Loan guarantees for consumer credit markets.
Athreya, Kartik B. ; Tam, Xuan S. ; Young, Eric R. 等
Decomposing the Effect of Taxes on Welfare
In this subsection we decompose the net effect of the loan
guarantee program. We consider two experiments, presented in Table 6,
where we ask how welfare changes if we confront an individual with the
pricing emerging from the presence of a loan guarantee, with and without
the taxes needed to finance the program. Starting in the top row of
Table 6 we display the effect of a move from the benchmark setting to
one in which a tax-free loan guarantee is provided. Welfare increases
quite substantially, again by least for the skilled and by most for the
unskilled. Since their income profile is flat, the NHS households
experience the largest gain because they use unsecured debt over most of
their life cycle. By contrast, the more-skilled types decrease unsecured
borrowing as they age (see Figure 5).
Turning next to the bottom row of Table 6, we present the welfare
implications of a move from a setting with a tax-free loan guarantee to
one where, including taxes, the program must now break even. As seen
earlier (Table 5), once taxes are imposed only the unskilled benefit
from a program this generous, and they lose proportionally more from
taxes than do the college types. Why are the costs of a small tax so
large in this model? With taxes, permanent income is reduced, leaving
households more exposed to the expenditure shock. As a result, they
"involuntarily" default more frequently, leading to more
deadweight loss and a much larger welfare loss than one would expect
from a tax of less than 4 percent. Due to the accumulation pattern of
net worth, on average NHS households are more exposed to this risk
(again, see Figure 5).
[FIGURE 5 OMITTED]
Table 7 decomposes the costs of the program by type. The loan
guarantee program transfers resources along two dimensions. First, loan
guarantees transfer resources from skilled households to less-skilled;
college types pay into the program, via taxes, significantly more than
they collect in terms of lower interest rates. Second, loan guarantees
transfer resources from individuals who pose little default risk (those
with low [lambda]) to those with a high value for [lambda], as the
latter pose more default risk, all else equal. This transfer occurs
because the high-risk types would pay substantially higher interest
rates without intervention and therefore gain a lot from the program.
Asymmetric Information
Returning to the problem noted at the outset of the previous
subsection, recall that the cost of limited access to unsecured credit
is likely largest for the least wealthy. This is particularly likely to
be true in a society that lacks the information storage, sharing, and
data analysis available in developed nations to effectively identify
credit risk at the time of loan origination (and then update it
regularly). As a first step in getting a sense of the quantitative
potential of loan guarantees to alter outcomes in such settings, we now
study stationary equilibria of our model under asymmetric information.
To remind the reader, in our economy, asymmetric information will
mean that the borrower will have characteristics that are not observable
to the lender; specifically, we assume neither current stigma, [lambda],
nor current net worth, a, can be directly observed. However, any
information about these variables that can be inferred from the
observable components of the state vector, as well as from the desired
borrowing level, b, is available to the lender. (23) We focus on two
representative examples: one that represents a relatively modest loan
guarantee program and results in welfare gains for all types under
symmetric information ([theta] = 0.1 and v = 0.1), and one that is more
generous and reduces the welfare of college-educated types ([theta] =
0.5 and v = 0.4). Our key finding is that the presence of asymmetric
information will increase the gains available from loan guarantees, no
matter how generous.
Allocations and Pricing
We first compare outcomes in the FI and PI economies. Table 8 shows
that a move from symmetric to asymmetric information has the following
effects. First, default falls for all types, and default skews more
strongly toward the high [lambda] type; these individuals are treated
relatively better under asymmetric information, since they get terms
that reflect the average default risk instead of their own, and
therefore end up borrowing amounts that induce relatively high default
rates. Second, overall the credit market shrinks, in the sense that we
observe fewer borrowers (of each type) and lower discharged debt
aggregates.
Figure 6 shows that pricing is significantly worse for the high
[lambda] (low bankruptcy cost) borrower and better for the low [lambda]
borrower. Under asymmetric information, the two types will be pooled
together, so that the default premium at a given debt level reflects the
average default risk. The result is that good borrowers face
significantly tighter credit limits and higher interest rates, while bad
borrowers face the same credit limit but lower interest rates. The shift
in pricing accounts for the smaller credit market size.
Third, as noted at the outset, our model features expenditure
shocks. These shocks take on a larger role in defaults under asymmetric
information (see Table 9). With tighter credit limits, big expenditure
shocks that hit when the household is young are hard to smooth, since
income is relatively low. The result is that essentially all defaults
are done by households who have received an expenditure shock, despite
this group being only 7.56 percent of the population. Information has
less of an impact on these defaults, since they are defaults on debt
that has been acquired involuntarily.
We now turn to the effects of loan guarantees under asymmetric
information. Table 8 shows that the change induced by the introduction
of the particular program is larger for all credit market aggregates
under asymmetric information, with the exception of the debt-to-income
ratio for college-educated households (in which case it is of only
slightly smaller magnitude). Figure 7 shows the increased access to
credit that guarantees provide in these two cases. Note that the
increase in the default rate is smaller under asymmetric information for
every education group. As a result, the taxes required to finance the
program are lower than under symmetric information.
[FIGURE 6 OMITTED]
Welfare
Table 10 displays the welfare effects of two different loan
guarantee programs. Relative to the symmetric information case, loan
guarantees are uniformly better when information is asymmetric; this
result holds for every case we have computed. The larger gain is partly
due to the lower tax burden required in the asymmetric information cases
and partly due to the severe pricing distortion caused by asymmetric
information evident in Figure 6.
To more directly describe the transfers between agents induced by
loan guarantees, Table 7 collects the proportion of costs paid by each
group. Now the loan guarantee program subsidizes the high [lambda] (low
stigma cost) types much more than under symmetric information. This
result is exactly what we would expect, given that this type is
receiving better credit terms under asymmetric information.
Targeted Loan Guarantees
Our results suggest that loan guarantees have the potential to
become primarily a means of transferring resources from the rich to the
poor. Moreover, our findings suggest that they may also lower welfare,
often of all types of agents, unless their generosity is modest. In our
results, default is disproportionately driven by those who have received
an expenditure shock. A natural question therefore is whether the
benefits of loan guarantees discussed at the outset can be preserved by
limiting compensation to lenders only when a borrower has suffered such
a shock. Expenditure shocks represent large increases in debts that are
rare and involuntarily acquired. As a result, a policy of guaranteeing
loans only under these conditions is unlikely to alter loan pricing
substantially (since these states are rare) but may substantially aid
households who find themselves in those rare states. Moreover, targeted
guarantees are unlikely to induce significant additional deadweight loss
because the default decision is more frequently heavily influenced by
expenditure shocks, which again, are rare.
To investigate this question, we study a case where v = 0.50 and
[theta] = 0.50, but where lenders only receive compensation in the event
that a bankruptcy coincides with a positive expenditure shock (x >
0). Table 11 shows that all groups gain from the introduction of a loan
guarantee program restricted in this manner. As before, the NHS
households gain most and the highly skilled gain the least. Nonetheless,
the ability of the conditionality of the program to overturn what was
initially a very large welfare loss to the skilled into a gain is
striking. (24)
To see the effect on aggregates more generally, we turn to Table
12. It is immediately clear that the tax rate needed to sustain the
restricted loan guarantee program is very small relative to the
unrestricted case, even though the debt discharged in bankruptcy is
similar to the unrestricted guarantee case. Nonetheless, the overall
level of debt responds to the restricted guarantee far more modestly
than the unrestricted case. For example, under restricted guarantees,
the mean debt-to-income ratio among high-school educated borrowers is
less than half that under unrestricted guarantees (0.2256 versus
0.4707). The central reason for the low tax rate is that the default
rate responds by far less than with an unrestricted program, even though
borrowing does increase nontrivially, relative to the benchmark case.
Under restricted guarantees, the bankruptcy rate roughly doubles, while
the unrestricted program implies a nearly ten-fold increase.
[FIGURE 7 OMITTED]
3. DISCUSSION
We have made a few assumptions in our model that require some
additional discussion. First, we have assumed that factor prices are
fixed. General equilibrium calculations would imply higher r and lower W
would prevail under loan guarantee systems, since they produce more
borrowing and less aggregate wealth (as well as increasing the amount of
transactions costs that works like a reduction in aggregate supply of
goods). Factor price movements of this sort are likely to make the
welfare costs larger (gains smaller), since the higher risk-free
interest rate would make borrowing more costly and the lower wages would
reduce mean consumption. Despite these effects, we choose to abstract
from equilibrium pricing because it is well known that income processes
representative of the vast majority of households will, in environments
such as ours, produce less wealth concentration than observed (see
Castaneda, Diaz-Gimenez, and Rios-Rull 2003), meaning that the mean
wealth position will be too similar to the median, implying larger
factor price changes than would occur if the distribution of wealth were
matched. Given the immense computational burden that matching the U.S.
Gini coefficient of wealth would impose on our OLG setup, and given that
the factor price adjustments should be small, we feel justified in
ignoring them. (25)
Second, we have financed the program using proportional labor
income taxes. An obvious alternative would be to finance the program
using progressive income taxes, where high income (college) types would
pay higher marginal tax rates. This approach would increase the gains to
the NHS types, who already gain substantially, and reduce (or even
eliminate) any gains to college types. We expect a similar result from
capital income taxation as well, since it will tend to tax the wealthier
college types more heavily. In contrast, a regressive income tax would
imply the types who benefit the most, the NHS, would pay a higher
marginal tax rate. Regressive tax systems seem unlikely to be
implemented on equity grounds, even if they are welfare-improving within
a specific model. We could also introduce separate programs for each
education group, so that the cross-subsidization that makes the program
so attractive to NHS types would be eliminated; we conjecture that this
case would result in larger gains for college types and smaller for NHS
types.
Third, there is a conceptual issue of the right benchmark
allocation. The U.S. corporate income tax rate is 35 percent and banks
are permitted to deduct losses due to nonperforming loans from their
taxable income. As a result, it may be that the appropriate benchmark is
a case where the loan guarantee program is not zero, but rather has a
large value of v and [theta] = 0.35. We can of course easily express the
welfare gains relative to this benchmark instead; a more detailed
investigation of this issue is part of ongoing work.
There are some natural extensions of our model that seem useful to
pursue. Given our results regarding the effect of loan guarantees to
redistribute toward the unskilled from the skilled, it would be
productive to know if the least skilled, for example, would benefit from
a loan guarantee program that was required for self-financing via taxes
on only the unskilled. Such an extension would be along the lines
explored in Gale (1991), who studies targeted loan guarantees designed
to facilitate credit access for certain identifiable subpopulations
(such as minority borrowers). Targeted programs would be related to the
regulations we mentioned earlier that require certain characteristics
not be reflected in credit terms; exactly how the dual goals of
encouraging access to these groups without allowing their
characteristics to alter credit terms would affect welfare is unknown
and worth studying. It would also be straightforward to investigate
loans targeted to individual borrowers who are deemed constrained by
competitive lenders. (26) In our model, since borrowers are at a
"cliff" in the pricing function, they would benefit from
government loans at their existing interest rate, provided the tax costs
are not "too high."
Also, our work is a step in the direction that, in the future, will
allow us to analyze the role of guarantees for mortgage lending.
However, the central role of aggregate risk in driving home-loan default
makes a full quantitative analysis that satisfactorily incorporates the
forces we do allow for here--asymmetric information and limited
commitment--currently infeasible. But we note that such a model would
have the same fundamental structure as that developed here.
4. CONCLUDING REMARKS
A significant share of the U.S. population appears credit
constrained. These households usually lack collateral and must therefore
rely on the unsecured credit market to help them smooth consumption in
the face of life-cycle and shock-related movements in income. However,
the unsecured credit market in the United States appears significantly
impeded by forces that keep the costs of unsecured debt default low, and
thereby make lending risky and, hence, expensive. Perhaps the most
widely used route to increase credit flows to target groups is via the
use of loan guarantees whereby public funds defray private lenders'
losses from default. Aside from their direct effects on credit access
and pricing, guarantees are likely to be particularly useful in
unsecured credit markets given limitations on the ability of policies to
directly influence borrowers' default incentives. In this article,
we assess the consequences of extending loan guarantees to unsecured
consumer lending to improve allocations.
Our article attempts to quantify the impact of loan guarantees in a
model that incorporates both meaningful private information and a
limited commitment problem into a rich life-cycle model of consumption
and savings. Our quantitative analysis focuses on evaluating the impact
of introducing loan guarantees into unsecured consumer credit markets.
These markets have large consequences for household welfare because they
influence the limits on smoothing faced by some of the least-equipped
subgroups in society, particularly the young and the unlucky.
Our calculations suggest first that, under symmetric information,
loan guarantees can actually improve the ex ante welfare of all
households if they are not too generous (meaning only small loans
qualify). This welfare gain is disproportionately experienced by
low-skilled households who face flat average income paths and relatively
large shocks. Indeed, such households gain from very generous programs,
but higher-skilled types rapidly begin to experience welfare losses as
loan guarantees are made more generous. These results arise because loan
guarantees induce a transfer from skilled to unskilled, and this
transfer can be substantial, while the gains to the skilled from seeing
loan pricing terms improve as a result of guarantees is relatively
small. Second, we find that allocations are quite sensitive to the size
of qualifying loans: Even modest limits on qualifying loan size invite
very large borrowing--as perhaps intended by proponents--but also spur
very large increases in default rates. As a result, loan guarantee
programs transfer resources in significant amounts from all households
to the lifetime poor. Under asymmetric information, the welfare gains
are larger for all households, as the taxes required to finance the
programs are smaller. Our work provides an answer for why, despite the
potential welfare gains from expanding guarantees to consumer credit
that thereby alleviate credit constraints for a marginalized population
otherwise lacking collateral, public guarantees on unsecured consumer
credit have not yet been implemented. The value of the program depends
on how elastically credit demand and supply respond to default risk,
which may be hard to estimate, and the programs are quite costly if too
generous. (27) As a practical matter, the forces at work in our model
may well be part of explaining why student loan default rates hit 25
percent in the early 1990s, at which point the government increased
monitoring and enforcement (recall also the similar findings of Lelarge,
Sraer, and Thesmar [2010] in the French entrepreneurship context).
The preceding intuition will likely carry over to markets beyond
the one for unsecured consumer credit, in particular for two areas that
have seen some form of loan guarantee: federal student loans and home
loans. It suggests that loans of the size guaranteed by a federal
student loan program would have been likely to default at high rates,
even under a relatively "partial" nature of the guarantee.
Similarly, the FHA/VA and others have historically provided loan
guarantees for mortgage loans. The calibrated costs of default measured
in our model suggest strongly that larger loans, especially if covered
more fully by a loan guarantee program, would lead to even greater debt
and default than that predicted for the consumer credit market.
Therefore, unless such loans are vetted carefully, one should expect a
high take-up rate, a high subsequent failure rate, and nontrivial
transfers from better-off households. Nonetheless, despite the risks
involved, a main result of the article is that a limited program,
specifically one where loan guarantees are made contingent on certain
rare but disastrous events, can deliver net gains for all households.
Such a policy seems worth exploring further. Of course, a caveat to the
conclusion that targeting guarantees to those who have suffered a bad
expense shock is that it may require additional resources to battle any
moral hazard that might be present, especially when default is allowed
upon getting any shock that is not a genuine catastrophe to households.
Taken as a whole, our results suggest that loan guarantees can help, but
care must be taken if policymakers intervene in credit markets through
the use of loan guarantees.
Lastly, because the results reported in this article suggest that
loan guarantees for household credit may be a powerful tool for altering
steady-state consumption, our work should be of help for future
examinations of the extent to which consumer lending and more
importantly, consumer willingness to borrow, can be amplified to spur
current consumption in business cycle contexts. The model of Gordon
(2015) could possibly be adapted to this question.
APPENDIX: QUANTITATIVE MODEL
We now provide a detailed description of the quantitative model
used here. As noted at the outset, it is essentially that of Athreya,
Tam, and Young (2012b), modified to accommodate changes in loan pricing
and taxes necessitated by loan guarantees.
Preferences
Households in the model economy live for a maximum of J <
[infinity] periods and face stochastic labor productivity and mortality
risk. Households supply labor inelastically. (28) Households differ
along several dimensions over their life cycles according to an index of
type, denoted y and defined in what follows. Each household of age j and
type y has a conditional probability [[psi].sub.j,y] of surviving to age
j + 1. Households retire exogenously at age [j.sup.*] < J. Let
[n.sub.j] denote the number of "effective" members in a
household. Households value consumption per effective household member
[c.sub.j]/n.sub.j]. They have identical additively separable isoelastic
felicity functions with parameter [sigma], and possess a common discount
factor [beta]. To smooth consumption, all households have access to
risk-free savings, and also debt that they may fully default on, subject
to some costs. These costs reflect the variety of consequences that
bankruptcy imposes on households, and need not be interpreted solely as
"stigma," but include any such costs. A portion of these costs
are represented by a nonpecuniary cost of filing for bankruptcy, denoted
by [[lambda].sub.j,y], which we also permit to depend on household type
y. Household preferences are therefore given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
where [d.sub.j] is the indicator function that equals unity when
the household chooses to default in the current period (in which case
[d.sub.j] = 1).
The existence of nonpecuniary costs of bankruptcy is strongly
suggested by the calculations and evidence in Fay, Hurst, and White
(1998) and Gross and Souleles (2002). The first article shows that a
large measure of households would have "financially benefited"
from filing for bankruptcy but did not, while both articles document
significant unexplained variability in the probability of default across
households after controlling for a large number of observables.
In this specification, a household with a relatively low value of
[[lambda].sub.j,y] will obtain low value from any given expenditure on
consumption ([c.sub.j]) in a period in which they file for bankruptcy.
This is meant to reflect the increased transactions cost associated with
obtaining utility via consumption expenditures in the period of a
bankruptcy. Examples include increased "shopping time" arising
from difficulty in obtaining short-term credit and payments services,
locating rental housing and car services, as well as any
stigma/psychological consequences. For convenience, we will sometimes
refer to [[lambda].sub.j,y] as stigma in what follows; we intend it to
be more encompassing. (29) Because of the breadth of costs that [lambda]
represents, we will allow it to vary stochastically over time and across
individuals as a function of their type y, according to a transition
function [p.sub.[lambda]].
At the time of obtaining a loan, a household who expects to have a
relatively low value of [lambda] next period will know that filing for
bankruptcy will result in a relatively high cost of obtaining any given
level of marginal utility in the next period. Given the current marginal
utility of consumption, consumption smoothing (i.e., keeping marginal
utility in accordance with the standard Euler equation) under bankruptcy
will therefore be costlier, all else equal, than for a household with a
high value of [lambda]. This is further amplified by the fact that
households are not allowed to borrow in the same period as when they
file for bankruptcy. For convenience, we will therefore refer to those
whose value of [[lambda].sub.j,y] is relatively low as
"low-risk" borrowers, and vice versa.
In addition to this nonpecuniary cost, there is an out-of-pocket
pecuniary resource cost A that represents all formal legal costs and
other procedural costs of bankruptcy. Lastly, households are not allowed
to borrow or save in the same period as a bankruptcy filing, to capture
provisions guarding against fraud that are routinely applied in court.
There are no other costs of bankruptcy in the model.
Endowments
Our focus on consumer credit makes it critical to allow for both
uninsurable idiosyncratic risk. Consumer default, and hence the value of
loan guarantees, is by all accounts strongly tied to individual-level
uninsurable risk (see, e.g., Sullivan, Warren, and Westbrook [1999,
2000] and Chatterjee et al. [2007]). (30) There are two sources of such
risk in our model. First, households face shocks to their labor
productivity, and because they are modeled as supplying labor
inelastically, face shocks to their labor earnings. Second, households
are susceptible to shocks to their net worth. The former represent
shocks arising in the labor market more generally, and the latter
represent sudden required expenditures arising from unplanned events
such as sickness, divorce, and legal expenses.
In addition to the use of credit to deal with stochastic
fluctuations in income and expenditures, consumer credit also likely
serves, as noted earlier, as a tool for longer-term, more purely
intertemporal smoothing in response to predictable, low-frequency
changes in labor income, such as those coming with increased age and
labor market experience. This leads us to specify, in addition to
transitory and persistent shocks to income, a deterministic evolution in
average labor productivity over the life cycle. This component of
earnings will reflect most obviously one's final level of
educational attainment, which is represented in the model as part of an
agent's type, y.
Specifically, log labor income will be determined as the sum of
four terms: the aggregate wage index W, a permanent shock y realized
prior to entry into the labor market, a deterministic age term
[[omega].sub.j,y], and a persistent shock e that evolves as an AR(1)
process. The log of income at age-j for type--y is therefore given by
log W + log [[omega].sub.j,y] + log y + log e + log v,
where
log (e') = [zeta] log (e) + [epsilon]', (4)
and a purely transitory shock log (u). Both [epsilon] and log (u)
are independent mean zero normal random variables with variances that
are y-dependent and have distributions [p.sub.e] and [p.sub.v],
respectively.
As for the risk of stochastic expenditures, we follow the
literature (e.g., Chatterjee et al. 2007 and Livshits, MacGee, and
Tertilt 2007), and specify a process [x.sub.j] to denote the expense
shock to net worth that takes on three possible values {0, [x.sub.1],
[x.sub.2]} from a probability distribution Px(*) with i.i.d.
probabilities {1 - [p.sub.x1] - [p.sub.x2],[p.sub.x1],[p.sub.x2]}.
We will take agents' permanent type y to reflect differences
between households with permanent differences in human capital.
Specifically, we will consider agents with three types of human capital:
those who did not graduate high school, those who graduated high school,
and those who graduated college. (31) This partition of households
follows Hubbard, Skinner, and Zeldes (1994). The central reason for
allowing this heterogeneity is that the observed differences in mean
life-cycle productivity for each of these types of agents gives them
different incentives to borrow over the life cycle. 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. They also face smaller shocks
than the other two education groups. The life-cycle aspect of our model
is key; in the data, while bankruptcies occur late into the life cycle
for some (see, e.g., Livshits, MacGee, and Tertilt 2007), defaults are
still skewed toward young households. (32)
Market Arrangement
As stated earlier, to smooth consumption and save for retirement,
households have access to both risk-free savings as well as one-period
defaultable debt. The issuance and pricing of debt is modeled as a
two-stage game in which households at any age j first announce their
desired asset position [b.sub.j], after which a continuum of lenders
simultaneously announces a loan price q. As a result, a household
issuing [b.sub.j] units of face value receives [qb.sub.j] units of the
consumption good today. A household who issues debt with face value
[b.sub.j] at age-j is agreeing to pay [b.sub.j] in the event that they
fully repay the loan, and pay zero otherwise (i.e., when they file for
bankruptcy). The fact that nonrepayment can occur with positive
probability in equilibrium means that lenders will not be willing to pay
the full face value, even after adjusting for one-period discounting.
Therefore, given any gross cost of funds [??], we must have q [less than
or equal to] 1/[??].
As we will allow for both symmetric and asymmetric information, we
introduce the following notation. Let I denote the information set for a
lender and [??] : b x I [right arrow] [0, 1] denote the function that
assigns a probability of default to a loan of size [b.sub.j] given
information I. Clearly, since default risk assessed by lenders will
depend in general on both their information and the size of the loan
taken by a household, so will loan prices. Therefore, let loan pricing
be given by the function q([b.sub.j], I). Under asymmetric information,
we allow lenders to use the information revealed by the size of the loan
request and lenders' knowledge of the distribution of household net
worth in the economy to update their assessment of all current
unobservables. Thus, lenders use their knowledge of both (i) optimal
household decision making (i.e., their decision rules as a function of
their state), and (ii) the endogenous distribution of households over
the state vector. We will describe the determination of this function in
detail below.
The household budget constraint during working life, as viewed
immediately after the decision to repay or default on debt has been
made, is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)
[a.sub.j] is net worth after the current-period default decision
[d.sub.j]. Therefore, [a.sub.j] = [b.sub.j-1] - [x.sub.j] if [d.sub.j] =
0 and 0 if [d.sub.j] = 1. Households' default decisions also
determine their available resources beyond removing debt, because
default consumes real resources [LAMBDA], arising from court costs and
legal fees. The last term, (1 - [[tau].sub.1] - [[tau].sub.2])
W[[omega].sub.j,y] yev, is the after-tax level of current labor income,
where r 1 is the flat-tax rate used to fund pensions and [[tau].sub.2]
is the rate used to finance the loan guarantee program. Keep in mind
also that implicit in the specification of the loan pricing function
q(*) is the fact that if the household borrows an amount in excess of
the guarantee limit, the price is that of an entirely nonguaranteed
loan.
The budget constraint during retirement is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)
where for simplicity we assume that pension benefits are composed
of a fraction v [member of] (0,1) of income in the last period of
working life plus a fraction [GAMMA] [member of] (0,1) of average income
W (we normalize average individual labor earnings to 1).
Consumer's Problem
The timing is as follows. In each period, all uncertainty is first
realized. Thus, income shocks e and v, the default cost [lambda], and
the current expense shock x are all known before any decisions within
the period are made. Following this, households must decide, if they
have debt that is due in the current period, to repay or default. This
decision, along with the realized shocks, then determines the resources
the household has available in the current period. Given this, the
household chooses current consumption and debt or asset holding with
which to enter the next period, and the period ends.
Prior to making the current-period bankruptcy decision, a household
can be fully described by [b.sub.j-1], the debt, if any, that is due in
the current period, their type y, the pair of currently realized income
shocks e and v, their cost of default [lambda], the current realization
of the shock to expenses, [x.sub.j], and their age j. (33)
Letting V(*) denote the household's value function prior to
the decision to default or repay, with primed variables denoting objects
one period ahead, we have the following recursive description. If the
household chooses to repay its debt [b.sub.j-i], and therefore sets
[d.sub.j] = 0, then the value they derive from state ([b.sub.j-1], y, e,
v, [lambda], x, j) is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)
subject to the budget constraint
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)
If the household has chosen bankruptcy for the current period
([d.sub.j] = 1), since we disallow credit market activity in the period
of bankruptcy, which implies [b.sub.j] = 0, we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
subject to the budget constraint:
[c.sub.j] + [LAMBDA] [less than or equal to] (1 - [[tau].sub.1] -
[[tau].sub.2]) W[[omega].sub.j,y]yev. (9)
Notice that both debt due in the current period, [b.sub.j-1], and
the current expenditure shock realization, [x.sub.j]; get removed by
bankruptcy, and hence disappear, when comparing the budget constraint
under bankruptcy to one under nonbankruptcy. By contrast, the resource-
and nonpecuniary costs, [LAMBDA], and [[lambda].sub.j,y], respectively,
both appear.
Given this, prior to the bankruptcy decision, the current-period
value function is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
For the full information setting we assume I contains the entire
state vector for the household; let I = (y, e, v, x, [lambda], j).
Abusing notation slightly, let d(*) now denote the decision rule
governing default. As described earlier, this function drives the
decision to repay a given debt or not, and hence depends on the full
household state vector. Letting non-primed objects represent current
period decisions, and using primed variables for objects dated one
period ahead, we have the following zero profit condition for the
intermediary. Simply put, it requires that the probability of default
used to price debt must be consistent with that observed in the
stationary equilibrium, implying that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)
Since d (b, e', v', x', [lambda]', j + 1)
specifies whether or not the agent will default in state e',
v', x', [lambda]') tomorrow at debt level b, integrating
over all such events one period hence produces the relevant estimated
default risk [[??].sup.fi]. This expression also makes clear that
knowledge of the persistent components (e, [lambda]) is relevant for
predicting default probabilities, and the more persistent these
characteristics are, the more useful they become in assessing default
risk.
Asymmetric Information
As we noted at the outset, earlier work, starting with Narajabad
(2012), and including the work of Sanchez (2009) and Athreya, Tam, and
Young (2012a), found that in past decades, unsecured credit market
outcomes may well have been affected by informational frictions. In the
latter article, asymmetric information governing individual-level costs
of bankruptcy were shown to be consistent with a variety of features of
the data from the 1980s and earlier. Thus, to evaluate the implications
of loan guarantees under asymmetric information, we assume that
nonpecuniary default costs, [[lambda].sub.j,y], is unobservable. With
the exception of current household net worth following the bankruptcy
decision in a period (which we denoted by a) all other household
attributes, including educational attainment, age, and the current
realization of the persistent component of income are assumed
observable. To be clear, using household decisions rules and the
distribution of households over the state space to infer a
borrower's current net worth, a, is not useful because the net
worth a is relevant to forecasting income, default risk, or anything
else; it is not. Rather, it is because lenders want to draw a more
precise inference on the current values of the persistent aspects of a
household's state. In this case the inference is about the current
realization of a household's [lambda], something that is clearly
relevant to assessing default risk.
Let [p.sup.*]([lambda]|b, y, e, v, x, j) denote the equilibrium
conditional probability of a household having a realized value of
[lambda], given that they have observable characteristics y, e, v, x, j,
and that they have issued bonds of b units of face value. To construct
the equilibrium assessment of default risk, [[pi].sup.*](*), lenders use
their knowledge of household decision making and the joint (conditional)
distribution of households over the state space to arrive at a
probability distribution for the current value of a household's
nonpecuniary default cost. (34) The best estimate of default risk is
then given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Equilibrium in the Credit Market
Here, we follow Athreya, Tam, and Young (2012b), and employ the
Perfect Bayesian Equilibrium (PBE) concept to define equilibrium in the
game between borrowers and lenders. Denote the state space for
households by [OMEGA] = B x Y x E x V x L x J x {0,1} [subset] [R.sup.6]
x[Z.sub.++] x{0,1} and space of information as I [subset] Y x E x V x L
x J x {0,1}. Let the stationary joint distribution of households over
the state be given by [GAMMA]([OMEGA]). Let the stationary equilibrium
joint distribution of households over the state space [OMEGA] and loan
requests b' be derived from the decision rules {[b'.sup.*]
(*), [d.sup.*](*)} and [GAMMA]([OMEGA]), and be denoted by [[PSI].sup.*]
([OMEGA],b'). Given [[PSI].sup.*]([OMEGA],b'), let
[[mu].sup.*](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, [OMEGA],
given b', be denoted by [[GAMMA].sup.*]([OMEGA]|b'). (35)
Definition 1 A PBE for the credit market game of incomplete
information consists of (i) household strategies for borrowing
[b'.sup.*] : [OMEGA] [right arrow] R and default [d.sup.*] :
[OMEGA] x [lambda] x E x V [right arrow] {0,1}, (ii) lenders'
strategies for loan pricing [q.sup.*] : R x I [right arrow] [0, 1/1+r]
such that [q.sup.*] is weakly decreasing in b', and (iii)
lenders' common beliefs about the borrower's state [OMEGA]
given a loan request of size b', [[GAMMA].sup.*] ([OMEGA]|b'),
that satisfy the following:
1. Households optimize: Given lenders' strategies, as
summarized in the locus of prices [q.sup.*] (b',I), decision rules
{[b'.sup.*] (*), [d.sup.*](*)} solve the household problem.
2. Lenders optimize given their beliefs: Given common beliefs
[[GAMMA].sup.*] ([OMEGA]| b'), [q'.sup.*] is the pure-strategy
Nash equilibrium under one-shot simultaneous-offer loan-price
competition.
3. Beliefs are consistent with Bayes' rule wherever possible:
[[GAMMA].sup.*]([OMEGA]|b') is derived from [[PSI].sup.*]([OMEGA],
b') and household decision rules using Bayes rule whenever b is
such that [[mu].sup.*](b') > 0.
Equilibria are located through an iterative procedure. The
interested reader is directed to the online appendix in Athreya, Tam,
and Young (2012b), where we discuss the computational procedure used to
solve for equilibria. As a quick summary, we define an iterative
procedure that maps a set of pricing functions back into themselves,
whose fixed points are PBE of the game between lenders and borrowers.
This procedure is monotonic, so starting from the upper limit yields
convergence to the largest fixed point. (36)
Government
The government's budget constraint is motivated by two
expenditures it must finance. Most importantly, it must finance payments
to a lender to honor the loan guarantee program. Letting [GAMMA](a, y,
e, u, x, [lambda],j) denote the invariant cumulative distribution
function of households over the states, this is given by tax
[[tau].sub.2], which must satisfy
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)
In addition to financing loan guarantees, the government funds
pension payments to retirees and to finance the loan guarantee system.
The government budget constraint for pensions is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)
Wage Determination
For both simplicity and substantive reasons, we assume constant and
exogenous factor prices in our welfare calculations. In particular, we
assume that the risk-free rate r is exogenous and determined by the
world market for credit. Our approach follows several articles in the
literature in abstracting from feedback effects onto risk-free rates of
saving coming from changes in borrowing in the unsecured credit market,
including Livshits, MacGee, and Tertilt (2007). This is a convenient
abstraction and will be reasonable as long as guarantee programs are not
inordinately generous.
Specifically, given r, profit maximization by domestic production
firms implies that
W = (1 - [alpha])[(r/[alpha]).sup.[alpha]/[alpha] - 1], (13)
where [alpha] is capital's share of income in a Cobb-Douglas
aggregate production technology.
Stationary Equilibrium
We have already given the definition of equilibrium for the game
between borrowers and lenders. The outcomes of that interaction were, of
course, part of a larger fixed-point problem that included, among other
things, the joint distribution of households over the state space,
[GAMMA](*), and the tax rates [[tau].sub.1] and [[tau].sub.2] needed to
fund transfers and loan guarantees, respectively. But this joint
distribution depended on household borrowing behavior, which in turn
influenced the construction of [GAMMA](*). Given this feedback, we will
focus throughout on stationary equilibria in which all aggregate objects
including, critically, the joint distribution [GAMMA](*), remain
constant over time under the decision rules that arise from household
and creditor optimization.
Computing stationary equilibria requires two layers of iteration.
We first specify the wage rate, interest rate, tax rates, and public
sector transfer and loan guarantee policies. This allows us to solve the
household's decision problem and locate the associated stationary
distribution of households over the state space--all for a given guess
of the equilibrium loan-pricing locus q(*). Our use of a risk-free
rate-taking open economy allows us to iterate on the function q(*)
without having to deal with any additional feedback from loan pricing to
risk-free interest rates and wages. Once we have located a price
function that is a fixed point under the stationary distribution induced
by optimal household decision making (which we can denote by
[q.sup.*](*)), we need to check if the government budget constraint
holds. Here, we must iterate again, this time on transfers and taxes. We
use Brent's method to solve for the tax rate that satisfies the
government budget constraint (re-solving for the fixed-point loan
pricing function [q.sup.*](*) each time); whenever Laffer curve
considerations arise, we choose the lower tax rate.
Parametrization
To assign values to model parameters, we proceed first by imposing
standard values from the literature for measures of income risk,
out-of-pocket expenses, risk aversion, and demographics. We then
calibrate the remaining model parameters, which are those governing
bankruptcy costs and the discount factor. The goal is to match, as well
as possible, key facts about bankruptcy and unsecured credit markets in
the United States, given income risk, risk aversion, and demographics.
As discussed earlier, we follow the literature by calibrating to recent
data and assuming symmetric information between borrowers and lenders.
The parametrization is relatively parsimonious and largely
standard. First, as mentioned above, we directly assign values to
household level income risk and risk aversion at values standard in the
literature. The model period is taken to be one year. The income process
is taken from Hubbard, Skinner, and Zeldes (1994), who estimate separate
processes for non-high school (NHS), high school (HS), and
college-educated (Coll) workers for the period 1982-1986. (37) Figure 3
displays the path [[omega].sub.j,y] for each type; the large hump
present in the profile for college-educated workers implies that they
will want to borrow early in life to a greater degree than the other
types (despite their effective discount factor being somewhat higher
because of higher survival probabilities). The process is discretized
with 15 points for e and 3 points for v. The resulting processes are
log (e') = 0.95 log (e) + [epsilon]'
[epsilon] ~ N (0, 0.033)
log (v) ~ N (0, 0.04)
for non-high school agents,
log (e') = 0.95 log (e) + [epsilon]'
[epsilon] ~ N (0, 0.025)
log (v) ~ N (0, 0.021)
for high school agents, and
log (e') = 0.95 log (e) + [epsilon]'
[epsilon] ~ N (0, 0.016)
log (v) ~ N (0, 0.014)
for college agents. We normalize average income to 1 in model
units, and in the data one unit roughly corresponds to $40,000 in
income. When we construct the invariant distribution of the model, we
assume households are born with zero assets and draw their first shocks
from the stationary distributions.
To assign values for the idiosyncratic risk of out-of-pocket
expenses, we choose the parameters for the expenditure shock [x.sub.j]
to be the annualized equivalent of those used in Livshits, MacGee, and
Tertilt (2007). For pensions, we set v = 0.35 and [GAMMA] = 0.2,
yielding an average replacement rate of 55 percent, and assume an
exogenous retirement age of [j.sup.*] = 45. Relative risk aversion is
set to [sigma] = 2, as is standard, and a value that also avoids
overstating the insurance problem faced by households. Lastly, with
respect to demographics, we set the measures of the college (Coll), high
school (HS), and non-high school (NHS) agents to 20, 58, and 22 percent,
respectively, and the maximum lifespan to J = 65, corresponding to a
calendar age of 85 years.
Table 1 in the main text displays the targeted moments and the
implied ones from the model. (38) Table 2 in the main text displays the
parameters associated with this calibration, along with the other
parameters of the model (such as the cost of default [LAMBDA], which is
set to match the observed $1, 200 filing cost). First, the default
rates, measured as filings for Chapter 7 bankruptcy, are very close to
the data. Second, the model does fairly well at matching the debt/income
ratios in the data, measured as credit card debt divided by income (from
the Survey of Consumer Finances 2004), although it reverses the order by
understating debt for college types and overstating it for non-high
school types. Lastly, the model generates a somewhat higher proportion
of the observed fraction of borrowers while yielding smaller value of
discharged debt to income ratio than currently measured. (39)
To parameterize the nonpecuniary costs of bankruptcy while limiting
free parameters, we represent [lambda] by a two-state Markov chain with
realizations {[[lambda].sub.L,y], [[lambda].sub.H,y]} that are
independent across households, but serially dependent with a symmetric
transition matrix [P.sub.[lambda]].
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The calibrated process suggests that nonpecuniary costs of
bankruptcy are largely in the nature of a "type" for any given
household. This interpretation arises because the benchmark calibration
reveals [lambda] to be very persistent, and therefore very unlikely to
change during the part of life where unsecured credit is useful. This
persistence is also what makes the model consistent with the observed
ability of households to borrow substantial amounts but still default at
a nontrivial rate. Despite this "implicit collateral," debts
discharged in bankruptcy are still higher in the data; however, the
discharge ratio from the data (obtained as the median debts discharged
in bankruptcy divided by the median income of filers taken from the
survey data of Sullivan, Warren, and Westbrook [2000]) is likely an
overestimate, as it includes small business defaults that are generally
large and not present in the model. The size of the values for [lambda]
are relatively large, implying that even the low cost types view default
as equivalent to a loss of nearly 10 percent of consumption; thus, the
primary source of implicit collateral in this model is stigma rather
than pecuniary costs.
Table 3 in the main text presents a decomposition of defaults
according to the various combinations of expense shock and stigma. The
median shock for x and the high value of [lambda] constitute only 3.55
percent of the population but are responsible for 58.11 percent of the
defaults under symmetric information, while the high shock for x and
high value for [lambda] are 0.23 percent of the population and 6.66
percent of the defaults. Thus, defaults are clearly skewed toward
households that experience an expenditure shock, consistent with the
model of Livshits, MacGee, and Tertilt (2007).
Lastly, while omitted from the tables for brevity, the other
relevant probability is that of the likelihood of default given the
receipt of an expenditure shock. This distribution yields two pieces of
information about the model. First, getting an expenditure shock,
particularly the largest one, greatly increases the likelihood of
default, all else equal. Second, the vast majority of households who
receive such a shock still do not default. The reason for this is that
the power of such shocks to drive default, while nontrivial, is still
naturally limited by the wealth positions households take on as they
move through the life cycle. Default is most likely to happen when one
has substantial debts at the same time that one receives such a shock.
This rules out relatively older households from being very susceptible;
as seen in Figure 6, they have, in the main, already begun saving for
retirement. (40)
REFERENCES
Aiyagari, S. Rao. 1994. "Uninsured Idiosyncratic Risk and
Aggregate Saving." Quarterly Journal of Economics 109 (August):
659--84.
Andolfatto, David. 2002. "A Theory of Inalienable Property
Rights." Journal of Political Economy 110 (April): 382-93.
Athreya, Kartik B. 2008. "Default, Insurance, and Debt over
the Life-Cycle." Journal of Monetary Economics 55 (May): 752-74.
Athreya, Kartik B., Juan M. Sanchez, Xuan S. Tam, and Eric R.
Young. 2012. "Bankruptcy and Delinquency in a Model of Unsecured
Credit." Federal Reserve Bank of St. Louis Working Paper 2012-042A.
Athreya, Kartik B., Xuan S. Tam, and Eric R. Young. 2009.
"Unsecured Credit Markets Are Not Insurance Markets." Journal
of Monetary Economics 55 (January): 83-103.
Athreya, Kartik B., Xuan S. Tam, and Eric R. Young. 2012a.
"Debt Default and the Insurance of Labor Income Risk." Federal
Reserve Bank of Richmond Economic Quarterly 98 (Fourth Quarter):
255-307.
Athreya, Kartik B., Xuan S. Tam, and Eric R. Young. 2012b. "A
Quantitative Theory of Information and Unsecured Credit." American
Economic Journal: Macroeconomics 4 (July): 153-83.
Becker, Gary S. 1967. Human Capital and the Personal Distribution
of Income: An Analytical Approach (Woytinsky lecture). Ann Arbor, Mich.:
Institute of Public Administration.
Calem, Paul S., Michael B. Gordy, and Loretta J. Mester. 2006.
"Switching Costs and Adverse Selection in the Market for Credit
Cards: New Evidence." Journal of Banking and Finance 30 (June):
1,653-85.
Castaneda, Ana, Javier Diaz-Gimenez, and Jose-Victor Rios-Rull.
2003. "Accounting for Earnings and Wealth Inequality." Journal
of Political Economy 111 (August): 818-57.
Chaney, Paul, and Anjan Thakor. 1985. "Incentive Effects of
Benevolent Intervention: The Case of Government Loan Guarantees."
Journal of Public Economics 26 (March): 169-89.
Chatterjee, Satyajit, P. Dean Corbae, Makoto Nakajima, and
Jose-Victor Rios-Rull. 2007. "A Quantitative Theory of Unsecured
Consumer Credit with Risk of Default." Econometrica 75 (November):
1,525-90.
Davila, Julio, Jay H. Hong, Per Krusell, and Jose-Victor Rios-Rull.
2012. "Constrained Efficiency in the Neoclassical Growth Model with
Uninsurable Idiosyncratic Shocks." Econometrica 80 (November):
2,431-67.
Dawsey, Amanda E., and Lawrence M. Ausubel. 2004. "Informal
Bankruptcy." Working Paper (April).
Dubey, Pradeep, John Geanakoplos, and Martin Shubik. 2005.
"Default and Punishment in General Equilibrium." Econometrica
73 (January): 1-37.
Fay, Scott A., Erik Hurst, and Michelle J. White. 1998. "The
Bankruptcy Decision: Does Stigma Matter?" University of Michigan
Working Paper 98-01 (January).
Gale, William G. 1990. "Federal Lending and the Market for
Credit." Journal of Public Economics 42 (July): 177-93.
Gale, William G. 1991. "Economic Effects of Federal Credit
Programs." American Economic Review 81 (March): 133-52.
Ghent, Andra C., and Marianna Kudlyak. 2011. "Recourse and
Residential Mortgage Default: Evidence from U.S. States." Review of
Financial Studies 24 (June): 3,139-86.
Gordon, Grey. 2015. "Evaluating Default Policy: The Business
Cycle Matters." Quantitative Economics. Available at:
www.qeconomics.org/ojs/forth/372/372-2.pdf.
Gross, David B., and Nicholas S. Souleles. 2002. "An Empirical
Analysis of Personal Bankruptcy and Delinquency." Review of
Financial Studies 15 (March): 319-47.
Hubbard, R. Glenn, Jonathan Skinner, and Stephen P. Zeldes. 1994.
"The Importance of Precautionary Motives for Explaining Individual
and Aggregate Saving." Carnegie-Rochester Conference Series on
Public Policy 40 (June): 59-125.
Innes, Robert. 1990. "Investment and Government Intervention
in Credit Markets when there is Asymmetric Information." Journal of
Public Economics 46 (December): 347-81.
Jappelli, Tullio. 1990. "Who is Credit Constrained in the U.S.
Economy?" Quarterly Journal of Economics 105 (February): 219-34.
Jeske, Karsten, Dirk Krueger, and Kurt Mitman. 2010. "Housing
and the Macroeconomy: The Role of Implicit Guarantees for Government
Sponsored Enterprises." Working Paper.
Jia, Ye. 2013. "Small Business Loan Guarantees as Insurance
Against Aggregate Risks." The B.E. Journal of Macroeconomics 13
(January): 455-79.
Lacker, Jeffrey M. 1994. "Does Adverse Selection Justify
Government Intervention in Loan Markets?" Federal Reserve Bank of
Richmond Economic Quarterly 80 (Winter): 61-95.
Lelarge, Claire, David Sraer, and David Thesmar. 2010.
"Entrepreneurship and Credit Constraints: Evidence from a French
Loan Guarantee Program." In International Differences in
Entrepreneurship, edited by Josh Lerner and Antoinette Schoar. Chicago:
University of Chicago Press, 243-73.
Li, Wenli. 1998. "Government Loan Guarantee and Grant
Programs: An Evaluation." Federal Reserve Bank of Richmond Economic
Quarterly 84 (Fall): 25-51.
Li, Wenli. 2002. "Entrepreneurship and Government Subsidies: A
General Equilibrium Analysis." Journal of Economic Dynamics and
Control 26 (September): 1,815-44.
Livshits, Igor, James MacGee, and Michele Tertilt. 2007.
"Consumer Bankruptcy: A Fresh Start." American Economic Review
97 (March): 402-18.
Malysheva, Nadezhda, and John R. Walter. 2013. "How Large Has
the Federal Financial Safety Net Become?" Federal Reserve Bank of
Richmond Working Paper 10-03R (March).
Narajabad, Borghan N. 2012. "Information Technology and the
Rise of Household Bankruptcy." Review of Economic Dynamics 15
(October): 526-50.
Pijoan-Mas, Josep. 2006. "Precautionary Savings or Working
Longer Hours?" Review of Economic Dynamics 9 (April): 326-52.
Sanchez, Juan. 2009. "The Role of Information in the Rise of
Consumer Bankruptcy." Federal Reserve Bank of Richmond Working
Paper 2009-4 (April).
Smith, Bruce D., and Michael J. Stutzer. 1989. "Credit
Rationing and Government Loan Programs: A Welfare Analysis." Real
Estate Economics 17 (2): 177-93.
Sullivan, Teresa A., Elizabeth Warren, and Jay Lawrence Westbrook.
1999. As We Forgive Our Debtors: Bankruptcy and Consumer Credit in
America. New York: Oxford University Press.
Sullivan, Teresa A., Elizabeth Warren, and Jay Lawrence Westbrook.
2000. The Fragile Middle Class: Americans in Debt. New Haven, Conn.:Yale
University Press.
Tam, Xuan S. 2009. "Long Term Contracts in Unsecured Credit
Markets." Manuscript.
Telyukova, Irina A. 2013. "Household Need for Liquidity and
the Credit Card Debt Puzzle." Review of Economic Studies 80
(January): 1,148-77.
Walter, John R., and John A. Weinberg. 2002. "How Large is the
Federal Financial Safety Net?" CATO Journal 21 (Winter): 369-94.
Williamson, Stephen. 1994. "Do Informational Frictions Justify
Federal Credit Market Programs?" Journal of Money, Credit and
Banking 26 (August): 523-44.
Zame, William R. 1993. "Efficiency and the Role of Default
when Security Markets are Incomplete." American Economic Review 83
(December): 1,142-64.
Zeldes, Stephen P. 1989. "Consumption and Liquidity
Constraints: An Empirical Investigation." Journal of Political
Economy 97 (April): 305-46.
(1) Source: www.census.gov/compendia/statab/2012/tables/12s1189.pdf. The unit is the family in the 2007 Survey of Consumer Finances, and
the sample is before either the deleveraging or Great Recession. All
dollars are 2007.
(2) In addition to these officially guaranteed loan programs, there
is one that dwarfs them all, and this is the one operated by the two
main government-sponsored enterprises, Fannie Mae and Freddie Mac. These
entities issue securities to investors that come with a guarantee
against default risk. The ultimate originators of mortgage credit taken
by homebuyers thereby receive, in essence, a loan guarantee. While such
guarantees have historically not been backed by the Treasury, they now
clearly are: mortgage-backed securities investors receive Fannie and
Freddie guarantees on loans with a face value of approximately $5
trillion, nearly half of the value of all household mortgage debt. See
Li, (2002), Walter and Weinberg (2002), and Malysheva and Walter (2013)
for more details. These articles show that the overall
contingent-liabilities of the U.S. government have grown substantially
over time. Lastly, beyond their sheer size, the scope of activities
receiving guarantees is noteworthy. Endeavors ranging from nuclear power
plant construction, trade credit, microenterprises, and support for
female entrepreneurs all currently receive loan guarantees.
(3) They are referred to as federal lines of credit. Details are
here: www personal.umich.edu/~mkimball/fiscal-bang-for-buck-29may12.pdf.
(4) In addition to decoupling risk and pricing, loan guarantees
will also reduce average interest rates, all else equal. This is
relevant for two reasons. First, concern with the consequences of
frequent repricing of consumer debt has already led to policy changes.
Most noticeably, the CARD Act of 2009 has responded by essentially
requiring longer-term commitments from lenders in an attempt to deter
frequent repricing. However, as studied by Tam (2009), such policies may
carry serious side effects. In particular, average interest rates are
predicted to rise substantially to offset the ability of a borrower to
"dilute" his debt (much as in the sovereign debt literature).
Second, average borrowing rates are likely important for welfare: Calem,
Gordy, and Mester (2006) show that many U.S. households appear to use
credit cards for relatively long-term financing, making the roughly
10-percentage-point cost differential between secured and unsecured
interest rates quantitatively important.
(5) Andolfatto (2002) develops a simple model to illustrate how
government policies (e.g., interest rate ceilings) may induce unintended
outcomes (e.g., credit constraints) that generate calls for further
policies to deal with these side effects (e.g., loan guarantees). A
related point is that to the extent that public insurance simply crowds
out familial or other forms of private insurance, the effects will be
overstated. This possibility is not addressed in our article, and so
should be kept in mind.
(6) With respect to the federal lines of credit noted earlier, and
the subsidy that will allow the scheme to affect allocations (unlike the
actuarially fair arrangement that we show is irrelevant), the
idea's originator, professor Miles Kimball of the University of
Michigan, argues as follows: "I am assuming the government will
lose money doing this--just not as much as if they handed the money away
as a tax rebate with no obligation of repayment. The losing money part
would stop private lenders cold [emphasis ours]."
(7) See, e.g., Federal Reserve Release G.19:
www.federalreserve.gov/releases/g19/Current/g19.pdf.
(8) Sometimes, these costs are primarily those arising from the
surrender of tangible collateral that, ex post, becomes less valuable
than reneging on the repayment obligation, e.g., as recent house price
declines have done (Ghent and Kudlyak 2011). In other cases, default
implies the destruction of intangible collateral, as described above.
But in all cases, loan guarantees fundamentally concern unsecured
lending.
9 In future work, we aim to analyze the role of guarantees for
mortgage lending. However, the central role of aggregate risk in driving
home-loan default makes a full quantitative analysis that satisfactorily
incorporates the forces we do allow for here--partially endogenously
asymmetric information and limited commitment--currently infeasible. But
that model would have the same fundamental structure as the one
developed here.
(10) In related work, Lacker (1994) investigates whether adverse
selection problems necessarily justify government intervention in credit
markets. When cross-subsidization between private contracts is not
feasible, intervention is generally welfare-improving.
(11) While substantially different than our model, it is important
to note the early work of Smith and Stutzer (1989), who provide a simple
argument for the use of loan guarantees in unsecured commercial credit
markets--compared to direct government loans or equity purchases, loan
guarantees are the only option that does not worsen the private
information problem. The interest rate reductions apply to all risk
types, so high-risk types do not find any particular advantage, beyond
what they already have, for pretending to be low risk. Other programs,
such as direct loans to those unable to obtain credit (who are low risk
in their model), will lead to additional incentives by high-risk
borrowers to claim the contracts intended for low-risk ones, a situation
that is harmful to efficiency. Two important distinctions between our
work and theirs are worth keeping in mind--the nature of the commitment
problem and the issue of government revenue balance. In Smith and
Stutzer (1989), limited commitment is a trivial consideration: Default
occurs when the borrower receives zero income and is costless (in terms
of direct costs). In contrast, U.S. bankruptcy procedures are voluntary
and clearly not costless: There is a filing fee in addition to
substantial time costs and some form of stigma/nonpecuniary costs appear
relevant as well (see Fay, Hurst, and White [1998] or Gross and Souleles
[2002]). Smith and Stutzer (1989) do not consider the financing of such
payments; any welfare gains from the guarantee could easily be wiped out
by the cost of taxation. In contrast, a central aspect of our analysis
is the requirement that transfers required to implement loan guarantees
be paid for via taxes.
(12) The figures represent outcomes under the following
parameterization for the endowments of each group. For the first group
of agents, three conditions hold: (i) the amount that can be feasibly
repaid in the bad state is large (that is, [e.sub.L] relatively big);
(ii) the household will default in both states under risk-free pricing
(in the case where [lambda] is small relative to [e.sub.L] and
[e.sub.H], and the latter are close together); and (iii) the household
would borrow if asset markets were complete ([beta] (1 + r) < 1 and
E([e.sub.2]) significantly larger than [e.sub.1]). This group is
(weakly) harmed by the intertemporal disruptions that default options
create; because the two states tomorrow are very similar, the household
would either default in both states and thus be unable to borrow at all
(q = 0), or it would not default in either state and thus care not at
all about default options. As a result, the outcome may be worse than if
bankruptcy were banned since, in the absence of a default option,
feasibility would permit borrowing against the (relatively high) value
of [e.sub.L].
For the second type's endowments, three conditions are
assumed: (i) the amount of debt that can feasibly be repaid in all
states is small (that is, [e.sub.H] is low); (ii) the household will
default only in the low state ([lambda] intermediate and [e.sub.L] and
[e.sub.H] far apart); and (iii) the household would borrow if asset
markets were complete ([beta] (1 + r) < 1 and E([e.sub.2]) =
[e.sub.1]). A member of this group can gain from the default option
because she actually can borrow more with a bankruptcy option, as she
does not intend to repay in the low state; thus, feasibility is limited
only by the amount that can be repaid in the high state and additional
consumption smoothing is feasible. This is a manifestation of what might
be referred to as a "supernatural" debt limit, as opposed to
the "natural" debt limit (e.g., Aiyagari 1994): Feasibility
involves what can be repaid in the best state instead of the worst.
(13) Although not shown in the figure, the typical indifference
curve turns upward at very low levels of b, but these lie well outside
the budget set.
(14) This program assumes that the household cannot obtain a
qualifying loan of size greater than v by visiting multiple lenders;
that is, we attach the qualification criterion to the borrower, not the
lender.
(15) We assume that no costless and credible signals are available
and that some additional hidden characteristic, such as initial wealth,
thwarts the lender's attempts to infer from b the exact value of
[pi].
(16) For example, the FHA loan guarantee fee structure is given
here: www.sba.gov/community/blogs/community-blogs/small-business-cents/understandingsba-7a-loan-fees.
(17) Our neutrality result holds in the asymmetric information
signaling model we study here. Whether it holds in a screening
environment (such as Sanchez [2009]) is unclear, since it may be
possible to offer (q, [tau]) pairs that separate types.
(18) Other related work includes Chatterjee et al. (2007) and
Livshits, MacGee, and Tertilt (2007), though the former uses an infinite
horizon.
(19) This restriction seems to be standard practice in markets
where some form of loan guarantee program exists. For example, FNMA
(Fannie Mae) will not issue guarantees on loans that do not conform to
their pre-set standards, which include a restriction on the
loan-to-value ratio.
(20) As we noted earlier, qualification actually applies to the
total debt of the borrower, not the total loan emanating from any one
lender. An implicit assumption is therefore that this debt burden is
observable.
(21) Specifically, we use the decomposition: var(log(c)) = var (E
[log (c)|j]) + E [var(log(c)|j)] .
(22) Davila et al. (2012) shows that utilitarian constrained
efficient allocations in a model with uninsurable idiosyncratic shocks
are skewed toward improving the welfare of "consumption-poor"
households (since they have higher marginal utility). While we do not
attempt to characterize constrained efficient allocations here, it seems
clear that this intuition would apply--thus, policies that raise the
utility of the least-skilled would seem to be preferable from a social
welfare perspective.
(23) We assume that credit markets are anonymous, so that past
borrowing is also not observable to the current lender. In Athreya, Tam,
and Young (2012b) we introduce a flag that tracks whether a household is
likely to have recently defaulted. Due to computational considerations
we do not examine this case here.
(24) We are implicitly assuming that expenditure shocks are likely
to be easy to observe; we doubt that agents could easily hide one from
the government, given the size and nature of these shocks. Our
calibration, as noted above, equates x to a combination of medical and
legal bills plus unplanned family costs; these expenses should be
relatively easy to monitor in practice.
(25) In Chatterjee et al. (2007), the model is calibrated to the
U.S. distribution of wealth; the resulting effects of an endogenous
risk-free rate are quantitatively unimportant.
(26) A stylized approach to this is taken in Smith and Stutzer
(1989).
(27) An important caveat here is that in our model, the costs of
default are assumed invariant to the level of default in the economy.
Thus, a major loan guarantee program may meaningfully affect default
costs. This is surely subject to at least some Lucas Critique-related
problems. Nonetheless, endogenizing these costs is beyond the scope of
the article.
(28) We abstract from elastic labor supply because it is known
(e.g., Pijoan-Mas 2006) that under incomplete markets, households
borrowing significant amounts tend to supply labor relatively
inelastically, and for our study, this margin is unlikely to be crucial.
It naturally implies that our welfare cost measurements may be biased,
but it is unclear which direction that bias would go.
(29) Another possibility is that these households gain the benefits
from bankruptcy without filing, as suggested by Dawsey and Ausubel
(2004). Athreya et al. (2012) extends the benchmark model to include a
delinquency state in which households do not formally file for
bankruptcy but also do not service their debt.
(30) In mortgage lending, loan guarantees protect lenders against
house price fluctuations, which in turn are strongly tied to aggregate
risk (or at least city-level risk). The full incorporation of the
aggregate risk, private information, and limited commitment needed to
analyze this specific class of guarantees remains an important topic for
future work.
(31) Mortality rates also differ by education, although this
heterogeneity is of no consequence for our questions.
(32) See Sullivan, Warren, and Westbrook (2000).
(33) To avoid repetition, we display only the value functions
during working life; retirement is entirely analogous.
(34) See Athreya, Tam, and Young (2012b) for details.
(35) Recall that the stationary distribution of households over the
state space alone is given by [GAMMA](-).
(36) Uniqueness cannot be ensured, since q = 0 is a fixed point of
our mapping. However, simple sufficient conditions exist to rule out q =
0 as the maximal fixed point; [LAMBDA] > 0 is enough to guarantee the
existence of an interval [-[LAMBDA], 0] of risk-free debt. Sufficient
conditions that ensure the existence of nontrivial default risk in
equilibrium are not known.
(37) In Athreya, Tam, and Young (2009) we study the effect of the
rise in the volatility of labor income in the United States and find the
effect on the unsecured credit market to be quantitatively small; the
key parameter for default is the persistence of the shocks. We would
find similar numbers if we adjusted the variance of the shocks upward to
conform to more recent data.
(38) The calibrated parameters are obtained by minimizing the
(equally weighted) sum of squared deviations between the data and
moments from the invariant distribution of the model. Since the model is
not linear, we cannot guarantee that there exists a set of parameters
that makes this criterion zero; indeed, we find that such a vector does
not seem to exist.
(39) If we had data on discharge by education type, we could permit
the persistence of [lambda] to vary by type and possibly match the
aggregates more closely.
(40) For agents with the relatively high value for [lambda] in the
model:
High expense shock: 26%
Median expense shock: 15%
Low expense shock (a value of zero) = 1%
For agents with the relatively low value for [lambda] in the model:
High expense shock: 17%
Median expense shock: 2%
Low expense shock (a value of zero) = 0%
The numbers are very similar under asymmetric information.
We thank Larry Ausubel, Dean Corbae, Martin Gervais, Kevin Reffett,
and participants in seminars at Arizona State, Georgia, Iowa, and the
Consumer Credit and Bankruptcy Conference at the University of Cambridge
for helpful comments and discussions. We also thank Brian Gaines for his
editorial assistance. We especially thank Borys Grochulski, Robert
Hetzel, and Miki Doan for detailed comments, and Edward Prescott for his
guidance as editor. The opinions expressed here are not necessarily
those of the Federal Reserve Bank of Richmond or the Federal Reserve
System. E-mail: kartik.athreya@rich.frb.org.
Table 1 Model Versus Data
Model Target/Data
Discharge/Income Ratio 0.2662 0.5600
Fraction of Borrowers 0.1720 0.1250
Debt/Income Ratio | NHS 0.1432 0.08
Debt/Income Ratio | HS 0.1229 0.11
Debt/Income Ratio | COLL 0.0966 0.15
Default Rate | NHS 1.237% 1.228%
Default Rate | HS 1.301% 1.314%
Default Rate | COLL 0.769% 0.819%
Table 2 Calibration
Parameter Value
[x.sub.low] 0.0000
[x.sub.median] 0.0888
[x.sub.high] 0.2740
[[lambda].sup.NHS.sub.low] 0.7675
[[lambda].sup.HS.sub.low] 0.7309
[[lambda].sup.Coll.sub.low] 0.7830
[[pi].sup.[lambda].sub.HH] = 0.9636
[[pi].sup.[lambda].sub.LL]
[j.sup.*] 45
[sigma] 2.0000
[alpha] 0.3000
Prob ([x.sub.low]) 0.9244
Prob ([x.sub.median]) 0.0710
Prob ([x.sub.high]) 0.0046
[[lambda].sup.NHS.sub.high] 0.9087
[[lambda].sup.HS.sub.high] 0.9320
[[lambda].sup.Coll.sub.high] 0.9017
J 65
[LAMBDA] 0.0300
[phi] 0.0300
r 0.0100
Table 3 Aggregate Effects of Loan Guarantee Program
[theta] = 0.50
v 0.00 0.10 0.30 0.60 0.70
[[tau].sub.2] 0.0000 0.0005 0.0174 0.0531 0.0664
Discharge/Income Ratio 0.2662 0.2691 0.5430 0.9907 1.1172
Fraction of Borrowers 0.1720 0.2039 0.2400 0.4023 0.4466
Debt/Income Ratio | NHS 0.1432 0.1648 0.4765 0.6562 0.7118
Debt/Income Ratio | HS 0.1229 0.1372 0.3707 0.5369 0.5934
Debt/Income Ratio | COLL 0.0966 0.1140 0.2532 0.3858 0.4124
Default Rate | NHS 1.237% 1.768% 11.651% 19.691% 20.877%
Default Rate | HS 1.301% 1.751% 11.658% 16.609% 17.836%
Default Rate | COLL 0.769% 0.987% 5.668% 11.569% 13.100%
Table 4 Optimal Generosity of Loan Guarantee Program
[theta]
= 0.50
COLL HS NHS
v = 0.00 [right arrow] v = 0.10 0.02% 0.08% 0.13%
v = 0.00 [right arrow] v = 0.20 -0.24% 0.20% 0.22%
v = 0.00 [right arrow] v = 0.30 -1.41% 0.27% 0.39%
v = 0.00 [right arrow] v = 0.40 -1.60% 0.19% 0.78%
v = 0.00 [right arrow] v = 0.50 -2.24% -0.11% 1.06%
v = 0.00 [right arrow] v = 0.60 -2.84% -0.35% 1.26%
v = 0.00 [right arrow] v = 0.70 -3.60% -0.44% 1.02%
Table 5 Distribution of Consumption
var E (var (log var (E (log
E(c) (log (c)) (c) |age)) (c) |age))
Aggregate
NO LG 0.8455 0.1894 0.1671 0.0223
LG v = 0.5, [theta] = 0.5 0.8016 0.1977 0.1755 0.0222
College
NO LG 1.0918 0.1776 0.1293 0.0481
LG v = 0.5, [theta] = 0.5 1.0521 0.3874 0.3354 0.0520
High School
NO LG 0.7767 0.2279 0.1907 0.0372
LG v = 0.5, [theta] = 0.5 0.7575 0.3926 0.3749 0.0180
Non-High School
NO LG 0.6579 0.2807 0.2582 0.0225
LG v = 0.5, [theta] = 0.5 0.6514 0.3932 0.3849 0.0083
Table 6 Welfare Decomposition, Symmetric Information
v = 0.5, [theta] = 0.50
COLL HS NHS
([q.sup.NLG], [[tau].sub.2] = 0.0) 4.86% 8.30% 10.69%
[right arrow] ([q.sup.LG],
[[tau].sub.2] = 0.0)
([q.sup.LG], [[tau].sup.2] = 0.0) -6.74% -7.76% -8.69%
[right arrow] ([q.sup.LG],
[[tau].sup.2] = 0.0386)
Table 7 Distribution of Net Costs Paid by Type
v = 0.50, [theta] = 0.50, FI
High [lambda] Low [lambda]
Taxes Transfer Taxes Transfer
Coll 0.1366 0.1050 0.1366 0.0384
HS 0.2995 0.5082 0.2995 0.1512
NHS 0.0639 0.1333 0.0639 0.0639
v = 0.50, [theta] = 0.50, PI
High [lambda] Low [lambda]
Taxes Transfer Taxes Transfer
Coll 0.1366 0.1155 0.1366 0.0341
HS 0.2995 0.4971 0.2995 0.1239
NHS 0.0639 0.1711 0.0639 0.0583
Table 8 Aggregate Effects of Loan Guarantees--Asymmetric
Information
v = 0.40
FI PI
[theta] = 0.0000 0.5000 0.0000 0.5000
[[tau].sup.2] 0.0000 0.0245 0.0000 0.0196
Discharge/Income Ratio 0.2662 0.6965 0.2021 0.6497
Fraction of Borrowers 0.1720 0.3109 0.1614 0.3036
Debt/Income Ratio | NHS 0.1432 0.4880 0.1209 0.4762
Debt/Income Ratio | HS 0.1229 0.3897 0.0909 0.3755
Debt/Income Ratio | COLL 0.0966 0.2691 0.0801 0.2389
Default Rate | NHS 1.237% 13.170% 0.956% 12.704%
Default Rate | HS 1.301% 12.310% 0.957% 11.407%
Default Rate | COLL 0.769% 6.304% 0.658% 5.412%
Table 9 Distribution of Default by State
FI PI
High [lambda] Low [lambda] High [lambda] Low [lambda]
Low x 0.2315 0.0092 0.0089 0.0000
Median x 0.5811 0.0670 0.8033 0.0399
High x 0.0666 0.0446 0.0890 0.0588
Table 10 Welfare Effects of Loan Guarantees
COLL HS NHS
NO LG [right arrow] [theta] = 0.50, -1.60% 0.19% 0.78%
v = 0.40,FI
NO LG [right arrow] [theta] = 0.50, -1.02% 0.98% 1.59%
v = 0.40,PI
NO LG [right arrow] [theta] = 0.10, 0.01% 0.02% 0.03%
v = 0.10,FI
NO LG [right arrow] [theta] = 0.10, 0.04% 0.08% 0.11%
v = 0.10,PI
Table 11 Welfare Effects of Restricted Loan Guarantees
v = 0.5, [theta] = 0.50
COLL HS NHS
NO LG [right arrow] Restricted LG 0.40% 0.77% 0.99%
Restricted LG [right arrow] Unrestricted LG -2.66% -0.88% -0.07%
Table 12 Aggregate Effects of Restricted Loan Guarantees
v = 0.50
[theta] = 0.00 0.50 0.50
No LG Restricted LG Unrestricted LG
[tau] LG 0.0000 0.0004 0.0386
Discharge/Income Ratio 0.2662 0.7208 0.8657
Fraction of Borrowers 0.1720 0.2408 0.3527
Debt/Income Ratio | NHS 0.1432 0.2649 0.5738
Debt/Income Ratio | HS 0.1229 0.2256 0.4707
Debt/Income Ratio | COLL 0.0966 0.1681 0.3285
Default Rate | NHS 1.237% 2.755% 16.797%
Default Rate | HS 1.301% 2.586% 14.619%
Default Rate | COLL 0.769% 1.643% 9.072%
