econometrica, vol. 75, no. 6 (november, 2007), 1525–1589 · 2007. 9. 21. · econometrica, vol....

Econometrica, Vol. 75, No. 6 (November, 2007), 1525–1589

A QUANTITATIVE THEORY OF UNSECURED CONSUMER CREDITWITH RISK OF DEFAULT

BY SATYAJIT CHATTERJEE, DEAN CORBAE, MAKOTO NAKAJIMA,AND JOSÉ-VÍCTOR RÍOS-RULL1

We study, theoretically and quantitatively, the general equilibrium of an economyin which households smooth consumption by means of both a riskless asset and un-secured loans with the option to default. The default option resembles a bankruptcyfiling under Chapter 7 of the U.S. Bankruptcy Code. Competitive financial intermedi-aries offer a menu of loan sizes and interest rates wherein each loan makes zero profits.We prove the existence of a steady-state equilibrium and characterize the circumstancesunder which a household defaults on its loans. We show that our model accounts for themain statistics regarding bankruptcy and unsecured credit while matching key macro-economic aggregates, and the earnings and wealth distributions. We use this model toaddress the implications of a recent policy change that introduces a form of “meanstesting” for households contemplating a Chapter 7 bankruptcy filing. We find that thispolicy change yields large welfare gains.

KEYWORDS: Bankruptcy, General equilibrium, Default risk.

1. INTRODUCTION

IN THIS PAPER, we analyze a general equilibrium model with unsecured con-sumer credit that incorporates the main characteristics of U.S. consumer bank-ruptcy law and replicates the key empirical characteristics of unsecured con-sumer borrowing in the United States. Specifically, we construct a model con-sistent with the following facts:• Borrowers can default on their loans by filing for bankruptcy under the rules

laid down in Chapter 7 of the U.S. Bankruptcy Code. In most cases, filing forbankruptcy results in seizure of all (nonexempt) assets and a full dischargeof household debt. Importantly, filing for bankruptcy protects a household’scurrent and future earnings from any collection actions by those to whomthe debts were owed.

1We wish to thank a co-editor and three anonymous referees for very helpful comments on anearlier version of this paper. We also wish to thank Nick Souleles and Karsten Jeske for helpfulconversations, and attendees at seminars at Complutense, Pompeu Fabra, Pittsburgh, Stanford,Yale, and Zaragoza universities and the Cleveland and Richmond Feds, Banco de Portugal, the2000 NBER Summer Institute, the Restud Spring 2001 Meeting, the 2001 Minnesota Workshopin Macro Theory, the 2001 SED, and the 2002 FRS Meeting on Macro and Econometric Soci-ety Conferences. Ríos-Rull thanks the National Science Foundation (Grant SES–0079504), theUniversity of Pennsylvania Research Foundation, and the Spanish Ministry of Education. In addi-tion, we wish to thank Sukanya Basu, Maria Canon, and Benjamin Roldán for detailed comments.Chatterjee and Corbae wish to thank Peet’s Coffee of Austin for inspiration. The views expressedin this paper are those of the authors and do not necessarily reflect those of the Federal ReserveBank of Philadelphia or of the Federal Reserve System. The working paper version of this paperis available free of charge at www.philadelphia.org/econ/wps/index.html.

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1526 CHATTERJEE, CORBAE, NAKAJIMA, AND RÍOS-RULL

• Post-bankruptcy, a household has difficulty getting new (unsecured) loansfor a period of about 10 years.2

• Households that default are typically in poor financial shape.3• There is free entry into the consumer loan industry and the industry behaves

competitively.4• There is a large amount of unsecured consumer credit.5• A large number of people who take out unsecured loans default each year.6

A key contribution of our paper is to establish a connection between therecent facts on household debt and bankruptcy filing rates with the theory ofhousehold behavior that macroeconomists routinely use. This connection is es-tablished by modifying the equilibrium models of Imrohoroglu (1989), Huggett(1993), and Aiyagari (1994) to include default and by organizing the facts onconsumer debt and bankruptcy in light of the model.

Turning first to the theory, we analyze an environment where householdswith infinitely long planning horizons choose how much to consume and howmuch to save or borrow. Households face uninsured idiosyncratic shocks toincome, preferences, and asset position, and therefore have a motive to accu-mulate assets and to sometimes borrow in order to smooth consumption. Wepermit households to default on their loans. This default option resembles aChapter 7 bankruptcy filing in which debts are discharged. We abstract fromthe out-of-pocket expenses of declaring bankruptcy, but assume that a bank-ruptcy remains on a household’s credit record for some (random) length oftime that, on average, is compatible with the length of time mandated by law.We assume that while the bankruptcy filing remains on a household’s record,it cannot borrow and incurs a (small) reduction in its earning capability.

It should be clear from this basic setup that an indebted household will weighthe benefit of maintaining access to the unsecured credit market against thebenefit of declaring default and having its debt discharged. Accordingly, creditsuppliers who make unsecured loans will have to price their loans to take intoaccount the likelihood of default. We assume a market arrangement wherecredit suppliers can link the price of their loans to the observable total debtposition of a household and to a household’s type. The first theoretical contri-bution of the paper is to prove the existence of a general equilibrium in which

2This is documented in Musto (1999).3This is documented in, for example, Flynn (1999).4See Chapter 10 of Evans and Schmalensee (2000) for a compelling defense of the view that

the unsecured consumer credit industry in the United States is competitive.5The Board of Governors of the Federal Reserve System constructs a measure of revolving

consumer debt that excludes debt secured by real estate, as well as automobile loans, loans formobile homes, trailers, or vacations. This measure is probably a subset of unsecured consumerdebt which amounted to $692 billion in 2001, or almost 7 percent of the $10.2 trillion that consti-tutes U.S. GDP.

6In 2001, 1.45 million people filed for bankruptcy in the United States, of which just over 1million were under Chapter 7 (as reported by the American Bankruptcy Institute) and the restfiled under Chapter 13.

UNSECURED CONSUMER CREDIT 1527

the price charged on a loan of a given size made to a household with givencharacteristics exactly compensates lenders for the objective default frequencyon loans of that size made to households with those characteristics. This proofis challenging because the default option may result in a discontinuity of thesteady-state wealth distribution with respect to the rental rate on capital andwages.

A second theoretical contribution of the paper is a characterization of de-fault behavior in terms of earnings for a given set of household characteristics.First we prove that for each level of debt, the set of earnings that triggers de-fault is an interval. Specifically, an earnings-rich household (one with earningsabove the upper threshold of the interval) is better off repaying its debt andsaving, while an earnings-poor household (one with earnings below the lowerthreshold of the interval) is better off repaying its debt and borrowing. Thisresult is important, because it reduces the task of computing default probabili-ties to that of computing default thresholds. Second, we prove that the defaultinterval expands with increasing indebtedness.

A third theoretical contribution is to show that our equilibrium loan priceschedules determine, endogenously, the borrowing limit facing each type ofhousehold. This is theoretically significant since borrowing constraints oftenplay a key role in empirical work regarding consumer spending. Thus, we be-lieve it is important to provide a theory of borrowing constraints that derivesfrom the institutional and legal features of the U.S. unsecured consumer creditmarket.

Turning to our quantitative work, we first organize facts on consumer earn-ings, wealth, and indebtedness from the 2001 Survey of Consumer Finances(SCF) in light of the reasons cited for bankruptcy by Panel Study of IncomeDynamics survey participants between 1984 and 1995. Our model successfullygenerates statistics that closely resemble these facts. To accomplish this, wemodel shocks that correspond to the reasons people give for filing for bank-ruptcy and that replicate the importance (for the filing frequency and debt) ofeach reason given. One of the three shocks is a standard earnings shock thatcaptures the job-loss and credit-misuse reasons. A second shock is a preferenceshock that captures the effects of marital disruptions. The third shock is a liabil-ity shock that captures motives related to unpaid health-care bills and lawsuits.This last shock is important because it captures events that create liabilitieswithout a person having actually borrowed from a financial intermediary—afact that turns out to be important for simultaneously generating large amountsof debt and default. To incorporate this liability, shocks to the model had to beexpanded to incorporate a hospital sector.

We use our calibrated model to study the effects of a recent change in bank-ruptcy law that discourages above-median-income households from filing un-der Chapter 7. We find that the policy change has a substantial impact. Thereis a roughly twofold increase in the level of debt extended without a significantincrease in the total number of defaults. We also find a significant welfare gain


from this policy: households are willing (on average) to pay around 1.6 percentof annual consumption to implement this policy.

Our paper is related to several recent strands of literature on unsecureddebt. One strand, starting with Kehoe and Levine (1993), and more re-cently Kocherlakota (1996), Kehoe and Levine (2001), and Alvarez and Jer-mann (2003), studies environments in which agents can write complete state-contingent contracts with the additional requirement that contracts satisfy aparticipation constraint—a constraint that comes from the option to perma-nently leave the economy (autarky) rather than meet one’s contractual oblig-ations. Since the participation constraint is always satisfied, there is no equi-librium default.7 To model equilibrium default on contractual obligations thatresembles a Chapter 7 filing, we depart from this literature in an importantway. In our framework, a loan contract between the lending institution anda household specifies the household’s next-period obligation, independent ofany future shock, but gives the household the option to default. The interestrate on the contract can, however, depend on such things as the household’scurrent total debt and socioeconomic characteristics that provide partial in-formation on a household’s earnings prospects. This assumption is motivatedby the typical credit card arrangement.8 Because of the limited dependence ofthe loan contract on future shocks, our framework is closer to the literatureon default with incomplete markets as in Dubey, Geanokoplos, and Shubik(2005), Zame (1993), and Zhang (1997). Zame’s work is particularly relevantbecause he shows that with incomplete markets, it may be efficient to allow abankruptcy option to debtors.

In innovative work, Athreya (2002) analyzed a model that includes a defaultoption with stochastic punishment spells, but in his model, financial interme-diaries charge the same interest rate on loans of different sizes even thougha large loan induces a higher probability of default than a small loan. As aresult, small borrowers end up subsidizing large borrowers, a form of cross-subsidization that is not sustainable with free entry of intermediaries.9 Enforc-ing zero profits on loans of varying sizes complicates our equilibrium analysisbecause there is now a schedule of loan prices to solve for rather than a singleinterest rate on loans.10

7Kehoe and Levine (2006) suggest that one can interpret the incentive constrained allocationas arising in an economy where individuals simultaneously lend and borrow from each other,where some agents default, and the bankruptcy penalty includes a mixture of a Chapter 7-styleseizure of assets and a Chapter 13-style garnishment of earnings.

8For more detail on the form of the standard credit card “contract,” see Section III of Grossand Souleles (2002).

9Lehnert and Maki (2000) have a model with competitive financial intermediaries who canprecommit to long-term credit contracts in which a similar type of cross-subsidization is alsopermitted.

10Livshits, MacGee, and Tertilt (2003) followed our approach where the zero profit conditionis applied to loans of varying size. However, they assumed that creditors can garnish wages of a


The paper is organized as follows. In Section 2, we describe our modeleconomy and characterize the behavior of agents. We then prove existenceof equilibrium in Section 3 and characterize properties of the equilibrium loanschedules. We describe and discuss our calibration targets in Section 4. We dis-cuss the properties of the calibrated economies in Section 5. In Section 6, wepose and answer our policy question. All proofs are given in the Appendix orthe supplement to this article (Chatterjee, Corbae, Nakajima, and Ríos-Rull(2007)).

2. THE MODEL ECONOMY

We begin by specifying the legal and physical environment of our modeleconomy. Then we describe a market arrangement for the economy. This isfollowed by a statement of the decision problems of households, firms, finan-cial intermediaries, and the hospital sector.

2.1. Legal Environment

We model the default option to resemble, in procedure and consequences, aChapter 7 bankruptcy filing. Consider a household that starts the period withsome unsecured debt. If the household files for bankruptcy (and we permita household to do so irrespective of its current income or past consumptionlevel), then the following things happen:

1. The household’s beginning of period liabilities are set to zero (i.e., itsdebts are discharged) and the household is not permitted to save in the currentperiod. The latter assumption is a simple way to recognize that U.S. bankruptcylaw does not permit those who invoke bankruptcy to simultaneously accumu-late assets: a bankrupt must relinquish all (nonexempt) assets to creditors atthe time that discharge of debt is granted by a bankruptcy court.11

2. The household begins next period with a record of bankruptcy. Let ht ∈01 denote the “bankruptcy flag” for a household in period t, where ht = 1indicates in period t a record of a bankruptcy filing in the past and ht = 0denotes the absence of any such record. In what follows, we will refer to h assimply the household’s credit record, with the record being either clean (h= 0)or tarnished (h = 1). Thus, a household that declares bankruptcy in period tstarts period t + 1 with ht+1 = 1

bankrupt person in the period in which that person files for bankruptcy and that a person hasunrestricted access to unsecured credit in the period immediately following default.

11Some assets are exempt in a Chapter 7 bankruptcy filing and can be retained by the filer.However, the nature and value of exempt assets vary across U.S. states, being quite low in somestates and quite high in others. For simplicity, we ignore this variation and assume that no ex-emptions are permitted. We also do not address Chapter 13 bankruptcy filings in this paper. In aChapter 13 filing, debtors can retain their assets in return for a promise to repay some portion ofthe total obligation from their future earnings.


3. A household that begins a period with a record of bankruptcy cannotget new loans.12 Also, a household with a record of bankruptcy experiences aloss equal to a fraction 0 < γ < 1 of earnings—a loss intended to capture thepecuniary costs of a bad credit record.13

4. There is an exogenous probability 1 − λ that a household with a recordof bankruptcy will have its record expunged in the following period. That is, ahousehold that starts period t with ht = 1 will start period t + 1 with ht+1 = 0with probability 1 − λ This is a simple, albeit idealized, way of modeling thefact that a bankruptcy flag remains on an individual’s credit history for only afinite number of years.

2.2. Preferences and Technologies

At any given time there is a unit mass of households. Each household isendowed with one unit of time. Households differ in their labor efficiencyet ∈ E = [emin emax] ⊂ R++ and in certain characteristics st ∈ S, where S is afinite set. There is a constant probability (1 − ρ) that any household will die atthe end of each period. Households that do not survive are replaced by new-borns who have a good credit record (ht = 0), zero assets (t = 0), and withlabor efficiency and characteristics drawn independently from the probabil-ity measure space (S × EB(S × E)ψ), where B(·) denotes the Borel sigmaalgebra and ψ denotes the joint probability measure. Surviving households in-dependently draw their labor efficiency and characteristics at time t from astochastic process defined on the measurable space (S × EB(S × E)) withtransition function Φ(et |st)Γ (st−1 st), where Φ(et |st) is a conditional densityfunction and Γ (st−1 st) is a Markov matrix. We assume that for all st the prob-ability measure defined by Φ(et |st) is atomless.

There is one composite good produced according to an aggregate productionfunction F(KtNt), where Kt is the aggregate capital stock that depreciates atrate δ and Nt is aggregate labor in efficiency units in period t. We make thefollowing assumptions about technology:

ASSUMPTION 1: For all KtNt ≥ 0 F satisfies (i) constant returns to scale,(ii) diminishing marginal returns with respect to the two factors, (iii) ∂2F/∂Kt ∂Nt > 0, (iv) Inada conditions with respect to Kt , namely, limKt→0 ∂F/∂Kt =∞ and limKt→∞ ∂F/∂Kt = 0, and (v) ∂F/∂Nt ≥ b > 0.

12We interpret the assumption that firms do not lend to households with a record of a bank-ruptcy filing in their credit history as a legal restraint on firm behavior. In the context of ourmodel, financial intermediaries value this restriction because it prevents individual lenders fromdiluting the punishment from default. In Chatterjee, Corbae, and Ríos-Rull (2004), we presenta dynamic adverse selection model in which default provides an imperfect signal about a house-hold’s type and, as a result, defaulters may find it prohibitively costly to borrow.

13There are pecuniary costs of a bad credit rating, such as higher auto insurance premia. Foran explicit analysis of pecuniary costs stemming from a loss of reputation in credit market, seeChatterjee, Corbae, and Ríos-Rull (2007).


The composite good can be transformed one-for-one into consumption, in-vestment, and medical services. As described in detail later, unforeseen med-ical expenditure is an oft-cited reason for Chapter 7 bankruptcy filing.

Taking into account the possibility of death, the preferences of a householdare given by the expected value of a discounted sum of momentary utility func-tions,

E0

∞∑t=0

(βρ)tU(ctη(st))

(1)

where 0< β < 1 is the discount factor, ct is consumption, and η(st) is a pref-erence shock in period t. We make the following assumptions on preferences.

ASSUMPTION 2: For any given s U(·η(s)) is strictly increasing, concave, anddifferentiable. Furthermore, there exist s and s in S such that for all c and sU(cη(s))≤U(cη(s))≤U(cη(s))

Consumption of medical services does not appear in the utility function be-cause we treat this consumption as nondiscretionary.14 Furthermore, we as-sume that consumption of medical services does not affect the productive ef-ficiency of the household. When they occur, the household is presented with ahospital bill ζ(st). We assume that each surviving household has a strictly pos-itive probability of experiencing a medical expense. Specifically, there existss ∈ S for which ζ(s) > 0 and Γ (st−1 s) > 0 for all st−1

2.3. Market Arrangements

We assume competitive factor markets. The real wage per efficiency unit isgiven bywt ∈W = [wminwmax] with wmin > 0. The rental rate on capital is givenby rt

The addition of a default option necessitates a departure from the conven-tional modeling of borrowing and lending opportunities. In particular, we posita market arrangement where unsecured loans of different sizes for differenttypes of households are treated as distinct financial assets. This expansion ofthe “asset space” is required to correctly handle the competitive pricing ofdefault risk, a risk that will vary with the size of the loan and household char-acteristics. In our model a household with characteristics st can borrow or saveby purchasing a single one-period pure discount bond with a face value in afinite set L⊂ R. The set L contains 0 and positive and negative elements. Wewill denote the largest and smallest elements of L by max > 0 and min < 0,respectively. We will assume that FK(max emin)− δ > 0

14Alternatively, we could assume that unless the medical expenditure is incurred, the house-hold receives −∞ utility.


A purchase of a discount bond in period t with a nonnegative face value t+1

means that the household has entered into a contract where it will receivet+1 ≥ 0 units of the consumption good in period t + 1. The purchase of adiscount bond with a negative face value t+1 and characteristics st means thatthe household receives qt+1st · (−t+1) units of the period-t consumption goodand promises to deliver, conditional on not declaring bankruptcy, −t+1 > 0units of the consumption good in period t + 1; if it declares bankruptcy, thehousehold delivers nothing. The total number of financial assets available tobe traded is NL ·NS where NX denotes the cardinality of the set X Let theentire set of NL ·NS prices in period t be denoted by the vector qt ∈ R

NL·NS+ .We restrict qt to lie in a compact set Q≡ [0 qmax]NL·NS , where 1 ≥ qmax ≥ 0 Inthe section on steady-state equilibrium, the upper bound on q will follow fromassumptions on fundamentals.

Households purchase these bonds from financial intermediaries. We assumethat both losses and gains resulting from death are absorbed by financial in-termediaries. That is, a household that purchases a negative face-value bondhonors its obligation only if it survives and does not declare bankruptcy and,symmetrically, an intermediary that sells a positive face-value discount bond isreleased from its obligation if the household to which the contract was sold isnot around to collect. We assume that there is a market where intermediariescan borrow or lend at the risk-free rate it Also, without loss of generality, weassume that physical capital is owned by intermediaries who rent it to com-posite goods producers. There is free entry into financial intermediation andintermediaries can exit costlessly by selling all their capital.

The hospital sector takes in the composite good as an intermediate inputand transforms it one-for-one into medical services. In our model, as in thereal world, some households may default and not pay their medical bills ζ(st).We assume that if some proportion of aggregate medical bills is not paid backdue to default, then hospitals supply medical services in the amount ζ(st)/mt

to households with characteristic st , where the markup mt > 1 is set to ensurezero profits.

2.4. Decision Problems

The timing of events in any period is (i) idiosyncratic shocks st and et aredrawn for survivors and newborns, (ii) capital and labor are rented and pro-duction of the composite good takes place, and (iii) household default andborrowing/saving decisions are made, and consumption of goods and servicestakes place. In what follows, we will focus on steady-state equilibria, wherewt =w rt = r it = i and qt = q2.4.1. Households

We now turn to a recursive formulation of a household’s decision problem.We denote any period-t variable xt by x and its period-(t + 1) value by x′


In addition to prices, the household’s current period budget correspondenceBhsd(e;qw) depends on its exogenous state variables s and e, its beginningof period asset position , and its credit record h. It will also depend on thehousehold’s default decision d ∈ 01, where d = 1 indicates that the house-hold is exercising its default option and d = 0 indicates that it is not. ThenBhsd(e;qw) has the following form:

1. If a household with characteristics s has a good credit record (h= 0) anddoes not exercise its default option (d = 0), then

B0s0(e;qw)= c ∈ R+ ′ ∈L : c+ q′s′ ≤ e ·w+ − ζ(s)(2)

We take into account the possibility that the budget correspondence may beempty in this case. In particular, if the household is deeply in debt, earningsare low, new loans are expensive, and/or medical bills are high, then the house-hold may not be able to afford nonnegative consumption. As discussed below,allowing the budget correspondence to be empty permits us to analyze bothvoluntary and “involuntary” default.

2. If a household with characteristics s has a good credit record (h= 0) andnet liabilities, (− ζ < 0) and exercises its default option (d = 1), then

B0s1(e;qw)= c ∈ R+ ′ = 0 : c ≤ e ·w(3)

In this case, net liabilities disappear from the budget constraint and no savingis possible in the default period. That is, we assume that during a bankruptcyproceeding a household cannot hide or divert funds owed to creditors.

3. If a household with characteristics s has a bad credit record (h= 1) andnet liabilities are nonnegative (− ζ ≥ 0), then

B1s0(e;qw)= c ∈ R+ ′ ∈L+ : c+q′s′ ≤ (1−γ)e ·w+−ζ(s)(4)

where L+ = L ∩ R+. With a bad credit record, the household is not permittedto borrow and is subject to pecuniary costs of a bad credit record.

4. If a household with characteristics s has a bad credit record (h= 1) and(− ζ < 0), then

B1s1(e;qw)= c ∈ R+ ′ = 0 : c ≤ (1 − γ)e ·w(5)

A household with a bad credit record and a net medical liability pays only upto its assets, cannot accumulate new assets, and begins the next period witha bad credit record. When the budget set is empty, these assumptions corre-spond to giving the household another Chapter 7 discharge. For this reason, we


denote this case by setting d = 1.15 For simplicity, we continue to make theseassumptions even when the budget set is not empty.16

To set up the household’s decision problem, define L to be all possible(h s) tuples, given that only households with a good credit record can havedebt and let NL be the cardinality of L. Then L ≡ L−− × 0 × S ∪ L+ ×01×S where L−− =L\L+ Let vhs(e;qw) denote the expected lifetimeutility of a household that starts with (h s) and e, and faces the prices q andw, and let v(e;qw) be the vector vhs(e;qw) : h s ∈ L in the set V ofall continuous (vector-valued) functions v :E ×Q×W → R

NL .The household’s optimization problem can be described in terms of a vector-

valued operator (T v)(e;qw) = (Tv)(h s e;qw) : (h s) ∈ L, whichyields the maximum lifetime utility achievable if the household’s future life-time utility is assessed according to a given function v(e;qw)

DEFINITION 1: For v ∈ V let (Tv)(h s e;qw) be defined as follows:1. For h= 0 and B0s0(e;qw)= ∅,

(Tv)(0 s e;qw)=U(e ·wη(s))+βρ

∫v01s′(e

′;qw)Φ(e′|s′)Γ (sds′)de′

2. For h= 0, B0s0(e;qw) = ∅, and − ζ(s) < 0,

(Tv)(0 s e;qw)

= max

maxc′∈B0s0

U(cη(s))

+βρ∫v′0s′(e


U(e ·wη(s))

+βρ∫v01s′(e


15U.S. law does not allow a household to file for Chapter 7 again within 6 years of having fileda Chapter 7 bankruptcy. Instead, such a household can only file for a Chapter 13. Since we donot consider Chapter 13 in this paper, we simply assume that these households receive anotherChapter 7 discharge of net liabilities.

16In the computational work that follows, this latter case almost never arises. That is, when− ζ(s) < 0, the budget set is invariably empty, because when medical liabilities occur, they arelarge relative to earnings of households with h= 1.


3. For h= 0, B0s0(e;qw) = ∅, and − ζ(s)≥ 0,

(Tv)(0 s e;qw)= maxc′∈B0s0

U(cη(s))

+βρ∫v′0s′(e


4. For h= 1 and − ζ(s) < 0,

(Tv)(1 s e;qw)= U(e ·w(1 − γ)η(s))+βρ

∫v01s′(e


5. For h= 1 and − ζ(s)≥ 0,

(Tv)(1 s e;qw)= max

c′∈B1s0U(cη(s))

+βρ[λ

∫v′1s′(e


+ (1 − λ)∫v′0s′(e

′;qw)Φ(e′|s′)Γ (sds′)de′]

The first part of this definition says that if the household has debt and thebudget set conditional on not defaulting is empty, the household must default.In this case, the expected lifetime utility of the household is simply the sumof the utility from consuming its current earnings and the discounted expectedutility of starting next period with no assets and a bad credit record. The secondpart says that if the household has net liabilities and the budget set conditionalon not defaulting is not empty, the household chooses whichever default optionyields higher lifetime utility. In the case where both options yield the sameutility, the household may choose either. The difference between default underpart 1 and default under part 2 is the distinction between “involuntary” and“voluntary” default. In the first case, default is the only option, while in thesecond case it is the best option. The third part applies when a household witha good credit record has no net liabilities. In this case, the household does nothave the default option and simply chooses how much to borrow/save.17 Thefinal two parts apply when the household has a bad credit record and hence nodebt. It distinguishes between the case where it has some net liability (which

17We do not permit households to default on liabilities ζ when − ζ ≥ 0. This is without lossof generality since all assets of a household can be seized during a bankruptcy filing (no exemptassets).


arises from a large enough liability shock) and the case where it does not. Inthe first case, the household is permitted to partially default on its liabilities asdescribed earlier. In the second case, the household simply chooses how muchto save.

THEOREM 1—Existence of a Recursive Solution to the Household Problem:There exists a unique v∗ ∈ V such that v∗ = T (v∗) Furthermore, (i) v∗ is boundedand increasing in and e, (ii) a bad credit record reduces v∗, (iii) the optimalpolicy correspondence implied by T (v∗) is compact-valued and upper hemicon-tinuous, and (iv) provided u(0 s) is sufficiently low, default is strictly preferable tozero consumption and optimal consumption is always strictly positive.

Because certain actions involve discrete choice, T (v∗) generally delivers anoptimal policy correspondence instead of a function. Given property (iii) ofTheorem 1, the measurable selection theorem (Theorem 7.6 of Stokey andLucas (1989)) guarantees the existence of measurable policy functions for con-sumption c∗

hs(e;qw) asset holdings ′∗hs(e;qw), and the default decision

d∗hs(e;qw).The default decision rule, along with the probabilistic erasure of a bank-

ruptcy flag on the household’s credit record, implies a mapping H∗(qw) : (L ×

E) × 01 → [01], which gives the probability that the household’s creditrecord next period is h′ The mapping H∗ is given by

H∗(qw)(h s eh

′ = 1)=

1 if d∗hs(e;qw)= 1,

λ if d∗hs(e;qw)= 0 and h= 1,

0 if d∗hs(e;qw)= 0 and h= 0;

H∗(qw)(h s eh

′ = 0)=

0 if d∗hs(e;qw)= 1,

1 − λ if d∗hs(e;qw)= 0 and h= 1,

1 if d∗hs(e;qw)= 0 and h= 0.

Then we can define a transition function GS∗(qw) : (L × E) × (2L × B(E))→

[01] for a surviving household’s state variables given by

GS∗(qw)((h s e)Z)(6)

=∫Zh×Zs×Ze

1′∗hs

(e;qw)∈ZH∗(qw)(h s edh

′)Φ(e′|s′)de′ Γ (sds′)

where Z ∈ 2L × B(E) and Zj denotes the projection of Z on j ∈ h s e,and where 1 is the indicator function. Since a household in state (h s e)could die and be replaced with a newborn, we can define a transition functionGN : (L×E)× (2L ×B(E))→ [01] to a newborn’s initial conditions given by

GN((h s e)Z)=∫Zs×Ze

1(′h′)=(00)ψ(ds′ de′)(7)


Combining these, we can define the transition functionG∗(qw) : (L×E)× (2L ×

B(E))→ [01] for the economy as a whole by

G∗(qw)((h s e)Z)(8)

= ρGS∗(qw)((h s e)Z)+ (1 − ρ)GN((h s e)Z)

Finally, given the transition function G∗, we can describe the evolution ofthe distribution of households µ across their state variables (h s e) for anygiven prices (qw) by use of an operator Υ . Specifically, let M(L × E2L ×B(E)) denote the space of probability measures. For any probability measureµ ∈M(L×E2L ×B(E)) and any Z ∈ 2L ×B(E) define

(Υ(qw)µ)(Z)=∫G∗(qw)(h s eZ)dµ(9)

THEOREM 2 —Existence of a Unique Invariant Distribution: For any (qw) ∈ Q × W and any measurable selection from the optimal policy correspon-dence, there exists a unique µ(qw) ∈ M(L × E2L × B(E)) such that µ(qw) =Υ(qw)µ(qw).

2.4.2. Characterization of the default decision

Since the option to default is the novel feature of this paper, it is useful to es-tablish some results on the manner in which the decision to default varies witha household’s level of earnings and with its level of debt. We will characterizethe default decision in terms of the maximal default set D

∗hs(qw) This set is

defined as follows: for h = 0 and − ζ(s) < 0, it consists of the set of e’s forwhich either the budget set B0s0(e;qw) is empty or the value from not de-faulting does not exceed the value from defaulting; for h= 1 and − ζ(s) < 0,it consists of the entire set E The maximal default set will coincide with theset of e for which d∗

hs(e;qw)= 1 if households that are indifferent betweendefaulting and not defaulting choose always to default.

THEOREM 3—The Maximal Default Set Is a Closed Interval: If D∗hs(qw)

is nonempty, it is a closed interval.

The intuition for this result can be seen in the following way. Suppose thatthere are two efficiency levels, say e1 and e2 with e1 < e2 for which it is opti-mal for the household to default on its debt. Now consider an efficiency levele that is intermediate between e1 and e2 Suppose that the household prefersto maintain access to the credit market at e even though it defaults at a higherearnings level e2 It seems intuitive then that the reason for not defaulting atthe lower earnings level associated with e must be that the household finds itoptimal to consume more than its earnings and incur even more debt. On the


other hand, the fact that the household defaults at the efficiency level e1 butmaintains access to the credit market at the higher efficiency level e suggeststhat the reason for not defaulting at the earnings level associated e must bethat the household finds it optimal to consume less than its earnings and re-duce its level of indebtedness. Since the household cannot simultaneously beconsuming more and less than the earnings level associated with e, it followsthat the household must default at the efficiency level e as well.

THEOREM 4—Maximal Default Set Expands with Indebtedness: If 0 > 1

then D∗0hs(qw)⊆D∗

1hs(qw)

The result follows from the property that v∗0s(e;qw) is increasing in and

the utility from default is independent of the level of net liabilities. Figure 1helps to visualize this for h = 0 and where v∗′d

hs(e;qw) denotes the valueconditional on actions (l′ d).

FIGURE 1.—Typical default sets conditional on household type.


2.4.3. Firms

Firms producing the composite good face a static optimization problem ofchoosing nonnegative quantities of labor and capital to maximize F(KtNt)−w ·Nt − r · Kt The necessary conditions for profit maximization imply (withequality if the optimal Nt and Kt are strictly positive) that

w≥ ∂F(KtNt)

∂Nt

and r ≥ ∂F(KtNt)

∂Kt

(10)

2.4.4. Financial intermediaries

The intermediary chooses the number at+1st ≥ 0 of type (t+1 st) contractsto sell and the quantity Kt+1 ≥ 0 of capital to own for each t to maximize thepresent discounted value of current and future cash flows

∞∑t=0

(1 + i)−tπt(11)

given K0 and a0s−1 = 0. The period-t cash flow is given by

πt = (1 − δ+ r)Kt −Kt+1 +∑

(t st−1)∈L×Sρ(1 −ptst−1)at st−1(−t)(12)

−∑

(t+1st )∈L×Sqt+1st at+1st (−t+1)

where pt+1st is the probability that a contract of type (t+1 st) where t+1 < 0experiences default and it is understood that pt+1st = 0 for t+1 ≥ 018 Notethat the calculation of cash flow takes into account that some borrowers willnot survive to repay their loans and some depositors will not survive to collecton their deposits.19

18Note that households with t+1 ≥ 0 may still default on their medical liabilities if those liabil-ities are sufficiently high.

19Here, and in the household’s decision problem, we assumed that a household enters into asingle contract with some firm. This simplifies the situation in that a household’s end-of-periodasset holding is the same as ′ the size of the single contract entered into by the household. How-ever, this is without loss of generality in the following sense. Let households write any collectionof contracts ′k ∈ L as long as ′ = ∑

k ′k ∈ L Consistent with the procedures of a Chapter 7

bankruptcy filing, assume that a household has the option to either (i) default on all negative face-value subcontracts (i.e., loans) or (ii) not default on any of them. In the case of default, assumethat creditor firms can liquidate any positive face-value subcontracts held by the household anduse the proceeds to recover their loans in proportion to the size of each loan. With these bank-ruptcy rules in place, the price charged on any subcontract in the collection ′k ∈L must be theprice that applies to the single contract of size ′ Consequently, as long as credit suppliers cancondition their loan price on total end-of-period debt position of a household, there is a market


If a solution to the intermediary’s problem exists, then optimization implies

i≥ r − δ(13)

(14)qt+1st

≤ ρ

1 + i if t+1 ≥ 0,

≥ ρ

1 + i (1 −pt+1st ) if t+1 < 0.(15)

If the optimal Kt+1 is strictly positive, then (13) holds with equality. Similarly,if any optimal at+1st is strictly positive, the associated condition (14) or (15)holds with equality. Furthermore, any nonnegative sequence Kt+1 at+1st ∞

t=0implies a sequence of risk-free bond holdings Bt+1∞

t=0 by the intermediarygiven by the recursion

Bt+1 = (1 + i)Bt +πt(16)

where B0 = 0

2.4.5. Hospital sector

Hospital revenue received from a household in state h s and e is givenby

(1 − d∗hs(e;qw))ζ(s)+ d∗

hs(e;qw)max0Observe that if a household has positive assets but negative net (after medicalshock) liabilities and defaults, the hospital receives If the household’s assetsare negative and it defaults, the hospital receives nothing. As noted before, fora bill of ζ the hospital’s resource cost is given by ζ/m Thus, hospital profitsin period t are given by∫ [

(1 − d∗hs(e;qw))ζ(s)+ d∗

hs(e;qw)max0 − ζ(s)/m]dµt(17)

where µt is the distribution of households over L×E at time t In steady state,m must be consistent with zero profits for the hospital sector.

3. STEADY-STATE EQUILIBRIUM

In this section we define and establish the existence of a steady-state equilib-rium and characterize some properties of the equilibrium loan price schedule.The proof of existence uses Brouwer’s FPT for a continuous function on a

arrangement in which the household is indifferent between writing a single contract or a collec-tion of subcontracts with the same total value. Parlour and Rajan (2001) analyzed equilibrium ina two-period model of unsecured consumer debt when such conditioning is not possible.


compact domain. Nevertheless, the proof is not straightforward. The nature ofthe difficulty—which is related to the possibility of default—is discussed laterin this section.

DEFINITION 2: A steady-state competitive equilibrium is a set of strictly pos-itive pricesw∗ r∗ i∗ a nonnegative loan-price vector q∗ a nonnegative defaultfrequency vector p∗ = (p∗

′s)′∈Ls∈S a nonnegative hospital markupm∗ strictlypositive quantities of aggregate labor and capitalN∗K∗ a nonnegative vectorof quantities of contracts a∗ = (a∗

′s)′∈Ls∈S bond holdings by the intermediaryB∗ decision rules ′∗

hs(e;q∗w∗)d∗hs(e;q∗w∗), and c∗

hs(e;q∗w∗), and aprobability measure µ∗ such that:

(i) ′∗hs(e;q∗w∗) d∗

hs(e;q∗w∗), and c∗hs(e;q∗w∗) solve the house-

hold’s optimization problem.(ii) N∗K∗ solve the firm’s static optimization problem.

(iii) K∗ a∗ solve the intermediary’s optimization problem.(iv) p∗

′s = ∫d∗′0s′(e

′;q∗w∗)Φ(e′|s′)Γ (s;ds′)de′ for ′ < 0 and p∗′s = 0

for ′ ≥ 0 (intermediary consistency).(v)

∫ [(1 − d∗hs(e;q∗w∗))ζ(s) + d∗

hs(e;q∗w∗)max0 − ζ(s)/m∗]dµ∗ = 0 (zero profits for the hospital sector).

(vi)∫edµ∗ =N∗ (the labor market clears).

(vii)∫

1(′∗hs

(e;q∗w∗))=′µ∗(ddh sde) = a∗′s, ∀(′ s) ∈ L × S (each loan

market clears).(viii) B∗ = 0 (the bond market clears).

(ix)∫c∗hs(e;q∗w∗)dµ∗ + ∫

ζ(s)

m∗ dµ∗ + δK∗ = F(K∗N∗)− γw∗ ∫

eµ∗(d1 dsde) (the goods market clears).

(x) µ∗ = µ(q∗w∗), where µ(q∗w∗) = Υ(q∗w∗)µ(q∗w∗) (µ∗ is an invariant proba-bility measure).

We will use the above definition to derive a set of price equations whosesolution implies the existence of a steady state. Conditions (ii) and (iii) in thedefinition imply the following equations. Since N∗ and K∗ are strictly positive,the first order conditions for the firm (10) and the intermediary (13) imply

w∗ = ∂F(K∗N∗)∂N∗ r∗ = ∂F(K∗N∗)

∂K∗ i∗ = r∗ − δ(18)

For a∗′s > 0 the intermediary first order conditions (14) or (15) imply

q∗′s =

ρ(1 −p∗′s)

1 + i∗ (19)

For a∗′s = 0, we will look for an equilibrium where the intermediary is indif-

ferent between selling and not selling the associated (′ s) contract. Then (19)holds for these contracts as well.


Condition (viii) implies the following equation. From the recursion (16),bond market clearing (viii) implies cash flow (12) can be written[

(1 − δ+ r∗)K∗ −∑

(s−1)∈L×Sρ(1 −p∗

s−1)a∗

s−1

]

−[K∗ −

∑(′s)∈L×S

q∗′sa

∗′s

′]

= 0

or, using (18) and (19),

(1 − δ+ r∗)[K∗ −

∑(s−1)∈L×S

q∗s−1

a∗s−1

]−

[K∗ −

∑(′s)∈L×S

q∗′sa

∗′s

′]

= 0

where ( s−1) denotes a loan size and characteristics pair from the previousperiod. Therefore, bond market clearing in steady state implies

K∗ =∑

(′s)∈L×Sq∗′sa

∗′s

′(20)

It can be shown that the goods market-clearing condition (ix) is implied bythe other conditions for an equilibrium.20 Thus, we can summarize an equilib-rium by the following set of four equations. The first two are price equationsthat incorporate household optimization (i), intermediary consistency (iv), la-bor market clearing (vi), loan market clearing (vii), and (20) into (18) and (19)to yield

w∗ = FN( ∑(′s)∈L×S

′q∗′s

∫1(′∗

hs(e;q∗w∗)=′µ∗(ddh sde)(21)

∫edµ∗

)

q∗′s =

ρ

1+FK(∑

(′s)∈L×S ′q∗′s

∫1′∗hs

(e;q∗w∗)=′µ∗(ddhsde)∫edµ∗)−δ

for ′ ≥ 0,ρ(1−∫

d∗′0s′ (e

′;q∗w∗)Φ(e′|s′)Γ (s;ds′)de′)1+FK

(∑(′s)∈L×S ′q∗

′s∫

1′∗hs

(e;q∗w∗)=′µ∗(ddhsde)∫edµ∗)−δ

for ′ < 0

(22)

20This is a nontrivial accounting exercise given that our environment admits default on loansand medical bills. For reasons of space, we omit a proof here. The proof is available in the Supple-ments to this paper on the Econometrica website (Chatterjee, Corbae, Nakajima, and Ríos-Rull(2007)).


The other two equations are given by (v) and (x).Proving the existence of a steady-state equilibrium reduces to proving that

there is a fixed point to equations (21) and (22), where the invariant distrib-ution µ∗ is itself a fixed point of a Markov process whose transition probabil-ities depend on the price vector. Provided the aggregate production functionhas continuous first derivatives (and these derivatives satisfy certain bound-ary conditions), a solution to this nested fixed point problem will exist (as asimple consequence of Brouwer’s FPT) if µ∗ is continuous with respect to theprice vector. Given a continuum of households, a sufficient condition for thecontinuity of µ∗ is that the set of households that are indifferent between anytwo courses of action be of (probability) measure zero. The assumption thatthe efficiency shock e is drawn from a distribution with a continuous cumu-lative distribution function (c.d.f.) goes a long way toward ensuring this, but,surprisingly, not all the way. Even with this assumption we cannot rule out thata continuum of households may be indifferent between defaulting and payingback.21 To work around this problem, in the Appendix we first establish the ex-istence of a steady-state equilibrium for an environment in which there is anadditional bankruptcy cost that is paid in the filing period. The form of thiscost ensures that the set of households that are indifferent between default-ing and paying back is finite and thereby restores the continuity of the invari-ant distribution with respect to the price vector.22 We then take a sequence ofsteady-state equilibria in which the filing-period bankruptcy cost converges tozero and establish that the limit of this sequence is a steady-state equilibriumfor the environment of this paper.

The equilibrium loan price vector has the property that all positive face-value loans (household deposits) bear the risk-free rate and negative face-value loans (household borrowings) bear a rate that reflects the risk-free rateand a premium for the objective default probability on the loan. Given therisk-free rate, which in equilibrium will depend on µ∗ default probabilities(and hence loan prices) do not depend on µ∗ This is because free entry into fi-nancial intermediation implies that subsidization across loans of different sizesis not possible; that is, it is not possible for intermediaries to charge more thanthe cost of funds on small low-risk loans to offset losses on large higher-riskloans. If there were positive profits in some contracts that were offsetting thelosses in others, intermediaries could enter the market for those profitableloans.

THEOREM 5—Existence: A steady-state competitive equilibrium exists.

21This case will occur if a household that is indifferent between defaulting and paying backfinds it optimal to consume its endowment when paying back. Then, ceteris paribus, householdswith slightly higher or slightly lower e’s will also be indifferent between defaulting and payingback.

22This form of additional bankruptcy cost is α · (e− emin) ·w, where α< 1.


For a finite r∗ it is possible that there are contracts (′ < 0 s) whose equi-librium price q∗

′s = 0 Even in this case, intermediaries are indifferent as tohow many loans of type (′ s) they “sell”; “selling” these loans does not costthe intermediary anything (since the price is zero) and it (rationally) expectsthe loans to generate no payoff in the following period. From the perspectiveof a household, taking out one of these free loans buys nothing in the currentperiod, but does saddle the household with a liability. Since the household cando better by choosing ′ = 0 in the current period, there is no demand for suchloans either.

We now deal with the limits of the set L for a given s. Models of precau-tionary savings have the property that when βρ(1 + r∗ − δ) < 1, there is anupper bound on the amount of assets a household will accumulate. This up-per bound arises because as wealth gets larger, the coefficient of variation ofincome goes to zero, and hence the role of consumption smoothing vanishes.23

Since ours is also a model of precautionary saving, the same argument appliesand max exists. With respect to the debt limit, min, it can be set to any valueless than or equal to [emax · wmax][(1 + r∗ − δ)/(1 − ρ+ r∗ − δ)]. This expres-sion is the largest debt level that could be paid back by the luckiest householdfacing the lowest possible interest rate and is the polar opposite of the onein Huggett (1993), Aiyagari (1994), and Athreya (2002). As we show in thenext theorem, for any s, a loan of this size or larger would have a price ofzero in any equilibrium. Hence, as long as the lower limit is at least as lowas −[emax · wmax][(1 + r∗ − δ)/(1 − ρ+ r∗ − δ)], it will not have any effect onthe equilibrium price schedule. We now turn to characterizing the equilibriumprice schedule.

THEOREM 6—Characterization of Equilibrium Prices: In any steady-statecompetitive equilibrium, (i) q∗

′s = ρ(1 + r∗ − δ)−1 for ′ ≥ 0, (ii) if the grid forL is sufficiently fine, there exists 0 < 0 such that q∗

0s= ρ(1+ r∗ −δ)−1, (iii) if the

set of efficiency levels for which a household is indifferent between defaulting andnot defaulting is of measure zero, 0> 1 > 2 implies q∗

1s≥ q∗

2s, and (iv) when

min ≤ −[emax ·wmax][(1 + r∗ − δ)/(1 − ρ+ r∗ − δ)] q∗mins

= 0

The first property simply says that firms charge the risk-free rate on deposits.The second property says that if the grid is taken to be fine enough, thereis always a level of debt for which it is never optimal for any household todefault. As a result, competition leads firms to charge the risk-free rate onthese loans as well. The third property says that the price on loans falls withthe size of loans, that is, the implied interest rate on loans rises with the sizeof the loan. The final property says that the price on loans eventually becomezero; that is, for any household, the price on a loan of size [emax · wmax][(1 +

23See Huggett (1993) and Aiyagari (1994) for a detailed argument.


r∗ − δ)/(1 − ρ+ r∗ − δ)] or larger is always zero in every equilibrium. In otherwords, the equilibrium delivers an endogenous credit limit for each householdwith characteristics s.

4. MAPPING THE MODEL TO U.S. DATA

We now establish that our framework gives a plausible account of the overallfacts on bankruptcy and credit. The challenging part is to account simultane-ously for the high frequency of default and for significant levels of unsecureddebt: high frequency of default makes unsecured debt very expensive, whichdeters consumer borrowing. We found that two realistic features account foraggregate default and credit statistics. First, not all unsecured consumer debt isa result of borrowing from financial intermediaries; some of it is in the natureof an “involuntary” loan resulting from reasons such as large medical bills. Sec-ond, marital disruption is often cited as a reason for filing: this is not necessarilyrelated to earnings shocks. In our model we take into account the possibility ofinvoluntary loans through our modeling of nondiscretionary medical expensesand we take into account private life events (such as divorce) as a possible trig-ger for default through the preference shock. There is a third feature of thereal world that we believe to be important as well, but we have not modeledit in the interest of keeping the dimensionality of the state space lower: thatmany U.S. households hold both unsecured debt and (nonexempt) assets—afact that no doubt lowers the default premium on unsecured loans and makesthem less expensive. We skirt this issue by focusing on the net asset positions ofhouseholds, but (as explained below) this impairs our ability to explain someaspects of the data.

We map two versions of our model economy to the data: they differ by whichidiosyncratic shocks are included. In both versions, the household characteris-tic s is simply the triplet (ξηζ), where ξ denotes a shock that controls theprobability distribution of labor efficiency e, η is a multiplicative preferenceshock, and ζ is the medical expense shock. We think of ξ as socioeconomic sta-tus (or occupation) upon which the distribution of household labor efficienciesdepend. In the first version, which we label baseline model economy, we restricts = (ξ10), so that the only idiosyncratic shocks are to socioeconomic status,efficiency, and death. We use the baseline model for illustration purposes be-cause it is simpler and in the vein of other incomplete market macro modelslike that of Aiyagari (1994). In the second version, labeled extended modeleconomy, we include the idiosyncratic shocks to preferences and medical lia-bilities. We use the extended model economy for the quantitative analysis. Weuse the reasons for bankruptcy cited by the Panel Study of Income Dynamics(PSID) survey participants to determine targets for the fraction of consumerdebt, the fraction of indebted households, and the fraction of people filing forbankruptcy that should plausibly be accounted for by the two versions. Plausi-bility in this context means that the model should explain the debt and default


statistics without generating counterfactual predictions for macroeconomic ag-gregates and for earnings and wealth distributions.

4.1. Model Specification

4.1.1. The baseline model economy (earnings shocks only)

The baseline model economy has 17 parameters. These parameters are listedbelow in separate categories; the number of parameters in each category ap-pears in parentheses.

Demographics (1): The probability of survival is ρ (which implies that themass of new entrants is 1 − ρ).

Preferences (2): We assume standard time-separable constant relative riskaversion preferences that are characterized by two parameters: the discountrate, β, and the risk aversion coefficient, σ .

Technology (3): There are two parameters that determine the properties ofthe production function: the exponent on labor in the Cobb–Douglas produc-tion function, θ, and the depreciation rate, δ. We also place in this categorythe fraction of lost earnings while a household has a bankruptcy on its creditrecord, γ.

Legal system (1): The legal system is characterized by the average durationof exclusion from access to credit, 1/(1 − λ).

Earnings process (10): The process for earnings requires the specification ofa Markov chain for ξ and of the distribution of e conditional on ξ. We use athree-state Markov chain Γ that we loosely identify with super rich (ξ1), whitecollar (ξ2), and blue collar (ξ3). The persistence of the latter two states ensuresthat earnings display a sizable positive autocorrelation. The first state providesthe opportunity and incentive for a high concentration of earnings and wealth(see Castañeda, Díaz-Giménez, and Ríos-Rull (2003)). This specification re-quires 6 parameters in the Markov transition matrix, but we reduce it to 4 bysetting the probability of moving from blue collar to super rich and vice versato zero. For the distribution of labor efficiency shocks, we need 6 more para-meters, 5 of which pertain to the upper and lower limits of the range of e foreach type (units do not matter and that frees up one parameter) and one addi-tional parameter to specify the shape of the c.d.f. of e. We assume the followingone-parameter functional form for the distribution function:∫ y

eξmin

Φ(e|ξ)= P[e≤ y|ξ] =[y − eξmin

eξmax − eξmin

]ϕ(23)


4.1.2. The extended model economy (earnings, preference, and liability shocks)

In this economy, we keep the baseline model shocks while adding a multi-plicative shock to the utility function and a medical liability shock. The prefer-ence shock takes two values η ∈ 1η > 1 and we assume that η = η cannothappen twice in a row. This implies the need to specify two parameters. Theliability shock ζ ∈ 0 ζ > 0 can take on only two values independent of allother shocks and is independent and identically distributed (i.i.d.) over time.Therefore, the Markov process Γ for s = (ξηζ) has 3 × 2 × 2 = 12 states.

4.2. Data Targets

We select model parameters to match three sets of statistics: aggregate sta-tistics, earnings and wealth distribution-related statistics, and statistics on debtand bankruptcy. The targets (which appear in Tables III and IV) contain rela-tively standard targets for macroeconomic variables and statistics of the earn-ings and wealth distribution obtained from the 2001 SCF (selected points ofLorenz curves, the Gini indices, and the mean to median ratios). We target theautocorrelation of earnings of all agents except those in the highest earningsgroup to 0.75.24

We now turn to the debt and bankruptcy targets, and discuss them in moredetail since they are novel. First, according to the Fair Credit Reporting Act, abankruptcy filing stays on a household’s credit record for 10 years. This fact isused in our model to calibrate the length of exclusion from the credit market.Second, according to the Administrative Office of the U.S. Courts, the totalnumber of filers for personal bankruptcy under Chapter 7 was 1.087 millionin 2002. According to the Census Bureau, the total population above age 20in 2002 was 201 million. Therefore, the ratio of people who file to total pop-ulation over age 20 is 0.54%. Third, since in our model households can onlysave or borrow, we use the 2001 Survey of Consumer Finances to obtain theconsolidated asset position of households. Only people with negative net worthare considered to be debtors. We exclude households with negative net worthlarger than 120% of average income since the debts are likely to be due to en-trepreneurial activity from which our model abstracts. These households are avery small fraction of the SCF (comprising only 0.13% of the original sampleof SCF 2001).25 The average net negative wealth of the remaining householdsis $63146, which divided by per household GDP of $94,077 is 0.0067. Thus,we take the target debt-to-income ratio to be 0.67%. Fourth, after excludingthe few households with debt more than 120% of average income, 6.7% of thehouseholds in the 2001 SCF had negative net worth.26

24The PSID data set that can be used to compute autocorrelation of earnings does not includethe highest earners.

25The average amount of debt for this group is $100,817, or 145% of the average income, andtheir income is relatively high.

26We also note that 2.6% of the households had zero wealth in the 2001 SCF.


TABLE I

REASONS (%) FOR FILING FOR BANKRUPTCY (PSID, 1984–1995)

1 Job loss 12.22 Credit misuse 41.33 Marital disruption 14.34 Health-care bills 16.45 Lawsuit/harassment 15.9

Source: Chakravarty and Rhee (1999)

The last three statistics are the relevant bankruptcy and debt targets for theextended model since it includes all important motives for bankruptcy. Thebaseline model does not include all motives; consequently its appropriate tar-get values are a fraction of their data values. According to Chakravarty andRhee (1999), the PSID classified the reasons for bankruptcy filings into five cat-egories and we report their findings in Table I. Among the five reasons listed,we associate the first two (job loss and credit misuse) with earnings shocks;marital disruption with preference shocks; and the final two (health-care billsand lawsuits/harassment) with liability shocks.27 Given these associations, weallocate the total volume of debt, the fraction of households in debt, and thefraction of filings according to the fraction of people citing the above reasons.For instance, given that 53.5% of households cited reasons we associate withearnings shocks, we assume that the fraction of filings corresponding to thisreason is 029% (i.e., 0535 × 00054 = 00029). We report these targets spe-cific to each of the model economies in Table II.

TABLE II

DEBT AND BANKRUPTCY TARGETS FOR EACH MODEL ECONOMY

Economy U.S. Baseline Extended

Reasons covereda 1–5 1, 2 1–5Percent of all bankruptcies 100 535 100Percent of households filing 054 029 054Debt-to-income ratio (%) 067 036 067Percent of households in debt 67 36 67

aThe numbers in this row are associated to the numbers in the previous table.

27We think of marital disruption as leading to higher nondiscretionary spending on the part ofeach partner, which in turn increases the marginal utility of discretionary spending. A high valuefor the multiplicative shock to preferences is meant to capture this effect.


4.3. Moments Matching Procedure

The baseline model economy has 17 parameters, which we classify into twogroups. The first group consists of 5 parameters, each of which can be pinneddown independently of all other parameters by one target. The survival rate ρis determined to match the average length of adult life, which we take to be40 years, a compromise for an economy without population growth or changesin marital status. The labor share of income is 064, which determines θ Thedepreciation rate δ is 010, which is consistent with a wealth-to-output ratioof 308 (its value according to the 2001 Survey of Consumer Finances) andthe consumption-to-output ratio of 070 The transition probability λ, whichgoverns the average length of time that a bankruptcy remains on a person’scredit record, is set to 09 consistent with the Fair Credit Reporting Act. Thecoefficient of relative risk aversion σ is fixed at 2.

The 12 parameters in the second group—including the discount rate β, thecost of having a bad credit record γ, 4 parameters governing the transition oftype characteristics Γ , 5 parameters defining the bounds of efficiency levelsfor the type characteristics, and 1 parameter characterizing the shape of thedistribution function of the efficiency shocks in (23) ϕ—are determined jointlyby minimizing the weighted sum of squared errors between the targets and thecorresponding statistics generated by the model. The targets that we chooseare listed in Table III and include the main macroeconomic aggregates, theproperties of earnings and wealth inequality, and the main statistics of debtand default (the percentage of defaulters, the percentage in debt, and the debt-to-output ratio). Our weighting matrix puts more emphasis on matching thedebt and bankruptcy filing targets than on earnings and wealth distributiontargets. The extended model economy has 16 parameters: the same 12 as thebaseline economy plus 2 preference shock parameters (size and probability)and 2 liability shock parameters (again size and probability).

4.4. Computation of the Steady State

Computation of the equilibrium requires three steps: an inner loop, wherethe decision problem of households given parameter values and prices (in-cluding loan prices) is solved; a middle loop, where market-clearing prices areobtained; and an outside loop (or estimation loop), where parameter valuesthat yield equilibrium allocations with the desired (target) properties are de-termined. We use a variety of (almost) off-the-shelf techniques, powerful soft-ware (Fortran 90) and hardware (26 processor Beowulf cluster) to accomplishour task. We confirmed our findings by recomputing equilibria with standardmethods that do not speed the process. The computational task of simulat-ing equilibrium model moments is extremely burdensome: each equilibriumrequires computing thousands of equilibrium loan prices—recall that we haveto solve for equilibrium loan price schedules—and it has taken thousands ofcomputed equilibria to find satisfactory configurations of parameter values.


TABLE III

BASELINE MODEL STATISTICS AND PARAMETER VALUES

Statistic Target Model Parameter Value

Targets determined independentlyAverage years of life 40 40 ρ 0975Coefficient of risk aversion 20 20 σ 2000Labor share of income 064 064 θ 0640Depreciation rate of capital 010 010 δ 0100Average years of punishment 10 10 λ 0900

Targets determined jointly: InequalityEarnings Gini index 061 061 Γ23 0219Earnings mean/median 157 195 Γ33 0964Percentage of earnings of 2nd quintile 40 43 ϕ 0470Percentage of earnings of 3rd quintile 130 105 e1

max/e3min 21,263.9

Percentage of earnings of 4th quintile 229 203 e1min/e

3min 14,335.7

Percentage of earnings of 5th quintile 602 635 e2max/e

3min 1163

Percentage of earnings of top 2–5% 158 173 e2min/e

3min 392

Percentage of earnings of top 1% 153 153 e3max/e

3min 305

Autocorrelation of earnings 075 074Wealth Gini index 080 073Wealth mean/median 403 330Percentage of wealth of 2nd quintile 13 30Percentage of wealth of 3rd quintile 50 63Percentage of wealth of 4th quintile 122 152Percentage of wealth of 5th quintile 817 751Percentage of wealth of top 2–5% 231 136Percentage of wealth of top 1% 347 342

Targets determined jointly: Savings, debt, and defaultProrated percentage of defaulters 029 029 β 0917Prorated percentage in debt 36 46 γ 0019Capital–output ratio 308 308 Γ11 0020Prorated debt–output ratio 00036 00036 Γ21 0001

4.5. Results

Table III reports the target statistics and their counterparts in the baselinemodel economy as well as the implied parameter values, while Table IV re-ports the same information for the extended model economy. Focusing on theextended economy, note that it successfully replicates the macro and distribu-tional targets. The capital–output ratio is exactly as targeted and so is the earn-ings Gini. The wealth Gini is somewhat lower than in the data but as Figures 2and 3 show, the overall fit of the model along these dimensions is quite good.Turning to the debt and bankruptcy targets, the extended model economy repli-cates successfully the percentage of defaulters and the debt-to-income ratio. Italso successfully replicates the relative importance of the reasons cited for de-fault. However, the percentage of households in debt is lower than the target.


TABLE IV

EXTENDED MODEL ECONOMY: STATISTICS AND PARAMETER VALUES

Statistic Target Model Parameter Value

Targets determined independentlyAverage years of life 40 40 ρ 0975Coefficient of risk aversion 20 20 σ 2000Labor share of income 064 064 θ 0640Depreciation rate of capital 010 010 δ 0100Average years of punishment 10 10 λ 0900

Targets determined jointly: InequalityEarnings Gini index 061 061 Γ11 0019Earnings mean/median 157 212 Γ21 0001Percentage of earnings in 2nd quintile 40 41 Γ23 0222Percentage of earnings in 3rd quintile 130 97 Γ33 0969Percentage of earnings in 4th quintile 229 202 ϕ 0387Percentage of earnings in 5th quintile 602 640 e1

max/e3min 14,599.2

Percentage of earnings in top 2–5% 158 180 e1min/e

3min 7,661.5

Percentage of earnings in top 1% 153 153 e2max/e

3min 709

Autocorrelation of earnings 075 074 e2min/e

3min 238

Wealth Gini index 080 073 e3max/e

3min 180

Wealth mean/median 403 322Percentage of wealth in 2nd quintile 13 30Percentage of wealth in 3rd quintile 50 63Percentage of wealth in 4th quintile 122 150Percentage of wealth in 5th quintile 817 754Percentage of wealth in top 2–5% 231 146Percentage of wealth in top 1% 347 323

Targets determined jointly: Savings, debt, and defaultPercentage of defaulters 054 054 β 0913Percentage in debt 67 50 γ 0035Percentage of defaults due to divorces 0077 0074 Prob of η 0012Percentage of defaults due to hospital bills 017 017 η 1680Capital–output ratio 308 308 Prob of ζ 0003Debt–output ratio 00067 00068 ζ 1150

This discrepancy is hard to eliminate because in the model, households thatborrow are very prone to default, implying a high default premium on loans.This increases interest rates on loans and reduces the participation of house-holds in the credit market.28 One important difference between the model andthe U.S. economy is that a typical indebted household has both liabilities and

28There are also some other factors that may account for this discrepancy. There are 3.2% ofhouseholds with exactly zero assets due in part to the discrete nature of periods (all newbornshave zero wealth); this makes the number of indebted people in the model lower than it ought tobe. Also, upon being hit by either a liability or a preference shock, households default immedi-ately, while in the data it takes longer.


FIGURE 2.—Earnings distributions for the U.S. and extended model.

assets. The presence of nonexempt assets reduces the incentives to default andlowers interest rates, thereby increasing borrowing.29 The baseline model econ-omy also closely replicates the relevant targets with the exception of the per-centage of households in debt.30

5. PROPERTIES OF THE MODEL ECONOMIES

5.1. Distributional Properties

Figure 4 shows a histogram of the wealth distribution in the extended model,excluding the long right tail, which comprises about 18% of the population.For households with a good credit record, the model generates a pattern ofwealth distribution that is typical of overlapping generations models. There is

29This difference is also the reason we do not target interest rate statistics. In the model, allborrowers have negative net worth, paying very high interest rates. In the real world, many bor-rowers also own nonexempt assets and the interest rate they pay presumably reflects this fact. Totarget interest rates in a meaningful way would require a model in which households hold bothassets and liabilities.

30In this case, the percentage is somewhat higher in the model relative to the target, but thisdiscrepancy could reflect an inaccurate target.


FIGURE 3.—Wealth distributions for the U.S. and extended model.

a significant fraction of households with zero wealth, many of whom are new-borns. After that, the measure of households with each level of assets grows fora while (most households accumulate some savings) before starting to slowlycome down for much larger levels of wealth. There is also a relatively largefraction of households with small amounts of debt relative to average incomeand there are some households with debt in the neighborhood of one-half ofaverage income. There are no households borrowing more than 60%, becausethe amount of current consumption derived from borrowing declines beyondthis level of debt due to steeply rising default premia.31 Households with a badcredit record consist mostly of households with very few assets. No one in thisgroup has debt, because these households are precluded from borrowing. Theright tail of this distribution is relatively long, indicating that some householdshave a bad credit record for many periods and have relatively high earningsrealizations.

5.2. Bankruptcy Filing Properties

Figure 5 shows default probabilities in the extended model on loans takenout in the current period, conditional on whether households are blue collar

31This point is discussed in more detail in Section 5.5.


FIGURE 4.—Wealth histogram in the extended model.

or white collar and on whether they are hit by the preference or the liabilityshock in the next period.32 We wish to make four points. First, the probabilityof filing for bankruptcy is higher for blue collar than white collar householdsfor every level of debt. This is a natural consequence of white collar house-holds receiving higher earnings on average than blue collar households. Forinstance, at a debt level of average income, no white collar worker is expectedto default, while more than 90% of blue collar workers are expected to default.Second, the default probabilities for both types of households are rising in thelevel of debt, which is consistent with Theorem 4. Third, no one is expectedto file for bankruptcy with a level of debt near zero, which is consistent withTheorem 6(ii). In particular, even the blue collar households are not expectedto default if their debt is less than about one-tenth of average income unlessthey are hit by the liability shock. Fourth, the threshold debt level below whichthere is no default for white collar households that are not hit by the liability orpreference shocks is 135% of average annual income, and a fraction of whitecollar households hit by the liability shock do not default.

32For instance, in Figure 5 the line associated with blue collar agents plots the value of∫d∗′0s′(e

′;q∗w∗)Φ(e′|s′)de′, where s′ = ξ′310; the line associated with white collar agents:

liability shock is the same expression with s′ = ξ′21 ζ.


FIGURE 5.—Default probabilities by household types in the extended economy.

Table V shows the number of people filing for bankruptcy by earning quin-tiles as a fraction of the entire population and as a fraction of those in debt.Across the two economies, the conditional probability of bankruptcy for house-holds in the lowest three earnings quintiles is very similar, but declines sharplyin the fourth quintile, and there are few defaulters in the top quintile (no-body defaults in the top quintile of the baseline economy, while some do inthe extended economy; recall that the liability shock is large and can hit allagents). The last two columns of Table V show the difference made by the lia-bility shocks by comparing the fraction of those in debt due to past debts alonethat default (fourth column) or the fraction of those in debt due to all rea-sons including this period’s liability shocks. In the extended model economy,0.27% of households get hit by the liability shock, of which 0.17% default. Theaggregate size of the liability shock (or aggregate medical services) is 0.58%of output, while actual medical expenditure is 0.31% of output (implying viaequation (17) that the markup m is 87%). An additional aspect of default be-havior that is not evident in these tables is that in every case, households be-low some earnings threshold default. Although the theory allows for a second(lower) threshold below which people pay back, that does not happen in theequilibrium of these calibrated economies.


TABLE V

EARNINGS AND BANKRUPTCIES: FRACTIONS (%) OF AGENTS THAT DEFAULT

Over Total Population Over Population in Debt

Extended ExtendedEconomy Baseline Extended Baseline w/o Hosp. with Hosp.

1st quintile 048 075 92 107 1372nd quintile 048 075 92 107 1373rd quintile 048 075 92 107 1364th quintile 003 036 42 70 1005th quintile 000 009 00 04 32

Total 029 054 64 79 108

5.3. Properties of Loan Prices

Since household type is quite persistent, the lower probabilities of bank-ruptcy of white collar households translate into their having a lower defaultpremium (higher q) than blue collar households. Figure 6 shows the price ofloans conditional on the loan size for white and blue collar households whohave not been hit with either the preference or liability shock.33 For a debt levelof less than one-tenth of average income, the price schedule appears flat. Asshown in Figure 5, for these levels of debt, blue collar households default withprobability 1 only if hit by a liability shock. Similarly, white collar householdsdefault only if hit by a liability shock, but they do not default with probabil-ity 1. Instead, the probability of default rises as the loan size increases fromzero. Because the probability of liability shocks is very small, the slight declinein loan prices implied by these default patterns is not evident in Figure 6. Forhigher levels of debt the loan price schedule for a white collar household isabove that of blue collar households. This is because type shocks are persistentand, as evident in Figure 5, white collar workers are less likely to default thanblue collar workers. For white collar households, the kink at a level of debtof 135% of average income results from the fact that for this and lower debtlevels, white collar households default only if hit by the preference or liabilityshock. For higher levels of debt, white collar households default also if earningsrealizations are low. The average interest rate on loans (weighted by the num-ber of households in debt) is 30.96%, implying an average default premium of29.27%.34

33For instance, the line labeled “White Collar Agents” is qs , where s = ξ210.34If we weight by the amount of debt for each debtor, the average loan interest rate is 55.97%,

substantially higher than the average rate paid per household because there are a small numberof households who borrow a large amount at very high interest rates. This is consistent with thehistogram of household wealth shown in Figure 4.


FIGURE 6.—Loan prices for blue and white collar households in the extended economy.

5.4. Accounting for Debt and Default

These properties of default and loan price schedules indicate different rolesof blue collar and white collar households in accounting for aggregate filing fre-quency and consumer debt. Blue collar households receive (on average) lowerearnings every period and frequently borrow to smooth consumption. On theother hand, if they receive a sequence of bad earnings shocks, they find it ben-eficial to file for bankruptcy and erase their debt. Since they are more likely todefault, blue collar households have to pay a relatively high default premiumand the premium soars as the size of the loan increases. As a result, blue collarhouseholds borrow relatively frequently in small amounts and constitute themajority of those who go bankrupt, but because they borrow small amounts,they account for only a small portion of aggregate consumer debt. In contrast,white collar households face a lower default premium on their loans becausethey earn more on average. Therefore, they borrow a lot more than blue collarhouseholds when they suffer a series of bad earnings shocks. The householdswith large amounts of debt in our extended model consist of these white collarhouseholds. As long as these households remain white collar, they maintainaccess to credit markets, but they file for bankruptcy if their socioeconomicstatus changes to blue collar because they then face an extremely high defaultpremium on their debt. This story resembles the plight of some members of the


American middle class, who borrowed a lot because they were considered to beearning a sufficient amount, but filed for bankruptcy following a big persistentadverse shock to their earning stream. To summarize, in our model blue collarhouseholds account for a large fraction of bankruptcies and a large fraction ofhouseholds in debt, while white collar households account for the large levelof aggregate consumer debt.

5.5. A Comparison with Standard Exogenous Borrowing Limits

We conclude this section by comparing our results with the two extremes typ-ically assumed in general equilibrium economies with heterogeneous agents:either agents are completely prevented from borrowing (the Bewley (1983)economy) or there is full commitment and hence agents can borrow up tothe amount they can repay with probability 1 (the Aiyagari (1994) economy).Table VI compares the steady states of the Bewley and Aiyagari economieswith our extended model economy. A critical difference between these threemodels is the form of the borrowing limit. The Bewley borrowing limit is ex-ogenously set at zero. For the Aiyagari economy, we assume that only thosehouseholds who are hit with a liability shock and cannot consume a positiveamount without default are permitted to default.35

As is apparent from the table, aggregate asset holdings in our economy arecloser to the Bewley model than the Aiyagari model. In terms of aggregatewealth-to-output ratio, the extended model accounts for (or economizes on)18% (=(3.12−3.08)/(3.12−2.90)) of the difference in the wealth-to-income ra-tios between the Bewley and Aiyagari economies. In terms of debt, there is

TABLE VI

THE EXTENDED MODEL VERSUS THE BEWLEY AND AIYAGARI ECONOMIES

Economy Extended Bewley Economy Aiyagari EconomyAvailability of Loans Yes No YesDefault Premium Yes — No

Output (normalization) 100 100 100Total asset 308 312 290Total debt 00068 — 013Percentage that file 054% — —Percentage with bad credit record 423% — —Percentage in debt 499% — 2684%Rate of return of capital 169% 155% 241%Avg loans rate (persons-weighted) 3096% — 241%

35The results are essentially the same under this assumption as in the case of the baselinemodel, where no one is permitted to default. In this alternative model, the wealth-to-output ratiois 2.88.


TABLE VII

COMPARISON OF BORROWING LIMITS

Earnings Type 1 (Super Rich) 2 (White Collar) 3 (Blue Collar)

Bewley economy 000 000 000Aiyagari economy 356 170 170Extended model B1(s) 1,280 1,280 535Extended model B2(s) 1208 100 014

1Unit is proportion to the average income of the respective economy.1For all economies, borrowing limits for agents that are not hit by either the preference shock or the hospital bills

shock are shown.

only about 5% of the debt of the Aiyagari economy; in terms of the percentageof households in debt, the extended model economy has 19% of the number ofborrowing households in the Aiyagari economy.

Table VII shows the endogenous borrowing limits for each of the three earn-ings types (super rich, white collar, and blue collar) for the model economies.In the extended model there are two kinds of borrowing limits. One is thesmallest loan size for which the corresponding price q is zero, conditionalon the type of household. We denote this borrowing limit B1(s). Formally,B1(s)= −max ∈ L− :qs = 0. The other borrowing limit, B2(s), is the levelof debt for which · qs is maximum. Formally, B2(s)= −arg max∈L− · qs.No one would want to agree to pay back more than this amount, because theywould receive less for such a commitment today: this happens because the de-fault premium on the loan rises rapidly enough to actually lower the product · qs. This borrowing limit for blue collar households (0.14 of average in-come) is less than one-tenth of Aiyagari’s limit (1.70 of average income). FromFigure 4 of the wealth distribution presented earlier, we know there is a massof borrowers at this debt level. This mass of households is constrained by theB2(s) limit.36 Since blue collar households are those most likely to be in needof loans, the extended economy imposes a stricter borrowing constraint for asubset of the population than the Aiyagari economy.

5.6. Borrowing Constraints and Consumption Inequality

Borrowing constraints have important implications for consumption in-equality. Table VIII shows the earnings and consumption inequality of theextended economy compared to the Bewley and Aiyagari economies. In alleconomies, the degree of consumption inequality is substantially lower thanthe degree of earnings inequality since the households use savings to smooth

36The B2(s) borrowing limit for the blue collar, which is 0.14 of average income, correspondsto the debt level of around 0.6 of average income in Figure 4. The former number (0.14) is thediscounted value of debt while the latter (0.16) is the face value of debt.


TABLE VIII

CONSUMPTION AND EARNINGS INEQUALITY (%)

Std. Dev.of log

Quintiles

1st 2nd 3rd 4th 5th

Extended modelEarnings 1.159 2.1 41 97 20.2 64.0Consumption 0.677 6.5 104 135 20.1 49.5

Bewley economyEarnings 1.159 2.1 41 97 20.2 64.0Consumption 0.691 6.4 105 136 20.1 49.4

Aiyagari economyEarnings 1.159 2.1 41 97 20.2 64.0Consumption 0.658 7.2 98 128 19.9 50.3

consumption fluctuations. However, there are some differences in consump-tion inequality across the three economies. The standard deviation of log con-sumption of the extended economy is about 2 percentage points higher thanfor the Aiyagari economy. On the other hand, the standard deviation of logconsumption in the extended economy is 1.4 percentage points lower than inthe Bewley economy. This shows that while the extended economy may lookmore like the Bewley economy in terms of its borrowing characteristics, it stillmanages to reduce close to half of the difference between the Bewley and theAiyagari economy in terms of consumption inequality.

6. POLICY EXPERIMENT

Given that our model matches the relevant U.S. statistics on consumer debtand bankruptcy, it is possible to examine the consequences of a change inregulation that affects unsecured consumer credit. Here we evaluate a recentchange to the bankruptcy law that limits “above-median-income” householdsfrom filing under Chapter 7. We assume that agents cannot file for bankruptcyif their income in the model economy is above the median (around 34% ofaverage output) and repaying the loan would not force consumption to be neg-ative.37 In our extended economy, this means that white collar agents cannot

37The law is more complicated than our experiment. A person cannot file under Chapter 7(and effectively would have to pursue Chapter 13) if all of the following three conditions aremet: (1) The filer’s income is at least 100 percent of the national median income for families ofthe same size up to four members; larger families use median income for a family of four plusan extra $583 for each additional member over four. (2) The minimum percentage of unsecureddebt that could be repaid over 5 years is 25% or $5,000, whichever is less. (3) The minimum dollaramount of unsecured debt that could be repaid over 5 years is $5,000 or 25%, whichever is less.We summarize these criteria by restricting filing to those with lower than median earnings as long


TABLE IX

ALLOCATION EFFECTS OF MEANS TESTING IN THE EXTENDED MODEL

Economy Extended Bankruptcy RestrictionMax Earnings for Filing ∞ Median IncomeGeneral Equilibrium Effect — No Yes

Output (normalization) 100 100 100Total asset 308 303 305Total debt 00068 00129 00128Percentage that file 054% 053% 053%Percentage with bad credit record 423% 414% 413%Percentage in debt 499% 824% 817%Rate of return of capital 169% 169% 180%Avg loans rate (persons-weighted) 3096% 1290% 1304%

default because their income is above the median. Table IX reports the changesin the model statistics with this policy for two cases. In the first case, the realreturn on capital is held fixed at the pre-policy-change level (which is what wemean by no general equilibrium effect), while in the second case, the real re-turn is consistent with a post-policy-change general equilibrium. We focus onthe numbers with the full general equilibrium effects, but note that such effectsare very small.

6.1. Effects on Allocations

The first thing to note is that the aggregate implications of the policy arevery small in terms of total savings and the number of filers, but quite sub-stantial in terms of total borrowing (which almost doubles) and the averageinterest rate charged on loans (which is reduced by more than one-half). Thefirst panel of Figure 7 shows the default probabilities in the extended modelfor blue and white collar workers. Notice that the units in the horizontal axisare much larger than the units in Figure 5, indicating the strong limits imposedon default by the means-testing policy. The second panel compares the defaultprobabilities for those that were not hit by either the liability or the preferenceshock in the extended economy with and without the means-testing policy. Thedefault probabilities for blue collar workers are reduced substantially and fora certain range are reduced to zero. The default probabilities for white collarhouseholds fall only for very large volumes of debt. As a result, the loan priceschedules shift up (the default premium schedules shift down) for both types ofhouseholds, as shown in Figure 8. Even though the change in default probability

as doing so does not result in negative consumption. The alternative would be to keep increasingthe liabilities for a few more periods. Our choices are consistent with the law, which allows, forinstance, a household with high medical liabilities to file even if their income is above median.


FIGURE 7.—Comparison of default probabilities in the economy with and without means test-ing.


FIGURE 8.—Loan prices in the extended model with and without means testing.

for the white collar households is not substantial, the default premia on loansto both white and blue collar households drop substantially. This is because forboth types of households there is a positive probability of being blue collar inthe next period.

Table IX presents the changes in aggregate statistics resulting from this pol-icy. Most interestingly, the number of bankruptcy filings barely changes eventhough the default probability schedule conditional on type shifts down foreach type of household. This occurs because the percentage of households indebt increases dramatically in response to lower interest rates on loans. Specif-ically, the percentage of households in debt goes up by more than 60%, from5.0% to 8.2%. Total debt almost doubles, implying that on average householdstake on bigger loans.

6.2. Effects on Welfare

In assessing the welfare effects of any policy change, one must take into ac-count the transition path to the new steady state. This is a daunting compu-tational task when taking into account the general equilibrium effects on therate of return and on wages (the closed economy assumption). Our findingsthat the long-run effects of either the small open economy assumption or the


closed economy assumption reported in Table IX indicate that the welfare cal-culation based on the small open economy assumption is not only interestingin itself, but also quite close to what would prevail in a closed economy. Ob-viously, our welfare analysis compares allocations with and without the imple-mentation of the means-testing policy under the same initial conditions—thewealth and credit record distribution in the steady state of the extended modeleconomy without means testing.

An important additional consideration arises in our environment when con-ducting welfare analysis: There are multiple agent types and, therefore, therewill not be agreement among types as to the desirability of a policy change.Consequently, some form of aggregation is necessary. We use two aggregationcriteria. The first criterion is the percentage of households that are made betteroff by the policy change and thus support it. The second criterion is the averagegain as measured by the average of the percentage increase in consumptioneach household would be willing to pay in all future periods and contingen-cies so that the expected utility from the current period in the initial steadystate equals that of the equilibrium associated with the new policy. Because ofour assumption on the functional form of the momentary utility function, theconsumption equivalent welfare gain for a household of type (h s) can becomputed as

100([∫

vhs(e;qw)Φ(e|s)de∫vhs(e;qw)Φ(e|s)de

]1/(1−σ)− 1

)(24)

where vhs is the value function in the equilibrium associated with the newpolicy.

Table X reports the desirability of the policy change for the two aggregationcriteria. First of all, we see that the welfare benefits of the policy reform arelarge—about 1.6% of average consumption—as measured by the utilitarianaverage of consumption-equivalent gains. We also see that the policy reformreceives almost unanimous support: around 0.01% of households oppose it.The largest gains accrue to those with a good credit record and debt. A pos-sible explanation of why this is such a good policy is that means testing takesaway the right to default in situations where the size of the utility gain from de-faulting is positive but not very high. Therefore, most people prefer to give thisoption up in exchange for lower interest rates (of course, people who opposeit are those with above-median earnings and a lot of debt, i.e., individuals forwhom the policy binds strongly, but there are not many of them).38

38In an earlier version of this paper, which employed a somewhat different calibration, weexplored another policy experiment where we reduced the mean exclusion time from borrowingfollowing default from 10 to 5 years. For details, see Appendix A.2 of http://www.phil.frb.org/files/wps/2005/wp05-18.pdf.


TABLE X

WELFARE EFFECT OF MEANS TESTING

Preference Shock Liability Shock

Shock Hit Not Hit Not Total

Proportion of households 0012 0988 0003 0997 1000

Average (%) gain in flow consumptionWith bad credit record 052 080 075 080 080With good credit record and debt 3235 664 035 695 693With good credit record and no debt 356 137 076 140 139Total 475 160 074 164 164

Households in favor of reform (%)With bad credit record 1000 1000 1000 1000 1000With good credit record and debt 914 1000 968 999 999With good credit record and no debt 1000 1000 980 1000 1000Total 996 1000 980 1000 1000

7. CONCLUSIONS

In this paper we accomplished four goals. First, we developed a theory ofdefault that is consistent with U.S. bankruptcy law. In the process we charac-terized some theoretical properties of the household’s decision problem andproved the existence of a steady-state competitive equilibrium. A key fea-ture of the model is that it treats different-sized consumer loans taken outby households with observably different characteristics as distinct financial as-sets with distinct prices. Second, we showed that the theory is quantitativelysound in that it is capable of accounting for the main facts regarding unsecuredconsumer debt and bankruptcy in the United States along with U.S. facts onmacroeconomic aggregates and facts on inequality characteristics of U.S. earn-ings and wealth distributions. Third, we explored the implications of an impor-tant recent change in the bankruptcy law that limits the Chapter 7 bankruptcyoption to households with below-median earnings. We showed that the likelyoutcome of this change will be a decrease in interest rates charged on unse-cured loans, and an increase in both the volume of debt and the number ofborrowers without necessarily having an increase in the number of bankrupt-cies. Furthermore, our measurements indicated that the changes will be big; forinstance, the volume of net unsecured debt may almost double. Finally, we con-structed measures of the welfare effects of the policy change. From the point ofview of average consumption, our calculations indicate that the benefits of thechange are large: on the order of 1.5 percent of average consumption. From thepoint of view of public support, we found that almost all households supportthe change. In terms of future research, two issues seem important. First, ana-lyzing environments in which households have some motive for simultaneouslyholding both assets and liabilities is likely to improve our understanding of the


unsecured consumer credit market. Second, incorporating unobserved differ-ences among households with regard to willingness to default is also likely toimprove our understanding of what happens to a household’s credit opportu-nities after bankruptcy and, therefore, to the costs of default, especially if wetake into account that individuals with bad credit scores can still have accessto credit. We are axploring this mechanism in Chatterjee, Corbae, and Ríos-Rull (2004).

Federal Reserve Bank of Philadelphia, Research, 10 Independence Mall,Philadelphia, PA 19106, U.S.A.; [email protected],

Dept. of Economics, University of Texas at Austin, Mail Code C 3100, Austin,TX 78712, U.S.A.; [email protected],

Dept. of Economics, University of Illinois at Urbana–Champaign, 1206 SouthSixth St., 330 Wohlers Hall, Champaign, IL 61820, U.S.A.; [email protected],

andUniversity of Pennsylvania, CAERP, CEPR, NBER, and Dept. of Economics,

University of Minnesota, 1035 Heller Hall, 271-19th Ave. South, Minneapolis, MN55455, U.S.A.; [email protected].

Manuscript received May, 2002; final revision received February, 2007.

APPENDIX

For reasons given in the text, the appendix generalizes the environment inthe paper to include a bankruptcy cost α · (e − emin) · w with α ∈ [0α] =A,where α < 1 to be paid only at the time of default. This requires us to expandthe space on which the operator T is defined to include A and modify theoperator T for case 2 (where the household chooses whether to default ornot) in Definition 1 to be

(Tv)(0 s e;αqw)

= max

maxc′∈B0s0

u(c s)+βρ∫v′0s′(e

′α;qw)Φ(e′|s′)

× Γ (sds′)de′

u([e− α · (e− emin)] ·ws)

+βρ∫v01s′(e

′α;qw)Φ(e′|s′)Γ (sds′)de′

where, to conserve on notation, we let u(c s) denote U(cη(s)).


A.1. Results for Theorems 1 and 2

The following restriction formalizes the assumption concerning u(0 s) inpart (iv) of Theorem 1.

ASSUMPTION A1: For every s ∈ Su((1 − γ)emin ·wmin s)− u(0 s)

>

(βρ

1 −βρ)

× [u(emax ·wmax + max − min s)− u((1 − γ)emin ·wmin s)

]

DEFINITION A1: Let V be the set of all continuous (vector-valued) functionsv :E ×A×Q×W →RNL such that

vhs(e;αqw)(25)

∈[u[emin ·wmin(1 − γ) s]

(1 −βρ) u(emax ·wmax + max − min s)

(1 −βρ)]

0 ≥ 1 ⇒ v0hs(e;αqw)≥ v1hs(e;αqw)(26)

e0 ≥ e1 ⇒ vhs(e0;αqw)≥ vhs(e1;αqw)(27)

v0s(e;αqw)≥ v1s(e;αqw)(28)

u(emin ·wmin(1 − γ) s)+βρ∫v01s′(e

′α;qw)Φ(e′|s′)Γ (sds′)de′(29)

> u(0 s)+βρ∫vmax0s′(e

′α;qw)Φ(e′|s′)Γ (sds′)de′

LEMMA A1: V is nonempty. With ‖v‖ = maxhssupeαqw∈E×A×Q×W |vhs(e;αqw)| as the norm, (V‖ · ‖) is a complete metric space.

PROOF: To prove V is nonempty, pick any constant vector-valued functionsatisfying (25). Such a function is continuous and clearly satisfies (26)–(28).Since it is a constant function, (29) reduces to the requirement that u(emin(1 −γ) ·wmin s)−u(0 s) > 0, which is satisfied by virtue of emin(1−γ) ·wmin > 0 andthe strict monotonicity of u(· s) To prove (V‖ · ‖) is complete, let C be theset of all continuous (vector-valued) functions from E ×A×Q×W → RNL Then (C‖·‖) is a complete metric space. Since any closed subset of a completemetric space is also a complete metric space, it is sufficient to show that V ⊂ Cis closed in the norm ‖·‖ Let vn be a sequence of functions in V converging tov that is, limn→∞ ‖vn−v∗‖ = 0 If v∗ violates any of the range and monotonicityproperties of V there must be some vn, for n sufficiently large, that violates


those properties as well. But that would contradict the assertion that vn belongsto V for all nHence, v∗ must satisfy all the range and monotonicity properties(25)–(28). To prove that v∗(e;αqw) is continuous, simply adapt the final partof the proof of Theorem 3.1 in Stokey and Lucas (1989) to a vector-valuedfunction. Q.E.D.

We now turn to properties of the operator T . It is convenient to have no-tation for the value of consumption for any action ′ d ∈ L × 01 (in-cluding actions that imply negative consumption). We will denote consump-tion for a household with h s who takes actions ′ d by c

′dhs(e;αqw).

Then c010s(e;αqw) ≡ [emin + (1 − α)(e − emin)] · w > 0 c01

1s(e;αqw) ≡e(1 − γ) ·w > 0 and c

′0hs(e;αqw)≡ w · e(1 − γh)+ − ζ(s)− q′s′ Ob-

serve that the value of consumption for ′0 can be negative.It is also convenient to define the expected utility of a person who starts

next period with ′h′ s′:ω′h′s(v)≡ ∫v′h′s′(e

′;αqw)Φ(e′|s′)Γ (sds′)de′Observe that ω is defined for a given v and so depends on αqw In whatfollows, we will sometimes make this dependence explicit.

Using similar notation, for any given pair of discrete actions ′ d ∈L × 01, we define lifetime utility as follows: (i) For h = 0 − ζ(s) ≥ 0φ

′00s(eαqw;ω(v)) ≡ u(maxc′00s(e;αqw)0 s) + βρω′0s(v). (ii) For

h = 0 − ζ(s) < 0 φ010s(eαqw;ω(v)) ≡ u(c01

0s(e;αqw) s) +βρω01s(v) and φ

′00s(eαqw;ω(v)) ≡ u(maxc′00s(e;αqw)0 s) +

βρω′0s(v). (iii) For h = 1 − ζ(s) ≥ 0 φ′01s(eαqw;ω(v)) ≡

u(maxc′01s(e;αqw)0 s) + βρ[λω′1s(αqw;v) + (1 − λ)ω′0s(v)].(iv) Finally, for h = 1 − ζ(s) < 0 φ01

1s(eαqw;ω(v)) ≡ u(c011s(e;αq

w) s)+βρω01s(v)Then we have the following assertions:

LEMMA A2: For any (′ d) φ′dhs(eαqw;ω(v)) is continuous in eαq

and w

PROOF: Observe that c′dhs(e;αqw) are each continuous functions of

eαq and w, and u is a continuous function in its first argument. Further,ω′h′s(v) is continuous in αq, and w because v ∈ V and integration preservescontinuity. Q.E.D.

LEMMA A3: For v ∈ V (T v)(e;αqw) is continuous in eαq and w

PROOF: By Lemma A2, φ′dhs(eαqw;ω(v)) is continuous. Hence,

max′d φ′dhs(eαqw;ω(v)) is also continuous in eαq and w Then it is

sufficient to establish that ∀h s ∈L

(Tv)(h s e;αqw)= max′d

φ′dhs(eαqw;ω(v))


If the maximum is over feasible (′ d), (Tv)(h s e;αqw) =max′d φ

′dhs(eαqw;ω(v)) Furthermore, for infeasible (′ d), the payoff

φ′dhs(eαqw;ω(v)) is assigned a value that is always (weakly) dominated

by some feasible ′ d This follows because by property (29), the utility fromconsuming nothing today and starting next period with max and a good creditrecord (the highest utility possible with an infeasible action) is less than theutility from consuming emin · wmin(1 − γ) today and starting next period withzero assets and a bad credit record (the lowest utility possible with a feasibleaction). Q.E.D.

COROLLARY TO LEMMA A3: For any v ∈ V the consumption implied by(Tv)(h s e;αqw) is strictly positive.

PROOF: The exact same argument as in Lemma A3 can be used to establishthat a feasible choice involving zero consumption is always strictly dominatedby a feasible choice involving positive consumption. Q.E.D.

LEMMA A4: Given Assumption A1, T is a contraction mapping with modulusβρ

PROOF: We first establish that T (V)⊂ VFor v ∈ V , T is continuous by Lemma A3.To establish that T preserves the boundedness property (25), note that since

qmins ∈ [01] consumption can never exceed emax ·wmax +max −min. Therefore,(Tv)(h s e;αqw) ≤ (1 −βρ)−1u(emax · wmax + max − min s) and sinceα < 1 c = (1 − γ)emin·wmin is feasible for all h s eαq and w Therefore,(Tv)(h s e;αqw)≥ 1

(1−βρ)u((1 − γ)emin ·wmin s) Hence

(T v)(e;αqw) ∈[

1(1 −βρ)u((1 − γ)emin ·wmin s)

1(1 −βρ)u(emax ·wmax + max − min s)

]NL

To establish that T preserves the monotonicity property (26), consider ahousehold with a given h s and two different asset holdings 0 > 1 (i) If0 > 0, then for d ∈ 01 B10sd(e;αqw) ⊆ B00sd(e;αqw) and theresult follows; (ii) if 0 ≥ 0 > 1 and 0 ≥ ζ(s), then B10sd(e;αqw) ⊆B00s0(e;αqw) and the result follows from using (28); (iii) if 0 ≥ 0> 1 and0 < ζ(s), then B10sd(e;αqw) ⊆ B00sd(e;αqw) and the result follows;(iv) if 1 ≥ 0 and 1 < ζ(s) ≤ 0, then B1hsd(e;αqw) ⊆ B0hs0(e;αqw)and the result follows from using (28); and (v) if 1 ≥ 0 and 1 ≥ ζ(s), thenB1hs0(e;αqw)⊆ B0hs0(e;αqw) and the result follows.


To establish that T preserves the monotonicity property (27), consider ahousehold with a given , h, s and two different efficiency levels e0 > e1 Sinceα< 1 Bhsd(e1;αqw)⊆ Bhsd(e0;αqw) and the result follows.

To establish that T preserves the monotonicity property (28), consider ahousehold with a given s and two different credit records. (i) If − ζ(s) < 0B1s1(e;αqw)⊆ B0sd(e;αqw) and the result follows; (ii) if −ζ(s)≥ 0B1s0(e;αqw)⊆ B0s0(e;αqw), the result follows from using (28).

To establish that T preserves the “default at zero consumption” property(29), by Assumption A1 and the fact that T satisfies the boundedness propertyit follows that

u((1 − γ)emin ·wmin s)− u(0 s)> βρ[(Tv)(max0 s e;αqw)− (Tv)(01 s e;αqw)]

Rearranging gives

u((1 − γ)emin ·wmin s)+βρ(Tv)(01 s e;αqw)> u(0 s)+βρ(Tv)(max0 s e;αqw)

Next we establish that T is a contraction with modulus βρ The first step isto establish the analogue of the Blackwell monotonicity and discounting prop-erties. Monotonicity: Let v v′ ∈ V and v(e;αqw)≤ v′(e;αqw) for all eαqw From the definition of the T operator, it is clear that (T v) ≤ (T v′).Discounting: It is also clear that for any κ ∈ RNL+ , [T (v+ κ)] (e;αqw) =(T v)(e;αqw)+βρκ. To prove that T is a contraction mapping, simply adaptthe final part of the proof of Theorem 3.3 in Stokey and Lucas (1989) to avector-valued function. This establishes that T is a contraction mapping withmodulus βρ Q.E.D.

THEOREM 1—Existence of a Recursive Solution to the Household Problem:There exists a unique v∗ ∈ V such that v∗ = T (v∗) Furthermore, (i) v∗ is boundedand increasing in and e, (ii) a bad credit record reduces v∗, (iii), the optimalpolicy correspondence implied by T (v∗) is compact-valued and upper hemicon-tinuous, and (iv) provided u(0 s) is sufficiently low, default is strictly preferable tozero consumption and consumption is strictly positive.

PROOF: Existence and uniqueness of v∗ as well as properties (i), (ii), and(iv) follow directly from Lemmas A1, A3, and A4, and the Corollary toLemma A3. To prove part (iii), define the optimal policy correspondence to be

χhs(e;αqw)= (c ′ d) ∈ Bhsd(e;αqw) :(c ′ d) attains v∗

hs(e;αqw)


To establish the first part of (iii), note that the correspondence χhs(e;αqw)is bounded because c is bounded between 0 and emax ·wmax + max − min and(′ d) ∈L×01. To prove that χhs(e;αqw) is closed, let cn ′

n dn be asequence in χhs(e;αqw) converging to (c

′ d) Since (′ d) are elements

of finite sets, ∃η such that ∀n > η, (′n dn) = (

′ d) Given that (cn

′ d) at-

tains v∗hs(e;αqw) ∀n > η we have cn = c

′d

hs(e;αqw). Therefore, c =c

′d

hs(e;αqw) and (c ′ d) ∈ χhs(e;αqw) To establish the second part

of property (iii), let nhn sn enαnqnwn → (h s eαqw) Since L isfinite, we can fix (nhn sn) = (h s) and simply consider enαnqnwn →eαqw Let cn ′

n dn ∈ χhs(en;αnqnwn) Since the correspondence iscompact-valued, there must exist a subsequence cnk ′

nk dnk converging to

(c ′ d) Furthermore, since ′ and d take on only a finite number of values,

∃η such that ∀nk > η we have (cnk ′nk dnk) = (c

′d

hs(enk;αnkqnkwnk)

′ d)

Then by optimality φ′d

hs(enkαnk qnkwnk;ω∗(αnk qnkwnk))= v∗

hs(enk;αnk

qnkwnk) and by continuity of φ′d

hs, v∗

hs, and ω∗ with respect to eαq,

and w, we have φ′d

hs(eαqw;ω∗(αqw)) = v∗

hs(e;αqw) Therefore,

(c = c′d

hs(e;αqw) ′

d) ∈ χhs(e;αqw) and the correspondence is up-per hemicontinuous. Q.E.D.

THEOREM 2—Existence of a Unique Invariant Distribution: For (αqw) ∈A × Q × W and any measurable selection from the optimal policy correspon-dence, there exists a unique µ(αqw) ∈M(L×E2L ×B(E)) such that µ(αqw) =Υ(αqw)µ(αqw).

PROOF: By the measurable selection theorem, there exists an optimal policyrule that is measurable with respect to any measure in M(L×E2L ×B(E))Therefore, G∗

(αqw) is well defined. To establish this lemma we then simplyneed to verify that G∗

(αqw) satisfies the conditions stipulated in Theorem 11.10of Stokey and Lucas (1989). The first condition is that G∗

(αqw) satisfies Doe-blin’s condition (which states that there is a finite measure ϕ on (L×E2L ×B(E)) an integer I ≥ 1, and a number ε > 0, such that if ϕ(Z) ≤ ε, thenG∗I(αqw)((h s e)Z) ≤ 1 − ε for all (h s e)). It is sufficient to show that

GN satisfies the Doeblin condition (see Exercise 11.4.g of Stokey and Lucas(1989)). Observe that since GN is independent of (h s e) we can pickϕ(Z)= GN((h s e)Z). Then GN satisfies the Doeblin condition for I = 1and ε < 1

2 Second, we need to show that if Z is any set of positive ϕ measure,then for each (h s e) there exists n≥ 1 such that G∗n

(αqw)((h s e)Z) >0 To see this, observe that if ϕ(Z) > 0 then GN((h s e)Z) > 0 for any(h s e) Therefore, G∗1

(αqw)((h s e)Z) > 0 Q.E.D.



We turn now to the proof of Theorem 3. We give a formal definition of themaximal default set and then establish two key lemmas. The maximal defaultD∗hs(αqw)= e : v∗

hs(e;αqw)=φ01hs(eαq;ω∗) where ω∗ is ω(v∗)

LEMMA A5: Let e ∈ E\D∗0s(0 qw) e > e and − ζ(s) < 0 If e ∈

D∗0s(0 qw) then c∗

0s(e;0 qw) > e ·w

PROOF: Since e ∈E\D∗0s(0 qw)

u(c∗0s(e;0 qw) s)+βρω∗

′∗0s (e;0qw)0s(0 qw)(30)

> u(e ·ws)+βρω∗01s(0 qw)

Let = (e− e) ·w> 0 The pair c = c∗0s(e;0 qw)+′ = ′∗

0s(e;0 qw)clearly belongs in B0s0(e;0 qw) Then by optimality, utility obtained by notdefaulting when labor efficiency is e must satisfy the inequality

u(c0s0(e;0 qw) s)+βρω∗′0s0(e;0qw)0s(0 qw)(31)

≥ u(c s)+βρω∗′0s(0 qw)

where c0s0(e;0 qw) and ′0s0(e;0 qw) are the optimal choices of c and

′ conditional on not defaulting. Since e ∈D ∗hs(0 qw)

u(c0s0(e;0 qw) s)+βρω′0s0(e;0qw)0s(0 qw)(32)

≤ u(e ·ws)+βρω∗01s(0 qw)

By (31) and the fact that e ·w+= e ·w, (32) can be rewritten

u(c s)+βρω∗′0s(0 qw)≤ u(e ·w+µ)+βρω∗

01s(0 qw)(33)

Then (33) minus (30) implies

u(c s)+βρω∗′0s(0 qw)(34)

− u(c∗0s(e;0 qw) s)−βρω∗

′∗0s (e;0qw)0s(0 qw)

< u(e ·w+ s)+βρω∗01s(0 qw)− u(e ·ws)

−βρω∗01s(0 qw)

or, by definition of (c ′)

u(c∗0s(e;0 qw)+ s)− u(c∗

0s(e;0 qw) s)

< u(e ·w+ s)− u(e ·ws)


Since u(· s) is strictly concave, the last inequality implies c∗0s(e;0 qw) >

e ·w. Q.E.D.

LEMMA A6: Let e ∈ E\D∗0s(0 qw) e < e and − ζ(s) < 0 If e ∈

D∗0s(0 qw) then c∗

0s(e;0 qw) < e ·w

PROOF: Since e ∈E\D∗0s(0 qw)

u(c∗0s(e;0 qw) s)+βρω∗

′∗0s (e;0qw)0s(0 qw)(35)

> u(e ·ws)+βρω∗01s(0 qw)

Let = (e − e) · w > 0 Consider the quantity c∗0s(e;0 qw) − If

c∗0s(e;0qw) − ≤ 0, then it must be the case that c∗

0s(e;0 qw) < e · wbecause e · w − = e · w > 0. So we only need to consider the casewhere c∗

0s(e;0 qw) − > 0 The pair c = c∗0s(e;0 qw) − ′ =

′∗0s(e;0 qw) clearly belongs in B0s0(0 qw). Then by optimality, utility

obtained by not defaulting when labor efficiency is emust satisfy the inequality

u(c0s0(e;0 qw) s)+βρω∗′0s0(e;0qw)0s

(0 qw)(36)

≥ u(c s)+βρω∗′0s(0 qw)

where, once again, c0s0(e;0 qw) and ′0s0(e;0 qw) are the optimal

choices of c and ′ conditional on not defaulting. Since e ∈D∗0s(0 qw),

u(c0s0(e;0 qw) s)+βρω∗′0s0(e;0qw)0s

(0 qw)(37)

≤ u(e ·ws)+βρω∗01s(0 qw)

By (36) and the fact that e ·w−= e ·w, (37) can be rewritten

u(c s)+βρω∗′0s(0 qw)≤ u(e ·w− s)+βρω∗

01s(0 qw)(38)

Then (38) minus (35) implies

u(c s)+βρω∗′0s(0 qw)− u(c∗

0s(e;0 qw) s)

−βρω∗′∗0s (e;0qw)0s(0 qw)

< u(e ·w− s)+βρω∗01s(0 qw)− u(e ·ws)

−βρω∗01s(0 qw)

or, by definition of (c ′),

u(c∗0s(e;0 qw) s)− u(c∗

0s(e;0 qw)− s)(39)

> u(e ·ws)− u(e · x− s)


Since u(· s) is strictly concave, the last inequality implies c∗0s(e;0 qw)−<

e ·w− or c∗0s(e;0 qw) < e ·w. Q.E.D.

THEOREM 3—The Maximal Default Set Is a Closed Interval: If D∗hs(0 q

w) is nonempty, it is a closed interval.

PROOF: First consider the case h= 0 If −ζ(s)≥ 0 thenD∗0s(0 qw)=

∅ If − ζ(s) < 0 let eL = infD∗0s(0 qw) and eU = supD

∗0s(0 qw) Since

D∗0s(0 qw) ⊂ E, which is bounded, both eL and eU exist by the complete-

ness property of R. If eL = eU the default set contains only one elemente= eL = eU and the result is trivially true. Suppose, then, that eL < eU Let e ∈(eL eU) and assume that e /∈D∗

0s(0 qw). Then there is an e ∈D∗0s(0 qw)

such that e > e (if not, then eU = e, which contradicts the assertion thate ∈ (eL eU)). Then, by Lemma A5, c∗

0s(e;0 qw) > e ·w Similarly, there is ane ∈D∗

0s(0 qw) such that e < e Then, by Lemma A6, c∗0s(e;0 qw) < e ·w

But c∗0s(e;0 qw) cannot be both greater and less than e · w. Hence, the

assertion e /∈ D∗0s(0 qw) must be false and (eL eU) ⊂ D

∗0s(0 qw). To

show that eU ∈ D∗0s(0 qw), pick a sequence en ⊂ (eL eU) converging to

eU Then, v∗0s(en;0 qw) − u(en · ws) = βρω∗

01s(0 qw) for all n SinceeU is clearly in E by the continuity of v∗

0s(e;0 qw) and u, it follows thatlimn→∞v∗

0s(en;0 qw)−u(en ·ws) = v∗0s(eU;0 qw)−u(eU ·ws). Since

every element of the sequence v∗0s(en;0 qw) − u(en · ws) is equal to

βρω∗01s(0 qw) it must be the case that v∗

0s(eU;0 qw) − u(eU · ws) =βρω∗

01s(0 qw) Hence, eU ∈ D∗0s(0 qw). By analogous reasoning, eL ∈

D∗0s(0 qw). Hence, [eL eU ] ⊆D∗

0s(0 qw). But by the definition of eL andeU D

∗0s(0 qw) ⊂ [eL eU ] Hence [eL eU ] = D

∗0s(0 qw). Next consider

the case h = 1 If − ζ(s) ≥ 0 then D∗1s(0 qw)= ∅ If − ζ(s) < 0 then

D∗1s(0 qw)=E Q.E.D.

THEOREM 4—Maximal Default Set Expands with Indebtedness: If 0 > 1

then D∗0hs(0 qw)⊆D

∗1hs(0 qw)

PROOF: First consider the case h = 0. Suppose e ∈ D∗00s(0 qw) Since

v∗0s(e;αqw) is increasing in v∗

00s(e;αqw) ≥ v∗10s(e;αqw) But

v∗00s(e;αqw)= u(e ·ws)+βρω∗

01s(αqw) Since default is also an optionat 1 it must be the case that v∗

10s(e;αqw)= u(e ·ws)+βρω∗01s(αqw).

Hence any e in D∗00s(0 qw) is also in D

∗10s(0 qw). Next consider the case

h = 1. If l′ − ξ(s) ≥ 0, then 0 − ξ(s) ≥ 0 and D∗01s(0 qw)=D

∗11s(0 qw).

If l0 − ξ(s) < 0, then l1 − ξ(s) < 0 and D∗01s(0 qw)=D

∗11s(0 qw).


Finally, if l0 − ξ(s) ≥ 0 and l1 − ξ(s) < 0, then D01s(0 qw) = ∅ andD11s(0 qw)= E. Q.E.D.


We now turn to the proof of existence of equilibrium. For the environmentwith α> 0 all conditions in Definition 2 remain the same except for the goodsmarket-clearing condition (ix), which we now call (ixA):∫

c∗hs(e;αq∗w∗)dµ∗ +K∗ +

∫ζ(s)

m∗ dµ∗

= F(N∗K∗)+ (1 − δ)K∗ − γw∗∫eµ∗(d1 dsde)

− αw∗∫(e− emin) · d∗

0s(e;αq∗w∗)µ∗(d0 dsde)

We state the following lemma without proof (the proof is available in theSupplements to this article on the Econometrica website (Chatterjee, Corbae,Nakajima, and Ríos-Rull (2007)):

LEMMA A7: The goods market-clearing condition (ixA) is implied by the otherconditions for an equilibrium in Definition 2.

Next, we establish the important result that for α > 0, the set of (h s e)for which a household is indifferent between two courses of action is finite.Since the probability measure associated with a finite set is zero, this resultallows us to ignore the behavior of households at “indifferent points” and sim-plifies the proof of existence of an equilibrium.

Given (h s) define the set of e for which a household is indifferent be-tween any two distinct feasible actions (′ d) and (

′ d) as I(

′d)(′d)hs (αqw)≡

e ∈ E :φ(′d)

hs (e;αqwω∗) = φ(′d)

hs (e;αqwω∗) ∩ e ∈ E : c(′d)

hs (e;αqw)≥ 0 c(

′d)

hs (e;αqw)≥ 0

LEMMA A8: (i) I(′0)(′0)

hs (αqw;ω∗) contains at most one element and (ii) ifα> 0, then I(

′0)(01)0s (αqw;ω∗) contains at most two elements.

PROOF: (i) Let e ∈ I(′0)(′0)hs (αqw) Since ω∗′0s(αqw) = ω∗

′0s(αq

w) it follows that ≡ u(w · e(1 − γh) + − ζ(s) − q′s′ s) − u(w · e(1 −

γh)+ − ζ(s)− q′s′ s) = 0 Therefore, consumption under each of the two

actions must be different. Since u(·) is strictly concave, an equal change inconsumption from these two different levels must lead to unequal changes in


utility. Therefore, for y = 0 we must have that u(w · [e+y](1−γh)+−ζ(s)−q′s

′ s)− u(w · [e+ y](1 − γh)+ − ζ(s)− q′s′ s) = Hence there can

be at most one e for which φ′00s(eqw;ω∗)=φ′00s(eqw;ω∗)

(ii) Let e ∈ I(′0)(01)0s (αqw) and let y > 0(a) Suppose that u(w · e + − ζ(s) − q′s

′ s) − u(w · [emin + (1 − α)(e −emin)] s) = ≥ 0 Given α > 0 it follows that u′(w · [e + y] + − ζ(s) −q′s

′ s) < u′(w · [emin + (1 −α)(e+ y − emin)] s) Now observe that u(w · [e+y]+ − ζ(s)−q′s′ s) is u(w · e+ − ζ(s)−q′s′ s)+ ∫ y

0 u′(w · [e+x]+ −

ζ(s)− q′s′ s)dx and u(w · [emin + (1 − α)(e+ y − emin)] s) is u(w · [emin +

(1 − α)(e− e)] s)+ ∫ y

0 u′(w · [emin + (1 − α)(e+ x− emin)] s)dx Therefore,

u(w · [e+ y]+ − ζ(s)−q′s′ s)−u(w · [emin + (1 −α)(e+ y− emin)] s) < Hence e+ y /∈ I(′0)(01)0s (αqw) On the other hand, it is possible that there isa z > 0 such that e− z ∈ I(′0)(01)0s (αqw) If so,

∫ z

0 u′(w · [e− x] + − ζ(s)−

q′s′ s)dx = ∫ z

0 u′(w · [emin + (1 − α)(e − x − emin)] s)dx Since ≥ 0 we

have u′(w · e+ − ζ(s)−q′s′ s) < u′(w · [emin + (1 −α)(e− emin)] s) There-fore, w · [e−z]+ −ζ(s)−q′s′ <w ·emin + (1−α)(e−z−emin) Then, givenα > 0, u(w · [e− z − y] + − ζ(s)− q′s

′ s)− u(w · [emin + (1 − α)(e− z −y − emin)] s) = because we would be taking more consumption away fromthe left-hand side than from the right-hand side when the left already has less.Therefore, I(

′0)(01)0s (αqw) can have at most two elements.

(b) Suppose that u(w · e + − ζ(s) − q′s′ s) − u(w · [emin + (1 − α)(e −

emin)] s) = < 0 Then, given α > 0 u′(w · [e − y] + − ζ(s) − q′s′ s) <

u′(w · [emin + (1 − α)(e− y − emin)] s) By an argument analogous to the firstpart of (a), we can establish that e− y /∈ I(′0)(01)0s (αqw) On the other hand,it is possible that there is a z > 0 such that e + z ∈ I(′0)(01)0s (αqw) If so,then u′(w · [e+ x] + − ζ(s)− q′s′ s)dx= ∫ z

0 u′(w · [emin + (1 − α)(e+ x−

emin)] s)dx By an argument analogous to the second part of (a), we can es-tablish that w · [e+ z] + − ζ(s)− q′s

′ > w · [emin + (1 − α)(e+ z − emin)]Therefore, given α > 0 u(w · [e+ z+ y] + − ζ(s)− q′s′ s)− u(w · [emin +(1 − α)(e+ z + y − emin)] s) = because we would be giving more consump-tion to the left-hand side than to the right-hand side when the left already hasmore. Therefore, I(

′0)(01)0s (αqw) can have at most two elements. Q.E.D.

Define E′dhs(αqw) ≡ e ∈ E :′∗

hs(e;qw) = ′ d∗hs(e;qw) = d to

be set of E that returns (′ d) as the optimal decision (when the house-hold has , h, s). Define ES

′dhs(αqw) ≡ e ∈ E : [φ′dhs(eqw;ω∗) −

max(′d) =(′d) φ′dhs(eqw;ω∗)] > 0 to be the set of e for which (′ d) is

strictly better than any other action.

LEMMA A9: For α > 0 E′dhs(αqw)\ES

′dhs(αqw) is a finite set.


PROOF: Observe that

E′dhs(αqw)\ES′dhs(αqw) ⊆

⋃(′d) =(′d)

I(′d)(′d)

hs (αqw)

Since the sets I(′d)(′d)

hs (αqw) are finite by Lemma A8, the result fol-lows. Q.E.D.

LEMMA A10: Let Z ∈ 2L ×B(E) and (nhn sn enαnqnwn)→ (h s eαqw) If α > 0, then for all but a finite set of (h s e): (i) ′∗

hs(en;αnqnwn) → ′∗

hs(e;αqw) and d∗hs(en;αnqnwn) → d∗

hs(e;αqw), and(ii) limn→∞G∗

(αnqnwn)(nhn sn enZ)=G∗

(αqw)(h s eZ).

PROOF: Since L is finite, it follows that there is some η such that for alln≥ η (nhn sn)= (h s)Without loss of generality, we simply consider thesequence (enαnqnwn)→ (eαqw). (i) Consider the set of efficiency levelsEShs(αqw) for which the household strictly prefers some action (′ d), thatis, EShs(αqw) = e :e ∈ ⋃

(′d)ES′dhs(αqw) For e ∈ EShs(αqw),

consider the sequence ′∗hs(en;αnqnwn)d∗

hs(en;αnqnwn) The se-quence lies in a compact and finite subset of R2 so we can extract a subse-quence nk converging to (

′ d) Furthermore, there is an η such that for

nk ≥ η (′∗hs(enk;αnkqnkwnk) d∗

hs(enk;αnkqnkwnk))= (′ d) Therefore,

for nk ≥ η φ′dhs(enk;αnkqnkwnkω∗) = v∗hs(enk;αnkqnkwnk) Taking lim-

its of both sides, and using the continuity of φ and v∗ established in Lemma A2and Theorem 1, respectively, φ

′d

hs(e;αqwω∗) = v∗hs(e;αqw) Since

e ∈ EShs(αqw) (′ d) = (′∗

hs(e;αqw)d∗hs(e;αqw)) Since the

set of efficiency levels for which there is indifference can be expressed as⋃(′d)E

′dhs(αqw)\ES

′dhs(αqw) by Lemma A9 this set is finite and the

result follows. (ii) Follows from the definition of G∗(αqw)(h s eZ) and

part (i). Q.E.D.

The next step is to establish the weak convergence of the invariant distrib-ution µ(αqw) with respect to αq and w Theorem 12.13 of Stokey and Lu-cas (1989) provides sufficient conditions under which this holds. However,if the household is indifferent between two courses of action at (h s e)the probability measure G∗

(αqw)((h s en) ·) need not converge weakly toG∗(αqw)((h s e) ·) as en → e, so condition (b) of the theorem is not sat-

isfied. To get around this problem, we use Theorem 12.13 to establish theweak convergence of an invariant distribution π(αqw)(h s) with the propertythat µ(αqw)(h s e)= π(αqw)(h s)Φ(e|s) Since Φ(e|s) is independent of(αqw) the weak convergence of µ(αqw) with respect to (αqw) follows.


We begin by constructing a finite state Markov chain over the space (h s)Let

P∗(αqw)[(h s) (′h′ s′)](40)

≡∫E

G∗(αqw)((h s e) (

′h′ s′E))Φ(e|s)de

Then by (8), (6), (7), and the fact that∫EΦ(e′|s′)de′ = 1 = ∫

EΦ(e|s)de we

have

P∗(αqw)[(h s) (′h′ s′)](41)

=[ρ

∫E

1′∗hs

(e;αqw)=′H∗(αqw)(h s eh

′)Γ (s s′)Φ(e|s)de

+ (1 − ρ)∫E

1(′h′)=(00)ψ(s′ de′)]

It is easy to verify that P∗(αqw) is a transition matrix and therefore defines a

Markov chain on the space (h s)

LEMMA A11: P∗(αqw) induces a unique invariant distribution π(αqw) on

(L2L)

PROOF: The proof follows by applying Theorem 11.4 from Stokey andLucas (1989). Let s ∈ S be such that ψ(sE) > 0 Since newborns mustbe of some type, such an s exists. Then P∗

(αqw)[(h s) (00 s)] ≥ (1 −ρ)ψ(sE) > 0 ∀h s Therefore ε = ∑

(′h′s′)min(hs) P[(h s) (′h′s′)] ≥ (1 − ρ)ψ(sE) > 0, which satisfies the requirement of Theorem 11.4(for N = 1). Q.E.D.

LEMMA A12: If (αnqnwn) ∈ A × Q × W is a sequence converging to(αqw) ∈A×Q×W , where αnα > 0 then the sequence π(αnqnwn) convergesweakly to π(αqw)

PROOF: The proof follows by applying Theorem 12.13 in Stokey and Lu-cas (1989). Part (a) of the requirements follows since L is compact. Part (b)requires that P∗

(αnqnwn)[(nhn sn) ·] converges weakly to P∗

(αqw)[(h s) ·]as (nhn snαnqnwn) → (h sαqw) By Theorem 12.3d of Stokeyand Lucas (1989), it is sufficient to show that for any (′h′ s′)limn→∞ P∗

(αnqnwn)[(nhn sn) (′h′ s′)] = P∗

(αqw)[(h s) (′h′ s′)] Since Lis finite, without loss of generality consider the sequence (αnqnwn) →(αqw). But from (40), limn P

∗(′h′ s′E) of e converges almost every-where to the measurable function G∗

(αqw)((h s e) (′h′ s′E)) of e Since

G∗(αnqnwn)

((h s e) (′h′ s′E)) ≤ 1, requirement (b) follows from the


Lebesgue dominated convergence theorem (Theorem 7.10 in Stokey and Lu-cas (1989)). Part (c) requires that for each (αqw) P∗

(αqw) induce a uniqueinvariant measure; this follows from Lemma A11. Q.E.D.

LEMMA A13: µ(αqw)(h s e)= π(αqw)(h s)Φ(e|s)PROOF: Define m(αqw)(h s) by µ(αqw)(h s e) ≡ m(αqw)(h s)Φ(e|

s) and let Z′ = ′ × h′ × s′ × J ′ Let

µ(αqw)(h s e)≡m(αqw)(h s)Φ(e|s)(42)

Then

µ(αqw)(Z′)(43)

= (Υ(qw)µ(qw))(Z′)

=∫ [

ρ

∫E

1′∗hs


′)

×Φ(de|s)Γ (s s′)∫J′Φ(e′|s′)de′

+ (1 − ρ)1(′h′)=(00)

∫J′ψ(s′ de′)

]m(αqw)(ddhds)

where the first equality follows as a consequence of µ∗ being a fixed point,and the second equality follows from the definitions in (9) and (42), and fromrecognizing

∫EΦ(de|s)= 1.

Letting J ′ =E in (43),

µ(αqw)(′h′ s′E)

=∫ [

ρ

∫E

1′∗hs


′)Φ(de|s)Γ (s s′)

+(1 − ρ)

∫E

1(′h′)=(00)ψ(s′ de′)]m(αqw)(ddhds)

=∫P∗(αqw)[(h s) (′h′ s′)]m(αqw)(ddhds)

where the first equality uses∫EΦ(e′|s′)de′ = 1 and the second follows by defi-

nition (41).But by (42),

µ(αqw)(′h′ s′E)≡m(αqw)(

′h′ s′)∫E

Φ(e′|s′)de′

=m(αqw)(′h′ s′)


Therefore,

m(αqw)(′h′ s′)=

∫P∗(αqw)[(h s) (′h′ s′)]m(αqw)(ddhds)

which implies that m(αqw)(′h′ s′) is the fixed point of the Markov chain

whose transition function is P∗(αqw) Hence m(αqw)(h s) = π(αqw)(h s)

and the result follows. Q.E.D.

LEMMA A14: If (αnqnwn) ∈ A × Q × W is a sequence converging to(αqw) ∈A×Q×W , where αnα > 0 then the sequence µ(αnqnwn) convergesweakly to µ(αqw)

PROOF: Since Φ(e|s) is independent of (αqw) the result follows fromLemmas A12 and A13. Q.E.D.

LEMMA A15: Let α > 0 K(αqw) ≡ ∑(′s)∈L×S

′q′s∫

1′∗hs

(e;αqw)=′ ×µ(αqw)(ddh sde) N(αqw) ≡

∫edµ(αqw) and p(αqw)(′ s)≡ ∫

d∗′0s′(e

′;αqw)Φ(e′|s′)Γ (s;ds′)de′ Then (i) K(αqw) (ii) N(αqw) and (iii) p(αqw)(′ s)are continuous with respect to (αqw)

PROOF: To prove (i), note that by Lemma A13,∫L×H×E 1′∗

hs(e;αnqnwn)=′ ×

µ(αnqnwn)(ddh sde) = ∑h

∫E

1′∗hs

(e;αnqnwn)=′Φ(de|s)π(αnqnwn)(h s).By Lemma A12, limn π(αnqnwn)(h s) = π(αqw)(h s) By Lemma A10,1′∗

hs(e;αnqnwn)=′ → 1′∗

hs(e;αqw)=′ except possibly for a finite number of

points in E By the Lebesgue dominated convergence theorem (Stokeyand Lucas (1989, Theorem 7.10)), limn

∫E

1′∗hs

(e;αnqnwn)=′Φ(de|s) =∫E

1′∗hs

(e;αqw)=′Φ(de|s) Then, since K(αnqnwn) is the sum of a finite num-ber of products, each of which converges, the sum converges as well. Toprove (ii), simply apply Lemma A14. To prove (iii), note that by Lemma A10,d∗hs(e;αnqnwn) → d∗

hs(e;αqw) except possibly for a finite number ofpoints in E By LDCT, limn

∫E×S d

∗′0s′(e

′;αnqnwn)Φ(e′|s′)Γ (s;ds′)de′ =∫E×S d

∗′0s′(e

′;αqw)Φ(e′|s′)Γ (s;ds′)de′ Q.E.D.

With these lemmas in hand, we are ready to prove the existence of a steady-state equilibrium for the case where α > 0 Once this is done, the existenceof equilibrium for the α= 0 case will be accomplished via a limiting argument.Define the vector-valued function whose fixed point gives us a candidate equi-librium price vector. At this point, we need to be explicit about the upper andlower bounds of the sets W and the upper bound of the set Q

ASSUMPTION A2: Assume that qmax = ρ(1 + FK(max emin)− δ)−1 wmin = b,and wmax = FN(max emin).


Note that our earlier assumption that max is such that FK(max emin) > δguarantees that qmax is strictly positive.

Let Ωα :Q×W →RNL·NS+1 be given by 39

Ωα(qw)≡Ωα

′≥0s(qw)

Ωα′<0s(qw)

Ωαw(qw)

(44)

where

Ωα′≥0s(qw)=

ρ(1 + FK(K(αqw)N(αqw))− δ)−1 for K(αqw) > 0,0 for K(αqw) ≤ 0,

Ωα′<0s(qw)

=ρ(1 −p(αqw)(′ s))(1 + FK(K(αqw)N(αqw))− δ)−1

for K(αqw) > 0,0 for K(αqw) ≤ 0,

and

Ωαw(qw)=

FN(K(αqw)N(αqw)) for K(αqw) > 0,FN(0N(αqw)) for K(αqw) ≤ 0.

A fixed point of this function is an equilibrium price vector provided anm∗ ≥ 1can be found for which condition (v) in Definition 2 is satisfied.

LEMMA A16: For α > 0 there exists (q∗w∗) ∈ Q ×W such that (q∗w∗) =Ωα(q∗w∗)

PROOF: The set Q×W is compact. By Assumption A2, Ωα(qw)⊂Q×W To see this, observe that by Assumption A1(iii), FK(max emin) is the lowestmarginal product of capital possible in this economy and, therefore, qmax isthe highest price on deposits possible. The lower bound on wages is the lowerbound on the marginal product of labor in Assumption A1(v) and the upperbound on wages is, by Assumption A1(iii) again, the highest marginal productof labor possible in this economy.

Next we need to establish that Ωα(qw) is continuous in q and w Note thatN(αqw) is always strictly positive since it is bounded below by emin. First, con-sider (αqw) such that K(αqw) > 0 and let (qnwn)→ (qw) By Lemma A15and continuity of FK and FN , it follows that Ωα(qnwn)→Ωα(qw) Second,consider (αqw) such that K(αqw) < 0 Then for any ε > 0 there exists η such

39w can always be made to exceed w by placing assumptions on the production technology.


that for all n≥ ηρ(1 + FK(K(αqnwn)N(αqnwn))− δ)−1

≤ ρ(1 + FK(K(αqnwn) emin)− δ)−1 < ε

Therefore, since ε can be made arbitrarily small, Ωα′<0s(qnwn) → 0 =

Ωα′<0s(qw) and Ωα

′≥0s(qnwn) → 0 = Ωα′≥0s(qw) Furthermore, there ex-

ists η such that for all n ≥ η Ωαw(qnwn) = FN(0N(αqnwn)) Therefore, by

Lemma A15 and continuity of FN , it follows that Ωαw(qnwn) → Ωα

w(qw)Third, consider (αqw) such that K(αqw) = 0 Then Ωα

′<0s(qnwn) → 0 =Ωα′<0s(qw) and Ωα

′≥0s(qnwn)→ 0 =Ωα′≥0s(qw) by an argument similar to

the above case where K(αqw) < 0 Furthermore, for any ε > 0, there exists ηsuch that for all n≥ η K(αqnwn) < ε and hence

FN(εN(αqnwn))≥Ωαw(qnwn)≥ FN(0N(αqnwn))

Therefore, by Lemma A15 and continuity of FN , it follows that

FN(εN(αqw))≥ limn→∞

Ωαw(qnwn)≥ FN(0N(αqw))

Since ε can be arbitrarily small, it follows that Ωαw(qnwn)→ FN(0N(αqw))=

Ωαw(qw)The result follows from Brouwer’s fixed point theorem. Q.E.D.

LEMMA A17: max ≥K(αq∗w∗) > 0

PROOF: If K(αq∗w∗) = 0 then q∗′s = 0 for all ′ by (44). Hence, the op-

timal decision for households with ≥ 0 is to choose ′ = max and the op-timal decision for households with < 0 is either to choose default todayand choose max tomorrow or to pay back and choose max today. Therefore,within at most one period the invariant distribution will have all its masson points with (maxh s e) Hence K(αq∗w∗) = max. But this implies thatΩα′≥0s(0w

∗) = ρ(1 + FK(maxN(0w∗)) − δ)−1 > 0 which yields a contradic-tion. HenceK(αq∗w∗) > 0 Since the asset holding of each household is boundedabove by max it follows that max ≥K(αq∗w∗) Q.E.D.

LEMMA A18: There exists a steady-state competitive equilibrium with α> 0

PROOF: For α > 0 we know there exists (q∗w∗) = Ωα(q∗w∗) by Lem-ma A16. Then provided (v) is satisfied, all the conditions for a competitivesteady-state equilibrium in Definition 2 are satisfied by construction of ΩαObserve that if the hospital sector has strictly positive revenue in the steady


state, that is∫ [(1 − d∗

hs(e;αq∗w∗))ζ(s)+ d∗hs(e;αq∗w∗)max0]dµ∗(45)

> 0

then we can always choose m∗ ≥ 1 to satisfy condition (v). Since we have as-sumed that every surviving household has a strictly positive probability of ex-periencing a medical expense and K(αq∗w∗) > 0 by Lemma A17, (45) is satis-fied. Q.E.D.

We now turn to the proof of existence of equilibrium when α= 0 This proofis constructive. We take a sequence of equilibrium steady states for strictly pos-itive but vanishing cost α and from this sequence construct equilibrium pricesand decision rules that work for the α= 0 case.

To do this, we will need the following definitions. For a given pair of op-timal decision rules (′∗

hs(e;αqw)d∗hs(e;αqw)), define the (optimal)

probability of choosing (′ d) given (h s) and (αqw) as x(′d)

hs (αqw)≡∫E

1′∗hs

(e;αqw)=′d∗hs

(e;αqw)=dΦ(de|s) Further, define the 2 ·NL ·NL element

vector of choice probabilities by x(αqw)≡ x(′d)hs (αqw) ∀(h s) ∈L and(′ d) ∈L× 01

Let a sequence of costs αn → 0 with αn > 0 For each αn, let (q∗nw

∗n) ∈Q×W

be an equilibrium price vector whose existence is guaranteed by Lemma A18.Since Q × W is compact, we can extract a subsequence (q∗

nkw∗

nk) converg-

ing to (qw) ∈ Q × W Let the corresponding sequence of measurable opti-mal decision rules and the sequence of optimal choice probability vectors be′∗hs(e;αnkq∗

nkw∗

nk) d∗

hs(e;αnkq∗nkw∗

nk), and x(αnk q

∗nkw∗

nk). Since each

term in the sequence x(αnk q∗nkw∗

nk) is in [01]2·NL·NL , we can extract another

convergent subsequence converging to some x ∈ [01]2·NL·NL Denote this sub-sequence of nk by m.

Thus, we have a sequence αmq∗mw

∗m, where q∗

m and w∗m are equilibrium

prices, converging to (0qw) and a corresponding sequence of optimal de-cision rules with choice probabilities x(

′d)hs (αmq

∗mw

∗m) converging to x(

′d)hs .

In Lemma A20, we construct, using information on choice probabilities alongthe sequence, measurable decision rules that are optimal for (0 qw) and thatdeliver the limiting choice probabilities x.

Recall that the set of e for which (d) is the optimal action given (αqw)is denoted E(

′d)hs (αqw) and the set of e for which (′ d) is the strictly optimal

action given (αqw) is denoted ES(′d)

hs (αqw). Let I(′d)

hs (αqw) be the setE(

′d)hs (αqw)\(ES(

′d)hs (αqw)), that is, the set of e for which (′ d) is an op-

timal action and for which there is also some other action that is equally good.Further, let ED(′d)

hs (αqw) be the set E\(ES(′d)

hs (αqw) ∪ I(′d)hs (αqw)),


that is, the set of e for which the action (′ d) is strictly dominated by someother action.

The next lemma bounds the measure of the sets ES(d)hs (0 qw) and

ED(′d)hs (0 qw). For convenience, denoteES(d)hs (0 qw) byES

(d),ED(d)

hs(0

qw) by ED(d)

, ES(d)hs (αmq∗mw

∗m) by ES(d)m , E(d)hs (αmq

∗mw

∗m) by E(d)m and

I(d)hs (0 qw) by I(d)

. And denote∫

1e∈A(e)Φ(de|s) byΦs(A). Then we havethe following assertions:

LEMMA A19: For all (h s) ∈ L, the following statements hold: (i)

Φs(ES(′d)

)≤ x(′d)hs , (ii)Φs(ED(′d)

)≤ [1 −x(′d)hs ], and (iii)∑

′∈LΦs(ES(′0)

)+Φs(ES

(01))+Φs(I

(01))= 1 = ∑

′∈L x(′0)hs + x(01)hs.

PROOF: See the supplements to this article on the Econometrica website(Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007)). Q.E.D.

LEMMA A20: For all (h s) ∈ L, there exist measurable functions chs(e),′hs(e), and dhs(e) for which the implied choice probabilities∫E

1′hs

(e)=′dhs(e)=dΦ(de|s) = x(′d)

(hs) and the triplet (chs(e) ′hs(e)

dhs(e)) ∈ χhs(e;0;qw)PROOF: See the supplements to this article on the Econometrica website

(Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007)).We now establish the analogue of Lemmas A12, A14, and A15 for the se-

quence αmq∗mw

∗m converging to (0 qw) Q.E.D.

LEMMA A21: Let π(0qw) be the invariant distribution of the Markov chain Pdefined by the decision rules (′

hs(e)dhs(e)) Then the sequence π(αmq∗mw

∗m)

converges weakly to π(0qw).

PROOF: See the supplements to this article on the Econometrica website(Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007)). Q.E.D.

LEMMA A22: Let µ(0qw) be the invariant distribution corresponding to the de-cision rules ′

hs(e) and dhs(e) Then the sequence µ(αmq∗mw

∗m) converges weakly

to µ(0qw)

PROOF: Since Φ(e|s) is independent of (αqw) the result follows fromLemmas A13 and A21. Q.E.D.

LEMMA A23: Let K(0qw) ≡ ∑(′s)∈L×S

′q′s∫

1′hs

(e)=′µ(0qw)(ddh sde) N(0qw) ≡

∫edµ(0qw) andp(0qw)(′ s)≡ ∫

d′0s′(e′)Φ(e′|s′)Γ (s;ds′)de′

Then (i) limmK(αmq∗mw

∗m)=K(0qw) (ii) limmN(αmq

∗mw

∗m)=N(0qw) and

(iii) limm p(αmq∗mw

∗m)(

′ s)= p(0qw)(′ s)


PROOF: See the supplements to this article on the Econometrica website(Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007)).

Since the choice probabilities along the sequence satisfy all equilibrium con-ditions and the constructed decision rules imply the limiting choice probabili-ties, it is straightforward to establish that all equilibrium conditions are satis-fied by the constructed decision rules as well. Therefore, the pair (qw) is anequilibrium price vector when α= 0. Q.E.D.

THEOREM 5—Existence: A steady-state competitive equilibrium exists.

PROOF: For the sequence q∗mw

∗m converging to (qw) let (′

hs(e)dhs(e) chs(e)) be the decision rules whose existence is guaranteed inLemma A20. Using qw′

hs(e)dhs(e) chs(e), we will construct a collec-tion

qw′hs(e;qw)dhs(e;qw)

chs(e;qw) r ipmNKaBµthat satisfies all the conditions of steady-state equilibrium in Definition 2.

Given qw the conditions we satisfy by construction are the following:(i) chs(e;qw)= chs(e)

′hs(e;qw)= ′

hs(e), and dhs(e;qw)=dhs(e) By Lemma A19 these decision rules solve the household’s optimiza-tion problem for α= 0 q= q and w=w

(x) µ = µ(qw) = Υ(qw)µ(qw) (where Υ is based on (′hs(e;q

w)dhs(e;qw));(vi) N = ∫

edµ;(vii) a′s =

∫1′hs(e;qw)=′µ(qw)(ddh sde).

(viii) K = ∑(′s)∈L×S q′s

′ ∫ 1′hs(e;qw)=′µ(qw)(ddh sde).

(v) m = [∫ [(1 − dhs(e;qw))ζ(s) + dhs(e;qw)max0]dµ(qw)]−1 ·∫ζ(s)dµ(qw).(iib) r = ∂F(KN)/∂K(iv) p′s =

∫d′0s′(e

′;qw)Φ(e′|s′)Γ (s;ds′)de′ for ′ < 0 and p′s = 0 for′ ≥ 0

The conditions we must verify are as follows:(iia) w= (∂F(KN))/∂N

Since (αmq∗mw

∗m) are equilibrium prices and K(αmq

∗mw

∗m) > 0 by Lemma A17,

then for all m,

f (w∗mK(αmq

∗mw

∗m)N(αmq

∗mw

∗m))

≡w∗m − FN(K(αmq

∗mw

∗m)N(αmq

∗mw

∗m))= 0


Observe that f is continuous in all arguments because FN is continuous. There-fore,

limm→∞

f (w∗mK(αmq

∗mw

∗m)N(αmq

∗mw

∗m))=w− FN(KN)= 0

since by Lemma A23 we know limm→∞(K(αmq∗mw

∗m)N(αmq

∗mw

∗m)) = (K(0qw)

N(0qw))= (KN) by construction.(iii) q′s = (ρ(1 −p′s))/(1 + r − δ)

Since (αmq∗mw

∗m) are equilibrium prices and K(αmq

∗mw

∗m) > 0 by Lemma A17,

then for all m and ′ ≥ 0,

f ((q∗′≥0s)mK(αmq

∗mw

∗m)N(αmq

∗mw

∗m))

≡ (q∗′≥0s)m − ρ(1 + FK(K(αmq

∗mw

∗m)N(αmq

∗mw

∗m))− δ)−1 = 0

Observe that f is continuous in all arguments because FK is continuous. There-fore, by Lemma A23 again,

limm→∞

f ((q∗′≥0s)mK(αmq

∗mw

∗m)N(αmq

∗mw

∗m))

= q′≥0s − ρ(1 + FK(KN)− δ)−1 = 0

Similarly,

f ((q∗′<0s)mK(αmq

∗mw

∗m)N(αmq

∗mw

∗m))

≡ (q∗′<0s)m

−ρ(1 − ∫

d∗′0s′(e

′αmq∗mw

∗m)Φ(de

′|s′)Γ (s;ds′))1 + FK(K(αmq

∗mw

∗m)N(αmq

∗mw

∗m))− δ = 0

By the choice of dhs(e;qw) and Lemma A20,

limm→∞

∫d∗′0s′(e

′αmq∗mw

∗m)Φ(de

′|s′)=∫dhs(e;qw)Φ(de′|s′)

Therefore, by Lemma A23,

limm→∞

f ((q∗′<0s)mK(αmq∗

mw∗m)N(αmq∗

mw∗nk))

= q′<0s −ρ(1 − ∫

dhs(e;qw)Φ(de′|s′)Γ (s;ds′))1 + FK(KN)− δ = 0

Finally, since the collection satisfies all conditions for an equilibrium ex-cept condition (ixA), it follows from Lemma A7 that (ixA) is satisfied aswell. Q.E.D.


THEOREM 6—Characterization of Equilibrium Prices: In any steady-statecompetitive equilibrium, (i) q∗

′s = ρ(1 + r∗ − δ)−1 for ′ ≥ 0, (ii) if the grid forL is sufficiently fine, there exists 0 < 0 such that q∗

0s= ρ(1+ r∗ −δ)−1, (iii) if the

set of efficiency levels for which a household is indifferent between defaulting andnot defaulting is of measure zero, 0> 1 > 2 implies q∗

1s≥ q∗

2s, and (iv) when

min ≤ −[emax ·wmax][(1 + r∗ − δ)/(1 − ρ+ r∗ − δ)] q∗mins

= 0

PROOF: (i) Follows from condition (iii) in the definition of competitive equi-librium.

(ii) Let the grid be fine enough so that there is at least one 0 < 0 for whichwmin · emin + 0 > 0 For a household the utility from defaulting on a loan ofsize 0 can be expressed as:

u(e ·ws)+βρ∫u(c∗

01s′(e′;q∗w∗) s′)Φ(de′|s′)Γ (sds′)

+(βρ)2

∫ [λω∗

′∗01s′ (e′;q∗w∗)1s′(q

∗w∗)

+ (1 − λ)ω∗′∗01s′ (e

′;q∗w∗)0s′(q∗w∗)

]×Φ(de′|s′)Γ (sds′)

Since wmin · emin + 0 > 0 an alternative to not defaulting is to pay off the loan,consume the remaining endowment, and in the following period set consump-tion equal to

c∗01s′(e

′;q∗w∗)+ γe′

The utility from this course of action is

u(e ·w+ 0 s)+βρ∫u(c∗

01s′(e′;q∗w∗)+ γe′ s′)Φ(de′|s′)Γ (sds′)

+ (βρ)2

∫ω∗′∗01s′ (e

′;q∗w∗)0s′(q∗w∗)Φ(de′|s′)Γ (sds′)

In view of (28), the utility gain from not defaulting must be at least as large as

u(e ·w+ 0 s)− u(e ·ws)(46)

+βρ∫ [u(c∗

01s′(e′;q∗w∗)+ γe′ s′)− u(c∗

01s′(e′;q∗w∗) s′)

]×Φ(de′|s′)Γ (sds′)


Since consumption is bounded above by emax · wmax + max − min and u(· s)is strictly concave for each s, the integral in the above expression is boundedbelow by ∫

[u(emax ·wmax + max − min + γe′ s′)

− u(emax ·wmax + max − min s′)]Φ(de′|s′)Γ (sds′)

Notice that the above integral is strictly positive and independent of the fine-ness of the grid for L Therefore, since u(· s) is continuous, the expression in(46) will be strictly positive if 0 is sufficiently close to zero. Hence, for a suffi-ciently fine grid there exists an 0 < 0 for which defaulting is not optimal andq∗0s

= ρ(1 + r∗ − δ)−1.(iii) If the set of efficiency levels for which a household is indifferent be-

tween defaulting and not defaulting is of measure zero, by Theorem 4 (themaximal default set expands with liabilities) it follows that d∗

20s(eq∗w∗) ≥

d∗10s(eq

∗w∗) for all e except, possibly, for those in a set of Φ(e|s) measurezero. Therefore,∫

d∗20s(eq

∗w∗)Φ(de|s)= p2s ≥ p1s

=∫d∗10s(eq

∗w∗)Φ(de|s)

and the result follows.(iv) Set min ≤ −[emax · wmax][(1 + r∗ − δ)/(1 − ρ + r∗ − δ)] If a house-

hold has characteristics s loan min, and endowment e · w, then its consump-tion, conditional on not defaulting, is bounded above by e ·w + min − ζ(s)+max′∈L−q∗

′s ·′ Since e ·w≤ emax ·wmax −ζ(s)≤ 0 and max′∈L−q′s ·′ ≤−ρ/(1 + r∗ − δ) · min consumption conditional on not defaulting is boundedabove by emax · wmax + min − ρ/(1 + r∗ − δ) · min ≤ 0 This means either thatthe set Bmin0η0(eq) is empty or that the only feasible consumption is zeroconsumption. In the first case, default is the only option; in the second case,it is the best option by (29). Therefore, in any competitive equilibrium q∗

mins

must be zero. Q.E.D.

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econometrica, vol. 75, no. 6 (november, 2007), 1525–1589 · 2007. 9. 21. · econometrica, vol....

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