or defaulting on the mortgage. In what follows we summarize key features of the choice problem and illustrate several cases that are central to our problem.

Figure 2 :

 2 Figure 2: Discrete Choice Functions (High Education Group).

Figure 3 :

 3 Figure 3: Mortgage Interest Rate by House Price Risk and Assets at Origination

  Initial Mortgage Rates for Low Educated Buyer.

  Initial Mortgage Rates for High Educated buyer.

Figure 4 :

 4 Figure 4: Median Interest Rates as a Function of Mortgage debt

Figure 5 :

 5 Figure 5: Event Study Around the Point of Bankruptcy Filing for High Education Individuals (a) Proportion receiving negative wage shocks (w = 1, 2, 3) versus positive (w = 4, 5) and labor income (y)

Figure 6

 6 Figure6reports the same information but for the lower education group. Two observations stand out.

Figure 6 :

 6 Figure 6: Event Study around the Point of Bankruptcy Filing for Low Education Individuals (a) Proportion receiving negative shocks (w = 1, 2, 3) versus positive (w = 4, 5) and labor income (y)

Figure

  Figure B.1: Wage Support and Age-Specific Transition Matrices for Wage Shock w

Figure C. 1 :

 1 Figure C.1: Lifecycle Profiles in Different Module Configurations with and without Bankruptcy Choices for the High Education Group

Table 1 :

 1 Hours and Ownership Summary Statistics

			Low Educ			High Educ
		mean	sd	median	mean	sd	median
	Hours Worked 1810.66 981.83 2024.00	2082.44 841.29	2100.00
	Ownership	0.56	0.50	1.00	0.71	0.45	1.00
	Age	42.05	10.34	42.00	43.17	10.63	43.00
	Source: PSID 2000-2006. All variables refer to the household head between age 25
	and 60. Observations: Low Education 10407; High Education 6961.

Table 2 :

 2 Average Bankruptcy and Default Rates

	Chapter 7 (‰) Chapter 13 (‰) Total bankruptcy (‰) Mortgage Default (%)
	7.634	2.210	9.222	2.130
	Notes: Number of observations=21822. Mortgage default is defined as 60 days or more past due. This is an
	average over county-level data for years 2000-2006. Authors' calculations based on CCP and Black Knight
	McDash data.

Table 3 :

 3 Association of Bankruptcy and Mortgage Default with Institutional and Other Characteristics

		Bankruptcies Chapter 7 Chapter 13	Default
	House Price	-0.060***	-0.052*** -0.121*** -0.101***
		(0.004)	(0.005)	(0.007)	(0.005)
	Default Rate	1.113***	0.677*	2.290***
		(0.259)	(0.308)	(0.504)
	Ownership Rate	-0.127	0.135	0.906*** -0.939***
		(0.114)	(0.125)	(0.194)	(0.124)
	Recourse	-0.048+	-0.051	-0.080	-0.107***
		(0.028)	(0.032)	(0.050)	(0.031)
	Homestead Exemption ($)	0.002	0.007**	-0.007*	0.002
		(0.002)	(0.002)	(0.004)	(0.002)
	Unlimited Exemption	-0.285	-0.727**	0.473	-0.093
		(0.207)	(0.229)	(0.351)	(0.229)
	% College or More	-0.026***	-0.028*** -0.043*** -0.037***
		(0.001)	(0.001)	(0.002)	(0.001)
	Bankruptcy Rate				0.057***
					(0.009)
	# Obs.	5846	5391	2642	4994
	R2	0.196	0.174	0.371	0.330
	Note: Annual County-level data between 2000 and 2006, hence an observation is a county-year pair. The
	dependent variables are the log-odds ratios of the respective bankruptcy or default rates by county and year.
	As a result, the analyses exclude county year observations with zero bankruptcies or defaults. House price is
	the Zillow house value index at county-year level; Ownership Rate is by county but fixed at 2010; Recourse
	classifies states into having mortgage recourse for lenders; Homestead Exemption is in 10000 dollars; and
	Unlimited exemption is a dummy equal to one if the state has unlimited homestead exemption and zero
	otherwise.

Authors' calculations based on CCP and Black Knight McDash data. * p < 0.05, ** p < 0.01, *** p < 0.001.

Table 4 :

 4 Predicted Rates of Bankruptcy and Default by Education

	Rate	High Education Low Education
	Total Bankrupty Filing Rates (‰)	7.72	11.40
	Chapter 7 Filing Rate(‰)	5.79	8.79
	Chapter 13 Filing Rate (‰)	2.08	3.96
	Mortgage Default Rate (%)	1.67	2.88

Table 5 :

 5 Preset Parameters

	Description	Symbol	Value	Source
	Prob of exit from bankruptcy state	δ	0.2	legal
	Risk free gross interest rate	1 + r	1.02	data
	Spending shock grid (1000 dollars)	a	[0.0, 32.00, 102.00] Livshits et al. (2007)
	Spending shock probabilities	π a	[0.925, 0.070, 0.005] Livshits et al. (2007)
	Discount factor	β	0.99	literature
	Rental price of housing	p r	0.0	normalization
	Proportional selling cost	φ	0.06	data
	Probability of deficiency judgment	ψ	0.1	Li and Oswald (2017)
	Homestead exemption modulo med. inc. ξ	1	Group 5 in table D.2
	Downpayment ratio	χ	0.1	data
	Constant relative risk aversion	γ	2.1	Mitman (2016)
	Foreclosure sale discount	δ d	0.78	Kaplan et al. (2020)
	Mortgage interest rate	r m	0.06	FRED average 30-year mortgage
	Annual hours worked full time	l L	2277	French (2005)
	Annual leisure endownment (hours)	L	4466	French (2005)
	House price shock persistence	ρ p	0.96	Mitman (2016)
	House price shocks SD	σ p	0.1	Mitman (2016)
	2003 Median household inc. (1000$)		43	Census Bureau
	Avergage Length of Ch. 13 repayment	T bk	5	legal
	Max. repayment to inc. ratio in Ch. 13		0.15	legal
	Bankruptcy technology	ζ	0.5	Mitman (2016)
	Ch. 7 filing cost post BAPCPA (1000$)	f 7	0.6	US Government Accountability Office (2008)
	Ch. 13 filing cost post BAPCPA (1000$) f 13	1.7	US Government Accountability Office (2008)
	Safety net: fraction of median income	c	0.1	assumption
	Unemployment benefit (1000$)	b	10	assumption

The estimation of the age-specific wage process after De

[START_REF] Nardi | Nonlinear household earnings dynamics, self-insurance, and welfare[END_REF]

is detailed in Appendix B. The remaining parameters are estimated using the simulated method of moments (SMM, see

[START_REF] Mcfadden | A method of simulated moments for estimation of discrete response models without numerical integration[END_REF][START_REF] Pakes | Simulation and the asymptotics of optimization estimators[END_REF]

combined with the Markov Chain Monte Carlo method (MCMC) adapted for SMM by

Table 6 :

 6 Parameter Estimates by Education Group

			High Education	Low Education
			Estimate Standard Error	Estimate Standard Error
	Utility: c, l and h
	Housing (warm glow)	µ	0.501	0.045	3.05	0.596
	Housing (multiplicative) θ H	-0.1	0.035	-0.9	0.895
	Consumption Exponent ω	0.4	0.608	0.44	82.903
	Fixed Cost of Work	θ P	998.541	1.051	2013.93	26.899
	Bequest Weight	θ W	7.91	0.006	6.614	0.272
	Utility: Bankruptcy and Default
	Stigma Ch. 7	λ 7	0.49	0.027	0.42	84.476
	Stigma Ch. 13	λ 13	0.695	0.006	0.535	59.267
	Stigma Default	λ de f	0.76	0.002	0.59	6.763
	Period T BK penalty	θ BK	0.0	0.234	213.9	5.701

Table 7 :

 7 Model Fit by Education Group

	High Education	Low Education
	Model Data	Model Data

Table 8 :

 8 Elasticities to House Price Risk and Wage Risk

		Elasticities to Risk Changes
		High Education	Low Education
		House Price (p) Wage (w)	House Price (p) Wage (w)
	Bankruptcy	0.135	-1.056	0.877	0.584
	Bankruptcy 7	0.167	-1.917	1.078	0.846
	Bankruptcy 13	0.066	0.835	0.707	0.362
	Interest	0.031	-0.101	0.04	0.083
	Ownership	0.043	0.12	-0.034	0.231
	Default	-0.038	-0.75	0.40	-0.858
	Hours	-0.012	-0.033	-0.002	-0.174
	Initial Mortgage Rate	-0.007	0.004	0.006	0.013
	Hours Renters	-0.005	0.001	0.01	0.025
	Hours Owners	-0.009	-0.04	-0.003	-0.231
	Hours Renters (in ch. 7)	-0.10	-0.081	0.026	0.275
	Hours Owners (in ch. 7)	-0.024	0.093	-0.046	-0.091
	Hours Renters (in ch. 13)	-0.0143	-0.0114	0.0003	0.286
	Hours Owners (in ch. 13)	-0.026	-0.19	0.059	-
	Note:

Table 9 :

 9 Abolishing Bankruptcy by Chapter

			High Education			Low Education
		Baseline No Ch. 7 No Ch. 13	Baseline No Ch. 7 No Ch. 13
	Bankruptcy ( )	10.078	0.529	9.648	14.987	3.752	13.166
	Bankruptcy 7 ( )	6.926	-	9.648	7.144	-	13.166
	Bankruptcy 13 ( )	3.151	0.529	-	8.446	3.752	-
	Interest (%)	12.63	6.97	16.28	25.31	27.15	36.27
	Ownership	0.712	0.694	0.711	0.573	0.547	0.571
	Default	1.208	0.693	1.197	1.661	1.254	1.654
	Hours	2186	2189	2187	2053	2022	2052
	Hours Renters	2242	2239	2245	2125	2064	2130
	Hours Owners	2184	2173	2184	2033	2004	2030
	Hours Renters (in ch. 13)	1666	1863	-	1687	1809	-
	Hours Owners (in ch. 13)	2194	-	-	1993	-	-
	Cons. Equivalent (%)	-	10.22	-1.74	-	-12.128	0.463

Table 10 :

 10 Impact of BAPCPA ReformWe analyze the effects of the BAPCPA reform by looking at two steady states of the model economy, before and after the reform. Throughout, we keep the stochastic shocks describing the economic environment constant. The policy equilibrium in the model is characterized by the implementation of the means test to access Chapter 7, the implementation of homestead exemption cap, and the increase of monetary filing costs.The results are shown in Table10by education group. These are comparisons of long run steady states and do not include any of the transition.

	Impact of BAPCPA Reform
		High Education	Low Education
		Baseline BAPCPA	Baseline BAPCPA
	Bankruptcy ( )	10.078	9.383	15.591	15.429
	Bankruptcy 7 ( )	6.926	6.521	7.144	8.493
	Bankruptcy 13 ( )	3.151	2.862	8.446	6.936
	Interest (%)	12.63	12.04	25.31	28.16
	Ownership	0.712	0.711	0.573	0.573
	Default (%)	1.208	1.206	1.661	1.653
	Hours	2186	2185	2053	2054
	Hours Renters	2242	2241	2124	2127
	Hours Owners	2184	2185	2033	2034
	Hours Renters (in ch. 13)	1666	1689	1687	1703
	Hours Owners (in ch. 13)	2194	2086	1993	2017
	Consumption Equivalent WTP (%)	-	-0.046	-	-0.603
	Starting with the high education group, we see that the reform has the expected effect of reducing both
	Chapter 7 and Chapter 13 cases, due to means tests requirement for Chapter 7 filing and higher filing costs

for bother Chapters. As a result, average unsecured interest falls. Aggregate ownership and default are barely affected by the reform. Aggregate hours worked do not change much either. However, renters in Chapter 13 work more while owner in Chapter 13 work less. Overall, higher educated individuals are almost indifferent

Table D

 D Table D.2: Grouping of US States by Legal Environment Concerning Bankruptcy and Mortgage Default State Deficiency Homestead Exemp. ($) Median Inc. ($) ξ Exemp./Med. Inc. group

			.1: Numerical Parameters
			Number of grid points chapter 13 repayment	4
	NC	No	Number of grid points assets 18500	45607.13	31	0.41	1
	WA	No	Number of grid points wage 40000	59951.18	5	0.67	1
	AK	No	Number of grid points p 54000	63456.71	5	0.85	2
	CA	No	Number of grid points mortgage 50000	58509.89	15	0.85	2
	MT ND AZ MN AL	No No No No Yes	Number of grid points house size 100000 Number of periods 80000 Levels of Labor supply 150000 Scale factor on asset grid 200000 5000	43752.43 51275.34 49907.10 59445.86 43445.55	2 35 5 1.3	2.29 1.56 3.01 3.36 0.12	2 2 3 3 4
	GA	Yes	Number of discrete choices renter (excluding buying) 10000 49418.75	3	0.20	4
	IL	Yes	Number of discrete choices owner in BK state 7500 54433.88	1	0.14	4
	IN	Yes	Number of discrete choices owner 7500	48301.03	6	0.16	4
	KY	Yes	Number of discrete choices owner in BK state 5000 42728.06	3	0.12	4
	MD OH TN VA WY	Yes Yes Yes Yes Yes	Initial unit price of house 0 Size of smallest house 5000 Size of a small house 5000 Size of a large house 5000 10000	68697.79 49214.44 43074.65 62967.78 53708.11	1 0.0 300 600	0.00 0.10 0.12 0.08 0.19	4 4 4 4 4
	AR	Yes	number of simulated individuals 17425	41227.34	15000	0.42	5
	CO	Yes	Mean of LogNormal initial asset distribution 45000 61377.39	-2.1	0.73	5
	DE	Yes	Std. Dev. of LogNormal initial asset distribution 50000 56565.67	0.9	0.88	5
	HI	Yes	17425	64089.82		0.27	5
	LA	Yes	25000	42654.21		0.59	5
	ME	Yes	35000	50249.51		0.70	5
	MI	Yes	17425	51084.04		0.34	5
	MO	Yes	15000	48774.10		0.31	5
	NE	Yes	12500	53861.02		0.23	5
	NJ	Yes	17425	68284.69		0.26	5
	NM	Yes	30000	45115.96		0.66	5
	OR	Yes	25000	52448.20		0.48	5
	PA	Yes	17425	51987.45		0.34	5
	SC	Yes	17425	44104.29		0.40	5
	SD	Yes	30000	49528.12		0.61	5
	UT	Yes	20000	60398.63		0.33	5
	WI	Yes	40000	53704.30		0.74	5
	WV	Yes	25000	42656.15		0.59	5
	CT	Yes	75000	67675.40		1.11	6
	ID	Yes	104471	50053.53		2.09	6
	MA	Yes	100000	63015.52		1.59	6
	MS	Yes	75000	38908.97		1.93	6
	NH	Yes	100000	68438.14		1.46	6
	NV	Yes	550000	54782.10		10.04	6
	NY	Yes	50000	52655.17		0.95	6
	RI	Yes	200000	55399.59		3.61	6
	VT	Yes	75000	55026.47		1.36	6
	FL	Yes	∞	47917.01			7
	IA	No	∞	52378.80			7
	KS	Yes	∞	48913.09			7
	OK	Yes	∞	46108.99			7
	TX	Yes	∞	48876.19			7

Note: Columns 2 and 3 are taken from table 7 of the online appendix of

[START_REF] Mitman | Macroeconomic Effects of Bankruptcy & Foreclosure Policies[END_REF]

. Median income in 2011 dollars. States are grouped by interquartile range of ξ and whether or not they allow Deficiencies.

Once in a Chapter 13 repayment plan, the individual is constrained to supply sufficient labor in order to be able to make the required payments.

Acknowledgements

This project was funded by the Sloan Foundation under grant 2011-5-24 ECON awarded in October 2011. We thank seminar participants at the NBER Summer Institute 2013, the Chicago Fed, UCL Implementation of Dynamic Models and SED Edinburgh conferences for comments and suggestions. Florian Oswald is grateful for additional funding under ESRC transformative research grant, grant number ES/M000486/1. The views expressed here are our own and they don't represent the views of the Federal Reserve Bank of Philadelphia or the Federal Reserve System.

* This project was funded by the Sloan Foundation under grant 2011-5-24 ECON awarded in October 2011. We thank seminar participants at the NBER Summer Institute 2013, the Chicago Fed, UCL Implementation of Dynamic Models and SED Edinburgh conferences for comments and suggestions. Florian Oswald is grateful for additional funding under ESRC transformative research grant, grant number ES/M000486/1. The views expressed here are our own and they don't represent the views of the Federal Reserve Bank of Philadelphia or the Federal Reserve System.

† Federal Reserve Bank of Philadelphia,

Appendices A The Complete Description of the Value Functions

A.1 The Choice of Renters

In what follows we describe the problem including the means test to cover the BACPCA period. let W represent the maximal expected lifetime utility for a renter of age t if not in a bankruptcy state and let Wj denote the utility for a renter in bankruptcy state j = 7, 13. Let s denote the end of period savings (i.e. s = a ) and med(y) the state median income. We then have: 13 if a < 0, yl < med(y), max W rent , W buy , W file. 13 if a < 0, yl ≥ med(y), max W rent , W buy if a ≥ 0.

(A.1)

The restriction on the discrete choice set of the renter in (A.1) makes explicit the fact that one only can file for bankruptcy if there are effectively unsecured debts to discharge. In addition we implement the BAPCPA means test by preventing individuals with labor income above a threshold med(y) to file for chapter 7. We define the conditional value functions next.

Value of Renting

The value of renting is given by

q a (a |w) as in ( 7), (A.6) where R is the current state space and R the state space as it evolves. Equation (A.3) is a standard budget constraint that requires expenditures (consumption c and saving/borrowing a ) to be equal to cash-on-hand (labour income plus assets minus rent, which is normalized to zero). Equations (A.4) and (A.5) show how the probability of bankruptcy for each case (π 7 and π 13 for Chapter 7 and 13 respectively) is calculated by the lender in order to define the price of debt q in equation (A.3). We use the convention that labelling a value function with -j means all other discrete choice values except j.

Renter Bankruptcy Chapter 7

The value of filing for bankruptcy under Chapter 7 as a renter is similar to the value of staying a renter with the exception that current assets are set to a = 0 in the budget constraint since all assets are used against the debt. Moreover, the various penalties are applied (psychic cost of bankruptcy λ 7,e ∈ (0, 1] and no borrowing; λ 7,e = 1 implies no punishment at all, λ 7,e = 0 would imply zero consumption as punishment.

A.2 The Problem of the Owner

The discrete choice problem of an owner not in a bankruptcy state is quite complex because of the large number of subcases, depending both on the asset position, the house value and the level of income. It is given by

where a ≥ 0 denotes someone with positive financial assets and hp tm t is the net equity in the house.

Again, not all discrete choices are available everywhere on the state space, as can be seen from the restrictions for each case. For example, the admissible chapter of bankruptcy depends on labor income lying below the threshold med(y), as before. The default choice is only an option if home equity is negative. In other words we assume that a person who has difficulty repaying a mortgage (say because of a negative income shock) but has positive equity in the house will always choose to sell. Owners with home equity in excess of the exemption level face eviction should they file for bankruptcy under Chapter 7. The level of homestead exemption determines whether an owner filing under Chapter 7, stays on in the house, or is evicted. We define the sub-problems in sequence below. Let S = (a, w, p, n, h, e, t) denote the current state space.

Value of Staying as Owner

The value of staying in the current home is

This problem is similar to the buyer's problem defined in the main text of the paper except that there is no down-payment in the budget constraint.

Value of Selling the Home

The value of selling depends on the renter's continuation value:

where (1φ)phm is the capital that can be recovered following the sale: φ is the proportion of capital lost by the process of selling due to administrative and marketing costs.

Value of Default

The default value, in turn, is similar to the value of selling with the exception that for a defaulter unsecured borrowing is impossible, and a one-time utility penalty is incurred. Regarding recourse legislation, we include a factor ψ ∈ [0, 1] here that relates to the fraction of negative equity ((1φ)(phm)) that is rolled over in post default life. For example ψ = 1 would mean that the entire remaining mortgage debt is rolled over into post default life. Notice that the future value is that of a renter, but the asset state takes into account any remaining mortgage debt d brought forward.

Owner Bankruptcy Chapter 7

The value of an owner who files for Chapter 7 while staying in the home is given by

This value is only defined if current assets are negative, a < 0. Crucially, the household may only stay in the house if net home equity lies below the homestead exemption level, i.e. iff (1φ)(phm) < ξ.

Value of Filing and Default

The value for the owner when filing for bankruptcy and defaulting on the mortgage at the same time is as follows:

assuming that any remaining mortgage debt is discharged in Chapter 7 bankruptcy.

Owner Bankruptcy Chapter 13

The main difference between Chapter 13 and Chapter 7 bankruptcy is that the owner may keep the house (and all other assets) no matter how much equity there is after signing up to a Chapter 13 repayment plan. Consequently we don't have to compute a value of eviction and we also rule out the possibility of filing for Chapter 13 and defaulting on the mortgage at the same time.

V file.13 (S) = max a >0,l∈L u (cλ 13,e , l, h) + βE w |w,p |p,a Ṽ13 ȳ(a, w), 0, w , p , n , h, e, t + 1 (A.16)

Owner Bankruptcy Punishment States

An owner in punishment state for either chapter has the discrete choice set "stay", "sell" and "default". His savings s cannot be negative (he cannot borrow). As in the case of the renter, exit from the state is governed by the Bernoulli random variable X ∼ Bernoulli (δ). Thus the value for this owner is

where the value for stay is given by

The value for sell is given by

Finally the value for default in the punishment state is given by

The amount of assets that the person carries over into the next period depends both on the extent of recourse in the specific state and on the amount of mortgage debt. In any case a cannot be negative since the person has already filed for bankruptcy and cannot borrow. However it can be positive if the person started saving after filing. In a recourse state the existing financial assets will be used to pay off the mortgage (under chapter 7). We assume that any remaining mortgage debt is then forgiven and a = 0.

Filing for Chapter 13 and defaulting at the same time is a particularly unrealistic choice, since the consumer assumes the increased burden of Chapter 13 (wage tax) without getting to enjoy the benefits (staying in the house).

We illustrate the evolution of the cross sectional average and standard deviation of simulated wages by age and by education group in

C More Model Evidence

In this appendix, we present life cycle profiles in asset holding, homeownership rate, mortgage borrowing, and hours worked for the baseline analyses as well as several counter factual policy analyses when we remove bankruptcy chapter-wise in figures C.1 for high and C.2 for low educated individuals. We include the profiles for experiment abolishment of chapter 7 light (labeled no7_light), where we restrict the means test to one half of state median income. Given we found very small welfare effects, we do not report those results in table 9, however the results are readily available. What stands out in those is the drastic change in aggregate behaviour induced by the complete abolishment of chapter 7, detering a lot of initial unsecured borrowing, and related to that, homeownership.

D.1 Supplementary Results: Changing the Level of Recourse

Mortgage lenders have different amounts of recourse to deficient mortgage contracts in different states. Our baseline assumes a probability of 10% that a deficiency judgment would be handed down to a borrower who defaults on the mortgage with negative equity. This parameter is informed by court records which were analyzed in Table A.1 of [START_REF] Li | Recourse and residential mortgages: the case of Nevada[END_REF], . In case a deficiency judgment is carried out (with probability 0.1), the borrower has to settle the remaining debt in the future, possibly leading them to file for bankruptcy.

The impact of varying recourse is displayed in Table D.3. Starting with the high education group, increasing recourse leads to declines in default as we would expect given the higher cost associated with default. It also leads to increases in bankruptcy because deficient mortgages turn into unsecured debt more often, increasing incentives to expunge the debt through bankruptcy. This has a direct impact on the average interest rate for unsecured debt, which is significantly lower in the no recourse scenario. However, aggregate ownership and hours worked are unaffected. We see a similar impact on the less educated group. Welfare declines steadily with increases in recourse, reflecting the reduction in insurance. This is particularly marked for the low education group, where an increase in recourse from 10% to 20% leads to a loss of welfare equivalent to a reduction in consumption of 3.5%. The complete abolition of recourse (by setting ψ = 0) would be equivalent to a 0.9% increase in consumption for this group. The higher educated group has a slightly positive willingness to pay for the no-recourse scenario (0.2%), and a slightly negative one for the high recourse one (-0.39 %). Note: The revel of recourse for mortgage lenders is governed by the parameter ψ, which controls the probability with which a mortgage in default will lead to a deficiency judgment. The Baseline value is ψ = 0.1.