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Working Paper Series

The Role of Information in the Rise
in Consumer Bankruptcies

WP 09-04

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Juan M. Sanchez
Federal Reserve Bank of Richmond

The Role of Information in the Rise
in Consumer Bankruptcies
Juan M. Sánchez∗
juan.m.sanchez@rich.frb.org

Federal Reserve Bank of Richmond†
April 2009
Working Paper No. 09-04
Abstract
Consumer bankruptcies rose sharply over the last 20 years in the U.S. economy.
During the same period, there was impressive technological progress in the information sector. This paper provides a theory to understand and quantify the role of
improvements in information technologies in consumer credit markets. Informational
frictions restrict the amount of debt that can be borrowed. In fact, in the equilibrium
in which investing in information is too expensive, many households borrow such small
amounts that the default risk is very low. When information costs drop and informational frictions vanish, those households borrow more and default is likely after a bad
shock. Quantitative exercises show that information costs have a significant effect on
the bankruptcy rate. Additionally, a drop in information costs generates changes in
other variables (e.g. interest rate dispersion) similar to what has occurred over the last
20 years.

Keywords: Consumer Debt, Bankruptcy, Informational Frictions
JEL classification: E43, E44, G33.
∗

My debt to Arpad Abraham, Jeremy Greenwood, and Jay Hong cannot be overstated. For helpful
discussions and insightful comments, I thank Mark Aguiar, Paulo Barelli, Mark Bils, Maria Canon, Harold
Cole, Emilio Espino, William Hawkins, Jim MacGee, Leonardo Martinez, Jose Mustre-del-Rio, Ronni Pavan,
Jose-Victor Rios-Rull, Balazs Szentes, Michele Tertilt, Rodrigo Velez; and participants in seminars at the
University of Rochester, Carlos III, Alicante, ITAM, Colegio de Mexico, ASU, Bank of Canada, FRB Richmond, and NYU; and conferences at Washington University (Money, Credit, and Policy), UPenn (Meetings
of Midwest Macro), Carnegie Mellon (Econometric Society), and MIT (SED). I also thank Julia Forneris for
editorial support. All remaining errors are mine.
†
The views expressed in this paper are those of the author, and do not necessarily reflect those of the
Federal Reserve Bank of Richmond or the Federal Reserve System.

1

1

Introduction

Consumer bankruptcy is a central issue today because of its explosive rise over the
last 20 years in the U.S. economy. Although many explanations have been proposed,
there is still no conclusive understanding of these trends. A possible reason is the
drop in information costs. This driving force may be important because during the
same period, there was impressive technological progress in the information sector—
often called the IT revolution—and the financial sector uses information intensively to
evaluate credit risk.1 The purpose of this paper is to provide a theory to understand
the effect of information on bankruptcy and to quantify its importance.
The number of annual bankruptcy filings increased by 1.3 million—from 286,444
to 1,563,145, almost 5.5 times—between 1983 and 2004, as depicted in Figure 1. Before the early 1980s, the rise in bankruptcy was moderate. According to Moss and
Johnson (1999), “from 1920 to 1985, the growth of consumer filings closely tracked the
growth of real consumer credit. Since then, however, the rate of increase of consumer
bankruptcies has far outpaced that of real consumer credit.” This conclusion is drawn
looking at the ratio of bankruptcies filings to real consumer credit.2 A similar measure,
the ratio of bankruptcy filings to the number of households in debt, will be used here.
This statistic, referred to as the bankruptcy rate hereafter, increased from 0.92% to 3%
between 1983 and 2004.
To study the role of the IT revolution on consumer bankruptcy, this paper extends
Chatterjee, Corbae, Nakajima, and Rios-Rull (2007) to incorporate informational frictions. The key ingredient is that the persistent component of income, referred to as
the productivity group hereafter, is unobservable. Lenders would like to know this
component because persistence implies that the productivity group will be useful to
estimate the risk of bankruptcy. To capture the existence of debt contracts with different intensity in the use of information technologies, two alternative contracts will be
considered. Screening contracts require the use of an screening technology to identify
a household’s permanent component of productivity. Think of a household answering
some questions to a credit card company’s employee (job characteristics, ZIP code,
total household income, monthly rent/mortgage payment, etc.) who enters the information into a computer and after a few minutes, tells the applicant about the offer’s
characteristics (mainly interest rate and amount that can be borrowed).3 Two things
1

For a careful description of the use of information technologies in the financial sector see the work of
Berger (2003). For an analysis of the effect of progress in monitoring technologies on the allocation of capital,
firms’ financing, and capital deepening, see the study of Greenwood, Sanchez, and Wang (2007).
2
See Figure 2 in Moss and Johnson (1999).
3
Similarly, think of a household filing a credit card application on Internet, or credit card companies

2

are important to notice about this contract: (i) the offer’s characteristics depend on
information about the applicant, and (ii) information is processed using information
technologies and therefore the state of information technologies will affect the cost of
this contract. The alternative contracts are called revelation contracts. Think of credit
markets populated by lenders who compete with one another in the design of contracts
intended to separate the different type of borrowers.4 These contracts do not require
information about a particular household nor do they require the use of information
technologies. Lenders design ex ante a menu with combinations of interest rates and
amounts of debt and it is up to the households to choose which pair of debt amount and
interest rate they prefer.5 The contract’s design induces the households to choose the
contract that was designed for them. The possibility of self-revelation using interest
rates and amounts borrowed in the Eaton and Gersovitz (1981) model of default has
not been noted before.6 This self-revelation is possible because the trade-off between
interest rates and borrowing amounts is different for households in different productivity groups. In particular, since low-productivity-group households are more likely
to file for bankruptcy, they accept a higher interest rate to be able to borrow more.
As a consequence, with revelation contracts, households above the lowest productivity
group obtain lower interest rates at the cost of borrowing smaller amounts.
Given the state of information technologies, which determines the cost of screening
contracts, households (lenders) decide which debt contract they prefer to use (offer).
The advantage of screening contracts is that households can borrow as much as they
want at their zero-profit interest rates. The main disadvantage of these contracts is that
they have an additional cost as they require the use of information technologies. As a
consequence, when screening costs are high enough, revelation contracts are preferred.
That is the key to understand how a drop in information costs generates more debt and
bankruptcy: Under revelation contracts some households cannot borrow as much as
they would like at their zero-profit interest rates, i.e., they are borrowing-constrained.
Then, as information costs drop, those households switch to screening contracts to
borrow more, and the number of bankruptcy filings rises. More debt generates more
bankruptcy because the benefit from bankruptcy—discharge of debts—is increasing in
the amount owed, while the costs—temporary exclusion from financial markets and
gathering information on a household and sending the offer by mail.
4
See chapter 11 in Hirshleifer and Riley (1992) for an excellent survey of theoretical results in this
environment.
5
Similarly, they could mail all the alternative offers to the households or offer all of them in a bank branch.
6
The applicability of this result goes beyond credit card markets. Notice that the Eaton and Gersovitz
(1981) model of default has been widely used in international finance (Aguiar and Gopinath, 2006; Arellano,
2008; Cuadra and Sapriza, 2006, 2008; Mendoza and Yue, 2008).

3

income lost—are independent of the household’s debt size.
This paper also evaluates the effect of information costs on bankruptcy quantitatively. First, the model is calibrated to account for relevant features of the U.S. data
for the year 1983. Specifically, it reproduces the bankruptcy rate, the debt-to-income
ratio, the capital-to-output ratio, and some moments of the joint distribution of debt
and income. The model also matches well non-targeted moments. Then, the model
is used to answer a quantitative question: Can the rise in the bankruptcy rate be
explained by the drop in information costs? Unfortunately, there is not a direct measure of the cost of obtaining and processing household information about the risk of
default. Therefore, to answer this question, the technology in the information sector is
recalibrated. A simple strategy is followed: This technological parameter is calibrated
to match the bankruptcy rate in 2004. Then, the model economy is computed with
all the parameters for 1983 but with the technology in the information sector for 2004.
The model is then challenged by analyzing implications of a drop in information costs
for: (i) the distribution of debt across income groups, (ii) the dispersion of interest
rates, and (iii) the projection of interest rates on income, are analyzed. The model
does reasonably well, even though these facts were not targeted.
The first quantitative paper studying the rise in consumer bankruptcies is Livshits,
MacGee, and Tertilt (2007a).

They argue that a drop in stigma (utility cost of

bankruptcy) together with a drop in transaction costs can explain the rise in bankruptcy.
Narajabad (2007) proposes an alternative explanation to understand the same fact: the
information technology revolution. Together with the current paper, Athreya, Tam,
and Young (2008); Drozd and Nosal (2008); and Livshits, MacGee, and Tertilt (2009)
present different models to evaluate this driving force.7
Narajabad (2007) evaluates the role of more informative credit rating technologies in
an environment with heterogeneity in the cost of bankruptcy. There are two important
differences between Narajabad (2007) and the current paper. First, households do
not know their type—their own cost of bankruptcy—when they sign a debt contract.
This assumption is crucial because it makes revelation contracts impossible and implies
that the key mechanism at work in this model is ruled out by assumption. Second,
restrictive assumptions make his model not very suitable for quantitative purposes. For
instance, by assuming that households cannot save, he makes any comparison between
the model and data on the distribution of assets—key in a model of debt—impossible.
7
Other recent papers on consumer debt and bankruptcy with informational frictions are Chatterjee,
Corbae, and Rios-Rull (2008, 2007). They incorporate asymmetric information in the Chatterjee, Corbae,
Nakajima, and Rios-Rull (2007) model of consumer debt and bankruptcy. They find that informational
frictions can generate exclusion after bankruptcy as an equilibrium outcome.

4

Athreya, Tam, and Young (2008) present a quantitative model of unsecured debt
with informational frictions. The equilibrium in the environment called “partial information” could be compared with the equilibrium in the current paper when the cost of
information is high enough that all lenders use revelation contracts. However, there are
two crucial differences. First, they consider signalling equilibria. Lenders’ beliefs are
crucial. Basically, beliefs must be used to select which equilibrium will be analyzed.8
Given the choice of beliefs, they find an equilibrium in which there is practically no
borrowing in the environment with partial information. This differs substantially from
the current paper in which, as explained above, lenders (partially) offset the lack of
information by designing debt contracts accordingly. This difference is very important
if we bear in mind that what was puzzling about the last 20 years is that the rate
of increase of consumer bankruptcies has far outpaced that of real consumer credit.
Second, only extreme cases in terms of information availability can be compared. That
is, only special cases of the model in the current paper can be studied.
Drozd and Nosal (2008) present a search model of the market for unsecured credit.
They study the effect of a drop in the cost of screening and soliciting credit customers
on debt and bankruptcy. The cost of screening is potentially close to the current paper.
However, Drozd and Nosal (2008) do not model asymmetric information, lenders have
no alternative to paying the cost of screening, and therefore their cost is more related
to a transaction cost as the one analyzed by Livshits, MacGee, and Tertilt (2007a).
Livshits, MacGee, and Tertilt (2009) present a very stylized model with informational
frictions in which lenders must pay a cost to design a contract. The model is used to
explore, qualitatively, the implications of technological progress in consumer lending.
The most remarkable prediction of the model that is supported by the data is that the
empirical density of credit card interest rates has become more dispersed since 1983.
The quantitative model in the current paper also has this prediction, and the results
are directly compared to the data.

2

The mechanism in a 2-period model

This section previews the main driving forces at work in the full model using a simple
2-period model. An additional simplification is that the analysis is in partial equilibrium; i.e., the risk-free interest rate, i, and wages, w, are given. All the proof of this
section’s lemmas are in the appendix.
8

That is why Hirshleifer and Riley (1992) conclude that the Nash equilibrium concept has little predictive
power in that environment.

5

Physical environment. The economy is populated by households and lenders. Households live for 2 periods, t = 1, 2, and they are endowed with a quantity of labor measured in efficiency units, ln , that can take 2 values, ln ∈ {lL , lH }, meaning low or high
productivity. The transition probability between state L and H is πL,H . Persistence
is also assumed: πH,H > πH,L and πL,L > πL,H . Importantly, it implies πH,H > πL,H .
Here, current productivity and “the persistent component of productivity,” or productivity group, coincide. Importantly, this does not need to be the case for the results in
this section to hold. For instance, all the results in this section go through if all the
households have the same income in the first period but some of them have a higher
probability of a transition to high productivity than others.
Credit markets. Lenders compete offering debt contracts. In particular, there are
two types of contracts. On the one hand, lenders sell revelation contracts. These contracts are designed to induce the households to reveal their productivity. The price
function in this case is qb, and depends only on the amount borrowed. On the other
hand, they sell screening contracts. These contracts require the use of information tech-

nologies to learn a household’s productivity. The price function in this case is qe and
depends on the household’s productivity and amount borrowed. The cost of screening
a household’s productivity (also referred to as information costs), C, is proportional to
the amount borrowed to simplify the analysis.9 It is important to bear in mind that
this cost is paid only if screening contracts are used.
Household’s problem. In period 1, households decide which type of contract they
prefer as well as how much to borrow.10 In period 2, after the realization of the
productivity shock, they decide whether to file for bankruptcy or to pay back the debt.
If they decide to file for bankruptcy, they lose a proportion of their income, τ . Thus,
the lifetime utility of a household born with assets a1 ∈ A, income yn = wln , and
facing a price function q is
U (a1 , y1,n ; q) = max u(y1,n + a1 − q(a2 , n; a1 )a2 )
a2 ∈A

(1)

+ βπn,H max{u(y2,H + a2 ), u(y2,H (1 − τ ))}
+ βπn,L max{u(y2,L + a2 ), u(y2,L (1 − τ ))}.
9

Later, in the quantitative general equilibrium model, this cost is independent of the amount borrowed.
The fact that households choose the contract (screening vs. revelation) is for exposition and without
loss of generality.
10

6

Here the price function q is used to represent qe or qb. Then, the choice of lender implies

that lifetime utility is U(a1 , y1,n ) = max{U (a1 − C, y1,n ; qe), U (a1 , y1,n ; qb)}. Similarly, it
is useful to define

U(a1 , y1,n ; a2 , q) ≡u(y1,n + a1 − qa2 ) + βπn,H max{u(y2,H + a2 ), u(y2,H (1 − τ ))}
+ βπn,L max{u(y2,L + a2 ), u(y2,L (1 − τ ))},
as the lifetime utility of a household born with assets a1 , income yn , borrowing −a2 at
the price q. This function will be used later to define self-revelation.
Zero-expected-profit prices. Lenders’ expected profits from a contract q(a2 , n) are
q(a2 , n)a2
| {z }

amount borrowers receive

− Pr(repayment | a2 , n)a2 (1 + i)−1 ,
|
{z
}
discounted amount lenders expect to recover

where Pr(repayment | a2 , n) is the lender’s expectation of repayment given the amount
borrowed (−a2 ) and the household’s type (n). Before the equilibrium prices for each
type of contracts (screening and revelation) are derived, it is useful to characterize the
prices implying zero-expected profits.
Lemma 1 Zero-expected-profit prices for each n = {L, H} are

if a2,L ≤ a2 ,
 (1 + i)−1
πn,H (1 + i)−1 if a2,H < a2 < a2,L ,
q(a2 , n) =

0
if a2 ≤ a2,H ,
where a2,L = −τ y2,L and a2,H = −τ y2,H .

Lemma 1 implies that zero-expected-profit prices vary as in Figure 2. The intuition
is simple. If debt is small enough (a2,L ≤ a2 ), households will prefer to pay back their
debt for both levels of next period income. This is because the benefit of bankruptcy—
discharge of debt—depends directly on the amount borrowed, while the cost—income
lost—is independent of it. For bigger debt (a2,H ≤ a2 < a2,L ), households will find beneficial to file for bankruptcy only if next period income is low. This is because the cost
of bankruptcy is directly related to the current income. Notice that in this range, prices
depend on current productivity. This is due to the fact that productivity is persistent,
implying that households with higher productivity have also higher expected productivity. For amounts of debt big enough, households will file bankruptcy for both level
of income. Clearly, this implies that prices must be zero to obtain zero-expected profits.

7

Preferences over debt and prices. Before the equilibrium can be characterized,
it is also useful to study the preferences of households over prices, q, and amounts of
assets for period 2, a2 . Specifically, it is important to characterize preferences over a2
and q in the range a2,H < a2 < a2,L , where borrowing implies risk of bankruptcy. In
this simple model, it is only there where lenders would like to charge different prices
to households with different current productivity.
Lemma 2 The slope of the indifference curves at q as a function of a2 ∈ [a2,H , a2,L ]
is


 < 0, for a2,H < a2 < a∗2 (q),
−M RSq,a2 (q, a2 )
= 0, for a2 = a∗2 (q),

> 0, for a∗2 (q) < a2 < a2,L ,

where a∗2 solves −u′ (y1,n + a1 − qa∗2 )q + βπn,H u′ (y2,H + a∗2 ) = 0.

The level of assets a∗2 (q) is the level of asset accumulation solving the first order
condition of the household’s problem given a price function constant at q. By construction, the slope is zero at that level of debt. Starting from there, it is simple to
understand the shape of the indifference curve. Any deviation from a∗2 (q) reduces the
household’s utility, implying that any deviation must be compensated with a higher q
to keep the household with the same utility.
Importantly, the range of a in which the slope is negative is not relevant to characterize equilibrium outcomes. This is because for any point in that range lenders and
households would be better off reducing the amount borrowed.11 Therefore, hereafter
(and in the figures) we focus on the increasing part of the indifference curves. More
importantly, indifference curves of households with different current productive have
different slopes at a given amount of debt. This result is crucial for the existence of
revelation contracts. Intuitively, this follows because (i) households with low productivity in period t are willing to borrow more than an households with high productivity,
and (ii) they are also more likely to file for bankruptcy in the second period, so they
are less affected by the interest rate.
Lemma 3 Take any value q and consider any a2 ∈ (a∗2 (q), a2,L ). Then, the slope is
bigger (steeper) for households with low productivity than for those with high productivity.
Equilibrium contracts. First, notice that the choice of contracts is irrelevant for
a2 > a2,L . This is because for those amounts of debt the probability of bankruptcy is
11

The household is better off by construction (decreasing indifference curve). The lender is better off
because the repayment probability is increasing (but not strictly) in a2 .

8

zero for both levels of current productivity. In that case, equilibrium prices are equal to
the zero-expected-profit prices described above. The rest of this subsection describes
equilibrium prices for screening and revelation contract for a2 ∈ [a2,H , a2,L ].
In equilibrium, competition between lenders implies that prices must imply zero
profits. First, focus on screening contracts. Zero-expected profits implies that prices
must take into account the cost of information.
Lemma 4 Equilibrium prices of screening contracts with a2 ∈ [a2,H , a2,L ] are q̃(a2 , n) =
q̄(a2 , n) − C.
Figure 3 shows an allocation in an economy with only screening contracts. It is
clear there that high-risk (low-productivity) households would be better off if they
could avoid paying C—they would be in a higher indifference curve. It is also clear
that lenders will be willing to offer q above the price of screening contracts (in that
range) and below zero-profit prices for low-productivity households, and do not pay for
information. They would obtain profits because the worst that can happen is that only
low-productivity households take it. This explains why contracts that do not require
the use of information technologies must be offered in equilibrium.
Now, focus on revelation contracts. Think that lenders design contracts under the
constraint that they must induce households to reveal their productivity. Using prices
and amounts of debt as instruments, it is possible to induce households to reveal their
type. It is indeed possible to separate households because in order to obtain more
debt, low-productivity households are willing to accept a bigger increase in interest
rates than high-productivity households; i.e., indifference curves are as described by
Lemma 3.
Suppose the price of a revelation contract, q, depend on the amount borrowed, −a2 ,
the household’s report on productivity, m, and the current stock of assets, a1 . Notice
that the price depends also on a1 because households’ willingness to borrow depends
on it. Then, we can define self-revelation.
Definition 1 A function q satisfies self-revelation if and only if for each given current
assets a1 , maxa2 U(a1 , y1,m ; a2 , q(a2 , m; a1 )) ≥ maxâ U(a1 , y1,m ; â, q(â, n; a1 )), n 6= m.
In words, q satisfies the self-revelation constraint if and only if households are better
off borrowing at the price designed for their productivity than misrepresenting their
productivity. Notice that for a given a1 , revelation implies that there must be at most
one value of q for each a2 —it cannot depend on the report. Otherwise, borrowers will

9

make the report that implies the highest q. Thus, q can be written only as a function
of (a2 , a1 ).
But, what are the equilibrium prices of revelation contracts? In general, equilibrium requires that lenders make zero profits and there is no other profitable contract
that borrowers would prefer.12 However, it is well known that using that definition
an equilibrium may not exist in this environment. Then, to guarantee existence, we
consider a similar concept, introduced by Riley (1979): prices of revelation contracts
are equilibrium prices if they imply zero profits and any other contract that is profitable and the borrowers would prefer imply subsidies across borrowers with different
risk. Under this definition, lenders do not deviate from the equilibrium allocation because those deviations will become unprofitable after a reaction that skims the cream
and produces losses for the defector.13 The following lemma characterizes equilibrium
prices of revelation contracts.
Lemma 5 The equilibrium prices of revelation contracts are

q̄(a2 , H) if a2 ≥ a2 (a1 ),
q̂(a2 ; a1 ) =
q̄(a2 , L) if a2 (a1 ) ≥ a2 ,
where a2 (a1 ) is such that U(a1 , y1,L ; a2 (a1 ), q̄(a2 (a1 ), H)) = maxâ U(a1 , y1,L ; â, q̄(â, L)).

Screening or revelation contract? Now, the choice of contracts can be described.
Given the cost of information, households compare the utility from both types of contracts and choose the one associated with higher utility.
Lemma 6 The choice of contracts is characterized by
1. Households with low productivity never prefer screening contracts.
2. There exists a cost of information c such that households with high productivity
are indifferent between screening and revelation contracts. Households with high
productivity prefer screening contracts if and only if C < c.

The effect of information costs on debt and bankruptcy. Lower information
costs allow high-productivity households to borrow more, making them more likely to
file for bankruptcy.
12

This would be a Nash equilibrium.
The existence problem of Nash equilibrium and the existence of a unique Reactive equilibrium—the
concept used here—are explained in details by Hirshleifer and Riley (1992), Chapter 11.
13

10

Lemma 7 If information costs fall from C0 to C1 and a household does not pay for
information at C0 but she pays for information at C1 , then:
1. This household’s debt increases.
2. This household’s probability of bankruptcy increases or stays constant.
Assume initially the cost of information is high enough, C0 > c. If the debt constraint associated with revelation contracts is tight enough, high-productivity households may prefer to borrow at risk-free price. Assume this is the case. In this allocation,
high-productivity households are borrowing-constrained: they would prefer to borrow
more at their zero-expected-profit price, but those prices cannot be offered because
low-productivity households will pretend to have higher productivity. Now, assume
technological progress occurred in the information sector implying that C1 < c. This
implies that high-productivity households prefer to pay the cost of information. If they
do so, it is because they borrow more in the new allocation. Actually, they would only
pay for information to borrow with some risk of bankruptcy. Thus, in this new allocation there is also more bankruptcy because this household now files for bankruptcy
with probability (1 − πH,H ).
These two results are important because they are qualitatively consistent with the
facts presented above for the U.S. economy. Hereafter a general equilibrium model is
developed in an attempt to quantify the importance of these results.

3
3.1

Quantitative General Equilibrium Model
The Model

Environment. Time is discrete and denoted by t = 0, 1, 2, . . . At any time there
is a unit mass of households. They discount the future at the rate β. Preferences of
households are given by the expected value of the discounted sum of momentary utility
"∞
#
X
E0
β t u(ct ) ,
t=0

where ct is consumption at period t. The utility function u is strictly increasing,
strictly concave, and twice differentiable. Let n ∈ N = {1, 2, . . . , N } denote the
productivity group of a household. Productivity groups are persistent, with transition
probability Π(nt , nt+1 ). Each household is endowed with one unit of time. Inside
each group, productivity is exogenously determined by labor endowments that come
from different group-specific intervals; for each n, l ∈ L(n) = [ln , ¯ln ]. Thus, labor

11

endowments and productivity groups at time t are correlated. The transition function
is φ(lt+1 | nt+1 )Π(nt , nt+1 ), where φ(lt+1 | nt+1 ) is a conditional density function.
Notice that the cumulative density function of lt+1 can be written as a function of nt
directly,
F (lt+1 | nt ) =

XZ

nt+1

n
l̄t+1

l0

φ(lt+1 | nt+1 )Π(nt , nt+1 ).

With this notation it is simpler to state the assumption on the transition function.
Assumption 1 nt is an index of first order stochastic dominance; i.e., if nt > n̂t ,
then F (lt+1 | nt ) ≤ F (lt+1 | n̂t ) for all lt+1 .
Assumption 1 is very important because it implies that individuals with higher nt
will have lower default risk. Thus, this infinite-horizon model resembles the 2-period
model presented above.

Information structure. There is asymmetric information between lenders and
borrowers about the latter’s productivity group, n. On one side, households know
their n. On the other side, if borrowers are not screened, then the productivity group
is private information. Nevertheless, each lender has access to a technology that can
be used to learn a household’s productivity group at a fixed cost. The stock of assets,
at , is publicly observable, as well as the credit flag indicating the bankruptcy record
(defined later).

Information firm’s problem. The information firm uses labor to produce information with the production function
zti (mt )1/γ ,
where zti is the productivity in information production and mt is labor demanded in
the information industry. This sector is simplified assuming it produces {0, 1}, where
0 means no information is produced and 1 means “a report with information about the
borrower’s productivity group is produced.” Then, zero-expected profits in this sector
implies that the cost of learning a borrower’s productivity group (or screening cost) is
C(zti , wt ) = wt (zti )−γ .

Production firm’s problem. It rents capital at the rate rt and hires labor at
the wage wt . With these factors the firm produces consumption goods in line with a
standard Cobb-Douglas production function. Thus, the firm’s problem is
n
o
max ztp (kt )1−θ (lt )θ − wt lt − rt kt ,
{lt ,kt }

12

where ztp is the technology in the production sector, and {lt , kt } are labor and capital
in this sector, respectively.

Credit industry. There are two types of debt contract: screening and revelation
contracts. There are many lenders competing among themselves offering debt contracts. They own the stock of capital, which they rent to the firms in the production
sector.
Lenders offering screening contracts. Borrowers have to pay the screening cost to be
able to use a screening contract. Think of borrowers buying a report at the information industry that proves their productivity group and submitting it to lenders. The
price charged is q̃(at+1 , nt ); i.e., a different price for each level of assets next period,
at+1 ∈ A, and productivity group, nt ∈ N. The price depends on at+1 because it
determines the debt the household will have to pay back next period, which in turn
affects her willingness to pay back the debt. It depends on nt because this determines
the transition probability to different productivity levels, and thereby the probability
of bankruptcy. Let deat+1 ,nt denote the number (measure) of contracts for households
e t+1 the stock of capital they accumulate for period
with {at+1 , nt } that lenders sell, K

t + 1, and Pr(repayment | at+1 , nt ) the repayment probability of this contract. Then,
period-t cash flow is given by

R
P
Pet = −
dea ,n Pr(repayment | at , nt−1 )at dat
nt−1
P R at e t t−1
+
nt at+1 dat+1 ,nt q̃(at+1 , nt )at+1 dat+1
et − K
e t+1 .
+ (1 − δ + r)K

e t+1 to maximize the present
Lenders design the contracts and choose deat+1 ,nt and K

discounted value of current and future cash flows,
∞
X
t=0

(1 + it )−t Pet ,

e 0 , and the
given the risk-free interest rate at period t, it , the initial stock of capital, K
number of each different contract initially sold, dea ,n .
0

−1

et+1 }∞ ,
The sequence of cash flows implies a sequence of risk-free bond holdings, {B
t=0

which can be obtained by the recursion

et+1 = (1 + it )B
et + Pet ,
B

e0 = 0. These bonds, which are issued by the lenders, are incorporated to allow
where B

for the accumulation of cash flows. They are not that important hereafter since they

13

et = B
e = 0.
will be zero in the stationary equilibrium defined later; i.e., B
Lenders offering revelation contracts. These lenders compete offering self-revelation
contracts. The condition a contract has to satisfy to be “self-revelation” is formally
stated later, after the household’s problem is introduced. That condition basically
states that, given the contract design, borrowers are better off revealing their productivity group. Since lenders offering revelation contracts do not observe n, prices depend
on the households’ reports on n. Additionally, since the current stock of assets affects
a household willingness to borrow, prices satisfying the revelation constraint depend
also on this variable.
Some notation in now introduced. Let dbat+1 ,nt ;at denote the number (measure) of
b t+1 the stock of
contracts uninformed lenders sell for households with {at+1 , nt , at }, K
capital they accumulate for period t + 1, and Pr(repayment | at+1 , n) the repayment

probability. Then, period-t cash flow is given by
R
R
P
b
Pbt = −
nt−1 at−1 at dat ,nt−1 ;at−1 Pr(repayment | at , nt−1 )at dat dat−1
P R R
b
+
b(at+1 , nt ; at )at+1 dat+1 dat
nt at at+1 dat+1 ,nt ;at q
b
b
+ (1 − δ + r)Kt − Kt+1 .
b t+1 to maximize
Lenders design the contract and choose dbat+1 ,nt ;at and K
∞
X
t=0

(1 + it )−t Pbt ,

b 0 , and dba ,n ,a . Again, a sequence of cash flows implies a sequence of
given it , K
0 −1 −1
bt+1 }∞ .
risk-free bond holdings, {B
t=0

Household’s problem. Hereafter, period-t variables will be expressed without
any subscripts or superscripts, and period-t + 1 variables will be represented with

superscripts ‘′ ’. Households decide on consumption, c, and asset accumulation, a′ . In
addition, they decide which kind of debt contract they would like to sign, and either
to file for bankruptcy or to pay back the debt. These decisions are made taking prices,
S = (q, w, i, r, qe(·), qb(·), C(·)), as given.

Several assumptions determine the advantages and disadvantages of bankruptcy.

The key advantage is the discharge of debts—assets in the period after bankruptcy
are set at zero. Thus, a household with too much debt may find it profitable to
file for bankruptcy. There are many disadvantages of doing so, however.14 In the
14

Here, disadvantages of filing bankruptcy are exogenous. Chatterjee, Corbae, and Rios-Rull (2008) show
how higher interest rates following default arise from the lender’s optimal response to limited information
about the household’s type and earnings realizations.

14

period of bankruptcy, consumption equals income, and neither saving nor borrowing
are allowed. Additionally, in the period right after bankruptcy, the defaulter will have
a bad credit flag. Having a bad credit flag implies that the household cannot borrow
and a proportion of income, τ , is lost.15 That flag remains in a household record for a
stochastic number of periods, meaning that the probability of a transition from bad to
good credit flag is λ ∈ (0, 1)—the fresh start probability. The use of λ is a simple way
of modeling a bankruptcy flag that remains on a household’s credit history for only a
finite number of years.
Lifetime utility for households in each possible state is defined as follows.
• Bad credit flag: Lifetime utility of a household excluded from credit markets is
R
P
B(n, l, a; S) = maxa′ ,c {u(c) + ρβ n′ Π(n, n′ ){λ l′ G(n′ , l′ , a′ ; S)φ(l′ | n′ )dl′
+(1 − λ)

subject to

R

l′

B(n′ , l′ , a′ ; S)φ(l′ | n′ )dl′ }},

(2)

c + q̄(a′ , n)a′ = a + lw(1 − τ ),
a′ ≥ 0, and c ≥ 0,
where G is the lifetime utility for households with good credit history (defined below),
which is a function of productivity, n, labor endowments, l, assets, a, and relevant
prices, S. Importantly, assets for the next period are restricted to be positive. Notice
that the household obtains utility in the next period just if she survives, and that
happens only with probability ρ. The utility from future periods depends on the probability of a fresh start, λ, while the utility from the current period depends on the
proportion of income lost because of bad credit status, τ . Denote the policy functions
for asset accumulation and consumption obtained from the solution to this problem as
A′b and Cb .
• Good credit flag: Lifetime utility is
G(n, l, a; S) = max{V (n, l, a; S), D(n, l; S)},
|
{z
} | {z }
pay back

(3)

bankruptcy

where V and D (defined below) are lifetime utilities for households paying back the debt
and filing bankruptcy, respectively. This means that a household with a good credit
flag has the choice of filing bankruptcy. The policy functions for asset accumulation and
15

Chatterjee, Corbae, and Rios-Rull (2008) build a model where no punishment is required after filing
bankruptcy. There, asymmetric information is crucial to create incentives for debt repayment, because
households signal their type by paying back their debt.

15

consumption are A′ and C, respectively. Additionally, the policy function R indicates
whether the household pays back the debt or not,

1 if V (n, l, a; S) ≥ D(n, l; S),
R(n, l, a; S) =
0 otherwise.
• Good credit flag and bankruptcy: Suppose the household chooses to file for
bankruptcy. Then, lifetime utility is
D(n, l; S) = u(lw(1 − τ )) + ρβ

P

n′

Π(n, n′ )

R

l′

B(n′ , l′ , 0; S)φ(l′ | n′ )dl′ .
(4)

Neither saving nor borrowing is allowed in this period. Therefore the household’s consumption equals net income (labor income minus the proportion
lost due to bankruptcy). In the period after bankruptcy, the household will
have a bad credit flag for sure and zero debt.
• Good credit flag and pay back the debt: Suppose the household decides to
pay back the debt. Then, she must decide which kind of contract to sign.
Thus, the value function is
V (n, l, a; S) = max{ Ve (n, l, a; S) ; Vb (n, l, a; S) },
{z
} |
{z
}
|

(5)

use information no information

where Ve (n, l, a; S) and Vb (n, l, a; S) (defined below) are lifetime associated

with borrowing using screening and revelation contracts, respectively. The
policy function U indicates whether the household borrows using revelation
contracts or not,
U (n, l, a; S) =



1 if Vb (n, l, a; S) ≥ Ve (n, l, a; S),
0 otherwise.

• Pay back the debt and screening debt contract: If the household
decides to sign a screening contract with a lender, then she faces
the debt price q̃(a′ , n), and her lifetime utility is
R
P
Ve (n, l, a; S) = maxa′ ,c {u(c) + ρβ n′ Π(n, n′ ) l′ G(n′ , l′ , a′ ; S)φ(l′ | n′ )dl′ },
subject to
c + q̃(a′ , n)a′ = a − C(z i , w) + lw,
and c ≥ 0,
(6)
where C(z i , w) is the cost of information. Notice that this cost is
independent of the amount borrowed, which is consistent with the

16

interpretation that the household buys a report about her type and
then presents it to the lender.
• Pay back the debt and revelation debt contract: Now suppose the
household prefers to use a revelation contract. Then, the relevant
debt price is qb(a′ , a) and there is no fixed cost to pay. Thus, her

lifetime utility is
R
P
Vb (n, l, a; S) = maxa′ ,c {u(c) + ρβ n′ Π(n, n′ ) l′ G(n′ , l′ , a′ ; S)φ(l′ | n′ )dl′ },
subject to
c + qb(a′ , a)a′ = a + lw,
and c ≥ 0.
(7)

3.2

The Equilibrium

Equilibrium prices for screening contracts must imply zero-expected profits. Therefore,
they can be written as
X
1
ρ
q̃(a , n) =
Π(n, n′ )
1+i
′
′

n

Z

R(n′ , l′ , a′ ; S)φ(l′ | n′ )dl′ .

(8)

l′

Here it is very clear why the price, q̃, depends on (a′ , n) and is independent of a. It
depends on a′ because it affects the bankruptcy decision, R, in each possible state. It
depends on n because it determines the transition probability to each n′ and therefore
the next period labor endowment, l′ . Finally, it is independent of a because it does not
affect either the transition probabilities or the bankruptcy decision in the next period.
The difference with screening contracts in the 2-period model (Lemma 4) is that the
(constant) cost of information that must be paid to use these contracts is now afforded
directly by the borrower.
To characterize revelation contracts, two differences with the 2-period example are
important. First, there are more than two types, so the limit for the zero-expectedprofit prices corresponding to a productivity group n will be the tighter of those set by
households with productivity group j < n. The second difference is that households
with any labor endowment l ∈ L(j) could pretend to be of productivity group j. Again,
the solution is to use the tighter limit among those set by l ∈ L(j). To write this
formally, additional notation must be introduced. First, consider the bankruptcy-free
limit,16
a = min{a : V (n, l, a) ≥ D(n, l), ∀n ∈ N, ∀l ∈ L(n)}.
a

16

This borrowing limit was first introduced by Zhang (1997) and Abraham and Carceles-Poveda (2006).

17

Then, let the function
R
P
V(n, l, a; a′ , j) = maxa′ ,c {u(c) + ρβ n′ Π(n, n′ ) l′ G(n′ , l′ , a′ ; S)φ(l′ | n′ )dl′ },
subject to
c + q̄(a′ , j)a′ = a + lw,
and c ≥ 0,

(9)

represent the lifetime utility if the amount of assets chosen for the next period is a′ and
the price used is the one satisfying the zero-expected-profit condition for households
in the productivity group j. The auxiliary limit for the price of households with
productivity group n set by households in the group j and with labor endowment l is
a(a, n; j, l) such that
max
V(j, l, a; a′ , j) = V(j, l, a; a(a, n; j, l), n).
′
a

(10)

Since l is unobservable, the limit for the price of households with productivity group
n set by households in the group j is
a(a, n; j) = max a(a, n; j, l).
l∈L(j)

(11)

Similarly, since we want to exclude all j < n from the price of households with productivity group n, the minimum a′ that can be offered at that price is
a(a, n) = min{a; max a(a, n; j)}.
j<n

(12)

Notice that the limit cannot be higher than the risk-free limit because information
about productivity is irrelevant for a > a. Now, equilibrium prices of screening contracts can be written as

 ρ
′

1+i if a ≥ a,


′

q̃(a , N ) if a > a′ ≥ a(a, N ),



q̃(a′ , N − 1) if a(a, N ) > a′ ≥ a(a, N − 1),
qb(a′ , a) =

...




q̃(a′ , 2) if a(a, 3) > a′ ≥ a(a, 2),


q̃(a′ , 1) if a(a, 2) > a′ .

(13)

These prices resemble those in the 2-period model. The only difference is that there
are model debt limits.17

Stationary equilibrium. Assume technologies in the information sector, z i , and
in the production sector, z p , are constant. Then, stationary equilibrium requires optimization together with aggregate conditions that guarantee markets clearing and
stationarity.
17

It is actually simple to see that those limits satisfy a(a, n) > a(a, n − 1). Two things are important for
this result. First, zero-expected-profit prices are increasing in n. This is clearly implied by Assumption 1.
Second, higher prices make borrowers better off, so tighter limits are required to keep them indifferent.

18

Definition 2 A stationary equilibrium with costly information is a set of policy functions A′b , Cb , A′ , C, R, U , l, m, and k, cumulative density functions Ψn (a, l), Ψgn (a, l),
Ψbn (a, l), and prices w, i, r, qe, qb, and C, such that the following conditions hold:

1. The functions A′b , Cb , A′ , C, R, and U solve the household’s problems, or satisfy
problems 2 to 7.

2. The function qe and qb are equilibrium prices, or satisfy 8 and 13, respectively.

3. The firm in the production sector maximizes profits given {w, r}, or
(1 − θ)z p (k)−θ (l)θ = r,
θz p (k)1−θ (l)θ−1 = w.

4. The function Ψn (a, l) is the stationary c.d.f. over (n, a, l), and Ψgn (a, l) and
Ψbn (a, l) are the stationary c.d.f. over (n, a, l) conditional on having good and
bad credit flags, respectively; or
Ψn (a, l) =Ψgn (a, l) + Ψbn (a, l),
XZ Z
g
′ ′
dΨn′ (a , l ) =
1{A(n,a,l)=a′ } Π(n, n′ )φ(l′ | n′ )R(n, a, l)dΨgn (a, l)
a

n

+λ
dΨbn (0, l′ ) =

l

XZ Z
a

n
XZ Z
a

n

l

Π(n, n′ )φ(l′ | n′ )(1 − R(n, a, l))dΨgn (a, l)

+ (1 − λ)
dΨbn (a′ , l′ ) =(1 − λ)

l

1{Ab (n,a,l)=a′ } Π(n, n′ )φ(l′ | n′ )dΨbn (a, l),

XZ Z

n
XZ Z
a

n

a

l

l

1{Ab (n,a,l)=0} Π(n, n′ )φ(l′ | n′ )dΨbn (a, l),

1{Ab (n,a,l)=a′ } Π(n, n′ )φ(l′ | n′ )dΨbn (a, l), a′ 6= 0.

5. The credit market clears, or
XZ Z
b
da′ ,n;a =
1{A(n,a,l)=a′ } U (n, a, l)dΨgn (a, l)
n

+

a

l

XZ Z
a

dea′ ,n =

n
XZ Z
a

n

l

l

1{Ab (n,a,l)=a′ } U (n, a, l)dΨbn (a, l),

1{A(n,a,l)=a′ } (1 − U (n, a, l))dΨgn (a, l).

6. The labor market clears, or

m+l=

XZ Z
n

a

ldΨn (a, l).

l

7. The goods market clears, or
XZ Z
XZ Z
z p (k)1−θ (l)θ =
C(n, a, l)dΨgn (a, l) +
Cb (n, a, l)dΨbn (a, l) + δk.
n

a

l

n

19

a

l

4

Calibration

The strategy here is to calibrate the model for 1983 and then evaluate the impact of
information costs on bankruptcy, debt, and other variables that can be used to test
the model with the data.

4.1

Calibration Strategy

The benchmark calibration, also referred as “1983 calibration,” is designed such that
the model represents the U.S. economy in the year 1983. The choice of this year is
very important. As mentioned by Moss and Johnson (1999), “from 1920 to 1985, the
growth of consumer filings closely tracked the growth of real consumer credit. Since
then, however, the rate of increase of consumer bankruptcies has far outpaced that
of real consumer credit.” The year 1983 will then represent a steady state before the
transformation of the market that started in 1985. The calibration consists in assigning
values to 25 parameters. Some of them can be determined using a priori information.
The others are determined jointly using the Nelder and Mead (1965) algorithm to
minimize the distance between key moments in the data and model. Parameters and
targets are explained in detail in the next subsections.

Parameters determined using a priori information (5). The survival probability, ρ, is determined to match a period of a financially active life of 40 years. The
utility function is
u(c) =

c1−σ
,
1−σ

where σ was chosen to match a coefficient of risk aversion of 2. The labor share of
income, 0.64, determines the value of the parameter in the production function, θ. The
depreciation rate, δ, is set at 7% annually. The probability of a fresh start, λ, is set to
match the average time of exclusion after bankruptcy (10 years).18

Parameters determined jointly (20). There are ten different productivity
groups or types, N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, where 1 and 10 are associated with
the lowest and highest labor endowments, respectively. The number of groups is an
important choice because it determines the maximum that lenders can learn about
households’ incomes when they decide to screen borrowers. Nevertheless, how much
information is contained in these ten types depends on their persistence, which is endogenously determined considering several moments from the data. Additionally, the
18

The same value is used by Chatterjee, Corbae, Nakajima, and Rios-Rull (2007).

20

proportion of total income that each group accounts for is obtained directly from the
Survey of Consumer Finance (SCF). The parameters to calibrate are described below.
- Transition matrix (6 parameters): Π. Several assumptions restrict the number
of parameters in this group. In particular, it is assumed that: (i) transitions
further than 2 types away than the current type are zero probability events, (ii)
persistence is the same for all the groups except {1, 2, 9, 10}, (iii) the transition
to one and two types higher and lower are proportionally the same for all the
types, and (iv) the highest type is a low probability state with very high labor
endowments, as in Castaneda, Diaz-Gimenez, and Rios-Rull (2003). After these
assumptions there are 6 parameters to calibrate.19 These parameters determine
the size of each productivity group which in turn affects the joint distribution of
debt and income.
- Technology in the information sector (1 parameter): z i . This parameter is important in determining the distribution of debt across income groups and the
bankruptcy rate.
- Discount factor (1 parameter): β. This parameter is crucial in determining the
economy capital-to-income and debt-to-income ratios and the bankruptcy rate.
- Income lost during bankruptcy (1 parameter): τ . Since this parameter represents
the fraction of earnings lost when households have a bad credit flag, it plays a very
important role determining the bankruptcy rate and the debt-to-income ratio.
- Labor endowments distribution exponent (1 parameter): ϕ . Given a type n, the
cumulative distribution for labor endowments, l ∈ [l n , ¯ln ], is
Z

x

ln

φ(l | n)dl =



x − ln
¯ln − ln

ϕ

,

where the exponent is one more parameter to calibrate. This parameter is particularly important in determining income inequality given the size of income
groups.
- Labor endowment intervals’ limits (10 parameters): l. The first one is normalized,
l1 = 1. The ending limit of each group is equal to the starting limit of the next
range; i.e., ¯l1 = l2 , ¯
l2 = l3 , ... , ¯
l9 = l10 . Thus, there are 10 parameters to be
calibrated: l = [¯l1 , ¯l2 , ... , ¯l10 ]. These limits determine the proportion of total
income in each group.
19

More details on the assumptions made on the transition matrix are provided in Appendix 7.2.

21

From these parameters, the last 10 will be taken directly from the data on income.20
The labor endowments distribution exponent will be calibrated in a separate step to
match the Gini coefficient of income. The remaining 9 parameters are chosen minimizing the distance between moments from the model and data. Specifically, there are
12 statistics used as targets. Most of them are very important in any model of debt
and bankruptcy. Others are relevant given the informational frictions in this paper.
The moments chosen as targets of the calibration are described in detail below. Their
values are presented in Table 1.
- Capital-to-output ratio (1 target): This target is fixed assets and consumer durable
goods over GDP, both obtained from the Bureau of Economic Analysis (BEA).
- Bankruptcy rate (1 target): The number of bankruptcy filings in a year, obtained
from the American Bankruptcy Institute, is prorated using 0.53 because income
shocks cause 53% of the cases of bankruptcy.21 Then, to construct the bankruptcy
rate, the prorated number of bankruptcy filings is divided by the number of
households with a credit card balance, obtained from the SCF.
- Earnings and wealth inequality (4 targets): The statistics in this group are all obtained from the SCF 1983. In particular, the targets used are the Gini coefficient
of income, the mean-to-median income ratio, the Gini coefficient of wealth, and
the mean-to-median wealth ratio.
- Debt-to-income ratio (1 target): Two alternative approaches have been followed
to compare the debt measure of the model and the data. The first one, followed by
Chatterjee, Corbae, Nakajima, and Rios-Rull (2007), is to define debt as minus
net worth when it is negative and zero otherwise. The alternative approach,
followed by Livshits, MacGee, and Tertilt (2007b), is to use credit card debt.
The “net worth approach” is followed here. The main advantage is that it is
consistent with a model with only one asset.22 The data is obtained from the
SCF.
- Debt across income groups (5 targets): Let Di denote the percentage of total debt
held by households with an income percentile lower than i. The moments D10 ,
D15 , D20 , D30 , and D40 are used in the calibration procedure. They are obtained
from the SCF using the definition of debt introduced above.
20

More details are discussed below.
This adjustment is necessary because the model only has income shocks and in reality other shocks
are also important. Chatterjee, Corbae, Nakajima, and Rios-Rull (2007) applied the same procedure when
they calibrated their model with only income shocks. Then, they show that the model can match the total
number of bankruptcies if other shocks are included.
22
As in Chatterjee, Corbae, Nakajima, and Rios-Rull (2007), debt-to-income is also prorated using 0.53.
21

22

Among the targets, the percentage of debt held by households with incomes lower
than the 10, 15, 20, 30, and 40 percentiles were not considered in previous literature.
They are important in the calibration because how much debt is held by households
in a productivity group depends on their expected income, which is determined by the
transition matrix parameters. More important, given all the other parameters, the cost
of information is crucial in determining the proportion of debt held by different income
groups, as explained at the end of Section 2. Thus, these targets will be useful for the
calibration of z i , too.
The next steps were followed to calibrate the model parameters minimizing the
distance between the moments from the data and model.
Step 1: Guess a value for 9 parameters {Π, z i , β, τ }.
Step 2: Given the value of Π, compute the measure of households in each type n
in the stationary distribution.
Step 3: Obtain 10 parameters, l = [¯l1 , ¯l2 , ¯l3 , ¯l4 , ¯l5 , ¯l6 , ¯l7 , ¯l8 , ¯l9 , ¯l10 ], using the SCF
1983 to match exactly the measure in each labor endowment’s interval.
Step 4: Given the measure in each interval and the limits l, search for the value
of ϕ that minimizes the distance between the Gini coefficient of income from the
model and data using the Nelder and Mead (1965) algorithm.
Step 5: At this point a value was assigned to each of the 20 parameters. Use
these parameters to compute the model and calculate the distance between the
moments from the data and model. If the distance to the targets is small enough,
end. Otherwise, choose a new value for {Π, z, β, τ } according to the Nelder and
Mead (1965) algorithm and return to step 2.
Step 3 is different from previous literature. For instance, Chatterjee, Corbae, Nakajima, and Rios-Rull (2007) also search on ¯li to match some targets. This procedure
can be better understood using Figure 7. First, households’ incomes from the SCF is
normalized such that the minimum non-zero income is 1. This will be the value of l1
in the calibration. Let ∆1 represent the measure of households in the first productivity
group given Π. Then, pick the upper limit of the first interval from the (normalized)
SCF data such that between 1 and ¯l1 , the measure of households is ∆1 . This is ¯l1 , in
Figure 7. The same procedure can be applied to determine the next ¯li taking ¯li−1 = li
and ∆i as given. Thus, all ¯li are obtained directly from the SCF.

23

4.2

Calibration Results

The results from the calibration strategy are presented on Table 1. How well does the
model fit the data? It can replicate key moments for the year 1983. In particular,
the first two statistics, the bankruptcy rate and the debt-to-income ratio, are closely
replicated. Despite the fact it overestimates the percentage of total debt held by
the poorest 10, 15, 20, 30, and 40 percentiles of the population, the power of the
model to explain debt across income groups is acceptable.23 Given that income is
directly obtained from the data, income distribution statistics are replicated very well.
The wealth inequality statistics considered as a target of calibration are also closely
reproduced. The success of the model reproducing wealth inequality is because the
income of rich households is well calibrated using the SCF and because of the structure
of the transition matrix.
Now the question is: What parameters are necessary to match the selected moments? The parameters obtained from the calibration are presented in Table 2. The
discount factor and the punishment after bankruptcy are similar in other quantitative
models of default and hard to compare with direct evidence. The parameter representing the technology in the information sector implies a cost of information around
5% of mean income. This is very high. Remember that if C is high, households will
avoid this cost by using revelation contracts. This in fact happens; more than 99% of
the borrowers use contracts that do not require the use of information. The parameters obtained for transition matrix parameters can be compared with data obtained
from the matched March supplement of the Current Population Survey (CPS). Table 3
presents the transition matrix from the data and the model for the first 6 types, which
accumulate 93% of the total population.24 Although there are several reasons that
can explain why the transition matrix from the CPS may be different,25 the transition
probabilities obtained in the calibration look relatively similar to those in the data. In
particular, the parameter in the main diagonal are quite similar.
While it is important to obtain a good fit of the targets with reasonable parameters,
it is perhaps more important to check that non-targeted moments are also reasonably
matched. Table 4 compares the data and model predictions for four non-targeted
moments. The charge-off rate, which is the value of loans removed from the books
23

If those target were not included, the model would do a terrible job. This is because, in general, only
the poorest individuals have debt in this type of model.
24
The comparison for the richest households is avoided because of the different treatment of households
on the top of the income distribution (top-coding) in the SCF and the CPS.
25
For instance, the model is calibrated using data on earnings from the SCF, which contains information
about households earnings, while the information from the CPS is about individuals’ earnings.

24

and charged against loss reserves as a percentage of average loans, and the number of
bankruptcy filings over the total population are bigger in the data than in the model.
However, given that they were not targeted, they are relatively similar. The proportion
of households with negative net worth is quite similar in the model and the data. More
importantly, the coefficient of variation of (paid) interest rates is very similar. This
is very important because the dispersion of interest rates is a measure of how much
information lenders use about borrowers’ characteristics.

5

Assessment of the IT Revolution

The quantitative model is now used to assess the role of information costs in the rise of
the bankruptcy rate over the last 20 years. Three questions are answered in this section:
(i) Can a drop in the cost of information replicate the rise in consumer bankruptcies
between 1983 and 2004?, (ii) What are the implications of that drop in information
costs for other variables?, and (iii) Does the mechanism in the infinite horizon model
resemble the one in the 2-period model? Finally, this section discusses distinguishing
implications of a drop in information costs.

5.1

Recalibrating z

To evaluate the role of the IT revolution, a new value for the technology in the information sector, z, is required. A simple methodology is followed here to obtain that
value: pick the new value of z such that the bankruptcy rate in the model matches
that in 2004.26
The value of z obtained in this exercise is 2.85, which implies a cost of information
(relative to mean income) smaller than 1%. Thus, the answer to the first question is
yes, the model can reproduce the rise in consumer bankruptcies between 1983 and 2004
if the cost of information drops from 5% to 1% of mean income.

5.2

Effect of z on other variables

The role of information costs in consumer debt and bankruptcy can be better understood by examining changes in other variables when information costs rise. The
results are presented in Table 5. In the comparison with the data for 2004, notice
26

Although this methodology has been widely used in macroeconomics, it has the disadvantage that the
value of z for 2004 is not identified. To do so, an exercise similar to the one done for 1983 should be performed
for 2004. However, in that exercise, all changes between 1983 and 2004 will have to be explained by the
parameters recalibrated. If other changes occurred during that period, the parameters obtained will also be
biased.

25

that all the statistics but the the bankruptcy rate were not targeted. By construction,
the bankruptcy rate is very similar in the data (1.60%) and the model for z = 2.85
(1.52%). Importantly, all the variables move in the same direction in the data (1983
vs. 2004) and the model (z = 1.26 vs. z = 2.85). More importantly, the changes in
three of the most relevant statistics—the charge-off rate, the ratio of bankruptcy filings
to the population, and the coefficient of variation of interest rates—are very similar
in the data and model. Changes in the distribution of debt across income percentiles
must be interpreted carefully. Since the statistic used before was the cumulative share
of debt, we must take differences to see which group is borrowing a bigger share in
2004 than in 1983. Notice that in the model, households between percentiles 10 and
40 are borrowing more. The same happens in the statistics computed from the SCF,27
although the magnitudes are much larger. Thus, changes in information costs alone
are not enough to explain changes in the distribution of debt across income groups.
However, notice that income inequality does not change by construction, while changes
in the data are very significant. These changes may be important in explaining changes
in the distribution of debt across income groups. Additionally, changes in information
costs do not modify wealth inequality. Finally, in the model, the rise in the debt-toincome ratio is significantly smaller (17%) than the rise in the bankruptcy filings over
the population (115%). Although this exaggerates what happens in the data (debt rose
81% and bankruptcy 149%), this result is important because, as mentioned by Moss
and Johnson (1999), what is special about this period is that “the rate of increase of
consumer bankruptcies has far outpaced that of real consumer credit.” Other driving
forces, such as a drop in transaction costs, have problems generating this asymmetric
response of debt and bankruptcy.28

5.3

The mechanism in the infinite horizon model

When information costs drop, households in income percentiles 10 to 40 increase their
share of total debt. Thus, as in the 2-period model, after technological progress in the
information sector, some households can borrow more and the default rate increases.
To analyze this effect in greater details, the distribution of debt (among those in debt)
is depicted in Figure 8. On the left, an economy with very high information costs
(z = 0.8) is presented. To understand the mechanism, the key is that there is almost
27

A similar result is reported in Figure 6 of Livshits, MacGee, and Tertilt (2009) following the alternative
approach to measure debt. Using their metric, the 40% poorest individuals had around 10% of the debt
in 1983 and 20% in 2000. Also in their case, this change is not explained by an increase in the bottom
10%—they had less than 3% of the debt in both years.
28
See Livshits, MacGee, and Tertilt (2007a).

26

no debt with risk of bankruptcy. In that economy, the default rate is really low (0.08%)
and there is almost no dispersion in the interest rate paid (the coefficient of variation is
1.9%, compared to 28% and 43% in 1983 and 2000 data, respectively), since almost all
households pay the risk free rate. On the right panel of Figure 8, an economy with very
low information costs (z = 10) is presented. Now, there are much more households
borrowing with risk of default. The bankruptcy rate in that economy is 2.2% and the
coefficient of variation of interest rates is 61.2%, both much higher than for the economy
with high information costs. Importantly, notice that while the bankruptcy rate and
the dispersion of interest rates increased dramatically, the debt-to-income ratio rose
from 0.35% to 0.50%. The key is not the rise in debt but the shift from non-risky to
risky debt that is generated by a drop in information costs.

5.4

Distinguishing implications

Finally, the model is challenged analyzing distinguishing implications that a drop in
information costs generates. This driving force has two distinguishing implications.

5.4.1

Changes in the dispersion of interest rates

Livshits, MacGee, and Tertilt (2009) point out that during the last 20 years the dispersion of interest rates increased significantly. Since the model was not calibrated to
match those moments, it is really challenging to compare the dispersion in the model
and the data. In the model, a drop in the cost of information production increases the
dispersion of interest rates. This is because when screening is too costly, lenders use
fewer contracts (only those satisfying self-revelation). Actually, this implies that most
of the households borrow at the risk free rate. When information costs drop, more
contracts are used and the dispersion of interest rates increase.
The dispersion of interest rates for the model with z = 1.26 and z = 2.85 and the
data for 1983 and 2004 was presented in Tables 4 and 5, respectively. The values are
surprisingly similar. In 1983, the coefficient of variation of interest rates was 28.3%
and the 1983 calibration’s number is 27.4%. In the model with low information costs
(z = 2.85) this coefficient is 43.7%, while in the data for 2004 it is 43.2%. Thus, this
driving force is very successful replicating the dispersion of interest rates.

5.4.2

Changes in the projection of income on interest rates

A very specific implication of this model is that the relation between income and
interest rates (after controlling by other variables) becomes stronger as the cost of
information drops. More precisely, given the amount of debt borrowed, interest rates

27

are decreasing in income only “when lenders use screening contracts.” Thus, another
test for information costs as the driving force behind the rise in consumer bankruptcies
is to compare the coefficient for income in an interest rates regression.29
The first two columns of Table 6 shows the projection of income on interest rates
using data from the SCF. In 1983 the effect of income on interest rates is not significantly different than zero. In contrast, in 2004, borrowers with higher income pay lower
interest rates.30 The last two columns present the results obtained using the model.
The value and the change in the coefficient between the different cases are quite similar.
This result not only supports the hypothesis that lenders use more information but also
indicates that it is appropriate to focus on information about households’ productivity.

6

Conclusions

How do information costs affect consumer bankruptcy? Asymmetric information and
costly screening are incorporated into a model of consumer debt and bankruptcy to
study this question. When screening is too expensive, uninformed lenders overcome
the lack of information by designing contracts to induce households to reveal their
income. The design of these contracts implies that low-risk households are borrowingconstrained. This is because contracts with low interest rates are linked to tight debt
limits to avoid high-risk households taking these contracts (they will prefer contracts
with higher interest rates and looser borrowing limits). With technological progress in
the IT sector, information costs drop and previously borrowing-constrained households
can now be screened and obtain more debt. Then, the rise in debt generates an increase
in bankruptcy filings because the benefits of filing bankruptcy increase with the amount
owed.
Can this model account for the changes in consumer credit markets over the last 20
years? Quantitative exercises are performed to answer this question. The parameters
are first calibrated to the year 1983. The model is successful replicating key (targeted
and non-targeted) moments for this year. Then, the cost of information is recalibrated
to the year 2004 to match the bankruptcy rate in that year. Without changing any other
parameter, the model can replicate many important features of the data. Importantly,
changes in the variables describing unsecured credit markets move in the same direction
as in the data. More importantly, in most of the cases changes are also quantitatively
similar. The fit of non-targeted moments—like the dispersion in interest rates or the
29
30

Thanks to Mark Bils for suggesting this exercise.
Edelberg (2006) studies risk-based pricing of interest rates for consumer loans and finds similar results.

28

charge-off rates—supports a drop in information costs as the driving force behind the
rise in bankruptcies over the last 20 years.

References
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in Incomplete Asset Markets,” Manuscript.
Aguiar, M., and G. Gopinath (2006): “Defaultable debt, interest rates and the
current account,” Journal of International Economics, 69, 64–83.
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“Default Risk and Income Fluctuations in Emerging

Economies,” American Economic Review, 98(3), 690–712.
Athreya, K., X. S. Tam, and E. R. Young (2008): “A Quantitative Theory of
Information and Unsecured Credit,” Federal Reserve bank of Richmond Working
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from the Banking Industry,” Journal of Money, Credit, and Banking, 35, 141–176.
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the U.S. Earnings and Wealth Inequality,” Journal of Political Economy, 111(4),
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Chatterjee, S., D. Corbae, M. Nakajima, and J. V. Rios-Rull (2007): “A
Quantitative Theory of Unsecured Consumer Credit with Risk of Default,” Econometrica, 75(6), 1525–1591.
Chatterjee, S., D. Corbae, and J. V. Rios-Rull (2007): “A Theory of Credit
Scoring and Competitive Pricing of Default Risk,” Manuscript.
(2008): “A Finite-Life Private-Information Theory of Unsecured Consumer
Debt,” Journal of Economic Theory, 142, 149–177.
Cuadra, G., and H. Sapriza (2006): “Sovereign default, terms of trade, and interest
rates in emerging markets,” Working Paper 2006-01, Banco de Mexico.
(2008): “Sovereign default, interest rates and political uncertainty in emerging
markets,” Journal of International Economics, forthcoming.

29

Drozd, L., and J. B. Nosal (2008): “Competing for Customers: A Search Model
of the Market for Unsecured Credit,” Manuscript.
Eaton, J., and M. Gersovitz (1981): “Debt with potential repudiation: theoretical
and empirical analysis,” Review of Economic Studies, 48, 289–309.
Edelberg, W. (2006): “Risk-based Pricing of Interest Rates for Consumer Loans,”
Journal of Monetary Economics, 53(8), 2283–2298.
Greenwood, J., J. M. Sanchez, and C. Wang (2007): “Financing Development:
The Role of Information Costs,” Economie davant garde, Research Report No. 14.
Hirshleifer, J., and J. G. Riley (1992): The Analytics of Uncertainty and Information. Cambridge University Press.
Livshits, I., J. MacGee, and M. Tertilt (2007a): “Accounting for the Rise in
Consumer Bankruptcies,” NBER Working Paper 13363.
(2007b): “Consumer Bankruptcy: A Fresh Start,” The American Economic
Review, 97(1), 402–418.
(2009): “Costly Contracts and Consumer Credit,” Manuscript.
Mendoza, E., and V. Yue (2008): “A Solution to the Default Risk-Business Cycle
Disconnect,” Manuscript, New York University.
Moss, D. A., and G. A. Johnson (1999): “The Rise of Consumer Bankruptcy:
Evolution, Revolution, or Both,” American Bankruptcy Law Journal, 73, 311–351.
Narajabad, B. N. (2007): “Information Technology and the Rise of Household
Bankruptcy,” Manuscript.
Nelder, J. A., and R. Mead (1965): “A Simplex Method for Function Minimization,” The Computer Journal, 7(4), 308–313.
Riley, J. G. (1979): “Informational Equilibrium,” Econometrica, 47(2), 331–59.
Zhang, H. H. (1997): “Endogenous Borrowing Constraints with Incomplete Markets,”
Journal of Finance, 52(5), 2187–2209.

30

7
7.1
7.1.1

Appendix
Proofs
Proof of Lemma 1

Proof: The model is solved backward. First, consider a household who has to make a
decision about bankruptcy. It is clear that this decision is characterized by
y2,n + a2 ≥ y2,n (1 − τ ), pay back,
y2,n + a2 < y2,n (1 − τ ), declare bankruptcy.
This implies simple threshold levels of assets for each level of income at which households are indifferent between filing bankruptcy and paying back the debt,
a2,L = −τ y2,L ,
a2,H = −τ y2,H ,
where a2,H < a2,L because y2,L < y2,H . Notice that if a household borrows less than
the limit for the low level of income, a2 ≥ a2,L , she will pay back the debt next period
if her income is low. Since a2,H < a2,L , she will also pay back if the level of income
next period is high. This implies that q(a2 , n) = (1 + i)−1 if a2,L ≤ a2 . Notice also
that for a2,H < a2 < a2,L the household will file for bankruptcy next period only after
a transition toward the low productivity. Since this happens with probability πn,L , we
have that q(a2 , n) = πn,L (1+i)−1 if a2,H < a2 < a2,L . Finally, notice that for a2 ≤ a2,H
the households will file for bankruptcy for sure. Thus, in this case, the price is equal
to 0.

7.1.2

Proof of Lemma 2

Proof: The slope of the indifference curve between q and a2 is
−M RSq,a2 (q, a2 ) =

−u′ (y1,n + a − qa2 )q + βπn,H u′ (y2,H + a2 )
.
u′ (y1,n + a1 − qa2 )a2

Notice that the denominator is always positive. The numerator is decreasing in a2 .
Define a∗2 (q) as the value of a2 that set the numerator equal to zero. Then, we have
that the slope will be

∗
 < 0, for aH
2 < a2 < a2 (q),
−M RSq,a2 (q, a2 )
= 0, for a2 = a∗2 (q),

> 0, for a∗2 (q) < a2 < aL
2.

31

7.1.3

Proof of Lemma 3

Proof: We need to show that the slope is bigger for households with low productivity
at period 1,
−u′ (y1,L + a1 − qa2 )q + βπL,H u′ (y2,H + a2 )
−u′ (y1,H + a1 − qa2 )q + βπH,H u′ (y2,H + a2 )
>
.
u′ (y1,L + a1 − qa2 )a2
u′ (y2,H + a1 − qa2 )a2
To see this, rewrite


−u′ (y1,n + a1 − qa2 )q + βπn,H u′ (y2,H + a2 )
βπn,H u′ (y2,H + a2 )
1
=
−q + ′
.
u′ (y1,n + a1 − qa2 )a2
a2
u (y1,n + a1 − qa2 )
Then, we only need to show that
βπL,H u′ (y2,H + a2 )
βπH,H u′ (y2,H + a2 )
>
.
u′ (y1,H + a1 − qa2 )
u′ (y1,L + a1 − qa2 )
This holds because πH,H > πL,H and u′ (y1,L + a1 − qa2 ) > u′ (y1,H + a1 − qa2 ).

7.1.4

Proof of Lemma 4

Proof: It is clear that q̃ implies zero profits given q̄. Also, it is clear that there is
no other contract that is profitable—and the borrowers would prefer—to q̃. For this,
just notice that a borrower would take q only if it is above q̃ (for a given a) and those
contracts imply negative profits if the cost of information is paid.

7.1.5

Proof of Lemma 5

Proof: To show that these prices are the Reactive equilibrium of this economy, we
need to prove that: (i) they imply zero-expected profits and (ii) any profitable deviation
implies cross-subsidization between borrowers with different risk of default. Figure 4
will be used to show these two points.
First, it is easy to see that these prices will imply zero-expected profits if borrowers
with low productivity borrow more than a2 (a1 ) and borrowers with high productivity
borrow less than a2 (a1 ). This is because for those ranges of a2 , prices q̂ are actually equal to q̄. Then, to see that borrowers separate themselves according to a2 (a1 ),
look at a2 (a1 ) in 4. By construction, households with low productivity are indifferent
borrowing more than a2 (a1 ) at the prices for low productivity households than borrowing a2 (a1 ) at the prices for high productivity. This implies that high productivity
households prefer to borrow less. Therefore, prices q̂ achieve separation.
Second, notice that any contract from which a low-productivity household would
deviate should have a q above q̄ for that level of a2 —remember that low productivity
borrowers are choosing a2 freely from q̄(·, L) in the equilibrium. But then, to be

32

profitable, this deviation should have a cross subsidy from high productivity borrowers.
Thus, any profitable deviation must have cross-subsidization.

7.1.6

Proof of Lemma 6

Proof: First, notice that type L households never prefer screening contracts. The
point eL in Figure 3 cannot be an equilibrium if lenders can offer contracts without
paying for information. In that allocation, the riskiest households are paying to reveal
their risk. For instance, some lenders could set the price at πL,H /(1 + i) − ǫ and do not
pay for information. Then, low productivity households would prefer that offer and
lenders will make profits.
To find the threshold cost of information, c, we use Figure 4. There, notice that the
point e′H , at which the high-income household is borrowing using screening contracts
if the cost is c, is on the same indifference curve, UH , as the point eH , at which the
high-income household is using revelation contracts. Thus, if the cost C is lower than
c, then high-income households prefer to take the screening contracts , and vice versa.

7.1.7

Proof of Lemma 7

Proof: There are two possibilities in which borrowers switch from not paying the cost
of information to paying it. Both cases are analyzed below:
- In Figure 5, the initial cost of information is high enough, C0 > c, and the type-H
household is borrowing-constrained at eH . When the cost of information drops
2 > U 1 , and debt
to C1 > c, these households prefer to pay for information, UH
H
H
increases. Since both levels of debt are in the range between aL
2 and a2 , the

default probability does not change. Notice that default does not change because
initially these households were already borrowing in the risky debt range.
- In Figure 6, the initial equilibrium allocation is at (eL , eH ). Notice that there,
high-type households are clearly borrowing-constrained. At this initial equilibrium allocation, default of H-types is actually zero. They borrow so little that
default is not optimal at any possible income level next period. When information costs fall to C0 , the new equilibrium allocation is (eL , elH ). There, type-H
households pay the cost of information, borrow more, and file for bankruptcy
with probability πH,L . Thus, in this case, H-type households’ debt rises and their
probability of bankruptcy increases from 0 to πH,L .

33

7.2

Transition matrix assumptions

The transition matrix is described in more detail in this subsection. This matrix can
be written as


̺1 χ1 ω χ2 ω
0
0
0
0
0
0
α−1 ̺2
χ
χ
0
0
0
0
0
1
2

χ−2 χ−1 ̺3
χ
χ2
0
0
0
0
1

 0
χ
χ
̺
χ
χ
0
0
0
−2
−1
3
1
2

 0
0
χ−2 χ−1 ̺3
χ1
χ2
0
0

 0
0
0
χ−2 χ−1 ̺3
χ1
χ2
0

 0
0
0
0
χ
χ
̺
χ
χ
−2
−1
3
1
2

 0
0
0
0
0 χ−2 χ−1 ̺4
χ1

 0
0
0
0
0
0 χ−2 χ−1 ̺5
1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9 1/9
which is equivalent to


̺1
R1 f1
R1 (1 − f1 )
̺2
R2 f1
 α−1
R (1 − f )
R3 f1
̺3
 3
1

0
R3 (1 − f1 )
R3 f1


0
0
R3 (1 − f1 )


0
0
0


0
0
0

0
0
0


0
0
0
1/9
1/9
1/9

0
0
R2 (1 − f1 )
0
R2 f1
R2 (1 − f1 )
̺3
R2 f1
R3 f1
̺3
R3 (1 − f1 )
R3 f1
0
R3 (1 − f1 )
0
0
0
0
1/9
1/9


ε
ε

ε

ε

ε
,
ε

ε

ε

ε
0

where Ri is obtained taking into account each row must sum up 1. Therefore, R1 , R2 ,
R3 , ̺4 and ̺5 are known, since they can be determined by
R1 = (1 − ̺1 − ε)
R2 = (1 − α−1 − ̺2 − ε)
R3 = (1 − ̺3 − R2 − ε)
̺4 = (1 − R3 − R2 f1 − ε)
̺5 = (1 − R3 − ε).
Then, only six parameter need to be calibrated : ρ1 , f1 , ε, α1 , ρ2 , and ρ3 .

34



...
0
0
ε
...
0
0
ε

...
0
0
ε
...
0
0
ε

...
0
0
ε
,
... R2 (1 − f1 )
0
ε

...
R2 f1
R2 (1 − f1 ) ε

...
̺4
R2 f1
ε
...
R3 f1
̺5
ε
...
1/9
1/9
0

Figure 1: The rise in consumer bankruptcies
2,200,000

2,000,000

1,800,000

1,600,000

1,400,000

1,200,000

filings by year

1,000,000

800,000

600,000

400,000

200,000

1980

1985

1990

1995

Year

Source: American Bankruptcy Institute.

35

2000

2005

Figure 2: Zero-expected-profit prices, q
Price
6


Assets, a2

..
..
..
..
..
..
..
..
..
..
..

1
..
1+i
..
.. . . . . . . . . . . 1 π
1+i H,H
..
..
..
..
..
..
..
.. . . . . . . . . . . 1 π
1+i L,H
..
..
.

a2,H

a2,L

?

Figure 3: Equilibrium prices of screening contracts, qe
Price
6

UH


Assets, a2

1
..
1+i
..
1
πH,H
.. . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . 1+i
1
.
..
.. . . . . . . . . . . 1+i πH,H − C
•
eH
..
..
..
.
UL ..
..
..
..
..
..
..
..
1
πL,H
.. . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . 1+i
1
..
.
.. . . . . . . . . . . 1+i πL,H − C
•
eL
..
..
-

a2,H

a2,L

36

?

Figure 4: Equilibrium prices of revelation contracts, q̂

UL (a1 )

Price
6

UH (a1 )


Assets, a2

..
..
..
..
..
..
..
..
..
..
..
a2,H

1
..
1+i
1
..•
.. . . . . . . . . . . 1+i
πH,H
eH
′
.
.
eH
1
.
.
•. . . . . . . . . . . .. . . . . . .. . . . . . . . . . . 1+i πH,H − c
..
..
..
..
..
..
..
..
..
..
..
.. . . . . . . . . . . 1 π
•
1+i L,H
.
..
eL
..
..
..
.

a2 (a1 )

37

a2,L

?

Figure 5: Cheaper information ⇒ more debt

UL1


Assets, a2

2
UH
1
UH

Discount price
6

1
..
1+i
.. . . . . . . . . . . 1 π
.. ′
•
1+i H,H
..
e
e
H
1
.. H•
πH,H − C1
. . . . . . . . . . . 1+i
.
1
..
. . . . . . . . . . . . . ... . . . . . . . . . . . 1+i
πH,H − c
..
..
..
..
..
..
..
..
..
..
1
πL,H
.. •
.. . . . . . . . . . . 1+i
e
.. L
..
..
..
-

aH
2

aL
2

?

Figure 6: Cheaper information ⇒ more debt and default
2
UH
1
UH

Discount price

61
..
1+i
e
.. H
..
.. . . . . . . . . . . 1 πH,H
1+i
..
1
πH,H − C1
.. . . . . . . . . . . 1+i
..
..
..
..
..
..
1
πL,H
. . . . . . . . . . . 1+i
..
..
.

•


Assets, a2

.. ′
.. eH•
..
..
..
..
..
..
..
..
..

UL
•

eL

aH
2

aL
2

38

?

Figure 7: Calibration of limits to income groups
f (l) 6

 ..
?
1

...
...
...
....
...
...
...
.
∆1 ...
...
.

l̄1

...
...
...
...
....
...
...
....
....
...
.
∆2 ...
...
.

-

¯l2

l

Figure 8: The effect of information costs on the debt distribution
.4

z = 10

.4

z = 0.8
.3732

.3

.3

.3318

.2795

Fraction
.2

Fraction
.2

.2772

.1901
.1595

.1632

.1

.1

.1082

.011

.0133 .0153 .0172 .0195

.0281

0

0

.0128

−.5

−.4

−.3
−.2
Assets over Mean Income

−.1

0

−.5

(a) High cost of information

−.4

−.3
−.2
Assets over Mean Income

−.1

(b) Low cost of information

39

0

Table 1: Goodness of fit for the year 1983

Targets
1983

“1983”
calibration

0.49%
0.33%
3.44

0.49%
0.36%
3.35

27.3%
30.9%
35.3%
45.3%
59.4%

28.7%
36.4%
44.0%
58.9%
73.9%

Gini coefficient of income
Mean-to-median income ratio

0.46
1.35

0.45
1.50

Gini coefficient of wealth
Mean-to-median wealth ratio

0.75
3.01

0.83
6.25

Statistics
Bankruptcy rate
Debt-to-income ratio
Capital-to-output ratio
Proportion
Proportion
Proportion
Proportion
Proportion

of
of
of
of
of

debt
debt
debt
debt
debt

held
held
held
held
held

by
by
by
by
by

income
income
income
income
income

poorest
poorest
poorest
poorest
poorest

10%
15%
20%
30%
40%

Table 2: Parameters for the “1983” calibration
Parameters

Values

Discount factor, β
Cost of bankruptcy, τ
Technology in the information sector, z i
Persistence of type n = 1
Persistence of type n = 2
Persistence of type n = {3, ..., 7}
Transition probability from type n = {1, ..., 9} to n = 10
Transition probability from type n = 2 to n = 1
Transition probability from type n = {2, ..., 8} to n + 1

0.921
0.189
1.261
0.442
0.766
0.416
0.0001
0.032
0.144

40

Table 3: Transition matrix types 1 to 6 (data and model)

[0, 6]
[6, 46]
[46, 65]
[65, 79]
[79, 87]
[87, 93]
...

[0, 6]
[6, 46]
[46, 65]
[65, 79]
[79, 87]
[87, 93]
...

[0, 6]
0.46
0.07
0.02
0.02
0.01
0.02

Data year 1983
Income groups between percentiles
[6, 46] [46, 65] [65, 79] [79, 87] [87, 93]
0.41
0.06
0.03
0.02
0.01
0.71
0.13
0.04
0.02
0.01
0.28
0.47
0.15
0.05
0.01
0.14
0.22
0.41
0.17
0.03
0.09
0.09
0.21
0.41
0.12
0.09
0.08
0.09
0.26
0.31

[0, 6]
0.44
0.03
0.11
0.00
0.00
0.00

Model’s “1983” calibration
[6, 46] [46, 65] [65, 79] [79, 87] [87, 93]
0.35
0.21
0.00
0.00
0.00
0.77
0.14
0.06
0.00
0.00
0.27
0.42
0.14
0.06
0.00
0.11
0.27
0.42
0.14
0.06
0.00
0.11
0.27
0.42
0.14
0.00
0.00
0.11
0.27
0.42

...

...

Source: Matched population from the Annual Demographic Income.
Supplement from the Current Population Survey.
The sample was restricted to men in working age.

Table 4: Goodness of fit to non-targeted moments

Moments
Charge-Off Rates
Bankruptcy filings over population*
Proportion of households with negative net worth
Coefficient of variation of interest rates

Data
1983
2.02%
0.06%
5.04%
28.3%

“1983”
Calibration
0.59%
0.03%
5.49%
27.4%

Source: SCF and Federal Reserve Statistical Release.
(*) Prorated using 0.53 as targeted moments for debt and bankruptcy.

41

Table 5: The effect of information on several variables

Statistics

Data
2004

(Diff ln data)*
’04 and ’83

Model with
z = 2.85

(Diff ln model)**
z = 2.85 and 1.26

Bankruptcy rate
Debt-to-income ratio
Capital-to-output ratio

1.60%
0.75%
3.44

118.0%
80.9%
-5.0%

1.52%
0.42%
3.35

113.2%
17.0%
-0.1%

10.0%
5.0%
10.0%
19.0%
16.7%

-100.8%
33.1%
81.7%
63.9%
17.7%

27.2%
7.8%
7.7%
15.2%
15.2%

-5.4%
1.6%
0.8%
2.0%
1.8%

0.53
1.63
0.81
4.70

15.2%
18.8%
7.3%
44.6%

0.45
1.50
0.83
6.25

0.0%
0.0%
0.0%
0.0%

3.67%
0.28%
5.38%
43.2%

98.0%
148.5%
31.8%
42.4%

1.64%
0.09%
3.67%
43.7%

102.9%
114.7%
1.5%
57.5%

Proportion
Proportion
Proportion
Proportion
Proportion

of
of
of
of
of

debt
debt
debt
debt
debt

held
held
held
held
held

by
by
by
by
by

income poorest 10%
percentiles 10 to 15%
percentiles 15 to 20%
percentiles 20 to 30%
percentiles 30 to 40%

Gini coefficient of income
Mean-to-median income ratio
Gini coefficient of wealth
Mean-to-median wealth ratio
Charge-Off Rates
Bankruptcy filings over population
Proportion of households with negative net worth
Coefficient of variation of interest rates

* The data for each year was first logged and then the difference was computed.
** The data for each value of z was first logged and then the difference was computed.

42

Table 6: The effect of income on interest rates
Dependent variable: interest rate

ln(income)

1983

Year
2004

“1983”

Model
“2004”

0.155
(0.219)

-0.768
(0.103)

0.063
(0.237)

-0.587
(0.175)

0.061
(0.102)
0.007
(0.012)
-0.001
(0.001)
0.720
(0.497)
-0.370
(0.439)
15.400
(2.114)

0.209
(0.052)
0.012
(0.008)
-0.001
(0.000)
-0.315
(0.242)
0.204
(0.215)
18.720
(1.076)

0.468
(0.124)

1.117
(0.091)

1115
0.010

6380
0.012

1078
0.013

6357
0.027

Controls
ln(credit card debt)
Age of the head of household
Age of the head of household squared
Male head of household
Married head of household
Constant

Observations
R squared

Note: the data from the SCF restricted is to households with credit card debt. The
difference between the coefficients for ln(income) is 0.923 and the standard deviation of
this difference is 0.379. The difference between the coefficients for ln(credit card debt)
is0.149 and the standard deviation of this difference is 0.178. The regressions with
the data generated with the model have a similar number of observations as those from
the SCF because a random sample of similar size was generated with the model.

43