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Working Paper Series

A Quantitative Theory of Information and
Unsecured Credit

WP 08-06R

Kartik Athreya
Federal Reserve Bank of Richmond
Xuan S. Tam
University of Cambridge
Eric R. Young
University of Virginia

This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/

A Quantitative Theory of Information and Unsecured
Credit∗
Kartik Athreya†

Xuan S. Tam‡

Federal Reserve Bank of Richmond

University of Cambridge

Eric R. Young§
University of Virginia
July 25, 2011

Working Paper No. 08-06R

∗

We thank Kosuke Aoki, V.V. Chari, Maxim Engers, Juan Carlos Hatchondo, Espen Henriksen, Mark
Huggett, Narayana Kocherlakota, Dirk Krueger, Nathan Larson, Igor Livshits, Jim MacGee, Leo Martinez,
Makoto Nakajima, Borghan Narajabad, Fabrizio Perri, Kevin Reffett, Vı́ctor Rı́os-Rull, Andy Skrzypacz,
Nick Souleles, and numerous seminar participants. We are particularly grateful to two anonymous referees
and the Editor, Steve Davis. We also thank Brian Gaines and Anne Davlin for excellent research assistance.
Tam thanks the John Olin Fellowship and the Bankard Fund for Political Economy for financial support.
Young thanks the Bankard Fund for Political Economy for financial support. The views expressed are those
of the authors and are not necessarily those of the Federal Reserve Bank of Richmond or the Federal Reserve
System. All errors are our responsibility.
†
Corresponding author.
Research Department, Federal Reserve Bank of Richmond, kartik.athreya@rich.frb.org.
‡
Centre for Financial Analysis & Policy, Judge Business School, University of Cambridge,
xst20@cam.ac.uk.
§
Department of Economics, University of Virginia, ey2d@virginia.edu.

1

Abstract
Over the past three decades six striking features of aggregates in the unsecured
credit market have been documented: (1) rising personal bankruptcy rates, (2) rising
dispersion in unsecured interest rates across borrowing households, (3) the emergence
of a discount for borrowers with good credit ratings, (4) a reduction in average interest
rates paid by borrowers, (5) an increase in aggregate debt relative to income, and (6) an
increase in the amount of debt discharged in bankruptcy relative to income. The main
contribution of this paper is to suggest that improvements in the ability of lenders
to observe borrower characteristics can help account for a substantial proportion of
most, though not all, of these observations. A central aspect of our findings is that the
power of signaling is likely to be weaker when it is non-pecuniary costs, rather than
the persistent component of income, that are unobservable. The ex ante welfare gains
from better information are positive but small.
Keywords: Unsecured Credit, Asymmetric Information, Bankruptcy
JEL Classification Codes: D82, D91, E21.

2

1

Introduction

For most of the postwar period the unsecured market for credit has been small. Direct
evidence from the Survey of Consumer Finances (SCF), as well as other sources (Diane Ellis
1998) shows that unsecured credit did not appear in any significant amount in the US until
the late 1960s. However, over the past three decades there have been dramatic changes
in this market. First, and perhaps the most well-known attribute of the unsecured credit
market in the period we consider has been the large increase in personal bankruptcy rates,
from less than 0.1 percent of households filing annually in the 1970s to more than 1 percent
annually since 2002. Teresa A. Sullivan, Elizabeth Warren, and Jay Lawrence Westbrook
(2000) also notes that not only are bankruptcies more common now than before, they are
also larger; as measured by ratio of median net worth-to-median US household income, the
size of bankruptcies grew from 0.19 in 1981 to approximately 0.26 by 1997. More generally,
the use of unsecured credit has intensified; it has nearly tripled, as measured by the ratio of
aggregate negative net worth to aggregate income, from 0.30 percent in 1983, to 0.67 percent
in 2001, to 0.80 percent in 2004 (as measured in the SCF).
Perhaps most dramatically, data from the SCF suggests that the distribution of interest
rates for unsecured credit was highly concentrated in 1983 and very diffuse by 2004. Measured
in terms of the variance of interest rates paid by those who report rolling over credit card
debt, we find that as of 1983 the variance was 7.90 percentage points, but by 2004 this
number had more than tripled to 26.63 percentage points. Interestingly, the change in the
variance of rates has not been accompanied by large changes in the mean spread on unsecured
credit relative to the risk-free rate (measured by the annualized 3-month t-bill rate): these
spreads have fallen by only slightly more than 1 percentage point.
The change in dispersion of interest rates appears to be a consequence of more explicit
pricing for borrower default risk. A variety of financial contracts, ranging from credit card

3

lines to auto loans to insurance, began to exhibit terms that depended nontrivially on regularly updated measures of default risk, particularly a household’s credit score and whether
the household had a delinquent account.1 Comparing data from 1983 and 2004, we find that
the distribution of interest rates for delinquent households shifted significantly to the right
of that of non-delinquent households, with the means of those with past delinquency being
over 200 basis points greater than those with no such events on their records.23
The purpose of this paper is to measure the extent to which these changes may reflect
improvements in the information directly available on borrowers’ default risk. Unsecured
credit markets are a particularly likely place for information to be relevant. Perfectly collateralized lending is, by definition, immune to changes in information: private information
is simply irrelevant. By contrast, in the case of unsecured debt, formal collateral is totally
absent. As a result, changes in the asymmetry of information will likely have consequences
for prices and allocations.
We emphasize that we do not think that information availability is the only thing likely
changed over this period. Many other things germane to unsecured credit market statistics
have changed over the past three decades (see Igor Livshits, James MacGee, and Michéle
Tertilt 2010 for a detailed discussion of the various factors). Our aim is simply to better
1

At least two related findings stand out from the literature. First, the sensitivity of credit card loan rates
to the conditional bankruptcy probability grew substantially after the mid-1990’s (Wendy Edelberg 2006).
Second, credit scores themselves became more informative: Mark Furletti (2003), for example, finds that
the spread between the rates paid by highest and lowest risk classifications grew from zero in 1992 to 800
basis points by 2002
2
All interest rate data is taken from the Survey of Consumer Finances. Specifically, the SCF question
regarding late payments is “Now thinking of all the various types of debts, were all the payments made the
way they were scheduled during the last year, or were payments on any of the loans sometimes made later
or missed?” The variable for SCF1983 is “V930” where a value of “1” means “all paid as scheduled” and
a value of “5” means ”sometimes got behind or missed payments.” For SCF2004 the variable is “X3004”
where the values of “1” and “5” are the same as for 1983.
3
In addition to the direct evidence noted above that information is likely better now than it was in the
earlier periods, there are indirect reasons to suspect so as well. The widespread use of computerized record
keeping of credit histories, and especially, of purchase histories, has likely improved lenders’ assessment of
the borrower’s state, and hence likely repayment probability. As a WalMart executive noted, “If you show
us who you are, we can tell you who you are, maybe even better than you know yourself.” (New York Times,
May 17, 2009)

4

understand the role played by improved information by keeping the environment fixed in all
other ways.
We develop a life-cycle model of consumption and savings in which borrowers vary in
their willingness to repay debts as a result of shocks to both income and non-pecuniary
costs of default. Importantly, lenders cannot always directly observe the state of a borrowing household. Nonetheless, agents’ borrowing and repayment decisions may convey
additional information about aspects of their condition that are otherwise private information. As the result, the gap between allocations under asymmetric information and their
symmetric-information counterparts may be narrowed. One of the answers we provide here
is a quantitative measure of this effect.
Our paper is most closely related to Juan M. Sánchez (2010) and Livshits, MacGee,
and Tertilt (2008). Sánchez (2010) posits that the increased size of the credit market can
be attributed to declines in the cost of offering contracts that separate households by risk
characteristics. Specifically, the era with contractual homogeneity and low levels of unsecured
risky credit is characterized by prohibitively expensive costs of offering screening contracts.
Relatedly, Livshits, MacGee, and Tertilt (2008) argue that in the past, high “overhead” costs
severely limited lenders from offering a wide menu of risky contracts. These papers therefore
both suggest that screening could not, and did not, play an important role in allowing lenders
to overcome the effect of asymmetric information on consumer default risk.4
The absence of screening as a route to escape the classical “lemons” problem does not,
however, rule out the separation of borrowers. In particular, credit market signaling by
borrowers, who are on the informed side of the market, still remains an option: borrowers
will transmit information via the size of the loan they request. The equilibrium level of
4

Our work is also related to Lukasz A. Drozd and Jaromir B. Nosal (2007), which offers a theory of
increased differentiation of borrowers based on declining contracting costs, and to Borghan Narajabad (2007)
who uses improvements in the quality of symmetric information about borrowers to induce changes in the
credit market. Both papers assume strong ex post commitment on the part of lenders, an assumption which
is hard to square with the flexibility credit card issuers have to change the terms at will (see Footnote 10).

5

activity in the consumer credit market then depends on the ability (and desire) of relatively
low-risk borrowers to use debt as a signal to separate themselves from those who pose higher
default risk.
We proceed by studying two settings. First, we allow lenders to observe all relevant
aspects of the state vector necessary to predict default risk. We intend this “Full Information” environment to represent the one currently prevalent (with outcomes compared against
the 2004 SCF, among other sources). Second, we compare the preceding allocation to one
where lenders are no longer able to observe all of these variables. We intend these “Partial
Information” settings to be representative of periods prior to the mid-1980s (and here, compare outcomes against data from the earliest (1983) wave of the SCF, among other sources).
The difference across these allocations is a quantitative measure of the effect of improved
information about shocks in unsecured credit markets.
Our findings suggest that improvements in the ability of lenders to observe borrower
characteristics can help account for a substantial proportion of most, though not all, of
these observations. In terms of bankruptcy, our model suggests that relative to the current
period, information by itself can account for approximately 46 percent of the total change
in bankruptcy seen in the data. In terms of interest rate dispersion, our model suggests that
the change in information captures 77 percent of the change in the variance of interest rates
paid by households, and a similar amount (73 percent) of the change in what we term a
“good borrower” discount. In terms of the change overall indebtedness, the model predicts
that information by itself would have led to an even greater change in borrowing than what
was observed, at 148 percent.
Two somewhat broader messages of our analysis are as follows. First, we find that
in general, asymmetric information is damaging, but not fatal to the functioning of the
unsecured credit market. Second, a central aspect of our findings is that the power of
signaling is likely to be weaker when it is costs germane to default decisions that are not
6

related to income, rather than the persistent component of income, that are unobservable.
The model also suggests that information may not be important in the growth of the average
size of bankruptcies, though it should be stated clearly that the model systematically fails
to generate bankruptcies as large as the data. In what follows, we describe our model and
parametrization scheme and then present results. The final sections provide more detail on
related work and then present some conclusions.

2

The Model

There is a continuum of ex ante identical households, who each live for a maximum of J < ∞
periods, supply labor inelastically until they retire at age j ∗ < J, and differ in their human
capital type, y. A household’s human capital type governs the mean of income at each age
over the life-cycle. A household of age j and human capital type y has a probability ψ j,y < 1
of surviving to age j + 1 and has a pure time discount factor β < 1. Households also vary
over their life-cycle in size. Let nj denote the number of adult-equivalent members present
when the head of the household is age-j. Consumption per person at age-j, cj , is a purely
private good and therefore produces less utility as the number of household members grows.
The economy is one in which all agents take a risk-free rate r as given, and where this
rate is invariant to aggregates generated by the households. There exists a competitive
market of intermediaries who offer one-period savings contracts that promise a deterministic
rate of return r, and also offer contracts for one-period loans that agents may default on by
invoking personal bankruptcy protection. Loans are modeled as arising from the purchase
of debt from households that is discounted at rate q, capturing both default risk and the
maturity of the loan. Specifically, if I ∈ I denotes the information available to lenders,
then let q (b, I) ∈ [0, q f ) be the discount applied to the bond issuance of face value b from a
household. In other words, if a household issues b and the market discounts this issuance at
7

q (b, I), the household then receives q (b, I) b today and owes, outside of bankruptcy, b units
next period.
Lenders utilize available information to assess default risk and offer individualized credit
pricing that is competitive given this risk. Bankruptcy is costly, and has both a pecuniary
cost ∆, and an individual-specific and stochastic non-pecuniary component (that will also
be persistent in the quantitative analysis) denoted by the term λ ∈ Λ ⊆ [0, 1]. The explicit
resource costs of bankruptcy represent legal fees, court costs, and other direct expenses
associated with filing. The existence of nonpecuniary costs of bankruptcy, represented by λ,
is strongly suggested by a range of recent work. First, Scott A. Fay, Erik Hurst, and Michelle
J. White (1998) find that a large measure of households would have “financially benefited”
from filing for bankruptcy but did not. Second, David B. Gross and Nicholas S. Souleles
(2002) and Fay, Hurst, and White (1998) document significant unexplained variability in
the probability of default across households, even after controlling for a large number of
observables. These results imply the presence of implicit collateral, which may or may not
be observable and heterogeneous across households; λ reflects any such collateral, including
(but not limited to) any stigma associated with bankruptcy. However, it also reflects a large
number of other costs that are not explicitly pecuniary in nature (as in Kartik Athreya
2002), such as the added transactions costs associated with a bad credit history (additional
difficulty in renting an apartment or obtaining a cell phone contract). We therefore model
λ as a multiplicative factor that alters the value of consumption in the period in which a
household files for bankruptcy. We will allow it to differ with human capital, to be persistent,
and to potentially vary over time.
Household preferences are represented by the expected utility function
XJ

j=1

X Yj
sj

i=0



βψ j,y Π s

8



j

"

nj
1−σ



ID (λj,y )cj
nj

1−σ #

(1)

where Π (sj ) is the probability of a given history of events sj , and σ ≥ 0 is the ArrowPratt coefficient of relative risk aversion. Letting the current period bankruptcy decision be
denoted by D ∈ {0, 1}, we set ID (λj,y ) = 1 if the household does not choose bankruptcy
(D =0), and ID (λj,y ) = λj,y < 1 otherwise. Our specification of non-pecuniary bankruptcy
costs is such that the higher the value of λ, the higher the risk of bankruptcy, because the
effective “tax” on consumption is smaller when λ takes a relatively high value than a low
one.5
The cost λ evolves stochastically. As a result, there are at least two interactions between
the household’s expectations of future income and the cost of bankruptcy it faces. First, to
the extent that consumption and expected future income are linked, the cost of bankruptcy
varies with income. Second, and more fundamentally, because of the signaling content of
debt those with low future income prospects and a low non-pecuniary cost of default will
again find the cost of bankruptcy, income, and consumption to be related.
By comparing the utility from not filing for bankruptcy with that arising from filing,
while holding consumption and family size fixed, our specification implies that the within1
period purely non-pecuniary cost of bankruptcy is given by c−σ ( 1−σ
−

λ1−σ
).
1−σ

For σ > 1,

(which is the case in our quantitative analysis), the cost of bankruptcy is decreasing in c.6
Thus, comparing two households with different levels of planned consumption in the current
period, the one with the higher level will face the smaller relative effective reduction in utility
arising from bankruptcy. However, there are two main points to note here regarding how
the costs of bankruptcy vary with consumption on the equilibrium path.
First, the decision to file for bankruptcy is made by comparing the value that can be
attained by paying the costs of bankruptcy in the current period with the value attainable
if a household chooses to repay fully. Critically, consumption in these two paths will in
5
6

Mnemonically: “high λ =⇒high risk,” “low λ =⇒low risk.”
We thank an anonymous referee for noting this point.

9

general not be the same, even in the current period. In particular, in equilibrium households
who pay the cost of bankruptcy will be those for whom the immediate relaxation in the
budget constraint arising from the discharge of debts is most valuable. All else equal, these
households will be ones whose current income is low and who expect future income to be
low as well. In this sense, our specification does not make the net benefits of bankruptcy
lower for the “poor” than the “rich.”7
Second, to the extent that those households with relatively good future income prospects
plan to consume more than others, starting in the current period onward they will actually
face a higher penalty (in terms of the absolute level of expenditures that their penalty is
equivalent to) than will those planning to consume less. On the other hand, since the
marginal value of consumption is lower for those planning to consume a relatively high
amount, the bite of this penalty is partially offset.
Our model allows us to study improvements in information arising from two places:
(i) improvements in lenders’ ability to forecast the future income of borrowers, and (ii)
improvements in the ability of lenders to assess factors orthogonal to predicted income,
but germane for the prediction of default risk. As to the latter, the now-common use of
detailed expenditure patterns and general “data-mining” by lenders suggests that they are
indeed interested in gleaning differences in default risk amongst groups whose observables,
perhaps including income, have already been forecasted as accurately by lenders as by the
borrower.8 As mentioned at the outset, our model will suggest that improvements in such
7

An additional, more minor, point here is that the variation in expected future income amongst
bankruptcy filers is not likely to be large, simply because for households with a given human capital
bankruptcy is not chosen when income is relatively high. Thus, the heterogeneity in planned current consumption among those expecting to file for bankruptcy is not likely to be high.
8
See
“What
Does
Your
Credit
Card
Company
Know
About
You?”
(http://www.nytimes.com/2009/05/17/magazine/17credit-t.html? r=1), for an interesting account of
recent practices in credit cards. Among others, the article describes the change in assessment of default
risk by lenders in response to bills indicating that a cardholder has utilized marriage counseling, or whether
cardholders were “logging in at 1 am (as it might indicate sleeplessness arising from anxiety)”, etc. Similarly,
a Canadian company, Canadian Tire, found that those visiting the dentist more frequently were far more
likely to repay than those who went to bars, all else equal. Lenders have also found that those who purchased

10

“softer” forms of information may indeed have played a role in generating the changes seen
in unsecured credit markets. Because we wish to avoid overstating the power of adverse
selection to unravel credit markets, we allow lenders to be perfectly informed about slowmoving components of the individual state vector, such as age and education, as these can
likely be relatively easily inferred even if not directly observed.9
Given that we will study settings with partial information, creditors will generally want
to track the history of default. We allow for tracking via a binary marker m ∈ {0, 1}, where
m = 1 indicates the presence of bankruptcy in a borrower’s past and m = 0 implies no record
of past default. This marker will reset in some future period, capturing the effect of current
regulations requiring that bankruptcy filings disappear from one’s credit score after 10 years
and agents are prohibited from declaring Chapter 7 bankruptcy more than once every 7
years. For tractability, we model the removal of the bankruptcy “flag” probabilistically. We
denote by ξ ∈ (0, 1) the likelihood of the bad credit market flag disappearing tomorrow;
having m = 0 does not prohibit the household from borrowing. This approach means that
some households in our model will be able to declare bankruptcy more than once every 7
years; however, since households in the US economy also have the option to declare Chapter
13 (once every 9 months) and typically have few non-exempt assets (the primary difference
between Chapters 7 and 13 is the dispensation of assets), this abstraction is reasonable and
avoids the need for a cumbersome state variable tracking “the number of periods since a
filing.”
Under partial information, the price charged to a household for issuing debt will generally
depend on m, so that households with recent defaults will receive different credit terms than
carbon monoxide indicators for their homes were far less likely to miss payments than those who purchased
“chrome-skull” car accessories, with the latter likely to be a luxury good. Finally, anecdotal evidence exists
of credit terms being adjusted when households made purchases that are indicative of financial trouble,
such as used tires; Annette Vissing-Jorgenson (2011) contains an investigation of this issue.
9
We ignore regulations that require certain characteristics not be reflected in credit terms, such as those
proscribed by the Equal Credit Opportunity Act.

11

households with “clean” credit. When information is symmetric, this flag is useless, though
it will, in general, be negatively correlated with debt (those with a documented past history
of bankruptcy m = 1 will borrow less on average). This point also illustrates that inferring
the extent to which bankruptcy affects future credit access is not clear cut; it can depend
critically on whether the bankruptcy reveals information relevant for future default risk.
Under asymmetric information, we make an anonymous markets assumption: no past
information about an individual (other than their current credit market status m) can be
used to price credit. This assumption rules out the creation of a credit score that encodes
past default behavior through a history of observed debt levels; since income shocks are
persistent, past borrowing would convey useful information, although it is an open question
how much. Given the difficulties encountered by other researchers in dealing with dynamic
credit scoring (see Satyajit Chatterjee, P. Dean Corbae, and José-Vı́ctor Rı́os-Rull 2011), we
think it useful to consider an environment for which we can compute equilibria.

2.1

Income, Consumption, and Financial Market Arrangement

The timing of decisions within a period is as follows. Agents first draw shocks to their
current period income. Log labor income is the sum of five terms: the aggregate wage index
W , permanent human capital level y ∈ Y (realized prior to entry into the labor market), a
deterministic age term ω j,y , a persistent shock e ∈ E that evolves as an AR(1)

log (ej ) = ρ log (ej−1 ) + ǫj ,

(2)

and a purely transitory shock ν∈ V. Both ǫ and log (ν) are independent mean zero normal
random variables with variances (σ 2ǫ and σ 2ν ) that are y-dependent. In the quantitative
exercises, we will interpret y as differentiating between non-high school, high school, and
college education levels, as in R. Glenn Hubbard, Jonathan Skinner, and Stephen P. Zeldes
12

(1994), and the differences in these life-cycle parameters will generate different incentives
to borrow across types. In particular, college workers will have higher survival rates and a
steeper hump in earnings; the second is critically important as it generates a strong desire to
borrow early in the life cycle, exactly when default is highest. The deterministic age-income
terms ω j,y (as well as the survival probabilities ψ j,y ) also differ according to the realization
of y.
All households then face a purely iid shock to their expenditures, denoted χ ∈ X , that
captures the effect of sudden changes in obligations that the household may not actively
“choose,” but nonetheless acquires. Examples include facing lawsuits, large out-of-pocket
health risk, and unexpected changes to the economies of scale or the legal assignment of
debts within the household, such as those arising from divorce (see Chatterjee et al. 2007 or
Livshits, MacGee, and Tertilt 2010). Lastly, households are required to pay a proportional
tax on labor earnings in each period, τ , to fund pension payments to retirees.
After receiving income and expenditure shocks and paying their taxes, the household
makes a decision regarding bankruptcy: if there is debt maturing in the current period, it
may be repudiated. Conditional on the default decision, the household makes a consumptionsaving decision and then the period ends.
Given the timing and our restriction to one-period debt, if lenders observe all household
attributes relevant for predicting default any bad household-level outcome that be observed
or inferred will immediately be reflected in the terms of credit (to the extent that information
is available or inferred by lenders). As a result consumption smoothing in response to bad
shocks more difficult all else equal: credit tightens exactly at times which it is most needed.
While this may not be an ideal abstraction, we make this assumption both for tractability
and because it keeps our model close to related benchmarks of Chatterjee et al. (2007), and
Livshits et al. (2007). More substantively, the level of commitment on the part of lenders to
not readjusting credit terms to be ex post optimal is not easily observed. What is observed,
13

however, is that until recently (since the CARD Act of 2009), credit contracts explicitly
permitted repricing by the lender at will.10
Denote by b ∈ B the face value of debt (b < 0) or savings (b > 0) that matures today.
Let primes denote one-period-ahead variables (that is, b′ is debt that will mature tomorrow).
If the household chooses bankruptcy, all their debts are removed )including the expense
stemming from the expenditure shock). After the bankruptcy decision, a household’s income
and asset position for the current period are fully determined. Given this vector, a household
of age-j chooses current consumption c, and savings or borrowing, b′ . If the household chose
bankruptcy at the beginning of the current period, they are prohibited for this period only
from borrowing or saving.11 Given the asset structure and timing described above, and using
D ∈ {0, 1} (defined earlier) to indicate whether an agent elected to file for bankruptcy in
the current period or not, the household budget constraint during working age and prior to
the bankruptcy decision is given by
c + q (b′ , I) b′ (1 − D) + ∆D ≤ (1 − D)b + (1 − τ ) W ω j,y eν + (1 − D)χ,

(3)

where q(·) is the locus of bond prices (a pricing function) that awaits an individual with
characteristics I that are directly observable (i.e. objects that do not need to be inferred
from behavior). When the household saves (b > 0) it receives the risk-free price q = 1/(1+r).
10

In an interview with Frontline in 2004, Edward Yingling, at the time the incoming president of the
American Bankers Association, notes that
the contract with the credit card company is that “We have a line of credit with you, but
we do have the right at any time to say we’re not going to extend that credit to you anymore,”
which, by the way, also includes, “We have the alternative to say you are now a riskier customer
than we had when we opened the agreement; we have the right to increase the interest rate,
because you now have become a riskier customer.”
Thus, credit card companies do appear to use this option on occasion.
Exclusion in the filing period follows the literature (Livshits, MacGee, and Tertilt 2007), and reflects the
legal practice of debtors facing judgements for fraudulent bankruptcies. Unlike this literature, however, we
do not impose exclusion in any subsequent period.
11

14

The budget constraint during retirement is
c + q (b′ , I) b′ (1 − D) + ∆D ≤ (1 − D)b + θW ω j ∗ −1,y ej ∗ −1 ν j ∗ −1 + ΘW + (1 − D)χ,

(4)

where for simplicity we assume that pension benefits are composed of a fraction θ ∈ (0, 1)
of income in the last period of working life plus a fraction Θ ∈ (0, 1) of average income
(which has been normalized to one). There are no markets for insurance against any of the
stochastic shocks.

2.2

Loan Pricing

We now detail the construction of the locus of equilibrium prices that will be quoted to
agents attempting to borrow b′ . All households take this locus as parametric, and we simply will refer to this locus as the function q(b′ , I). Recall that I denotes the information
directly observable to a lender. In the full information case, I includes all components of
the household state vector I = (y, e, ν, χ, λ, j, m), while only a subset of these variables are
directly observed under asymmetric information. Under both symmetric and asymmetric
information, we focus on competitive lending arrangements in which lenders must have zero
profit opportunities.
With full information, a variety of pricing arrangements will, under competitive conditions, lead to the same price function. Full information is also the case previously studied
in the literature (see Chatterjee et al. 2007 or Livshits, MacGee, and Tertilt 2010), and is
a special case of our model. In contrast, under asymmetric information it is well known
that outcomes often depend on the particular “microstructure” being used to model the
interaction of lenders and borrowers (Martin Hellwig 1989). Specifically, since we have modeled households as issuing debt to the credit market, we must take into account the fact
that the size of any debt issuance itself conveys information about the household’s current
15

state. In other words, we study a signaling game in which loan size b′ is the signal. As we
will detail further below, the lender’s task is to form estimates of the current realizations
of the two persistent shocks (e,λ), given this signal. Given an estimate and knowledge of
household decision-making, lenders can then compute the likelihood of default, and in turn,
the conditional expectation of profits obtaining from any loan price q they may ask of the
borrower.
Lenders and borrowers play a two-stage game. In the first stage, borrowers name a level of
debt b′ that they wish to issue in the current period. Second, a continuum of lenders compete
in an auction where they simultaneously post a price for the desired debt issuance of the
household and are committed to delivering the amount b′ in the event their “bid” is accepted;
that is, the lenders are engaging in Bertrand competition for borrowers. In equilibrium
borrowers choose the lender who posts the highest q (lowest interest rate, r = q −1 − 1) for
the desired amount of borrowing. Thus, households view the pricing functions as schedules
and understand how changes in their desired borrowing will alter the terms of credit because
they compute the locus of Nash equilibria under price competition.
The Bertrand competition spelled out above leads to equilibrium prices that must not
permit lenders to do better than break even on loans, given their (common) estimate of
′

default risk, once that estimate is updated to reflect the signal sent by households. Let π
bb :
b′|I → [0, 1] denote the function that provides the best estimate of the probability of default,
conditional on surviving, to a loan of size b′ under information regime I. Since default
′

is irrelevant for savers, π
bb is identically zero for positive levels of net worth. In contrast,
′

π
bb is equal to 1 for all debt levels exceeding some sufficiently large threshold. Given the
exogenous risk-free saving rate r, let φ denote the proportional transaction cost associated

with lending, so that r + φ is the risk-free borrowing rate; the pricing function takes into
account the automatic default by those households that die at the end of the period (and
we implicitly assume any positive accidental bequests are used to finance some wasteful
16

government spending).
′

Given any π
bb , the break-even pricing function must satisfy
q (b′ , I) =

2.2.1

Full Information







1
1+r 

′
1−b
π b ψ j,y
1+r+φ

if b ≥ 0

(5)

if b < 0

In the full information setting, given debt issuance b′ and knowledge of the two persistent
shocks e and λ, the lender does not actually need to know the current realizations of the
transitory shock and expenditure shock as they will not help forecast next period’s realization
of the household’s state. We include them here to maintain consistent notation with the
partial information setting to be detailed further below.
Zero profit for the intermediary requires that the probability of default used to price debt
must be consistent with that observed in the stationary equilibrium, implying that
′

π
bb =

X

e′ ,ν ′ ,λ′ ,χ′

π χ (χ′ ) π e (e′ |e) π ν (ν ′ ) π λ (λ′ |λ) d (b′ , e′ , ν ′ , χ′ , λ′ ) .

(6)

Since d (b′ , e′ , ν ′ , χ′ , λ′ ) is the probability that the agent will default in state (e′ , ν ′ , χ′ , λ′ )
tomorrow given current loan request b′ , integrating over all such events tomorrow is what is
necessary to estimate relevant default risk. This expression also makes clear that knowledge
of the persistent components e and λ is critical for predicting default probabilities; the more
persistent they are, the more useful knowledge of their current values becomes in assessing
default risk. We now turn to loan pricing under partial information.

17

2.2.2

Partial Information

“Partial Information” in our model refers to cases in which lenders cannot directly observe
at least one of (i) the current realization of the persistent component of earnings, e, or (ii)
the non-pecuniary cost of bankruptcy, λ. The objects e and λ, because they are the only
persistent components, are the two stochastic elements necessary and sufficient to perfectly
forecast default risk for any given loan request b′ . That is, knowledge of e and λ would, given
b′ , collapse the model to the “Full-Information” case.
To capture the effects of limits on information held by creditors about households, we
also rule out the ability of lenders to directly observe objects that through household decision
rules, would be completely informative about a household’s current e and λ, even when these
objects are not themselves useful for forecasting default risk. These variables are the current
realization of the transitory shock ν, current net worth b, and total income.12 The only
exception to this limit on observability is for the expenditure shocks, χ, which, given the
interpretations it has in the literature, we allow to always be directly observable.
The signaling approach adopted here introduces problems with multiplicity of equilibria:
there will in general be many outcomes satisfying the requirements of equilibrium. Multiplicity arises because the standard solution concept appropriate for games of incomplete
information, Perfect Bayesian Equilibrium, does not fully discipline the beliefs players hold
off the path of equilibrium play. As a result, given enough freedom to pick beliefs the
modeler can deliver many outcomes. In our context, this freedom means that even in cases
when the nature of private information is known a priori to be genuinely irrelevant – for
12

In our setting, if total income, W ω j,y eν, and the transitory shock ν were observable, e would be observable (since W and ω j,y are always assumed observable). An alternative would be to allow total income to
be observed, but not either of its stochastic components (e, ν). This arrangement endows lenders with more
information than we allow them. We will show, however, that even in this case, additional information on
income is actually not very powerful in altering allocations, and so would be even less relevant under the
alternative in which lenders were allowed perfect observability of total income. Lastly, note that we always
assume that lenders can observe age and education, and hence that lenders always know the expectation of
a borrower’s income conditioned on both these objects.

18

example, when all currently privately-held information governing the costs and benefits of
default costs one period ahead involves iid random variables) – one can still deliver equilibria
in which such private information is “led” to matter simply by imposing off-equilibrium-path
beliefs (in)appropriately.
Since our contribution is partly linked to the manner in which we select equilibria, something dependent on the iterative procedure we employ, it is important to check the nature of
outcomes that are selected and the nature of the off-equilibrium-path beliefs that are induced
by our algorithm. We check if our iterative procedure leads to an outcome in which private
information is falsely led to matter in the iid case detailed above, and find that it does not.
This test, along with a proof of convergence to the largest pricing function that satisfies zero
profit for the lenders, provides us with confidence that we are selecting off-equilibrium beliefs
in a reasonable manner.
The lender’s problem is to infer e, λ, or possibly both, as these are the two householdlevel state variables useful for forecasting the bankruptcy decision, one period hence, of a
household who has requested a loan of b′ . We will describe the inference problem for the
case in which neither e nor λ is directly observable. The modifications for the case where
only one of these objects is unobservable are obvious. If lenders cannot directly observe
these items, however, they must resort to constructing an estimate of the value of the pair
(e, λ) received by a household requesting b′ . To make the best possible inference of (e, λ),
lenders will also use any knowledge they have of the decision rules for households and the
distribution of households over the states.
To deal with the inference problem tractably, we will restrict attention throughout to stationary equilibria. Let this stationary joint distribution be denoted by Γ (b, y, e, ν, χ, λ, j, m),
and let the household decision rule for optimal borrowing be given by b′ = g (b, y, e, ν, χ, λ, j, m).
For any given e and λ, not all values of the state vector are consistent with the observables,
particularly the loan request b′ . Therefore, let S{e,λ} denote the set of values for remaining
19

household characteristics, b and ν, that, for a household with given e and λ, could be consistent with the observable vector (b′ , y, χ, j, m). This set is therefore the (possibly set-valued)
pre-image of g(·) for a given e, λ. Notice that the state variables b and ν are both unobservable and useless for directly predicting e′ and λ′ , but nevertheless contain useful information
on current e and λ via the household’s decision rule.
Let Pr (e, λ|b′ , y, j, m, χ) denote the probability of a borrower with observables (b′ , y, j, m, χ)
having current state (e, λ) based on knowledge of the decision rules of agents. In the partial information environment the calculation of Pr (e, λ|b′ , y, j, m, χ) is nontrivial because it
involves the distribution of endogenous variables. In a stationary equilibrium, the conditional probability of a household having any particular (e, λ) pair, given both observables
and lender inference from decision rules, is as follows.
′

Pr (e, λ|b , y, j, m, χ) =

Z

dΓ (b, y, e, ν, χ, λ, j, m)

(7)

S{e,λ}

Given this assessment, the lender can compute the likelihood of default on a loan of size
b′ :
′

π
bb =
2.2.3

X

e,λ

hX

i
′
′
′
′
′
′ ′
′
′
π
(χ
)π
(e
|e)
π
(ν
)
π
(λ
|λ)
d
(b
,
e
,
ν
,
χ
,
λ
)
Pr (e, λ|b′ , y, j, m, χ) .
χ
e
ν
λ
′

e′ ,ν ′ ,χ′ ,λ

(8)

Equilibrium

Our specification of the game between borrowers and lenders makes it essentially identical
to the standard education-signaling model as described in Andreu Mas-Colell, Michael D.
Whinston, and Jerry R. Green (1995), though with an important difference: in our model
the signal is productive. Namely, debt not only may convey information about one’s type,
it allows for intertemporal and inter-state transfers of purchasing power that agents would

20

have chosen even under completely symmetric information. The game described above is a
standard game of incomplete information, and as such, comes with a standard equilibrium
concept: Perfect Bayesian Equilibrium (PBE).
Intuitively, a PBE is given by a set of beliefs and strategies for households and lenders,
where the strategies for each are each optimal given the (common) beliefs they hold, and the
beliefs they each hold are, in turn, derived rationally from the strategies they each (correctly)
anticipate their opponents will use. In our model, optimal household strategy concerns the
amount of debt to issue, and beliefs are over the prices they believe the loan market will
charge them for all debt levels they may consider. In the case of lenders, optimal strategy
concerns deciding what price to post for any debt request given their beliefs and the Bertrand
game they each play with all other lenders. Lenders are required to use the decision rule for
households’ optimal debt issuance and Bayes’ rule wherever possible to arrive at a posterior
probability over a household’s type, given the observables.
A Perfect Bayesian Equilibrium (PBE) (see Mas-Colell, Whinston, Green, Definition
9.C.3, p. 285, and the additional requirement given on p. 452) is as follows. Denote the
state space for households by Ω = B × Y × E × V × L × J × {0, 1} ⊂ R4 × Z++ × {0, 1}
and space of information as I ⊂ Y × E × V × L × J × {0, 1}. Let the stationary joint
distribution of households over the state be given by Γ(Ω). Let the stationary equilibrium
joint distribution of households over the state space Ω and loan requests b′ be derived from
the decision rules {b′∗ (·), d∗ (·)} and Γ(Ω), and be denoted by Ψ∗ (Ω, b′ ). Given Ψ∗ (Ω, b′ ), let
µ∗ (b′ ) be the fraction of households (i.e., the marginal distribution of b′ ) requesting a loan
of size b′ . Lastly, let the common beliefs of lenders on the household’s state, Ω, given b′ , be
denoted by Υ∗ (Ω|b′ ).13
Definition A PBE for the credit market game of incomplete information consists of (i)
13

Recall that the stationary distribution of households over the state space alone is given by Γ(·).

21

household strategies for borrowing b′∗ : Ω → R and default d∗ : Ω × λ × E × V → {0, 1}, (ii)
 1 
such that q ∗ is weakly decreasing
lenders’ strategies for loan pricing q ∗ : R × I → 0, 1+r
in b′ , and (iii) lenders’ common beliefs about the borrower’s state Ω given a loan request of

size b′ , Υ∗ (Ω|b′ ), that satisfy the following:
1. Households optimize: Given lenders’ strategies, as summarized in the locus of prices
q ∗ (b′ , I), decision rules {b′∗ (·), d∗ (·)} solve the household problem.
2. Lenders optimize given their beliefs: Given common beliefs Υ∗ (Ω|b′ ), q ′∗ is the
pure-strategy Nash equilibrium under one-shot simultaneous-offer loan-price competition.
3. Beliefs are consistent with Bayes’ rule wherever possible: Υ∗ (Ω|b′ ), is derived
from Ψ∗ (Ω, b′ ) and household decision rules using Bayes rule whenever b is such that
µ∗ (b′ ) > 0.
2.2.4

Locating Equilibria

We now describe how we find outcomes that satisfy the conditions laid out above.
Off-Equilibrium-Path Beliefs From the household’s perspective, however, it is essential
that they know the price they will face at any debt level they might contemplate; it is only
then that they can solve a well-posed optimization problem. So what is a household to
expect that a lender will infer about them should they contemplate issuing a debt level no
one is expected to choose in a proposed equilibrium? A trivial example in our context is
the PBE in which lenders believe that all borrowers will default with probability one on
any debt, and no one borrows. Given the ability of such pessimism to be self-fulfilling in
signaling models, we structure our iterative procedure to avoid limiting borrowing in such
a manner. Since our process for locating equilibrium price functions is one that generates
22

a monotone sequence of functions, points at which pricing functions reach zero will remain
there in all subsequent iterations.
The component of our equilibrium for which off equilibrium-path beliefs are germane is
the pricing function q ∗ . This function is derived as the fixed point of a mapping that we
describe further below. The locus q ∗ describes for each agent type (where “type” denotes
all directly observable characteristics of a borrower) what pricing they can expect, and what
default risk lenders expect, at all debt levels. However, this pricing function, even if taken as
given by all participants, will not necessarily lead to all debt levels being chosen by any given
agent type. Thus, in each of these places, any value taken by the function q ∗ necessarily
reflects some off-equilibrium-path beliefs on the part of participants.
Finding Equilibrium Loan Prices The only tractable way to locate equilibria is iteratively. Any iterative procedure, in turn, necessarily requires, and ultimately influences, a set
of off equilibrium-path beliefs, and we now describe how our approach does so. We begin
our iterations, for reasons described in more detail below, by positing that lenders hold the
most optimistic views they could have about a borrower: they literally ignore the possibility
of default until it becomes a certainty. That is, the implied beliefs for the intermediary are
such that default is predicted to never occur except when it must (to allow the household to
obtain positive consumption) in every state of the world. As a result, credit availability at
the outset of our iterations, defined in terms of the interest rate for any loan size that one
might obtain, is maximized under this specification.14 . This is because debt levels in excess
of what is observed under this initial pricing function will not be chosen when prices are
(weakly) higher, which they will be in any equilibrium. While we make no formal qualitative
14

It is useful to compare our initial pricing function with the natural borrowing limit, the limit implied
by requiring consumption to be positive with probability 1 in the absence of default. Our initial debt limit
is larger than the natural borrowing limit, as agents can use default to keep consumption positive in some
states of the world; we only require that they not need to do this in every state of the world. This point is
also made in Chatterjee et al. (2007).

23

claims about the extent of credit availability in equilibrium, the manner in which update
pricing for debt helps us preserve credit access along the path to, as well as under, the equilibrium pricing function. The algorithm discussed below is described in more detail in the
Online Appendix.
Given a pricing function, households make decisions about debt. For all debt levels
selected by positive fraction of households, we can then compute the likelihood of default as
a function of their observable type. These are Steps 1-5 in the algorithm. Next, for debt
levels that lie in any range not chosen by any households, we proceed as follows. Let k be
the index of iteration, let q (k) and be the current price function, and µ(k) be the associated
distribution of debt requests induced by agent decisions, given q (k) . For any given debt
request b′ such that µ(k) (b′ ) = 0, define b′ < b− < 0 as the nearest lower debt level relative
to b′ at which µ(k) (b′ ) > 0. b− is therefore the upper end (or right-endpoint) of a segment of
debt levels no household requests, given q (k) . Define qb (b− , I) to be the actuarially fair price
′

at b− ; this is where we use optimal inference by lenders to construct π
bb .

Next, define b+ < b′ < 0 to be the nearest higher debt level (i.e. a more negative value

for assets) relative to b′ at which µ(k) (b+ ) > 0. b+ is therefore the lower end (or left-endpoint)
of the same segment of debt levels, again given q (k) . Denote by qb(b+ , I) the actuarially fair

price at b+ . Thus, at any b, qb (b, I) is the actuarially fair price (see ) at any debt level that is

observed under µ(k) , and equal to qb(b− , I)∀b ∈ (b+ , b− ]. The collection of these segments is

then denoted qb(·, I).15 With qb in hand, as well as the current iterate for the pricing function

q (k) , we construct the next iterate of the pricing function, q (k+1) as a convex combination of
q (k) and qb(·, I) that places weight Ξ on q (k) . The preceding is Step 6.

We want to point our here that we do not confront the agent with the function qb; we

cannot guarantee that the sequence of qb functions is monotone. Instead, we update the
pricing functions extremely slowly using a weight Ξ on the current iterate q (k) close to
15

Note that qb is a (lower semicontinuous) step-function.

24

one (we use Ξ =0.985), and thereby obtain a monotone sequence of pricing functions (see
the proof of convergence in the Appendix where this property is used). This procedure
constitutes Steps 7 and 8 in the algorithm. Repeating this procedure to convergence, we
obtain the equilibrium pricing function and loan request distribution. In addition, note that
the segments starting at b− will, in subsequent iterations, lead those there to potentially
lower their borrowing slightly in the next iteration, as they receive discontinuously better
pricing for doing so—even under the heavily convexified pricing function we will face them
with at that point.16 As a result, some pooling can occur in equilibrium as agents of different
types are inframarginal at any discrete jumps in loan pricing. We will describe some steps
to measure pooling in more detail below.
Given the definition of equilibrium for the game, a stationary equilibrium for the overall
model is standard: we simply require that the pricing functions and distribution of households over the state space are invariant under the optimal decision-making described above,
and that the tax rate τ allows the government to meet its budget constraint.
2.2.5

Government Budget Constraint

The only purpose of government in this model is to fund pension payments to retirees using
a proportional tax on labor earnings, τ . The government budget constraint is

τW

X

y,e,v,χ,λ,j,m

= W

X

y,e,ν,χ,λ,j,m
16

Z

Z

ω j,y eνdΓ (b, y, e, ν, χ, λ, j < j ∗ , m)

b

!

(θω j ∗ −1,y ej ∗ −1 ν j ∗ −1 + Θ) dΓ (b, y, e, ν, χ, λ, j ≥ j ∗ , m)

b

We thank a referee for bringing this to our attention.

25

(9)

3

Calibration

In what follows, we describe our strategy to assign parameters to the model. In terms of
notation, we will use the abbreviation “PI(x)” to denote the partial information settings in
which one or more objects x are not directly observable, and “FI” to denote the setting with
full information. Thus, the case where neither the current e nor λ are visible is denoted
“PI(e, λ),” the case where only e is not observable by “PI(e),” and lastly, the case where
only λ is not visible, by “PI(λ).”
Our calibration assigns numerical values to model parameters under the maintained assumption that the current setting is one of full information. This assumption requires explanation; it is clear that there is asymmetric information between unsecured borrowers and
lenders in the data due to the regulatory constraints we mentioned earlier. Our approach is
driven primarily by two considerations. First, it is computationally intractable to calibrate
our asymmetric information model.17 Second, it turns out that the FI outcomes are actually
close to PI outcomes when the only information that is not directly observable is the current
persistent component of income: FI and PI(e) outcomes are close. In this sense, one could
consider PI(λ) as a benchmark. Anticipating the results, this finding of our model suggests
that it is improvements in the ability of lenders to forecast or assess household characteristics
relevant to bankruptcy decisions that are not summarized by income alone that have been
important in driving the changes seen in unsecured credit markets.
Our targets are (1) the ratio of median debt discharged in bankruptcy to median US
household income, (2) the total fraction of US households with negative net worth, (3) the
conditional mean of debt for those who hold debt of each of the three educational groups and
17

The PI cases simply take too long for us to explore much of the parameter space (one equilibrium takes
10 days even when efficiently parallelized on a 32-processor cluster). Moreover, modifications intended to
make the convergence more rapid have generally led to equilibria in which credit is nearly shut down. Given
that the beliefs for lenders required to sustain those equilibria are somewhat unreasonable, we are stuck with
slow methods.

26

(4) the personal bankruptcy rate for each of the three educational groups. We therefore have
a total of eight targets, so we calibrate eight model parameters. These are (1) the discount
factor β, (2) two values for the non-pecuniary cost of bankruptcy for each educational level
hi
lo
hi
lo
hi
(6 parameters–(λlo
N HS , λN HS , λHS , λHS , λCOLL , λN HS )) and (3) the (common) persistence

parameter for this cost, ρλ .
The statistics on default rates are based on the measures from Sullivan, Warren, and
Westbrook (2000). The debt targets are all from the SCF (2004), and use exactly the same
definitions employed by Chatterjee et al. (2007) and Sánchez (2010), presented in Table
1; the maintained assumption within the model is that households have a single asset with
which to smooth consumption. Given our life-cycle setting, this assumption is not likely
to be crucial for the question we pose. As noted above, in the data (as well as in the
model), defaults are largely the province of the young (Sullivan, Warren, and Westbrook
2000); young households also have few gross assets, implying that negative net worth and
unsecured debt largely coincide (see also Table 2.4 in Sullivan, Warren, and Westbrook 2000).
As in Chatterjee et al. (2007) and Sánchez (2010), we identify debt with negative net worth,
and the specific value we target for aggregate-debt-to-income in the FI setting is 0.67 percent,
the same as theirs. For the measure of borrowers, we choose a target of 12.5 percent, as
it lies in middle of the interval defined by the estimate of 6.7 percent in Chatterjee et al.
(2007) and 17.6 percent in Edward N. Wolff (2006).
To parameterize “expense” shocks, χ, we follow Livshits, MacGee, and Tertilt (2007),
and specify an iid random variable that is allowed to take three values {0, χl , χh }. The shock
χl is set to be 7 percent of mean income, and χh is set at 27 percent of mean income. The
relative likelihoods are Pr(χ = 0)=0.9244, Pr(χ = χh )=0.0710, and Pr(χ = χl )= 0.0046.
As a result, only a minority of household receive shocks, and of these, most do not receive
a catastrophic one. Our calibration target for the model’s bankruptcy rate is 1.2 percent,
in line with the average over the period 2000-2006. The median discharge to median US
27

household income ratio that we use takes the ratio of median debt discharged in Chapter 7
bankruptcy (taken from Sullivan, Warren, and Westbrook (2000), Table 2.4) and compares
this relative to median US household income from the Census Bureau, and is therefore set
to 0.27.18
For earnings risk, we do not calibrate any parameters. We instead use values consistent
with those previously obtained in the literature. We set θ = 0.35 at an exogenous retirement
(model) age of 45 and Θ = 0.2, yielding an overall replacement rate for retirement earnings of
approximately 55 percent. The income process is of the “Restricted Income Profiles (RIP)”
type and is taken from Hubbard, Skinner, and Zeldes (1994), which estimates separate
processes for non-high school, high school, and college-educated workers for the period 19821986.19
In Athreya, Xuan S. Tam, and Eric R. Young (2009) we study the effect of the rise
in the volatility of labor income in the US and find it to be quantitatively unimportant.
Therefore, we use the process estimated on the earlier data even though we compute the FI
case assuming it applies to 2004. The process for ω j,y displays a more pronounced hump for
college types than the others; details are available in the Online Appendix. The shocks are
discretized with 15 points for e and 3 points for ν. The resulting processes have a common
ρ = 0.95, with σ 2ǫ = (0.033, 0.025, 0.016) and σ 2ν = (0.04, 0.021, 0.014) for non-high-school,
high school, and college agents; the measures of the three groups are 16, 59, and 25 percent,
respectively. The risk-free rate on savings is set to 1 percent to reflect the assumption that
households have access to a risk-free and liquid savings instrument. For lending costs, we
18

Alternatively, since we identify defaultable debt with negative net worth in the model, we could use
net worth among filers as our measure of debt discharged. Sullivan, Warren, and Westbrook (2000), p. 72,
reports median net worth among filers in 1997 data as −$10, 542. Comparing this number with US median
household income yields a very similar target. The near-equivalence of gross unsecured debt and negative
net worth reflects the fact that nearly all uncollateralized debts are dischargeable for nearly all filers, because
few have significant positive asset positions (including home equity).
19
Figure 6 in the Online Appendix displays the relative incomes over the life-cycle for all three human
capital groups.

28

set φ = 0.03 to generate a 3 percent spread between rates of return on broader measures
of capital and the risk-free borrowing rate, consistent with transactions costs measured by
David S. Evans and Richard L. Schmalensee (1999). ∆ is set equal to 0.03; if one unit
of model output is interpreted as $40, 000 – roughly median income in the US – then the
filing cost is equal to $1200, and is an estimate inclusive of filing fees, lawyer costs, and the
value of time. Finally, ξ = 0.2589 implies that 95 percent of households who do not file for
bankruptcy again will have clean credit after 10 years.
The calibration generates a stochastic process for the non-pecuniary cost λ with the
following properties. We specify that it follows a Markov chain, and the calibration assigns
this process very high persistence. As a result, it appears that such costs are likely partly
in the nature of a household-level “type,” as they will remain generally unchanged for any
given household over time. However, the variance of the cost is non-trivial, and as a result,
households will differ substantially from each other in their willingness to repay debt, all else
fixed. Lastly, we set risk aversion/inverse elasticity of intertemporal substitution at σ = 2
for all households. The calibrated parameter values are collected in Table 2.

4

Results

We evaluate the implications of changes in information on six aggregates of interest. These
are (1) the overall bankruptcy rate, measured by dividing the American Bankruptcy Institute’s aggregate Chapter 7 filing rate by the number of US households in 2004 (FI case)
and in 1983 (PI cases), (2) the variance of interest rates on unsecured debt in the 2004
SCF (FI case) and 1983 SCF (PI cases), (3) the difference in average credit-card-to-risk-free
rate spreads paid by those with a past bankruptcy on their record relative to those without
one, measured as the difference between the average real credit card interest rate on which
balances were paid and the ex-post real (using PCE deflator) annualized 3-month t-bill rate,
29

(4) the mean interest rate spread paid by borrowers under PI and FI, (5) the average debt
in the economy, as measured by the ratio of total negative net worth to total economywide
income (from the SCF 1983 and 2004), and (6) the median unsecured debt (as measured by
Sullivan, Warren, and Westbrook (2000), Table 2.4) at the time of bankruptcy relative to
US median household income from US Census data from 2004 (under FI) and 1983 (under
PI).20
We now address our main question: how do the six aggregates above change when information is systematically withdrawn from lenders? These change measure the role better
information plays in determining credit market outcomes.

4.1

Full Information

In the full information case, lenders form forecasts of default risk that coincide with the
borrowing household’s own conditional expectation. For brevity, we omit any detailed discussion of the Full Information setting here (the reader is directed to Athreya, Tam, and
Young 2009 for a complete presentation). As a brief summary, Table 1 presents the aggregate
unsecured credit market statistics from the model under full information.21 The calibration
procedure matches closely the targets we set in terms of the fraction of those with negative
net worth, the overall level of unsecured debt relative to income for each education group,
and the bankruptcy rate for each education group. However, the benchmark model is not
able to fully capture the ratio of median discharge to median income. Part of the difference
20

We use the SCF sample weights to obtain a representative measure. An alternative debt-to-income target
is the debt-to-income ratio for households where a member files. However, while debt at the time of filing is
well measured, household income is not. Instead, for a large fraction of filings, only the filer’s income, and
not the household’s income, is recorded. We therefore normalize by US household income instead. While we
do not report it here, the results are very similar when we target these other ratios. In particular, as we will
show, the model does well along a number of dimensions, but fails to capture the size of bankruptcies seen
in the data. To measure the variance of interest rates, we use only those accounts on which interest is paid,
and we do not weight observations by debt levels.
21
Figure 9 in the Online Appendix contains more detail on loan pricing for this case.

30

between the model and the data is that the latter employ a substantially broader measure
by including the large debts associated with small business failures (see Sullivan, Warren,
and Westbrook 2000), a feature which is absent from the model. Unfortunately, the data
that clearly separate discharge according to the nature of the filing (purely personal or small
business) do not exist.

4.2

The Effect of Asymmetric Information

The central goal of our paper is to assess the extent to which improvements in information
held by creditors can help account for the large changes seen in unsecured credit aggregates
over the past three decades. Our results suggest that along several, but not all, dimensions,
the answer is “a substantial amount.” We focus our attention on the behavior of the six
unsecured credit-market aggregates laid out earlier, and measure the extent to which moving
from the FI case to a partial information setting generates changes in these aggregates. We
take the benchmark partial information environment to be PI(e, λ).
4.2.1

Information and Aggregates

The first two columns of Table 3 compare the levels of each of the six items of interest in
the data as of 1983 against the predictions of the PI(e, λ) model, while the third and fourth
column represent recent (2004) data and the predictions of the model under FI. Given that
we have calibrated the model to match the 2004 data as closely as possible, what is relevant
is the extent to which the change in information accounts for differences seen between 1983
and 2004. For ease of exposition, the bottom half of the table displays changes in the levels
of the variables of interest, and the fraction of these changes generated by our model when
we move from an FI setting to the PI(e, λ) environment.
In terms of bankruptcy, our model suggests that relative to the current period, informa-

31

tion by itself can account for approximately 46 percent of the total change in bankruptcy
seen in the data. In terms of interest rate dispersion, our model suggests that the change in
information captures 77 percent of the change in the variance of interest rates paid by households, and a similar amount (73 percent) of the change in the “good borrower” discount.
The model also captures the observed change in the spread of mean unsecured interest rates
relative to the risk-free rate reasonably well (97 percent). In terms of the change overall indebtedness, the model predicts that information by itself would have led to an even greater
change in borrowing than what was observed, at 148 percent.
Finally, we turn to the size of bankruptcies. As stated earlier, we measure the model’s
predictions for the ratio of median debt discharged in bankruptcy to median US household
income. Given this normalization, it is clear that bankruptcies have grown larger over time.
As seen in Table 1, the benchmark calibration is unable to match the size of the median
bankruptcy, and instead generates bankruptcies of approximately half the size. Moreover,
the model predicts that improved information by itself should have left the size of equilibrium
debt-discharge essentially time invariant. One mechanism potentially at play in limiting the
size of debts discharged, even when aggregate borrowing increases, is seen in Table 8 in the
Online Appendix. Nearly 40 percent of bankruptcy filers are those with no expense shock
(“Low χ”), and a high risk of default (“High λ”). Under FI, a switch in λ from low to high
will lead to an increase in the cost of borrowing, and thereby limit the size of debt that an
agent can accumulate. In a PI setting, however, the borrower can continue to accumulate
debts for some time before defaulting. The preceding statements are about the sensitivity of
borrowing terms to personal circumstances. But the average cost of borrowing also matters,
and works in an offsetting direction with respect to the size of bankruptcies: The FI setting is
one in which, in general, credit is cheaper and more households can borrow at any given rate,
relative to a given PI setting. As a result, more households are prone to having bankruptcies
generated via expense shocks, simply because more households can borrow (especially when
32

comparing PI(λ), where only λ is unobservable, to FI).22
One part of the model’s inability to generate large enough bankruptcies likely stems
from our use of relatively flexible one-period debt. As mentioned earlier, not only can such
a setting hinder smoothing to a greater extent than one in which contracts take the form of
credit lines that lenders are implicitly committed to honoring (perhaps due to reputational
concerns), it also may – for the very same reason – be part of why bankruptcies are small.
The difference can certainly be regarded as a shortcoming of the benchmark model with
respect to its ability to generate bankruptcies as large as in the data.
There are two other asymmetric information economies one could consider within the
framework of our model. These are the cases in which only persistent income shocks are
unobservable, PI(e), and the one in which only non-pecuniary costs are unobservable, PI(λ).
Table 4 displays the outcomes under these two information regimes. Two results are worth
noting. First, the ability to observe income is actually not vital to allocations. Comparing
the first two columns of Table 4 shows that allocations are largely similar across these two
information regimes. One reason for this can be seen in Figure 1. Denote by PI(e|λlo
COLL ) the
pricing in the PI(e) case for a college-educated household with non-pecuniary cost λ = λlo
COLL .
Define PI(e|λhi
COLL ) analogously. We see that under either value for the observable λ, the
pricing function confronting a household with the modal persistent income shock (which is
the median e) in a FI setting is quite similar to the single price function that households face
under PI. In other words, for any given observable value of λ, the pricing most households
receive will not differ substantially between FI and PI(e).
The second finding is that the ability to assess a household’s cost of bankruptcy is important. Table 4 shows that when λ is the only unobservable, allocations change quite
substantially relative to FI. While the size of bankruptcies and the measure of borrowers
remains stable, we see that debt/income ratios and bankruptcy rates fall very close to the
22

We thank an anonymous referee for highlighting this mechanism.

33

benchmark private information setting. As a result, the model suggests that not all asymmetric information is equally important, and in particular that asymmetry of information
on income by itself does not lead to large reductions in credit supply. Our findings also
are consistent with the fact that credit card lenders rarely, if ever, follow up on cardholder
income over time, and typically ask for only self-reported income at the time of application.23
4.2.2

How Does Our Approach Matter for the Outcomes Selected?

Both our initial choice of pricing function and our procedure for “filling in the holes” at
intermediate iterates of the pricing function are “optimistic” in the sense that lenders believe
that borrowers are no riskier than anyone with the next lower observed debt level. As a
result, within the interval defined by any two observed debt levels, borrowing costs will not
rise with debt, even though higher debts will generally be associated with greater default
risk.24 Thus, if there are “holes” in the equilibrium set of borrowing levels, it will be despite
the fact that we allowed households to select borrowing levels that were “subsidized” from
their perspective at iterations leading up to the converged pricing function (though of course,
not in equilibrium).
Our algorithm also gives relatively low risk (e.g. high current income) agents relatively
higher incentives to borrow, since our initial pricing function will affect the behavior of the
low risk agents more than high risk households. This is because high risk agents are willing
to pay a premium to borrow that the low risk would not. As a result, our updating is set to
work against going to a much lower pricing function q and remaining there (that is, protected
against much higher interest rates for any given debt issuance).
A second point to cover is the role played by the off-equilibrium-path beliefs that are
23

As reported on a credit card industry website, current practice is to use credit history and
self-reported income at the time of application to estimate income, without verification of income
(http://www.creditcards.com/credit-card-news/credit-card-application-income-check-1282.php).
24
Of course, in equilibrium, lenders will earn zero profits so they are indifferent across different specifications of these beliefs. Our approach just aims to sustain as much lending as possible.

34

induced by our algorithm, and reflected in the converged price function q ∗ (b) facing any
particular type of agent. First, one can see from Figure 1 that the pricing functions are
monotone in the face value of debt. Therefore, one immediate restriction on off-equilibrium
beliefs is that households never believe that lenders would view them as less risky if they
increased the size of the loan they request. This restriction plays a role in the convergence
of the algorithm, and is shown by applying Theorem 6 from Chatterjee et al. (2007).
Next, consider the bottom-right panel in Figure 3, which is the PI(λ, e) case, for a young
(age 29) college-educated household. This type is representative of a group who has the
desire, all else equal, to borrow. For this group, we see that the distribution of debts for
this class of households contains a region where no one borrows (i.e., debt levels in excess of
approximately -0.08 units). Looking at the pricing for these regions indicates that the pricing
never improves with debt, and also inherits the property from the updating procedure that
it does not get worse until one hits the next higher level of debt that attracts a positive
measure of borrowers.
As to the question of how much pooling the model generates, we proceed in two steps.
First, as a general matter, pooling will occur under asymmetric information when the relatively low-risk agent in a given pool finds it not very valuable to lower their requested debt
amount in return for a lower interest rate (higher q). As a way of assessing the overall extent
of pooling, we study a representative set of decision rules across the unobserved components
of the household’s state (b, λ, e). In Figure 2 we plot the law of motion b (which, in a slight
abuse of notation, nests the decision rule for today’s debt issuance and the effect of declaring
bankruptcy today, which forces b′ = 0). Note the close proximity of the decision rules to
each other across values of λ for many debt and savings levels. However, notice also that
these households clearly differ in their default incentives. In particular, for the household
with the median income shock, the level of debt that triggers default (seen in the jump to
lo
b′ = 0) is substantially lower for households with λ = λhi
COLL than for those with λ = λCOLL .

35

The pattern in Figure 2 is representative of what is seen across education levels for most
households in the early part of the life-cycle (e.g. below age 40). Given the distribution of
initial wealth b (more on this below), we conclude that a nontrivial proportion of households
with differing default risk send essentially the same signals in equilibrium. Of course, when
income is low (“lo e”) we see that decisions under either value of λ are similar, as households are driven to default more directly by low income than by the non-pecuniary cost that
bankruptcy carries for them. By definition, however, such realizations of income are rare,
and hence do not carry as much weight in determining aggregate outcomes. The proximity
of decision rules across households with differing λ at relatively low initial debt levels, and
their divergence at higher debt levels, is precisely why the PI(λ) and PI(λ, e) environments
look substantially different from either the PI(e) or FI cases.
The fact that decision rules overlap in many places, however, may still be misleading
because it does not display the fraction of households located at points of the state space
that lead them to send similar signals. To address this issue, we now return to Figure 3.
Throughout this figure we hold λ fixed at λhi
COLL . Beginning with the top-right panel, we
see that there are indeed several combinations of initial wealth b and persistent shock e
that lead households to send identical signals. The horizontal dotted line in the top-right
panel shows one particular signal b′ that is sent by multiple types of households who are
observably indistinguishable, and the top-left panel takes this level of debt (on the y-axis)
and then displays on the x-axis the equilibrium price q for this particular debt level.
The next question is how many households exist at this debt level. To answer this
question we move to the bottom-right panel, which displays the distribution of households
over initial debt (net worth) levels b, and see that households closer to the median e (here, the
line labeled e6 ) are relatively more populous (unsurprisingly) amongst those who borrow b′
than are the other classes of agents who issue the same level of b′ . The latter are, naturally,
those households whose current persistent shock realization is not as good (e.g. e1 ), but
36

whose initial wealth b was relatively high. These households wish to smooth consumption
and so will choose to issue the same debt as their luckier (in terms of e) counterparts.
Lastly, the bottom-left panel displays the fraction of households paying this particular
bond price for their issuances. Note that this price reflects the information we just discussed:
lenders understand that a given signal b′ is not equally-likely to have come from the various
combinations of b, e (and λ in the PI(λ) and PI(λ, e) cases); instead, they will use the
distribution plotted in the bottom-right panel to assess the odds.
Two final points are worth emphasizing. First, there is only one systematic way to
assess the role of the algorithm in equilibrium selection and allocations, which is to check
its performance against a PI setting in which one already has a direct method for solving
for an equilibrium, and then check if iterations lead to the same outcomes. This test is
intractable as a general matter of course, and is why we employ the numerical approach we
do in the first place. Second, there is still one case where we can check the algorithm: when
all unobserved shocks are iid. In this case we know already what the PI outcome is: it is
the FI outcome because the private information concerns variables that the lender has no
interest in calculating. Therefore, one test that our algorithm should pass is that allocations
and prices converge to the FI outcomes. We find that our algorithm passes this test.
4.2.3

Information and Credit Supply

We turn now to evaluating the role that asymmetric information plays on the supply side of
the unsecured credit market. This role is measured by comparing the pricing of unsecured
credit arising from a given asymmetric information setting with others, or the most natural
counterpart from the Full Information setting. Figure 1 contains at least three messages.
First, quite naturally, private information of every variety considered here constrains credit
supply: ceteris paribus, loan prices are high (q is lower) for a given borrowing level under
private information. For example, as we move from FI to PI(e), notice that q under either
37

value for the non-pecuniary cost of bankruptcy shifts credit supply inward. Second, under
all information regimes, unsecured debt in excess of about 5 percent of economywide average income begins to carry default risk: the loan pricing function begins at this point to
fall, and implies an interest rate higher than the risk-free borrowing rate. Third, as emphasized already, the model suggests that private information about bankruptcy costs is
more important than private information about income, for a relatively natural reason. In
an environment where only the persistent shock was unobservable, agents have a way of
separating themselves: an agent with a poor current realization will be willing to pay more
for a large loan to smooth consumption than will one who has received a good shock. As
a result, those asking for large loans in this case will be “correctly” identified as relatively
high risk borrowers, who, because of currently low income, will be willing to pay a price that
is roughly actuarially-fair. As a result, aggregate allocations under PI(e) are relatively close
to those under FI.25
In sharp contrast to the ability of the unsecured market to function relatively smoothly
in the absence of direct information on the current component of persistent income risk,
when non-pecuniary costs are unobservable outcomes are more sharply restricted relative to
FI. Looking again at Figure 1, and holding fixed the current persistent shock at its median
value, we see that a household faces a much higher interest rate for any given debt level
they might issue; if further compounded by unobservable income risk, credit supply shrinks
substantially. In the latter case, this contraction is because an interaction between the desire
to borrow when faced with a particularly bad persistent shock and a low non-pecuniary cost
of default makes it very attractive to borrow, if possible. The strength of adverse selection
incentives can be seen by looking at the FI case. Here, we can see that the difference in loan
pricing for those with high and low λ is very substantial, which further explains why, when
combined with asymmetric information on income e, credit is relatively restricted.
25

We are indebted to an anonymous referee for clarifying this mechanism to us.

38

4.2.4

Information, Interest Rates, and Dispersion

As seen in Table 5, our model makes predictions for the effect of improved information
on the mean and variance of interest rates paid by borrowers who have an indicator of
past bankruptcy (m = 1) on their record. As documented in Edelberg (2006) and Furletti
(2003), past default appears substantially more correlated with credit terms now than in past
decades. In particular, the positive correlation of interest rates with past defaults may be seen
as a form of “punitive” sanctions imposed by creditors. However, under competitive lending
and full information, such penalties will not be viable. Nonetheless, given the persistence
of shocks, the income events that trigger default may well persist, and therefore justify risk
premia on lending. Indeed, we will argue that this view is a plausible interpretation of the
data.
Note first that even under FI, there will generally be a non-zero correlation between the
terms of credit and m; even though the flag is irrelevant under FI, it turns out that many
who file do so when they have been hit by a bad persistent shock. As a result the pricing
they face under FI is worse than that faced by the average household. Table 5 shows how
the flag is related to credit pricing under two PI cases relative to FI.26 The most useful thing
to note is that in either case, the entries for V ar(r|m = 1) and V ar(r|m = 0) show that
the model does fairly well at replicating the level of the dispersion under FI, and also the
fact that dispersion is higher for those with the flag m = 1. However, we also see that the
reduction in information to either the PI(λ,e) case or the PI(λ) case leads to a substantially
smaller reduction in the variance than occurred in the data. For instance, we see that the
reduction in variance predicted by the model (taking those with m = 1) for example, is from
34.55 to 17.07, while in the data, the reduction was almost twice as large, going from 33.88
to 8.68. A related prediction of the model also suggests that information is not the whole
26

See also Table 9 in the Online Appendix.

39

story: the model fails to explain why the difference between the mean interest rates paid by
those with and without a bad flag is close to zero (−22 basis points), instead predicting that
the difference should be a positive spread of 121 basis points.27 The model suggests thus
suggests that information can help explain the difference in dispersion seen amongst those
with a bad flag and those without. This finding is of interest because interest rate dispersion
was never targeted.
4.2.5

Information and Welfare

Having found that changes in information may have been important in explaining changes
in the behavior of the U.S. unsecured credit market, we now address the normative question
of whether these changes are likely to have improved household welfare. The reason that
it is not obvious that ex ante welfare will rise with more precise information is that some
households may receive a subsidy from others by successfully pooling with them under partial
information. If this subsidy is large for households who draw an initial type that will be
worse off than average, then the suppression of information helps the credit market provide
a form of insurance.28
Given our model specification, there are five ways in which the information held by lenders
can improve. Starting from the PI(λ,e) case, lenders can move to one of three information
settings: PI(λ), PI(e), and FI. And starting from PI(λ) and PI(e), lenders’ information can
improve to FI. In addition, because the economy is one with a fixed risk-free rate (and a
tax rate that only depends on the invariant age distribution), the welfare gains are additive:
there are no spillovers across agents as a result of the differential credit constraints they face
across informational regimes. For example, the welfare gain from moving from the PI(λ, e)
27

The Online Appendix contains additional results on the relationship between mean interest rates paid
by borrowers by education-type when m = 1 and m = 0.
28
The preceding is not merely a theoretical curiosity, it is the reasoning behind recent changes in statutory
restrictions on the information that lenders may explicitly use, such as the Equal Credit Opportunity Act.
The literature refers to this outcome as the Hirshleifer effect.

40

case to FI is the sum of three items: the welfare gain/loss from a move from PI(λ, e) to PI(λ)
plus the gain/loss from a move from PI(λ) to PI(e), and lastly from PI(e) to FI. For brevity,
therefore, we present only the minimal number of cases needed to examine all five welfare
improvements. The results are given in Table 6, and report how much welfare would change
if we moved according to each specific improvement in information. Given the additivity,
one can directly add the row entries in any given column to see the gains from particular
improvement in information.
In terms of average the gains or losses from better information, we find that, ex ante,
households are always better off under full information.29 In fact, summing the entries in the
‘NHS’ column of Table 6, we find that these households gain the most from improvements in
information, while the college educated gain the least.30 Part of the reason that the welfare
gains are not huge, and benefit the more skilled by less, is that the PI environments still
allow for considerable amounts of credit, and the relative contraction in credit supply (see
Figure 4, for a picture that displays pricing function for the NHS households, which can be
compared with Figure 1) is smaller for the skilled than the unskilled. From Figure 1 we see
that, holding λ fixed, a move from FI to PI(λ, e), college educated households can still borrow
roughly 10 percent of income at nearly the risk-free rate, while NHS households, as seen in
Figure 4, will pay a default premium for essentially any borrowing they might attempt.
Table 6 also assesses the welfare role played by the non-observability of non-pecuniary costs
λ alone, by comparing welfare in the move from PI(λ, e) to PI(e) and FI (for the particular
value of λ under consideration). The punchline to the preceding results is that the gains in
ex ante household welfare from improvements in information are positive but small.
We also present some welfare calculations involving the elimination of bankruptcy. As
noted in Athreya, Tam, and Young (2009), the asymmetric information problem disappears
29

These welfare gains are computed after the newborn household learns y but before e, ν, χ, λ are observed.
These results run in the opposite way from the gains of eliminating default; Athreya, Tam, and Young
(2009) show that banning default benefits the college types more than the NHS types.
30

41

if borrowers can commit to repayment; after all, if the borrower pays back in all states
tomorrow, the lender has no incentive to estimate the current state. With expenditure
shocks, a no-bankruptcy regime may be infeasible, so we present two related but feasible cases
– we permit bankruptcy only if the household receives an expenditure shock, or we permit
bankruptcy only if consumption would otherwise be negative (or zero). It is obvious that
solving the asymmetric information problem is far less important than fixing the commitment
problem for borrowers, for the reasons discussed in our earlier paper.

5

Concluding Remarks

In this paper, we have shown that improved information held by unsecured creditors on factors relevant for the prediction of default can account for a nontrivial portion of the changes
seen in unsecured credit markets. Given the prohibitive costs of overcoming private information via rich arrays of screening contracts, we show that the remaining route of borrower
signaling could have been effective at separating borrowers according to risk characteristics.
Quantitatively, a central aspect of our findings is that the power of signaling is likely to be
weaker when it is non-pecuniary costs, rather than the persistent component of income, that
are unobservable. A technical contribution of our work is an algorithm to compute equilibria
with individualized pricing and asymmetric information; the algorithm is general and could
easily be applied to alternative settings.
As mentioned in the paper, a feature of recent work on consumer default, including
the present paper, is that it imposes a type of debt product that may not mimic all the
features of a standard unsecured contract offered by real-world credit markets. Given that
credit conditions are typically only adjusted by two events – default or entering the market to
either purchase more credit or to retire existing lines – credit lines may be a more appropriate
abstraction. This issue is tackled in symmetric information settings by Xavier Mateos-Planas
42

(2007) and Tam (2009).
Lastly, the “RIP” income process we use is standard, but future work on bankruptcy
will likely benefit from allowing for learning about income. Both consumption dynamics and
usefulness of bankruptcy ultimately hinge on the persistence of risk faced by households.
Using the “HIP” specification of Fatih Guvenen (2007), where estimated persistence is lower,
is therefore a natural next step in assessing the role of consumer default and bankruptcy,
and the role of information in altering unsecured credit use.

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Econometrica 55(3), pp. 647-662.
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(2007), ”A Quantitative Theory of Unsecured Consumer Credit with Risk of Default,”
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[16] Gross, David B. and Nicholas S. Souleles (2002), ”An Empirical Analysis of Personal
Bankruptcy and Delinquency,” Review of Financial Studies 15(1), pp. 319-47.

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[17] Guvenen, Fatih (2007), “Learning Your Earning: Are Labor Income Shocks Really Very
Persistent?” American Economic Review, 97, pp. 687-712
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Institutions: Where Are We Currently Going?” European Economic Review 33, pp.
277-85.
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Fresh Start,” American Economic Review 97(1), pp. 402-18.
[21] Livshits, Igor, James MacGee, and Michèle Tertilt (2008), ”Costly Contracts and Consumer Credit,” mimeo, University of Western Ontario and Stanford University.
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45

[26] Press, William H., Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery
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Table 1: Calibration Targets and Model Performance
Target
Med(Discharge)/Med(US HH Income)
Fraction(Net Worth < 0)
Agg. NW(NW < 0)/Agg. Income | NHS
Agg. NW(NW < 0)/Agg. Income | HS
Agg. NW(NW < 0)/Agg. Income | COLL
Default Rate | NHS
Default Rate | HS
Default Rate | Coll

Model
0.1329
0.1720

Source
Sullivan et al. (2000)
Chatterjee et al. (2007) &
Wolff (2006)
0.1432 0.0800 the 2004 SCF
0.1229 0.1100 the 2004 SCF
0.0966 0.1500 the 2004 SCF
1.237% 1.228% Sullivan et al. (2000)
1.301% 1.314% Sullivan et al. (2000)
0.769% 0.819% Sullivan et al. (2000)

46

Data
0.2688
0.1250

Table 2: Calibrated Parameters
Parameter
β
ρλ
λlo
N HS
λhi
N HS
λlo
HS
λhi
HS
λlo
N HS
λhi
COLL

Interpretation
discount rate
persist. of non-pec. cost
low val. of λ for Non-High-School
high val. of λ for Non-High-School
Low val. of λ for High-School
high val. of λ for High-School
low val. of λ for College
high val. of λ for College

Value
0.9628
0.9636
0.7675
0.9088
0.7310
0.9320
0.7831
0.9071

Figure 1: Information and Credit Supply
pricing functions, age=29, COLL, m=0
0.96

median e, PI(λ)
0.94

lo

PI(e|λCOLL)
0.92

FI(median e|λlo
)
COLL

0.9

0.88

q

PI(λ, e)
PI(e|λhi

0.86

)

COLL

FI(median e|λhi
)
COLL

0.84

0.82

0.8

0.78
−0.22

−0.2

−0.18

−0.16

−0.14

−0.12

b′

47

−0.1

−0.08

−0.06

−0.04

−0.02

Table 3: Model Performance for Aggregates
1983
2004
Data
Model
Data Model
0.20%
0.72% 1.21% 1.18%
7.90%
15.53% 26.63% 29.01%
−0.22%
1.21% 2.23% 3.01%
10.08%
8.21% 11.07% 9.19%
0.40%
0.42% 0.80% 1.04%
0.192
0.134
0.269
0.133
Data
Model Explain Change
−1.01% −0.46%
45.54%
−18.73% −14.48%
77.31%
−2.45% −1.80%
73.47%
−0.99% −0.96%
96.97%
−0.40% −0.62%
147.62%
−0.077
0.001
−1.35%

Levels
BK Rate
Rate Variation
Good Borrower Discount
Mean Rate Spread
Agg. NW(NW < 0)
Med(Discharge)/Med(US HH Income)
Changes(1983-2004)
BK Rate
Rate Variation
Good Borrower Discount
Mean Rate Spread
Agg. NW(NW < 0)
Med(Discharge)/Med(US HH Income)

Table 4: Unsecured Credit Market Aggregates

Aggregate Debt
Med(Discharge)/Med(US HH Income)
Fraction of Borrowers
Debt/Income Ratio | NHS
Debt/Income Ratio | HS
Debt/Income Ratio | COLL
Default Rate | NHS
Default Rate | HS
Default Rate | COLL

48

FI
PI(e)
PI(λ) PI(λ, e)
0.0104 0.0094 0.0053 0.0042
0.1329 0.1351 0.1371 0.1342
0.1720 0.1709 0.1711 0.1493
0.1432 0.1339 0.1206 0.1203
0.1229 0.1182 0.0964 0.9063
0.0966 0.0944 0.0863 0.0768
1.237% 1.018% 0.809% 0.778%
1.301% 1.197% 0.819% 0.789%
0.769% 0.728% 0.638% 0.463%

Table 5: Dispersion and Credit Sensitivity: PI and FI
1983
2004
Levels
Data PI(λ, e) PI(λ) Data Model
E(r − r f )
10.08
8.21
8.24 11.07
9.17
f
E(r − r |m = 1)
9.86
9.41
9.47 12.85 12.18
E(r − r f |m = 0)
10.08
8.20
8.23 10.68
9.15
V ar(r)
7.90
15.53 15.51 26.63 29.01
V ar(r|m = 1)
8.68
17.07 17.41 33.88 34.55
V ar(r|m = 0)
7.53
15.51 15.49 25.60 29.18
Changes
Data PI(λ, e) PI(λ) Data Model
f
f
E(r − r |1983)-E(r − r |2004)
−0.99 −0.96
E(r|m = 1)-E(r|m = 0)
−0.22
1.21
1.24
2.23
3.03
V ar(r|m = 1)-V ar(r|m = 0)
1.15
1.56
1.92
7.28
5.37
V ar(r|1983)-V ar(r|2004)
18.73 14.48
V ar(r|m = 1, 1983)-V ar(r|m = 1, 2004)
25.20 17.48
V ar(r|m = 0, 1983)-V ar(r|m = 0, 2004)
18.07 13.67

Table 6: Information and Ex Ante Welfare
COLL
PI (λ, e) → PI (λ)
0.029%
PI (λ) → PI (e)
0.023%
PI (e) → FI
0.034%
FI → NBK (if χ = 0)
1.403%
NBK(if χ = 0) → NBK (if c > 0) 3.242%

HS
0.052%
0.030%
0.035%
1.205%
3.432%

NHS
0.097%
0.047%
0.065%
1.103%
5.045%

Note: χ = 0 represents no adverse expenditure shock.
NBK (if χ = 0) : Can only file for bankruptcy if χ > 0
NBK (if c > 0) : Can only file for bankruptcy if c ≤ 0.

49

Figure 2: Evidence of Pooling in Decision Rules
age=29, coll, m=0
0.02
0.01

median e, hi λ

median e, lo λ

0
−0.01

b′

−0.02

lo e, lo λ

−0.03
−0.04

lo e, hi λ

−0.05
−0.06
−0.07
−0.08
−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

b

Figure 3: Measuring Pooling
0

0
−0.02

−0.04

−0.04

−0.06
−0.072
−0.08

−0.06
−0.072
−0.08

e6
e3
e1

b′

−0.02

e9

−0.1
0.85

0.9

0.95

−0.1
−0.1

1

−0.05

0

0.05

0.1

0

0.05

0.1

Ψ(Ω, b′ | b′ < 0)

0.2
0.3
0.15
0.2

0.1

0.1

0
0.85

0.05

q(b=−0.072)

0
−0.1

1

q

−0.05

b

50

Figure 4: Information, Education, and Credit Supply
pricing functions, age=29, NHS, m=0
0.94

λlo
, median e, FI
NHS

0.92

λlo
, PI(e)
NHS

q

0.9

0.88

NHS, median e, PI(λ)
0.86

NHS, PI(λ, e)
0.84

0.82

−0.12

−0.1

−0.08

b

51

−0.06

−0.04

−0.02

6

Appendix: Not For Publication

6.1

Computing Partial Information Equilibria

The imposition of conditions on beliefs off-the-equilibrium path makes the computational
algorithm we employ relevant for outcomes, so we discuss in some detail our algorithm for
computing partial information competitive equilibria. The computation of the full information equilibrium is straightforward using backward induction; since the default probabilities
are determined by the value function in the next period, we can solve for the entire equilibrium, including pricing functions, with one pass. The partial information equilibrium is not
as simple, since the lender beliefs regarding the state of borrowers influence decisions and
are in turn determined by them; an iterative approach is therefore needed.
1. Fix an agent type by observables (y, j, m, χ)
2. Guess the initial function q (0) (b′ , y, j, m, χ) discussed above
3. Solve household problem to obtain b′ = g (b, y, e, ν, χ, λ, j, m), S{e,λ} , and d (b′ , e′ , ν ′ , χ′ , λ′ ) ,
and Γ(·).
4. For all b′ observed, use S{e,λ} and Γ(·) to compute Pr (e, λ|b′ , y, j, m, χ) the probability that an individual is in (e, λ) given observed (b′ , y, j, m, χ) . Knowledge of
Pr (e, λ|b′ , y, j, m, χ) and the distribution of households over the remaining observable
state variables implies Υ(Ω|b′ ).
5. Compute
π
bb (b′ , y, j, m, χ) =

X X X X
e

ν

λ

′

χ′

π χ (χ′ )π b (b′ , y, e, ν, χ, λ, j, m) Pr (e, λ|b′ , y, j, m, χ) ,
(10)

the expected probability of default for an individual in observed state (b′ , y, j, m, χ);
52

′

6. Fill in the “holes” in π
bb for all b′ not observed by applying the interim off-equilibrium
beliefs as described in the main text.

7. Compute an “intermediate” price function qb for all b′ , that is actuarially fair (compet′

itive) given the preceding estimate π
bb :
qb (b′ , y, j, m, χ) =



b′



′

1−π
b (b , y, j, m, χ) ψ j
1+r+φ

for all b ≥ bmin (b, y, j, m, χ) ;

(11)

8. Set
q (1) (b′ , y, j, m, χ) = Ξq (0) (b′ , y, j, m, χ) + (1 − Ξ) qb (b′ , y, j, m, χ)

where Ξ is set very close to 1 (we use 0.985), return to Step 1 and repeat until the
pricing function converges.
Because the household value function is continuous but not differentiable or concave, we
solve the household problem on a finite grid for b′ , using linear interpolation to evaluate
the value function at points off the grid. Similarly, we use linear interpolation to evaluate
q at points off the grid for b′ . To compute the optimal savings behavior we use golden
section search (see William H. Press et al. 1993 for details of the golden section algorithm)
after bracketing with a coarse grid search; we occasionally adjust the brackets of the golden
section search to avoid the local maximum generated by the nonconcave region of the value
function. To calibrate the model we use a derivative-free method to minimize the sum of
squared deviations from the targets; the entire program is implemented using OpenMPI
instructions in Fortran95 on a 32-processor Mac cluster. We make no claims about the
program working for parameter values we have not explored. The computational cost of the
PI equilibria is very large (roughly ten days of computing time per equilibrium), and our
attempts to speed it up have led to equilibria with substantially less borrowing.

53

We first provide a proof that the numerical procedure has a maximal fixed point, and
that this fixed point is the one we converge to. Because the machinery used for this proof
may not familiar to all readers, we provide some basic definitions as well.

6.2

Proof of Convergence

Define Q to be a collection of nondecreasing step functions defined over a finite set of points
D ⊂ R; the number of steps is therefore necessarily finite, so Q itself is finite. Let Q contain
a maximal element 1 and a minimal element 0, using the pointwise ordering of functions
(f - g if and only if f (b) ≤ g (b) ∀b ∈ D); suppose all such members can be ordered.
Endow Q with the Scott topology, which defines order-continuity: a function Φ : Q → Q is
order-continuous (or Scott-continuous) if sup {Φ (R)} = Φ {sup (R)} for each R ⊂ Q.
Since Q is finite, order-continuity reduces to pointwise convergence in Rn .
Define Φ : Q → Q as the mapping defined in the algorithm; heuristically, Φ takes a given
loan pricing function q 0 , constructs the break-even pricing function qb implied by behavior

of agents confronted with q 0 , and then produces a new pricing function q 1 as a convex
combination of the two functions:




Φ q 0 = Ξb
q q 0 + (1 − Ξ) q 0
for some Ξ ∈ [0, 1]. Define Φn (q 0 ) as Φ composed n times.
Lemma 6.1 (Q, -) is a complete lattice.

Proof (Q, -) is clearly a lattice, as it is a collection of lower-semicontinuous step functions
that are nonincreasing and ordered pointwise.

To see that it is a complete lattice, note

that an arbitrary collection of lower-semicontinuous functions has a pointwise supremum,

54

and 0 bounds all collections from below. Therefore, following Brian A. Davey and Hilary
A. Priestley (2002) (Q, -) is a complete lattice.
We will apply the following theorem (see Andrzej Granas and James Dugundji 1986).
Theorem 6.2 (Tarski-Kantorovitch) Let (Q, -) be a partially-ordered set and Φ : Q → Q
be order-continuous.

Assume there exists q ∈ Q such that (i) q ≥ Φ (q) and (ii) every

countable chain in {x|x ≤ q} has an infimum.

Then the set of fixed points of Φ is not

empty. Furthermore, q ∗ = inf n {Φn (q)} is a fixed point and q ∗ is the maximum of the set
of fixed points of Φ in {x|x ≤ q}.
Any complete lattice is a partially-ordered set.
Let d = 1 denote the decision to default next period, and Pr (d = 1|b) denote the probability of default next period conditional on issuing debt b today.
Proposition 6.3 Pr (d = 1|b) is nondecreasing given q (b). That is, the probability of default weakly rises with the amount of debt, holding fixed pricing.
Proof See Chatterjee et al. (2007), Theorem 6.
Conjecture 6.4 Pr (d = 1|q) is nonincreasing at each b. That is, the probability of default
on a given amount of debt is weakly falling in the price of that debt.
This conjecture rules out the possibility that two iterates q n and q n+1 ”cross” each other,
in that q n (b) > q n+1 (b) for some b and q n (b) < q n+1 (b) for other b; that is, it implies that
{b
q (q n )} is a monotone chain. The economic content of the conjecture is that the pool of
borrowers who choose a particular b level in equilibrium cannot improve as q falls, implying
that the equilibrium default rate on that debt level must weakly rise. The conjecture does
not appear to be provable in general, but it is satisfied by the numerical procedure we use.
55

Fortunately, the proof of convergence below does not require {b
q (q n )} to be a monotone
chain, only {q n }; with careful choice of Ξ we have been able to guarantee monotonicity of
{q n } in all cases we examined.
Theorem 6.5 Φ has a maximal fixed point q ∗ = Φ (q ∗ ). Furthermore, {Φn (1)} → q ∗ .
Proof Under Conjecture 6.4, Φ is a monotone nonincreasing mapping in the pointwise
ordering (q n  q n+1 ). Φ is order-continuous because the sequence {q n (b)} is monotone for
each b and confined to a compact set [0, 1] . By the Tarski-Kantorovitch theorem, the set
of fixed points is nonempty and the chain {Φn (1)} → q ∗ , the maximal element of the set of
fixed points.
Uniqueness is not generally assured, since q = 0 is a fixed point; uniqueness therefore
only obtains when there does not exist any fixed point with q ≥ 0. A sufficient condition
for q ∗ 6= 0 is that Λ > 0; in that case, no default will occur at b > −Λ so q = 0 is never
the maximal fixed point.

A separate sufficient condition is λ > 0, since again there will

exist a small enough debt level that will never be defaulted on, although it is not possible
to characterize analytically where this debt level lies. Necessary conditions for q ∗ 6= 0 are
unknown.
Any equilibrium q must be a fixed point of Φ.

Since our program converges to the

maximal fixed point, it converges to the competitive equilibrium with the lowest interest
rate functions. This equilibrium has the property that, with an exogenous risk-free rate r,
budget sets are largest under full information (and therefore consumer welfare is maximized).
Under asymmetric information the first statement still holds (budget sets are larger when
interest rates are lower), but utility may not be maximized for all individuals due to the
potential for pooling; nevertheless, we think that this equilibrium is the natural one to
study.
56

6.3

More on Pooling vs Separating

We now discuss the nature of the equilibrium that we compute.

We define a pooling

equilibrium to be one in which the choice of b is the same for two observably-equivalent
but different households; that is, for PI(λ) we say an equilibrium pools borrowers if either
(i) both the low and high-λ individuals choose the same b given the same remaining state
variables; or (ii) given the same λ two different a individuals choose the same b, given that
they face the same q function.

For PI(e, λ) the definition extends in the natural way.

Inspecting the law of motion for net worth shows that pooling is possible, since a horizontal
line at some b values intersects more than one decision rule, but the existence of possible
pooling does not guarantee it will emerge on the equilibrium path.
Pooling is most likely to occur in the model when (i) the pricing function is flat to the
right of the equilibrium decision and (ii) the indifference curve in (b, q) space is steep to the
right of the equilibrium decision. In the model, almost all borrowers are “constrained” in
the sense that their MRS is not equal to the slope of the pricing function (which is zero at
all points of continuity and undefined elsewhere); that is, borrowers locate at the edge of
“cliffs” in the pricing function that occur when a new state tomorrow is added to the default
set.31

If the flat segment is sufficiently long just to the right of the equilibrium decision

and the indifference curve is steep, the gain from reducing borrowing (in terms of improved
interest rates) is too small, as it requires a very large drop in current consumption to obtain
a small increase in q. As a result, multiple “types” might pool at a given cliff point.
Examining the pricing functions in Figure 1, we see two things. First, the pricing function is relatively steep, so pooling is an unlikely outcome; the relative steepness follows from
standard results that a single asset is sufficient to make the utility cost of fluctuations small,
31
This property is not generic. Because indifference curves are U-shaped eventually, it could be the case
that households choose values for b on the interior of a flat segment of q. We do not find any such outcomes
in our model, however.

57

meaning that households do not mind saving a little more. Forward induction arguments
and/or the test of equilibrium dominance (see In-Koo Cho and David M. Kreps 1987) typically destroy pooling equilibria on the grounds that they are sustained by unreasonable
beliefs about the identity of types who would choose certain off-equilibrium signals; indeed,
with two types their Intuitive Criterion guarantees that agents arrive at the best separating
allocation (the Riley equilibrium), and with three or more types the D2 criterion of Jeffrey
S. Banks and Joel Sobel (1987) delivers the same. In the figure we see that the PI pricing
function closely tracks the pricing function for the low-λ type at low debt levels but the
high-λ type for high debt levels; that is, the lender correctly interprets a low debt issuance
level as likely coming from a relatively-safe borrower.

Thus, while we cannot prove that

our algorithm selects equilibria according to any forward induction argument, it does seem
to capture the flavor of these refinements.

However, as noted in the main body of the

paper, pooling does occur on the equilibrium path, so our procedure does not deliver the
best separating equilibrium as D2 would.
6.3.1

The Roles of λ and ∆ Under Full Information

The non-pecuniary cost of bankruptcy, λ, plays an important role in allocations. It is meant
to capture various aspects of deadweight costs borne by households in bankruptcy. There
are two key aspects to this process – the coefficient of variation and the persistence. If
one forces λ to remain constant and uniform across households when it is chosen to match
the observed filing rate, the model produces a too-small discharge-income ratio and can no
longer capture the heterogeneity in default costs implied by the estimates of Fay, Hurst, and
White (1998). The calibrated value of λ in this case is also too small, in the sense that
the model generates counterfactually-small bankruptcies, and as a result will understate the
welfare costs of frequent default. If λ differs across households but is iid, discharge rates
remain too low as the average λ needs to be small. Without persistence, no household’s
58

implicit collateral is expected to be particularly valuable in the next period and thus cannot
support large debts. Thus, our calibration allows for the high cross-sectional dispersion and
high persistence in λ needed in order to jointly support (i) large risky debts on which default
premia are paid, (ii) frequent default, and (iii) relatively large discharges. If we change the
average λ in the economy, the effect is to move default rates and discharge levels in opposite
directions (see Athreya 2004). Furthermore, changing merely the average λ delivers little
change in the dispersion in the terms. Thus, changes in stigma can be dismissed as the force
driving all of the changes in the unsecured credit market.
Lastly, we discuss the roles played by the two main transactions costs, ∆ and φ. As
noted in Livshits, MacGee, and Tertilt (2007) and Athreya (2004), dropping transactions
costs can potentially deliver the trends in the default rates and debts observed in the data, so
these changes are worth examining as competitor stories. No household in the model would
default on any debt less than this cost (i.e, when b > −∆), so higher values of ∆ can support
larger debts in general. Changes in ∆ only alter the length of the initial flat segment where
risk-free borrowing is sustained (Figure 5 plots price q as a function of borrowing b). Changes
in φ only shift the pricing functions up and down (see Figure 5), altering the cost of issuing
any given amount of debt. Thus, neither change will alter the variance of interest rates that
agents receive, as they affect all agents symmetrically. To get a change in the distribution of
interest rates, one needs to generate changes in the slope of the pricing functions. As a result,
stories that place falling transactions costs at the heart of the changes in the unsecured credit
market cannot account for the homogeneity observed in the earlier period. More details on
experiments with (λ, ∆, φ) are available upon request.
Having given a flavor of how pricing works in the model, we turn now to credit availability
under full information; the ‘supply side’ of the credit market is seen most clearly in the pricing
of debt facing households in varying states of income. For questions regarding unsecured
credit, the young are the most relevant population, and we therefore focus on their access to
59

credit. Figure 1 displays the pricing functions for college types at age 29 given both the low
and high value of λ; as would be expected the higher the realization of e the more credit is
available (at any given interest rate). For low realizations of e the pricing functions look like
credit lines – borrowing can occur at a fixed rate (in this case, the risk-free rate) up to some
specified level of debt, after which the interest rate goes to ∞. For higher realizations the
increase in the interest rate is more gradual, meaning that some risky borrowing will occur in
equilibrium; for some borrowers, the marginal gain from issuing debt is sufficiently high that
they are willing to pay a default premium to do it. The pricing functions for noncollege types
look similar but involve higher interest rates at any given level of debt. Similar pictures arise
for older agents – they are weakly decreasing in debt with more gradual increases in interest
rates for luckier agents. Middle-aged agents (say, age 45) can borrow significantly more than
their younger counterparts, although they choose not to do so in equilibrium because they
are saving for retirement. Given a high value of λ agents can borrow a lot more (as would
be expected).
Lastly, a reason for our focus on improved information as a candidate explanation for the
facts is the increase in dispersion of credit terms observed (and so paid in equilibrium). Figure
7 displays model’s implications for the evolution of the variance of equilibrium borrowing
rates over the life-cycle across the two main information regimes we consider. These rates
are not weighted by the amount of debt, they are direct measures of dispersion computed
identically to what we measure from the SCF. The dispersion in interest rates is fairly flat
over the life-cycle, and the variance is systematically higher for the less educated groups, since
those groups are the ones who borrow and default on the equilibrium path. In the model,
agents are willing to pay fair premiums for the option to default and do so. Furthermore,
since all pricing is actuarially-fair with respect to default risk, agents who pose higher risk
will pay higher prices to borrow. We also note that in general, the less well educated pay
higher interest rates throughout life than their better educated counterparts. Given the
60

earnings process, this is not surprising. However, what is interesting here is that the model
suggests that partial information may lead to lower average interest rates, or alternatively,
that improvements in information may well coincide with the observation of more households
borrowing at higher interest rates. The reason is intuitive, and reflects the pricing functions
displayed earlier. In essence, loan interest rates under partial information are frequently high
enough to discourage borrowing.
6.3.2

Information and the “Causes” of Bankruptcy

Here, we detail the joint role played by expenditure shocks and non-pecuniary costs of
bankruptcy. The results are given in Table 7. Each cell in which the information regime
is held fixed presents the joint distribution of expenditure shocks and non-pecuniary costs
of those who have filed for bankruptcy. What is clear is that under all information regimes
the bulk of filers have low non-pecuniary costs (high λ). More interestingly, defaulting
households typically have not received the largest expenditure shock. Specifically, the median
expenditure shock is zero, and yet households with this realization account for more than half
of all filers. Perhaps the most interesting finding here is that as that information becomes
more limited, we see that the fraction of households in bankruptcy who have a low nonpecuniary cost of filing and have received no expenditure shock grow dramatically. This
result suggests that “strategic bankruptcy” is certainly a real possibility under PI regimes
in a way that is more severely restricted under FI. In particular, while the latter group
accounts for 46 percent under FI, they are 76 percent of all filers under PI(λ, e). With
respect to income, Figure 8 shows that most defaulting households have not experienced
extremely bad transitory shocks.
Despite the possibility that the joint distribution of expenditure shocks and non-pecuniary
costs suggests some strategic filing, what is still true is that the high expenditure shock is
rare, and so may not be seen often among filers for that reason alone. Nonetheless, such
61

a shock may well “push” someone into bankruptcy when it occurs. Table 8 presents the
conditional probability of bankruptcy, given a particular constellation of the two variables of
interest. It is clear here that the high shock does indeed “cause” a disproportionate amount
of bankruptcy, relative to its unconditional likelihood. Moreover, table 8 makes clear the
power of this shock: conditional on getting this bad expenditure shock, the probability
of bankruptcy is not highly sensitive to either the non-pecuniary cost or the information
regime that prevails. By contrast, filing probabilities vary much more substantially across
information and non-pecuniary costs when the expenditure shocks are smaller.

6.4

Tables
Table 7: Expenditure Shocks and Non-Pecuniary Costs Among Filers
FI
PI
High λ Low λ High λ
Low χ
0.3776 0.0118 0.1664
Median χ 0.4621 0.0623 0.6299
High χ
0.0515 0.0346 0.0725

(e)
PI
Low λ High λ
0.0065 0.0405
0.0755 0.7793
0.0492 0.0849

(λ)
PI (λ, e)
Low λ High λ Low λ
0.0000 0.0496 0.0000
0.0393 0.7608 0.0476
0.0561 0.0853 0.0567

Table 8: Expenditure Shocks and Non-Pecuniary Costs as Triggers

Low χ
Median χ
High χ

6.5

FI
PI (e)
PI (λ)
PI (λ, e)
High λ
Low λ
High λ
Low λ
High λ
Low λ
High λ
Low λ
0.964% 0.030% 0.385% 0.015% 0.068% 0.000% 0.077% 0.000%
15.380% 2.071% 18.986% 2.276% 17.123% 0.863% 15.430% 0.965%
26.422% 17.751% 33.728% 22.889% 28.792% 19.025% 26.702% 17.750%

Figures

62

Table 9: Credit Sensitivity

Mean
COLL
HS
NHS

b < 0, m = 0
b < 0, m = 1
b
r
b
r
0.0718 7.33% 0.0688 9.74%
0.0402 9.58% 0.0341 12.55%
0.0310 12.16% 0.0285 13.71%

Figure 5: The Role of Transactions Costs
pricing functions, age = 29, λlo

, w/ m = 0

COLL

0.98

0.96

low φ

0.94
baseline

0.92

0.9

q

low ∆
0.88

0.86

0.84

0.82

0.8
−0.22

−0.2

−0.18

−0.16

−0.14

−0.12

b’

63

−0.1

−0.08

−0.06

−0.04

−0.02

Figure 6: Labor Productivity over the Life-cycle
2.0

1.8

Efficientcy Units

1.6

COLL

1.4

HS

1.2

1.0
NHS
0.8

0.6

0.4
20

25

30

35

40

45

50

55

60

65

age

Figure 7: Variance of Equilibrium Borrowing Rates over the Life-Cycle
−3

x 10

5

NHS, PI(λ, e)

NHS, FI

COLL, FI

COLL, PI(λ, e)

HS, PI(λ, e)

HS, FI

var(r | b<0, age)

4

3

2

1

0
25

30

35

40

45

age

64

50

55

60

Figure 8: The Income Shocks of Defaulters
0.14

median ν

0.12

prob(e,ν | B=1)

0.1

0.08

0.06

high ν

0.04

0.02
low ν
0

1

3

6

9

12

15

e

Figure 9: Credit Supply Under Full Information
pricing functions, age = 29, FI
0.96

lo

0.95

λNHS, hi e

λhi

, hi e

hi
λNHS,

hi e

COLL

lo
λCOLL,

0.94

hi e

0.93
λhi

q

0.92

, mid e

COLL

0.91

lo

λCOLL, mid e

0.9

λlo
, mid e
NHS

0.89
λhi , mid e
NHS

0.88

0.87

0.86
−0.6

−0.5

−0.4

−0.3

b’

65

−0.2

−0.1