Default, insurance, and debt over the life-cycle

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Contents lists available at ScienceDirect

Journal of Monetary Economics

Journal of Monetary Economics 55 (2008) 752– 774

0304-39

doi:10.1

$ I am

Akyol, Y

Andrea� Tel.

E-m

journal homepage: www.elsevier.com/locate/jme

Default, insurance, and debt over the life-cycle$

Kartik B. Athreya �

Research Department, Federal Reserve Bank of Richmond, P.O. Box 27622, Richmond, VA 23261, USA

a r t i c l e i n f o

Article history:

Received 30 November 2006

Received in revised form

1 May 2008

Accepted 2 May 2008Available online 15 May 2008

JEL classification:

D52

G33

J64

Keywords:

Default

Unsecured debt

Life-cycle consumption inequality

32/$ - see front matter & 2008 Elsevier B.V

016/j.jmoneco.2008.05.006

grateful to both the editor, Janice Eberly, a

ongsung Chang, Huberto Ennis, Juan Carlos

Waddle, and Steve Williamson for discussio

: +1804 697 8225; fax: +1804 697 8217.

ail address: [email protected]

a b s t r a c t

The widespread use of debt and default suggests that unsecured credit markets play an

important role in consumption smoothing. In this paper, I address two previously

unanswered questions. First, how does policy towards debt default affect the evolution of

consumption and net worth over the life-cycle? Second, how does debt default policy

interact with social insurance over the life-cycle? The findings are as follows. First, US

default policy appears ‘‘lax’’, in the sense that it creates severe credit constraints,

especially for the young. Second, eliminating default will lower consumption inequality

among the young, but will increase it among the old. Third, social insurance alters default

risk and, in turn, loan pricing, and therefore matters for purely intertemporal smoothing.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

The ability to issue defaultable debt, such as credit-card debt, and other unsecured liabilities, appears important to UShouseholds. At an individual level, indebtedness is a ubiquitous feature of the household balance sheet. Hurst and Willen(2007) measure the median unsecured debt holdings of borrowing households at $5,600 dollars, and find that over half(62%) of all households under 40 years old hold unsecured debt. Data from the Survey of Consumer Finances (2001) showsalso that net worth remains very low for many households over most of the life-cycle. For example, households in the 25thpercentile of US net worth hold less than $60,000 over their entire life-cycle; even when home equity is included. At anaggregate level, the Federal Reserve in 2005 measured the ratio of unsecured debt to GDP at roughly 10%, or one trilliondollars.

The option to default also appears important to US households: not only do many US households acquire largeunsecured debts, they also frequently refuse to repay them. For example, 15% of American households filed for personalbankruptcy in the decade between 1995 and 2005. Two-thirds of the households who filed did so in the wake of adisruption to earnings (Sullivan et al., 2000), indicating that default serves as a safety valve for distressed debtors. At anational level, the fraction of all unsecured loans lost to default has averaged nearly 5%, or $40 billion annually, over thepast decade.

. All rights reserved.

nd an anonymous referee for very valuable suggestions and extremely careful reading. I thank Ahmet

Hatchondo, Chris Herrington, Mark Huggett, Igor Livshits, Jim MacGee, Brian Minton, Victor Rios-Rull,

ns. I thank Hubert Janicki, Andrea Waddle, and Brian Gaines for outstanding research assistance.

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www.elsevier.com/locate/jme

dx.doi.org/10.1016/j.jmoneco.2008.05.006

mailto:[email protected]

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K.B. Athreya / Journal of Monetary Economics 55 (2008) 752–774 753

Theory attributes both the use of credit markets, and the use of default, to the consumption-smoothing motivations ofhouseholds. The hardship seen among bankruptcy filers and other defaulters therefore suggests two things. First, that debtwas taken in an attempt to smooth consumption, and second, that ‘‘enough’’ bad luck occurred to prevent households fromborrowing any more, and/or repaying existing debt. The ability to borrow is governed, ultimately, by incentives to repay,which are in turn driven by the variety of penalties levied on defaulters.1 The exposure to bad luck depends on the totalityof insurance held by the household, especially against income risk. Most income-insurance programs, such as welfare,medicaid, and the unemployment insurance (UI) system are publicly provided. A natural conjecture, therefore, is thatpenalties for default and social insurance policies together play a key role in explaining credit use.

Given the data above and the theoretical explanation of debts as an outcome of consumption smoothing, a very strikingfeature of US data is that cross-sectional consumption inequality rises rapidly over the entire life-cycle. As documented, forexample, in Deaton and Paxson (1994) and Storesletten et al. (2004), between the ages of 20 and 60, the cross-sectionalvariance of (log) consumption inherits the growth of life-cycle income inequality, and more than doubles. Similarly, networth inequality grows large over the life-cycle. Moreover, these features are robust: earnings inequality grows even whenmeasured net-of-transfers, and consumption and income inequality rise rapidly among even those who ex ante appearsimilar. As a result, Storesletten et al. (2004) argue convincingly that uninsurable risks are very important.

In this paper, I address the following question. How do policies towards debt default and social insurance jointly affectthe life-cycle evolution of household net worth and consumption, and especially their inequality? I construct a life-cyclemodel of consumption and savings that features uninsurable income risk, well-defined transfer policies, and procedures fordebt default. The main findings are as follows. First, default policy is important for explaining household net worth over thelife-cycle. In particular, I show that current US default policy creates severe credit constraints for households, especially theyoung. Second, default policies strongly interact with social insurance policies and from a welfare perspective, the twoshould be coordinated. Lastly, I show that measured consumption inequality can be misleading for assessing creditconstraints, and household well-being. In particular, eliminating default relaxes credit constraints, and induces householdsto borrow more when young. In turn, consumption inequality falls among the young. However, the inability to default onpreviously acquired debts, especially for those who have been unlucky ex post, turns out to increase consumptioninequality at older ages. Ex ante welfare of households nonetheless improves.

The present work is related most directly to four papers (Hubbard et al., 1995; Storesletten et al., 2004; Athreya andSimpson, 2006; Livshits et al., 2007). The distinctions between this paper and the preceding work are as follows. The workof Hubbard et al. (1995) is a landmark in its quantitative analyses of wealth accumulation under means-tested socialinsurance, and finds that means-tested programs are influential in lowering the savings of the income and wealth–poor(see also Huggett and Ventura, 2000). Such programs are therefore promising candidates for explaining the use ofunsecured debt. However, their model disallows borrowing. One contribution of the present work is to show that betterrisk-sharing via social insurance yields better intertemporal smoothing precisely by increasing borrowing among theyoung. More recently, Storesletten et al. (2004) innovates by representing income risk in a manner consistent with thegrowth in cross-sectional income inequality over the life-cycle. However, its focus is not on credit use, so it not only rulesout defaultable debt, but also abstracts from all asset-tested transfer schemes. On the other hand, Athreya and Simpson(2006) is similar in spirit to the present work, as it evaluates the interaction of UI with default policy. However, it ignoreslife-cycle smoothing. Moreover, it abstracts from the role of means-tested transfers, focusing instead on high-frequencyrisk-sharing and incentive problems.2

Livshits et al. (2007) is very novel in focusing on the reform of bankruptcy law in a life-cycle model. Relative to theirwork, the important innovations of this paper are twofold. First, Livshits et al. (2007) abstracts from the central issuestudied here: the joint roles of default policies and social insurance in explaining credit use and inequality. The absence ofany social insurance in their model also implies that the elimination of default would eliminate all borrowing. Thepreceding feature, and these authors’ focus on bankruptcy law, leads them to focus on the trade-off between the fulldischarge of debts, and a very specific debt rescheduling procedure. In contrast, the present model allows not only for‘‘intermediate’’ policy changes, but also for the study of gains available from the total elimination of default. Second,Livshits et al. (2007) employs an income risk process that generates a constant cross-sectional variance in income over thelife-cycle. However, this process is at odds with the evidence (e.g. Storesletten et al., 2004), and for my investigation, notinnocuous. Notably, it prevents the accurate evaluation of how default and social insurance alter the transmission ofincome risk to consumption inequality. Therefore, I employ instead an income process that generates the observed life-cycle evolution of income inequality, and calibrate the benchmark model to approximate the observed life-cycle evolutionof wealth inequality. The discipline imposed by this restriction turns out to matter substantially for the quantitativeimplications of credit market and insurance policies.

1 In this paper, the term default is used to describe any procedure that allows a household to successfully cease repayment of debts. Personal

bankruptcy is one important process by which the latter occurs. However, if the debt is collateralized, neither bankruptcy nor default can shelter the

wealth, as the law allows for a variety of actions to be taken by creditors, all of which may be costly to borrowers. Therefore, default is most applicable to

unsecured debts and is, for my purposes, interchangeable with bankruptcy.2 Similarly, Chatterjee et al. (2007) and Zha (2001) also study infinite-horizon settings, making them inappropriate for the questions posed here. The

present work also differs from the approach of Krueger and Perri (2006) where default is prohibited, and risk-sharing is impeded only by limited

commitment. In that setting, increases in income risk mitigate this problem.

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K.B. Athreya / Journal of Monetary Economics 55 (2008) 752–774754

My study is also related to a literature in law and economics. Notably, Rea (1984) formally addresses the conflictsbetween incentives and insurance created (and resolved) by default. Posner (1995) has recognized that the incentives forrisk-taking may increase under generous social insurance. In particular, when social insurance is combined with moralhazard, there may be another compelling reason to avoid prolonged repayment schemes such as wage garnishing, as theymight simply send people into the welfare rolls. Recently, however, Han and Li (2007) find little support for the strength ofsuch incentives in the data. Lastly, I am influenced by Robe et al. (2006), which is valuable for its documentation of thewide variety of default policies employed by societies through time, and across geographic areas.

In what follows, Sections 2 and 3 present the model and equilibrium, Sections 4 and 5, the parameterization and results,and Section 6 concludes.

2. Model

The economy consists of a continuum of J overlapping generations of working households. Households valueconsumption, do not value leisure, and therefore supply labor inelastically.

2.1. Preferences

Preferences are represented by a standard time-separable CRRA utility function over consumption during working life,and a ‘‘retirement felicity function’’, f, that is defined on wealth xR taken into retirement. Households have a commondiscount factor b and discount exponentially. The general problem for the household is to choose consumption fcjg

Jj¼1, and

retirement wealth xR, to maximize lifetime utility. Let PðC0Þ denote the space of all feasible combinations ðfcjg, xRÞ, giveninitial state C0. The household optimization problem is then

supðfcjg;xRÞ2PðC0Þ

E0

XJ

j¼1

bjc1�m

j

1� mþ fðxRÞ (1)

Retirement felicity as a function of retirement wealth takes the same form as preferences over consumption in workinglife:

fðxRÞ ¼x1�a

R

1� a(2)

2.2. Endowments

Households receive endowments in the form of labor income, and from their entitlement to means-tested publictransfers.

2.2.1. Labor income

Households face stochastic productivity shocks to their labor supply. Because households do not value leisure, they aremodeled as simply receiving stochastic endowments of the single consumption good in each period. The income processfaced by households in the model is intended to represent precisely those risks which remain, net of (i) all private insurancemechanisms and (ii) all non-means-tested public insurance programs, such as the US UI system. I closely follow theliterature (e.g. Hubbard et al., 1995; Storesletten et al., 2004; Huggett and Ventura, 2000), and disaggregate logendowments into three components: an age-specific mean of log income mj, persistent shocks, zj, and transitory shocks, uj.Log income therefore evolves as

ln yj ¼ mj þ zj þ uj (3)

where

zj ¼ gzj�1 þ Zj; gp1; jX2 (4)

uj�i.i.d. Nð0; s2uÞ; Zj�i.i.d. Nð0; s2

Z Þ; uj; Zj independent (5)

To reflect heterogeneity prior to any direct exposure to labor market risk, households draw their first realization of thepersistent shock from a distribution with a different variance than at all other ages. That is,

z0 ¼ 0 and Z1�Nð0; s2Z1Þ (6)

In subsequent periods, the log of household (non-asset) income is determined as the sum of the unconditional mean oflog income mj, the persistent shock ln Zj and the transitory shock, ln uj. The restriction to inelastic labor supply meansabstracting from a potentially useful smoothing mechanism, as seen in the infinite-horizon model of Pijoan-Mas (2006).However, it keeps the model parsimonious and easy to interpret.

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2.2.2. Means-tested transfer income

Social insurance in the US is aimed at providing a floor on consumption. Therefore, the constituent programs not onlyreplace income (e.g. social security disability, TANF, etc.), but also pay for large ‘‘expenses’’ such as health care and nursingcare via Medicaid and Medicare. The latter are particularly expansive, with large populations being guaranteed eligibility.My focus on income risk and ‘‘voluntary’’ debt accumulation leads me to avoid the possibility of involuntary expenditureswhich some private creditors are forced to finance.3 Means-tested transfers, tð�Þ, are represented as a function of currentage j, net assets xj, and non-asset income level yj. In the benchmark model, transfers will not depend explicitly on age.However, to illustrate the mechanisms by which transfers alter life-cycle net worth, I also study age-dependent transferschemes. Parametrically, transfers are specified exactly as in the seminal work of Hubbard et al. (1995):

tðj; xj; yjÞ ¼maxf0; t�ðmaxð0; xjÞ þ yjÞg (7)

In Eq. (7), total pre-transfer resources are given by ðmaxð0; xjÞ þ yjÞ. The means-testing restriction is represented in theinner term ‘‘t�ðmaxð0; xjÞ þ yjÞ’’, whereby these resources are deducted in order to provide a minimal income level t.If for example, xj40, yj40, and xj þ yj4 t, then the household gets no public transfer. By contrast, if xj40 and xj þ yjo t,the transfer scheme provides the difference, leaving the household with t units of the consumption good at thebeginning of the period. Similarly, households do not receive additional resources simply to cover debts, which requiresimposing the term ‘‘maxð0; xjÞ’’. In addition, I require that transfers are non-negative, which necessitates the outer ‘‘max’’operator.

2.2.3. Cash transfers vs in-kind transfers

In the US, means-tested social insurance is implemented through a combination of cash and in-kind transfers. However,the relative size of cash and in-kind transfers depend on the nature and size of the shock received by a household. Forexample, if a household suffers prolonged disability, then both cash transfers (via the Social Security Administration’s SSI-Disability program), and in-kind transfers (via Medicaid) will be made. Conversely, if the household is able to work, but isrelatively unproductive, the predominant transfer will be in cash, via the earned-income-tax-credit (EITC). Making thecomposition of transfers dependent on the shock itself is beyond the scope of the model. Moreover, even if the fraction ofin-kind transfers remains invariant to the shock, solving the household’s problem requires adding the extra constraint thatconsumption always remain weakly greater than transfers, i.e., cj4tðj; xj; yjÞ 8j; x; y. The model simplifies matters by usingpure cash transfers. Nonetheless, for the case where the proportion of cash to in-kind transfers is independent of the shockreceived, the key findings are completely robust to this simplification. Specifically, under the aggregate ratio of cash toin-kind transfers consistent with estimates of Garfinkel et al. (2006), allocations are identical to the benchmarkmodel economy.4 This is perhaps natural; the model’s focus on consumption and savings during working age and theabstraction from out-of-pocket medical expenses imply that the most relevant set of transfer programs are cash-transferprograms.5

2.2.4. Retirement income

In the last period of working life J, households evaluate retirement savings according to the function fðxRÞ, and saveaccordingly. Households aged J þ 1 are guaranteed to have a minimal standard of living given by the threshold tR. Therepresentation of social insurance policy after working life is aimed at capturing the sum of welfare programs and the sumof social security and medicare. Transfers during retirement are therefore not means-tested, and are given instead by asingle lump-sum transfer xt to all retiring households. My approach follows Huggett (1996) and Gourinchas and Parker(2002). A household’s wealth level at retirement is then the sum of the household personal savings xJþ1 and the baselineretirement benefit xtR

xR ¼ xJþ1 þ xtR (8)

The amount xtR is the wealth level that, when annuitized at the discount rate Rf , and adjusted for the probability ofsurvival for k periods, pk, yields a flow of income each period equal to the societal minimum retirement consumption floor

3 However, expense shocks still appear related to financial distress, and sometimes, to default, suggesting that coverage is imperfect. See Himmelstein

et al. (2005) and Johnson et al. (2006) for empirical analyses of the specific role of health expense shocks in financial distress. Incorporating such shocks is

not trivial, however. Any model in which liabilities exogenously appear on a household’s balance sheet implicitly assumes the existence of a class of

involuntary creditors. Future work on default and social insurance must provide a coherent account of how these gaps in coverage occur.4 I have studied two alternatives to the benchmark economy; (i) pure in-kind transfers and (ii) a mixture aimed at a simple approximation of current

US policy. They are omitted here for space considerations, but all results are available upon request. Perhaps the only difference worth mentioning is that

under the counterfactual experiment in which all transfers are in-kind, the age at which consumption inequality under the ‘‘no-default’’ regime finally

exceeds that under the other two regimes is delayed to age-50, relative to around age-40 under the cash-transfers economy. However, even this difference

disappears under lower transfer regimes, reflecting the fact that in these cases, the transfer level did not make the ‘‘savings’’ constraint bind for the

household.5 Notably, Medicaid transfers, the largest in-kind transfer scheme, are disproportionately consumed by the elderly. See e.g. Moffitt (2003) for a

detailed account of mean-tested transfer programs in the US. For working-age households, the key transfer programs are cash programs, such as (i) the UI

system, (ii) temporary aid to needy families (TANF), and (iii) the EITC.

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tR. That is, minimal retirement wealth xt solves

XK

k¼1

pktR

ðRfÞk¼ xtR (9)

2.3. Technology and market arrangement

Households of age-j choose savings, or a level of one-period defaultable debt, denoted xjþ1, to smooth consumption inthe face of uncertainty. A value for xjþ140 is interpreted as savings, which earns the return Rf40.6 To remain close to theliteratures on both life-cycle consumption and household default, I assume a small open-economy setting whereby Rf isexogenous.7 If xjþ1o0, households have borrowed in the current period, and may choose in the next period to repay theirdebt, or to default. When borrowing, the interest rate is denoted by Rð�Þ. The interest rate will reflect credit risk arising fromthe default option, and will also include transaction costs, c, arising from resources used in intermediation. Default iscostly, and all explicit and implicit costs of defaulting are summarized by the parameter l, which reduces household utilityin the period where debts are repudiated. This cost represents, among other things, the cost of legal representation andcourt fees, as well as any other relevant costs.

It is worth providing some intuition at this point regarding the use of a single, all-encompassing cost of default, asopposed to having several costs that apply separately to court costs, ‘‘stigma’’, and the increased post-default cost ofborrowing. In addition to the substantial advantage in simplicity that the current approach provides, it also captures theessential nature of the preceding penalties. In particular, stigma and punitive credit exclusion both involve ex postdeadweight loss, as does l. Second, and more subtly, the extent to which defaulters do face subsequent difficulties inborrowing may be due more to the release of information relevant to calculating default risk, rather than reflecting punitiveactions by creditors. Notably, the competitive intermediation structure assumed here, and arguably most relevant to UShouseholds, would not sustain punitive exclusion, as it is not renegotiation-proof. Allowing for unobserved heterogeneity istherefore critical to deliver a coherent model of competitive exclusion, and is well beyond the scope of this paper.8

2.4. Recursive formulation

The household’s problem is recursive in a state vector that is defined as follows. During working life, a household’sfeasible set for consumption and savings is determined by its age j, beginning-of-period net worth xj, current-periodrealization of the persistent shock zj, and current-period realization of transitory income uj. Notice that the state vectorcannot be collapsed in this model to a vector of (i) age, (ii) a simple ‘‘cash-in-hand’’ (e.g. Deaton, 1992) variable, and (iii) thecurrent persistent shock. This is because the payoff to bankruptcy depends on the precise composition of resources held bythe household at the beginning of the period. To see this, consider any two households i ¼ 1;2, whose net worth xðiÞ, andincome yðiÞ, differ such that (i) xð1Þo0 and xð2Þ40 and (ii) xðiÞ þ yðiÞ ¼ bY for some constant bY . Even though these householdshave the same ‘‘cash-in-hand’’ they will not, in general, face the same decision problem. In particular, since xð1Þo0, defaultis a possibility for the first household, but not the second. Moreover, since both the persistent shock z and total income y arenecessary for the household to optimize, the state vector must be either be written as ðj; x; z;uÞ, or ðj; x; z; yÞ, and I choose theformer.

2.4.1. Value functions

As the household enters a period, they first make the decision to default, if they enter with debt. Households are notallowed to issue debt within the period in which they default. Households may save, however, and can borrow in allsubsequent periods. The default decision determines current-period resources, and the consumption/savings (possiblydebt) decision is then made. Once debt or savings is chosen, the period ends.9 Let WR

ð�Þ denote the value of choosing torepay any debt that comes due at age-j, and WD

ð�Þ the value of eliminating debts via default. The beginning-of-period valuefunction must therefore satisfy

Vðj; xj; zj;ujÞ ¼ max½WRðj; xj; zj;ujÞ;W

Dðj; xj; zj;ujÞ� (10)

6 Since shocks are realized once-per-period, the restriction to one-period debt is natural in any setting in which lenders cannot commit themselves

against renegotiating the terms of debt in the future. In most credit-card arrangements, for example, the right to reprice at will is made explicit. However,

it is important to note that longer-term debt may allow households more flexibility in borrowing. Moreover, to the extent the default premia make

unsecured borrowing expensive, households may ultimately ‘‘overuse’’ collateralized debt (e.g. home equity) as an alternative method to smooth

consumption. Both points suggest that multi-period debt should be allowed for and studied in future work.7 Examples include Livshits et al. (2007), Carroll and Samwick (1997), and Hubbard et al. (1995). This abstraction is also reasonable in the present

context, as the measure of households most responsive to the policy experiments conducted here hold very low net worth in the aggregate.8 Chatterjee et al. (2006) is a first attempt at generating endogenous exclusion. As a quantitative matter, I have computed alternatives that explicitly

model exclusion, and find that it turns out to be unimportant. Results are available upon request.9 Notice again that transfers are cash, as we have not imposed the extra constraint that cjXtðj; xj ; yjÞ.

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Given that default removes debt and is irrelevant otherwise, WR and WD must, in turn, solve the following problems. Thevalue function when repaying debts is

WRðj; xj; zj;ujÞ ¼ sup

xjþ1

c1�mj

1� mþ bEzjþ1jzj

Vðjþ 1; xjþ1; zjþ1;ujþ1Þ

( )(11)

Subject to

cj þxjþ1

Rðj; xjþ1; zjÞpyj þ tðj; xj; yjÞ þ xj (12)

where the function Rðj; xjþ1; zjÞ specifies the interest rate associated with the level of savings or borrowing, xjþ1, chosen bythe household with current age j, and current realization of the persistent shock zj. Given that debt has one-periodmaturity, the interest rate will depend only on the three preceding factors.

On the other hand, the value of defaulting is given by

WDðj; xj; yj;ujÞ ¼ sup

xjþ1

c1�mj

1� m� lþ bEzjþ1jzj

Vðjþ 1; xjþ1; zjþ1;ujþ1Þ

( )(13)

Subject to

cj þxjþ1

Rfpyj þ tðj; xj; yjÞ (14)

xjþ1X0 (15)

In particular, the debt obligations present in the RHS of (12) disappear in (14), but the household pays the cost, l. Networth chosen in the current period must be non-negative, which implies using the (scalar) risk-free interest rate Rf in thebudget constraint.

2.5. Loan pricing

To determine the interest rate function Rðj; xjþ1; zjÞ, I proceed as follows. Creditors are competitive and hold a sufficientlylarge number of loans of any given size for a law of large numbers to guarantee them a deterministic rate of return on loansof that size. In particular, while individual loans are risky because of the possibility of default, the total losses to a givenlender in any period are deterministic. Creditors must expect to break even on each loan of a given size by pricingcontingent on all observable aspects of borrowers. Creditors are assumed to observe all factors relevant to forecasting one-period-ahead default risk on household debt. In the model, these factors are age j, persistent shock zj, and the face value ofdebt issued by the household, xj. In observing these items, lenders incur a proportional intermediation cost c. Theobservability of a household’s state vector implies that households differing in age and current productivity will also facedifferent prices for credit.

To compute zero-profit loan pricing, let pDðj; xjþ1; zjÞ denote the probability of default on a loan of size xjþ1 made to ahousehold currently of age j, with current-period persistent income shock zj. To compute pDðj; xjþ1; zjÞ, let the functionIðjþ 1; xjþ1; zjþ1;ujþ1Þ be the indicator function over whether a household entering next period with debt of face value xjþ1

will find default superior to repayment if they receive shocks zjþ1 and ujþ1. That is,

Iðjþ 1; xjþ1; zjþ1;ujþ1Þ ¼ 1 iff WDðjþ 1; xjþ1; zjþ1;ujþ1Þ4WR

ðjþ 1; xjþ1; zjþ1;ujþ1Þ

Therefore, pDð�Þ is calculated at each age j as follows:

pDðj; xjþ1; zjÞ ¼

ZZIðjþ 1; xjþ1; zjþ1;ujþ1Þf ðzjþ1;ujþ1jzjÞdzjþ1 dujþ1 (16)

Given pDð�Þ, the zero-profit interest rate is then given by

Rðj; xjþ1; zjÞ ¼Rfþ c

ð1� pDðj; xjþ1; zjÞÞ(17)

3. Equilibrium

I use the standard notion of stationary recursive partial competitive equilibrium (RCE). Let O � J � x� z� u denote thestate space, and let o denote the borel s-algebra on O. Given the risk-free rate Rf , and the state-contingent transfer functiontðj; xj; yjÞ, a RCE is a collection of:

1.
o-measurable pricing and default functions R�ð�Þ and pDð�Þ; 2. o-measurable value functions WD
ð�Þ, WRð�Þ, and Vð�Þ;

3.
o-measurable decision rule for assets gð�Þ, such that xjþ1 ¼ gðj; xj; zj;ujÞ;

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4.

har

exc

Ven

The

a transition function Pðb;BÞ giving the conditional probability that a household currently in state b will be in a subsetB 2 o next period;

5.
a measure, m� : o! R, of households over subsets of the state space;
such that:

1.
The value functions satisfy the recursions (11) and (13), and the maximization (10), subject to the (equilibrium) budgetconstraints (12) and (14).
2.
The decision rule gð�Þ attains the supremum of the RHS of the value function (10) for all values of the current state:(j; xj; yj).
3.
Loan prices and default probabilities are consistent, whereby lenders earn zero expected profits on each loan of size xjþ1,given by Eqs. (16) and (17). R
4.
The measure, m� : o! R, is stationary. That is, m� satisfies m�ðbÞ ¼ Pðb;BÞdm�.
4. Parameterization

The model period is one calendar year. Households begin working life at age 21, and retire at age 65, implying J ¼ 44.Risk aversion and discounting are set at standard values of m ¼ 2 and b ¼ 0:96, respectively. I set the (gross) risk-freeinterest rate on savings following Mehra and Prescott (1985), whereby Rf

¼ 1:01%. The transactions cost on consumerlending is set whereby c ¼ 3:4%, following Evans and Schmalensee (1999), who use data from credit-card banks. Lastly,following Storesletten et al. (2004), households are born with zero financial wealth, i.e., x1 ¼ 0.

To understand the implications of default policy, I use Robe et al. (2006) to motivate a natural partition of default costsinto three stylized categories. Each category will be defined by the credit limits that arise as a result. First, the credit limitobtaining when default is strictly prohibited ðl ¼ 1Þ will be referred to as the ‘‘natural-borrowing-limit’’ (see e.g.Ljungqvist and Sargent, 2000).10 Second, at the polar opposite, the case where default carries no adverse consequenceswhatsoever (l ¼ 0Þ is therefore one where no unsecured debts are possible. The intermediate case is set at lUS

¼ 0:9, avalue, that along with all other parameters, allows the model to approximate the observed life-cycle paths of default andnet worth.

The benchmark parameterization of the income process sets g ¼ 0:99, s2u ¼ 0:063, s2

Z1¼ 0:22, s2

Z ¼ 0:0275. These valuesallow the model to match three important targets. First, the variances of the transitory shock and initial persistent shockallow the model to match the variance of log income among the youngest working-age households in the data. Second, thenear-unit root in the persistent shock of g ¼ 0:99 generates the essentially linear life-cycle growth of cross-sectionalvariance in log income documented in Storesletten et al. (2004). Third, the variance of the persistent shock beyond theyoungest age captures the total increase in cross-sectional (log) income variance over the life-cycle, from approximately0.28 among 21-year-olds, to approximately 0.90 among new retirees. To parameterize the profile of the mean of logendowments over the life-cycle, I use the data on median earnings from Census (2000) on US males. Since endowments arelog normal, the mean of log endowments equals the logarithm of median endowments. Therefore, I take logs of thepreceding estimates of median earnings, and generate a profile fmjg

Jj¼1.11 When solving the household’s problem

numerically, I use a 256 point discretization for the income process, and employ the approximation of Tauchen (1986) torepresent the stochastic components of the income process. To obtain decision rules and value functions, I use standarddiscrete-state space dynamic programming.12

The fact that the income process generates an increasing cross-sectional variance with age is crucial. It is this prominentfeature of the data that will make precise the role played by credit markets in moderating the effects of income risk overthe life-cycle. I discipline the model by targeting the distribution of US net worth, which I measure using the Survey ofConsumer Finances (2001). I employ a smoothed version of a variety of percentiles, by age, of this distribution, whichgenerates paths similar to those of Kennickell (2002).

Default targets in the model are guided by the data generated by Chapter 7 ‘‘total-liquidation’’ bankruptcy. This form ofbankruptcy is the predominant form of non-business bankruptcy, accounting for over 70% of filings in each of the past twodecades. I set l to target the Chapter 7 filing rate as of 1991 0.5%, as well as the mean net worth of Chapter 7 bankruptcyfilers, as of 1991, of $16,815. The relative frequency of bankruptcy by age is based on data from Sullivan et al. (2000). I targetthe filing rate as of 1991 as it is contemporaneous with the income and consumption data that is targeted by the

10 While l ¼ 1 is useful primarily as a benchmark polar case, Robe et al. (2006) documents the widespread use, until fairly recently, of extremely

sh penalties against debt default. Notably, debtors in the US faced lengthy imprisonment as recently as the early 20th century. This constituted an

eptionally harsh punishment as such sentences carried with them the real risk of disease and death. See also Moss (2002).11 The resulting profile is very similar to the profile of mean efficiency units estimated by Hansen (1993). A third process is that of Huggett and

tura (2000). The latter process is steeper at both ends of the life-cycle, as it is adjusted for labor force participation (treated as an exogenous process).

results are robust to the all three choices of the age-profile for the mean of log income, and are available upon request.12 All codes are available from the author on request.

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datamodel

Fig. 1. Net worth percentiles, model and data.


benchmark model (e.g. Storesletten et al., 2004). The preceding is also part of a relatively stable unsecured creditenvironment, and precedes the transitional outcomes obtaining in the mid to late 1990s, due especially to importanttechnological changes in the intermediation of unsecured credit.13

The final objects for parameterization are the means-tested transfer function and retirement benefit. With respect topreferences for retirement wealth, I set a ¼ m ¼ 2. Turning first to the means-tested transfer function, tðj; xj; yjÞ, theinterpretation is that households are eligible for a transfer, subject to the sum of the current income and wealth fallingbelow a threshold t, deemed necessary by society. I denote the transfer under current US policy as tUS, and settUS ffi $7;600. The dollar value of this income floor is less than the inflation-adjusted value of Hubbard et al. (1995) ofapproximately $10,800 in constant 1991 dollars per household annually, but allows the benchmark model to much bettercapture the observed asset accumulation of households in the lower percentiles of the wealth distribution.14 The policy

13 See e.g. Edelberg (2006), Furletti (2003), and Narajabad (2007).14 The probabilistic receipt of some classes of transfers (especially housing assistance) is part of what is being captured in this reduction. Notably,

Hubbard et al. (1995) assign households the expected value of the transfer. However, this will overestimate the floor, as the value of the expectation will

be strictly greater than the value of the lottery to the household. Lastly, the $10,800 is arrived at by adjusting for inflation the $7000 income floor of

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Fig. 2. Default policy and life-cycle consumption inequality.


experiments will then compare this benchmark with three alternatives, t ¼ t0 ¼ $500, t ¼ tL ¼ $3;500, and lastly,t ¼ tH ¼ $14;000. Similarly, the baseline retirement benefit tR is set at $8,600 in constant 1991 dollars, again slightly lowerthan the Hubbard et al. (1995) estimate.

4.1. The fit of the benchmark model

Fig. 1 shows that the model provides a reasonable approximation to the wealth distribution at a variety of wealthpercentiles, for essentially all of the life-cycle. Most importantly, the benchmark accurately captures the use of unsecureddebt by the lowest two wealth groups under consideration: the 5th and 10th percentiles. There is an initial increase in debtfor the first 10–12 years of adult life, followed by a flattening of savings at close to zero. The absence of life-cycleaccumulation in anticipation of retirement among the lower percentiles is a striking feature of the data. Hubbard et al.(1995) found that this observation could be accounted for by the implicit tax on wealth accumulation created by asset-tested transfer programs. For the periods preceding retirement, this mechanism applies in the present set-up. To see this,one can compare outcomes for low-wealth households against those for households in high wealth percentiles. Thesehouseholds are more likely to have received high income realizations, and given the high persistence of shocks in thebenchmark economy, make them less likely to qualify for transfers at retirement. Therefore, these households save atrelatively high rates in the model, as in the data.

Under current US means-tested transfer programs and income risk, the model is also able to generate the observedevolution of cross-sectional consumption inequality with age, measured by the variance of the logarithm of householdconsumption. Specifically, as seen in Fig. 2, the model captures the estimates from Deaton and Paxson (1994), whereby thevariance of log consumption more than doubles from approximately 0.25 at age 20 to near 0.65 by age 65.15 Relative to thefinding of Livshits et al. (2007), Fig. 3B, a key payoff to the incorporation of an income process that generates growth inincome variability is the ability of my model to accurately capture the consumption risk faced by households. By contrast,in the former, consumption inequality remains essentially constant over the life-cycle at the much lower level of 0.3.

(footnote continued)

Hubbard et al. (1995), which was measured in 1984 dollars, using the CPI ‘‘All Items’’ index. Moreover, as the referee has noted, less-than-100% utilization

may also arise from various transactions costs, lowering the value of the transfer.15 More recently, Heathcote et al. (2005) argue that consumption inequality may grow less over the life-cycle than estimated by Deaton and Paxson

(1994). I have re-computed the experiments to match this alternative view of the data, and find that the general results all go through. Complete details

are available on request.

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Fig. 3. Default policy and net worth.


5. Results

The first main result of this paper is that default policy is important for explaining household net worth andconsumption over the life-cycle. Specifically, US-style ‘‘Fresh-Start’’ policy towards debt default appears to impose severerestrictions on the use of unsecured credit, and in turn, create an undesirably large amount of inequality early in workinglife. Most surprisingly perhaps is that cross-sectional consumption inequality and net worth both evolve over the life-cyclefor most households much as it would if unsecured credit were completely unavailable. Conversely, the prohibition ofdefault, holding all else fixed, results in dramatic changes in levels and distributions of consumption and wealth over thelife-cycle.

The second main result is that default policy and social insurance policy interact strongly, with each exerting substantialinfluence on the effectiveness of the other. Specifically, I show first that social insurance policy can significantly alter theability of default policy to change outcomes. Second, as a converse, I show that default policy is pivotal in determining theimpact of US mean-tested social insurance on consumption and wealth distribution.

From a normative perspective, the results suggest first that households would prefer the elimination of default, evenunder very low minimal social insurance programs. This is notable precisely because the relatively low costs of personalbankruptcy in the US were in large part justified by appeals to the presence of large uninsurable risks.16 Second, andperhaps more surprisingly, household welfare and consumption inequality are not directly related. A robust finding for ourenvironment is that the prohibition of default leads to lower consumption inequality early in working life, but relativelyhigher inequality later in life. This path of inequality is precisely the result of households being able to borrow againstuncertain future income.

The remainder of the results are organized as follows. Section 5.1 demonstrates the importance of default policy forconsumption and unsecured credit markets over the life-cycle. Sections 5.2 and 5.3 then show that default policy and social

16 See e.g. Moss (2002).

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Table 1Default policy, credit use, and consumption smoothing

fl; tg Debt

Income

Fraction with negative net worth cvðconsÞ

cvðyÞ

f0; tUSg 0 0 0.7356

flUS; tUSg �0.0915 0.1954 0.7307

f1; tUSg �0.3863 0.3986 0.7230


insurance policy interact to jointly determine household net worth and consumption over the life-cycle. Section 5.4discusses the implications of the model for default and Section 5.5 provides welfare implications.

5.1. Default policy matters for household wealth and consumption

Table 1 summarizes the implications of default policy on aggregate indebtedness, consumption smoothing, and welfare.Starting with the aggregate unsecured-debt-to-income ratio, it is clear that default policy is important for explaining debts.When default is so easy that unsecured borrowing is impossible, debt-to-income ratios are trivially zero. However, thisratio rises substantially to 9.15% under current US debt and social insurance policy. This increase, however, is dwarfed bythe jump to 38.63% when default is ruled out. Are the changes in aggregate indebtedness primarily changes in the intensivemargin, whereby a given set of households simply borrows more under high default penalties, or does the set of borrowinghouseholds change? Comparing the first and second columns of Table 1 shows that both margins are operative, as thefraction of households with negative net worth is seen to grow rapidly from 19.54% under US default policy, to 39.86%under the prohibition of default.

What do the constraints identified above imply for the unconditional volatility of consumption relative to income? Thethird column of Table 1 presents the ratio of the unconditional coefficient of variation of consumption to that of income.Two findings are noteworthy. First, default policy appears to have minimal importance for aggregate measures ofconsumption smoothing. While smoothing does improve (i.e., the ratio falls) as default is made more costly, the change isminimal, on the order of one percentage-point.

The absence of a larger impact of default policy on the unconditional coefficient of variation of consumption is at firstpuzzling. The resolution lies in the aggregation of age-specific variation that is inherent in the unconditional coefficient ofvariation. Fig. 2 records the variance of log consumption of households by age obtaining under the three default policiesunder consideration. Most importantly, eliminating default produces substantial reductions in inequality for approximatelythe first 20 years of working life, followed by an even longer period in which inequality is slightly larger than under USdefault policy. Any average over households of different ages will then produce similar results. Moreover, Fig. 2 also makesclear that current US default policy leads to a life-cycle evolution of consumption inequality that appears extremely similarto that obtaining if all borrowing were ruled out.

The results above are a direct consequence of the effect of default on the ability of households to smooth consumptionvia borrowing. Fig. 3 tracks net worth over working life for the 5th, 25th and 40th percentiles of the conditional (on age)wealth distribution. Thus, default policy seriously alters debt levels for the entire life-cycle. For example, in the 5thpercentile, US default policy generates an increase in borrowing relative to the no-borrowing case that lasts untilapproximately age 40. Beyond that age, however, US policy is literally isomorphic to a total prohibition on unsecured debt.For the 10th and 25th percentiles, a similar story holds. In these cases, however, households have received large enoughincome shocks to lead them to accumulate very little debt under US default policy. That is, the path of debts, and later,assets, under l ¼ lUS for the 10th and 25th percentiles are closer to the ‘‘no-borrowing’’ case than for the 5th percentile.Moreover, though not shown here, the same qualitative feature holds true at all higher wealth percentiles as well. Bycontrast, there is a striking difference in debts over the life-cycle when default is ruled out. For example, at even the 40thpercentile of net worth by age, we see the large increases in debt made possible by being able to borrow at risk-free interestrates.17 In sum, the model suggests that US default policy generates quantitatively important credit constraints which alternet worth and consumption over the entire life-cycle.

The strong effects of default policy on outcomes, in particular the interpretation of current policy as ‘‘lax’’, doespotentially depend on the specification of risk in the current model. Notably, the model focuses on earnings shocks, andabstracts from other types of risks that might dissuade households from borrowing, even when they could obtain relativelylarge levels of credit at low interest rates. Notably, the model abstracts from aggregate shocks, and health risks. However,the model also abstracts from labor supply as an additional smoothing mechanism, and thereby limits the ability ofhouseholds to deal with the shocks that are currently specified. Such an abstraction can have (at least) two effects. First,knowing that they can supply labor if they are indebted and unlucky may lead to more borrowing under strict default

17 The results also contradict the commonly expressed views that (a) US households are ‘‘too’’ indebted, and (b) that generous bankruptcy laws are to

‘‘blame’’. My results suggests instead that (1) US households are not nearly as indebted as they want to be, at least when young, and (2) it is a tightening of

default policy that would encourage indebtedness.

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Fig. 4. The effect of default policy on net worth depends on social insurance.


policy. Conversely, a model with valued-leisure would mean that indebtedness today could mean undesirable fluctuationsin future labor supply.

5.2. The effect of default policy depends on social insurance provision

The effects above are striking. However, they may depend on social insurance policy. Consider a move, for example, frompresent US default policy to the elimination of default, i.e., a move from l ¼ lUS to1. If social insurance is very minimal,e.g. t ¼ t0; default risk may be high because households face the possibility of realizing very low incomes. Moreover, the

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household’s ability to borrow will not increase by much when default is ruled out. By contrast, if social insurance isgenerous to begin with, the elimination of default may not confer large benefits in the terms of credit, because its effect ondefault risk may be small. However, households’ ability to borrow may grow substantially when default is ruled out.

Each panel in Fig. 4 contains wealth accumulation over the life-cycle for a given percentile of the wealth distributionacross the three default regimes under consideration. Each row holds fixed the level of social insurance. The top row of thefigure collects outcomes under the lowest transfer policy under consideration, t ¼ t0. For the 5th percentile of wealth, seenin the top left-hand panel of the figure, the elimination of default increases the level of debt held by households. Relative tocurrent US default policy, the increase in debt is between 10,000 and 15,000 dollars for the first 20 years of working life. Forhouseholds in the 40th percentile, seen in the top, right-hand panel of the figure, the increases in debt arising from rulingout default are somewhat smaller. However, I show next that these increases in debt are much smaller than those obtainingwhen default is ruled out under more generous social insurance.

Fig. 5. Default policy, social insurance, and loan pricing.

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Fig. 6. Transfer policy and life-cycle consumption inequality.


The implications of eliminating default when t ¼ tH are given in the bottom row of Fig. 4. In this case, the amount ofborrowing increases dramatically. For example, the increase in indebtedness arising from the elimination of default isapproximately 40,000 dollars for the first two decades of working life among the 5th percentile of households, and roughly20,000 dollars for households as high as the 40th percentile. Thus, the response of debt to the elimination of default undert ¼ tH is more than double than under t ¼ t0.

To understand the role of social insurance for households’ ability to issue debt, consider Fig. 5. In each row, I hold fixedthe current component of persistent labor productivity and present the equilibrium zero-profit loan interest rate schedulefor households of ages 21 and 35 across all four social insurance regimes. In each of the panels, we see clearly how lowertransfers make borrowing more expensive by holding fixed a level of debt, and comparing interest rates across socialinsurance policies. Since the prohibition of default can only lower the interest rate faced by households, their ‘‘ability’’ toborrow grows as default is prohibited.

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Table 2Aggregates: transfers to all

fl; tg Debt

Income

Fraction with negative net worth cvðconsÞ

cvðyÞ

Welfare gain (FÞ

f0; t0g 0 0 0.7865 �0.0058

flUS; t0g �0.1080 0.1102 0.7824 0

f1; t0g �0.2176 0.2175 0.7792 0.0120

f0; tLg 0 0 0.7622 �0.0067

flUS; tLg �0.1069 0.1288 0.7579 0

f1; tLg �0.2741 0.2964 0.7533 0.0185

f0; tUSg 0 0 0.7356 �0.0097

flUS; tUSg �0.0915 0.1954 0.7307 0

f1; tUSg �0.3863 0.3986 0.7230 0.0215

f0; tHg 0 0 0.6690 �0.013

flUS; tHg �0.1318 0.4223 0.6606 0

f1; tHg �0.6627 0.5871 0.6442 0.0145


To see how important eliminating default can be in expanding the ability of households to use risk-free debt, define the‘‘natural debt limit’’ as the debt level that, if rolled over, would exhaust a household’s income in the worst state of theworld, i.e., xNBL � ðtþminj yjÞ=ðR

fþ c� 1Þ.18 Under the most stringent transfers t ¼ t0, this limit exceeds $20,000!

Furthermore, as lower minimal transfers seem implausible, the elimination of default can essentially always be expected toallow households to feasibly borrow very large amounts of debt without explicit collateral. Therefore, the fact thathouseholds do not borrow more when default is ruled out is direct evidence of the idea that under very low socialinsurance floors, households may simply be unwilling, not unable, to borrow.

Does the response of consumption inequality to changes in default policy also depend on social insurance provision?Fig. 6 shows consumption inequality over the life across all default and social insurance regimes. Specifically, each panelholds fixed transfers and displays the paths of consumption inequality under all three default regimes. In all four panels, wesee that the prohibition of default produces the same qualitative result: less inequality when young, more inequality whenolder. Thus the mechanism by which default alters smoothing is robust to the specification of transfers. Quantitatively,though, two findings are worth mentioning. First, the prohibition of default generates smaller reductions in inequalityunder generous social insurance than under the other cases, as seen by comparing the top-left and bottom-right panels.Second, the change in the path of inequality induced by the elimination of default appears dependent on the socialinsurance regime in place. In particular, the greater the level of social insurance provision, the longer is the time for whichconsumption inequality under the ‘‘no-default’’ regime remains below that for the other default regimes. Under the lowesttransfer regime, we see that by approximately age 28, consumption inequality under no-default exceeds that of otherdefault policies. By contrast, under US social insurance, the corresponding age is closer to 40. When transfers are set to thehighest level, t ¼ tH, consumption inequality remains lower for essentially the entire life-cycle.

5.3. The effect of social insurance provision depends on default policy

Does the converse to the question posed above also hold? That is, does the impact of social insurance provision dependon default policy? To see why it might, consider a move from present US social insurance provision to a high minimalincome floor, i.e., a move in the model from t ¼ tUS to tH. When default is allowed, an increase in social insurance lowersdefault risk. If this effect is relatively large, the resulting ‘‘cheap’’ credit may result in higher equilibrium borrowing. On theother hand, this change in social insurance policy may result in lower equilibrium borrowing, as households face fewerdifficult circumstances in which they need to borrow to smooth consumption. By contrast, when default is prohibited,social insurance cannot affect credit pricing. However, in this case, the same increase in social insurance as above maymake households more willing to borrow on debts that they must always repay, since future income cannot fall to very lowlevels.

Beginning with the behavior of aggregates, we see in Table 2 that under US-style default policy, while household debtsdo not depend strongly in the social insurance policy in place, the fraction of borrowers does. Notice that when l ¼ lUS, thedebt–income ratio remains within the relatively small interval (9.15–13.18%) across all four social insurance regimes, whilethe corresponding increase in the fraction of households with negative net worth increases from 11.02% to 42.23%. Bycontrast, the elimination of default dramatically increases the response of aggregate debt to social insurance. For example,consider a policy change that raises the minimal income floor from t ¼ t0 to tH. Under US default policy, the debt–incomeratio rises from 10.8% to 13.18%. The same change in social insurance policy also triples the aggregate debt–income ratiofrom 21.76% to 66.27%. Eliminating default allows households to most fully leverage their risky future income for current

18 See e.g. Ljungqvist and Sargent (2000, p. 371).

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Fig. 7. The effect of social insurance policy on net worth depends on default policy.


consumption, and when the worst possible income realization is relatively large, so is the ‘‘natural borrowing limit’’implied by solvency alone.

At the level of individual households, Fig. 7 illustrates the joint effects of social insurance and default policy. Within eachpanel, social insurance is varied, holding fixed default policy. Two findings emerge. First, wealth holdings fall as minimaltransfer levels grow. Second, and much more striking, is that the reduction in life-cycle accumulation depends strongly onthe ability of households to borrow. As seen in the top row of the figure when borrowing is ruled out (l ¼ 0), there is asubstantial impact on wealth accumulation for households in the 5th percentile, and a significant, but relatively muchsmaller impact on wealth accumulation of households in the 40th percentile. Moreover, the often binding lower bound ofzero under the no-borrowing case limits the absolute size of changes in net worth arising as a result of means-testedtransfers. In sharp contrast, in the bottom row of the figure, we see that when default is ruled out, under the most generoustransfer regime (t ¼ tH), net worth from the 5th percentile all the way to the 40th percentile remains negative for more

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Fig. 8. The timing of transfers can matter for life-cycle net worth.


than half of working life, often significantly so. Similar patterns hold for the other transfer regimes as well. With respect toconsumption inequality, the ability of increased transfers to slow the growth of cross-sectional consumption inequality isgreatly amplified when default is made prohibitively costly. Notably, even under generous transfers, default policy mattersfundamentally, as seen in the fact that for the young, the path of inequality in the absence of default always lies below thepath under current US default policy.

5.3.1. Does the timing of transfers matter?

To better understand the mechanics by which social insurance policy alters net worth over the life-cycle, I presentevidence from two counterfactual exercises. First, I deny all but the most minimal transfers (i.e., I set t ¼ t0) to allhouseholds above the age of 35. Second, I deny all but the most minimal transfers to all those below the age of 35. In theformer case, households will be concerned, especially when default is prohibited, about borrowing a ‘‘large’’ amount, asthey must pay back these debts even in states of the world where their income, net-of-transfers, is very low. In the latter,households may suffer from very low incomes when young, which may lead them to borrow, especially when they are

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Table 3Bankruptcy statistics

Transfer structure Bankruptcy rate (%) my j bankruptcy Debt discharged in bankruptcy

t ¼ tL; l ¼ lUS

All 0.15 $8665 $3880

Young 0.15 $8275 $11,200

Old 0.15 $8520 $6630

t ¼ tUS; l ¼ lUS

All 0.10 $11,050 $5800

Young 0.12 $9585 $13,840

Old 0.15 $9225 $6100

t ¼ tH; l ¼ lUS

All 0.04 $14,800 $11,780

Young 0.07 $11,775 $14,210

Old 0.13 $10,105 $7810


eligible for transfers when old. Fig. 8 presents the changes to life-cycle wealth accumulation occurring in a move, forexample, from present US social insurance policy t ¼ tUS to the higher transfer regime t ¼ tH. Under current policy, itappears that the timing of transfers is not particularly important. However, under the high-transfer regime, when transfersare given only to the young (i.e., the first column) households save substantially more than before, given US default policy.By contrast, the figure shows that when transfers are taken away from the young, there is very little effect of timing, evenwhen t ¼ tH. These findings are therefore consistent with transfers in middle- and late-working life emboldeninghouseholds to borrow for intertemporal purposes.

5.4. Default

The classification of default policies used here implies that there is positive default in equilibrium only when l ¼ lUS.For aggregate default-related outcomes, Table 3 contains three statistics for each social insurance transfer level: (i) theunconditional default rate, (ii) the mean income among defaulters, and (iii) the mean level of debt discharged in default.Conditional on each transfer level, I present outcomes from the three different timing structures for transfers introducedearlier. For life-cycle default-related outcomes, the results are summarized in Fig. 9.

The model produces the qualitatively correct age-profile of default, with the highest filing rates occurring early in life.Quantitatively, the benchmark model performs well early in the life-cycle, but does not capture the incidence of defaultlater in working life. As a result, the aggregate filing rate at 0.1%, seen in Table 3, is lower than the 0.5% seen in the data.19

The model performs well in approximating the mean net worth of those in Chapter 7 bankruptcy, with the modelgenerating a value of approximately $13,800, as compared to the $16,800 measured in the data. It also turns out that theoverall implications for net worth, borrowing, and consumption inequality survive a variety of modifications which allowthe model to generate more equilibrium default.20 Intuitively, the reason that the results are robust is because default israre, and because the benchmark economy captures fairly well the wealth accumulation of especially the young poor. Mostof the effects of default policy operate by changing prices off the equilibrium path, by significantly modifying how muchhouseholds must pay for various levels of debt. For aggregate statistics, the rarity of default also means that directcontribution of the default rate on aggregated measures of consumption smoothing will remain small.

The ‘‘steepness’’ of the pricing functions seen in Fig. 5 means that the marginal benefit to debt issuance along the steepportion is very low. This is simply because a greater discount on marginal debt issuance leads to a commensurately smallerincrease in the current resources generated by additional borrowing. However, as the ‘‘face’’ value of debt rises steadily, sodoes the likelihood of future default and the expected cost coming from the associated penalty. In turn, in equilibrium, fewhouseholds borrow at levels where they pay a significant risk-premium on debt. Note, however, that this outcome isdistinct from the fact that the aggregate default rate in the model is smaller than in the data. Specifically, in a model wherehouseholds were subject to large, but transitory, ‘‘expense’’ shocks, borrowers would always face some ‘‘baseline’’ defaultrisk that would be essentially independent of their borrowing levels and current state vector. In turn, all borrowing would

19 This shortcoming is a more general property of current models of consumer default. For example, in a model without social insurance, Livshits et al.

(2007) employ the assumption that households are hit by potentially very large ‘‘expense’’ shocks, such as health care expenditures that directly reduce

assets. These shocks play a crucial role in generating default in their model, being present in nearly 90% of simulated bankruptcies. By contrast, Sullivan et

al. (2000) estimate that only 19% of bankruptcies arise from health related expenses, for example. However, given that expense shocks for the wealth poor

are at least partially insured via social insurance, what is required is an explicit model of precisely how in eligibility for such transfers is determined, and

what role, in turn, such ‘‘holes’’ in the safety net play. This is well beyond the scope of the present paper, but is a key issue for future research.20 These include the steeper age-earnings profile used by Huggett and Ventura (2000), which encourages more borrowing. I retain the process based

on census data, as it is very similar to that of Hansen (1993) which is a benchmark in the literature.

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Fig. 9. Social insurance and bankruptcy over the life-cycle.


exhibit a risk-premium, even if the aggregate default rate were low. The model therefore represents the equilibrium cost ofcapital that could be expected to arise from the default risk created by earnings-related income shocks, which arepredominantly persistent in nature.

5.4.1. Social insurance and default

With respect to social insurance policy and the aggregate default rate, the general tendency of default rates is to fall associal insurance floors rise. This provides some support for the argument that the lack of social insurance may be animportant determinant of consumer default.21 However, the relationship is not monotonic, with the aggregate default rateinitially falling with transfers, but subsequently rising. Nonetheless, as seen in both Table 3 and Fig. 9, the rates of defaultare an order of magnitude higher under the two lowest transfer policies (t ¼ t0 and tL), than in the other two cases. The risein aggregate default rates with a move to t ¼ tH is related to the substantial increase in the fraction of households withnegative net worth, seen earlier in Table 1.

Social insurance policy also appears to change the circumstances prevailing among defaulters. In particular, we see thatmean income at the time of default increases systematically with the generosity of transfers. Fig. 5 reflects the reduction indefault risk created by increased transfers. Lastly, as transfers grow, so does the level of debt in default. In summary,generous social insurance is predicted to result in a lower default rate, but with each default being exercised byprogressively higher income households with larger unsecured debts.

5.4.2. Default and wealth dynamics

The option to default may have implications for wealth dynamics, both by making borrowing more expensive, and alsoby allowing households to discharge large sums of debts in a single operation. Table 4 collects one-step wealth transitionprobabilities as a function of default regime. The topmost panel of Table 4 corresponds to the case l ¼ 0, the middle table tol ¼ lUS, and the bottom table to l ¼ 1. A priori, it is reasonable to expect that these effects are most likely to beconcentrated among the parts of the wealth distribution where default is relatively likely. This turns out to be correct. Thesimilarity of the right-most column of each of the three panels of Table 4 makes clear that default policy has very littleimpact on the probability of entering any percentiles above the median in one-step, given that a household started below

21 See, e.g. Sullivan et al. (2000).

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Table 4Wealth transition probabilities

Wealth %-ile tnt þ 1 10th 20th 30th 40th 50th 60th–100th

t ¼ tUS; l ¼ l0

10th 0.7444 0.1636 0.0442 0.0357 0.0097 0.0024

20th 0.1415 0.5689 0.2212 0.0147 0.0464 0.0072

30th 0.0797 0.045 0.6418 0.1694 0.0332 0.0309

40th 0.0293 0.0279 0.0935 0.6096 0.1945 0.0452

50th 0.009 0.0072 0.018 0.0866 0.6945 0.1847

t ¼ tUS; l ¼ lUS

10th 0.7871 0.078 0.0577 0.0286 0.0319 0.0167

20th 0.3468 0.3696 0.2197 0.0378 0.0232 0.0029

30th 0.0633 0.0614 0.671 0.1469 0.0417 0.0158

40th 0.0452 0.008 0.0996 0.6384 0.1678 0.0410

50th 0.0124 0.0059 0.0109 0.1103 0.6801 0.1804

t ¼ tUS; l ¼ 110th 0.8587 0.0906 0.0197 0.0174 0.0085 0.0051

20th 0.1115 0.6947 0.1291 0.0271 0.0253 0.0124

30th 0.0252 0.2024 0.5671 0.1522 0.0351 0.0181

40th 0.0065 0.0195 0.2354 0.5376 0.1664 0.0346

50th 0.0012 0.0037 0.015 0.1024 0.6817 0.1960


the median. Moreover, for most households below the median, the absolute probability of moving beyond the 50thpercentile is low. In sum, default policy under current US social insurance policy has essentially no impact on the wealthdynamics of those beyond the 50th percentile.

For those below the median in terms of wealth, default policy has significant, and non-monotonic, effects on wealthdynamics. In particular, it is clear that there is a substantial increase in the persistence of wealth for the bottom two decileswhen default is ruled out. The second notable difference is the relatively large increase in the likelihood that a household inthe second-lowest decile will move down to the lowest decile under US-style policy than under either of the other tworegimes. This reflects the ability of households to borrow, and then default if necessary. When borrowing is disallowed, orwhen default is prohibited, having low wealth is particularly unattractive, and leads households to remain where they are.One caution, however, is that the objects being compared are relative wealth levels, and it is clear from the earlier resultsthat absolute levels of accumulation can differ substantially across default regimes. Nonetheless, in the US, the results doallow for the possibility that default may be a non-trivial force altering the wealth dynamics of the poor.

5.5. Welfare

The sharp changes documented above in net worth and consumption smoothing arising from various default and socialinsurance policies suggest that households will not be indifferent between them. Let VB and VP be the indirect utilityfunctions associated with the benchmark and any proposed policy, respectively. I measure changes in welfare bycomputing the percentage increment to benchmark consumption required by newly entering working household with zeronet assets (i.e., j ¼ 1, x1 ¼ 0) to be indifferent between the benchmark economy and the proposed policy. Let f z1

ð�Þ and f uð�Þ

be the unconditional marginal densities of age-1 persistent and transitory shocks, respectively. Denoting the welfarechange by F, we have

F ¼

RRVPð1;0; z1;uÞf z1

ðzÞf uðuÞdz1 duRRVBð1;0; z1;uÞf z1

ðzÞf uðuÞdz1 du

" #1=ð1�mÞ

� 1 (18)

Table 2 contains three main findings. First, the welfare consequences of lowering the cost of default below current levels isfairly minimal, which is simply a further reflection of the laxity of current policy. Second, eliminating default alwaysproduces welfare gains, irrespective of the generosity of transfers. Third, for all but the highest transfer level, the gains toruling out debt default are increasing in transfers. This is suggestive of a more fundamental trade-off between socialinsurance and default. In Athreya and Simpson (2006), a similar result was obtained but for reasons related to increasedmoral hazard in a model of UI. In the present setting, welfare gains from eliminating default are driven by the improvedintertemporal smoothing made possible by generous transfers and low borrowing costs. From a welfare perspective, theresults make clear that default policy is a tool that is critical to allowing society to realize the gains from extant socialinsurance policy. In this vein, though I do not study the specific differences between them, the findings are suggestive withrespect to the stylized observation that continental Europe, which has relatively generous social insurance, also makespersonal bankruptcy very costly.

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Fig. 10. Default policy, social insurance policy, and welfare over the life-cycle.


5.5.1. Welfare over the life-cycle

If default penalties are suboptimally low, in the sense that they lower ex ante welfare, why do they persist? Aninteresting possibility is that such policies are differentially attractive to households of different ages. In particular, bothshocks and the stationary distributions of assets differ systematically by age, as does the value of future access to credit. Fig.10 shows that the welfare implications of default policy do indeed depend on age in a striking way: only the very youngprefer strict default policy. That is, for all but the very young, expected utility under the age-specific invariant distributionis higher under lax default policy than when default is eliminated. The same figure also shows that this result obtainsacross all three social insurance levels on display. The intuition for this is that both those who are still young, and thosewho are unlucky, have often accumulated debts (recall Fig. 1). Given the extremely high persistence of shocks, those whoare currently indebted because of prior bad luck must adjust consumption downward, and so do not expect to use credit inthe near future. This makes them willing, ex post, to support a policy that allows for debt relief. By the same token, thosewho have been lucky thus far are also relatively unlikely to need credit in the future, and so are essentially indifferent todefault policy. The net effect is that for most households, expected utility rises as default penalties fall. As a result, it maynot be surprising that generous debt repudiation policies such as Chapter 7 ‘‘liquidation’’ bankruptcy continue to prevail inthe US.

5.5.2. Additional considerations

With respect to the interpretation of the welfare effects documented above, some caveats are in order. First, I haveabstracted from the presence of extremely urgent needs to consume along with the class of involuntary creditors needed tokeep consumption non-negative in all states of the world. In the work of Livshits et al. (2007), these shocks and the abilityto finance them are assumed, and shown to be necessary to generate a welfare-improving role for debt relief. Second,households in the model do not face aggregate risk. The presence of such risk, if it dramatically lowers the lowest possibleearnings level (net of transfers) that a household can generate, will make the ‘‘natural borrowing limit’’ (e.g. Aiyagari, 1994)very low. In this case, debt use would remain low irrespective of default risk. Third, I have employed very standard time-consistent preferences and rational expectations for households. An interesting aspect of the legal justifications forendowing all unsecured debtors with a non-waivable bankruptcy option is the explicit mention of the inability of

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households to exercise ‘‘self-control’’, or forecast rationally.22 The quantitative relevance of such considerations remains atopic for resolution prior to arriving at a definitive view on the effects of default and social insurance policies.

6. Conclusion and future work

The widespread use of debt and default suggests that unsecured credit markets play an important role in themanagement of household income variations, both over time, and across states-of-nature. In this paper I ask two questions.First, how does policy towards debt default affect the life-cycle evolution of household consumption and net worth?Second, given the use of debt primarily by the young, and default primarily by the unlucky, how does debt default policyinteract with social insurance provision to determine life-cycle credit use? The main findings are as follows. First, I find thatchanges in default policy can generate large changes in the life-cycle path of consumption and wealth, especially theirinequality. In particular, I identify current US default policy to be ‘‘lax’’, in the sense that it creates severe credit constraintsfor households, especially the young. Second, default policy and social insurance policy interact, and should therefore becoordinated. Lastly, a useful finding is that while the elimination of default reduces consumption inequality among theyoung by relaxing binding borrowing constraints, the resultant accumulation of non-defaultable debt leads those who areunlucky when older to compromise on consumption. In turn, consumption inequality among the old is higher under relaxedcredit constraints than when households cannot borrow.

With respect to future work, an important challenge remains. This is to provide a compelling reconciliation of theexistence of minimal social insurance, especially in terms of medical coverage for the poor, with the ability of households tobe pushed in bankruptcy and default by large, or catastrophic, expenses. A richer model of eligibility and provision ofexpense-related social insurance is needed before more definitive statements can be made on the ideal mixture of socialinsurance policy and default policy. Nevertheless, the current model strongly suggests that the gains to such coordinationmay be quite large.

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Default, insurance, and debt over the life-cycle