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Federal Reserve Bank of Chicago

The Dynamics of Work and Debt
Jeffrey R. Campbell and Zvi Hercowitz

WP 2004-05

The Dynamics of Work and Debt∗
Jeffrey R. Campbell† Zvi Hercowitz‡
December 2003

Abstract
This paper characterizes the labor supply and borrowing of a household facing collateral requirements that limit its debt and compel it
to accumulate equity in its durable goods stock. The household’s
discount rate exceeds the market rate of interest, so it would otherwise finance increased current consumption by borrowing against
future wages. Collateral constraints generate a positive comovement
between the household’s debt, the stock of durable goods and labor
supply following wage or interest rate shocks–as the household’s labor supply adjusts to finance downpayments on new durable good
purchases and the subsequent debt repayment. Increasing the speed
of debt repayment amplifies these movements.

∗

We are grateful to Gadi Barlevy, Eric French, and Matthew Shaprio for their valuable input. Please direct correspondence to Campbell at Economic Research, Federal
Reserve Bank of Chicago, 230 South LaSalle Street, Chicago, IL 60604-1413. e-mail jcampbell@frbchi.org
†
Federal Reserve Bank of Chicago and NBER
‡
Tel Aviv University

1

1

Introduction

This paper characterizes the consumption and labor supply of a household
facing collateral requirements that limit its debt. The household’s rate of
time preference is higher than the average interest rate, so it wishes to trade
future earnings for current consumption. However, the need for collateral
implies that the household can borrow only for purchasing durable goods,
including housing. If the collateral requirements impose a minimum downpayment and accelerated amortization of the debt, then the choices of hours
worked and durable purchases become directly linked to each other through
the household’s flow budget constraint.
In the face of stochastic wages and interest rates, the choices of this household differ substantially from those of a standard unconstrained household:
First, a permanent wage increase induces an unconstrained household to
finance a one-time increase in its durable goods stock by borrowing against
future earnings (or by reducing financial assets). Assuming that household’s
preferences are consistent with balanced growth as in King Plosser, and Rebelo (1988), the wage’s income and substitution effects on leisure offset each
other and leave labor supply unchanged. Under collateral constraints building up the durable stock takes time, during which debt accumulates and
labor supply is higher.
Second, a temporary decline in the interest rate reduces hours worked
of an unconstrained household, as consumption and leisure become cheaper
relative to their future counterparts (Barro and King (1984)). The collateral
constraint on debt makes these choices infeasible, as the household cannot
borrow to finance leisure and nondurable consumption. Instead, the constrained household increases its labor supply to acquire the down payment
required for the desired additional purchases of durable goods.
Clearly, the household under consideration here cannot be a representative consumer in a closed economy, because the market for loaned funds must
clear, along with the labor and the commodity markets. The funds should
come from a more patient household, as in Becker (1980) and Krusell and
Smith (1998), which is likely to be wealthier than the household considered
here–as the patient household accumulates assets. If the wealth differential
is large, the choices of the representative debtor will disproportionately affect
aggregate hours worked.
The present model may be used to interpret the marked increase in the
stock of consumer debt since the middle 1980’s. The model predicts that as
2

the repayment of principal slows down–which in practice can follow from less
costly access to refinancing–hours worked become both less responsive to
wage or interest rate shocks and less persistent. Thus, our model is consistent
with some aspects of the decline in business cycle volatility following 1984
documented by McConnell and Quiros (2000) and Stock and Watson (2003),
given financial innovation that reduces consumers’ equity in their durable
goods.
Our analysis builds on the literature on the implications of capital market
imperfections for household behavior. Regarding consumption, the empirical
failure of Euler equations based on unconstrained intertemporal substitution
was documented by Zeldes (1989) and others. More directly related to this
paper are the results in Fortín (1995) and Del Boca and Lusardi (2003),
who found that married women’s labor supply increases with their household’s mortgage debt, using Canadian and Italian data. Here, we model this
connection as arising from an empirically relevant collateral constraint, and
consider its implications for the household’s responses to wage and interest
rate shocks.
Rupert, Rogerson, and Wright (2000) stress that the accumulation of
durable goods plays in general no role for the dynamic labor supply decisions of a household facing complete markets–or even incomplete markets
but with access to a one period bond.1 In contrast, with collateral constraints the accumulation of durable goods is central for labor supply, given
the interaction generated through the budget and borrowing constraints.
The remainder of this paper proceeds as follows. In the next section, we
present evidence on the comovement of hours worked, consumer debt, and
the stock of consumer durables at business cycle frequencies and over the long
run. In Section 3, we specify the household’s preferences, endowments and
constraints, and characterize the solution to its utility maximization problem. To build intuition for the quantitative results, we consider in Section
4 a simpler version of the model without accelerated debt repayment. Section 5 presents the household’s dynamic responses to wage and interest rate
1

In their setup, no interaction in utility between home capital and leisure is introduced.
Greenwood and Hercowitz (1991) consider a setup with such an interaction, stressing
complementarity of home capital and time spent in non-work activities. These differences,
however, are related to the specification of home production, which is another channel of
generating a link between durable purchases and labor supply.

3

shocks in a calibrated version of the model, and it assesses how reducing collateral requirements affects the volatility and persistence of the household’s
choices. The disaggregation of household durables is also addressed. Section
6 contains concluding remarks.

2

Some Evidence on Households’ Debt, Work,
and Durables

To illustrate the potential importance of collateral constraints on households,
we present here aggregate evidence on the comovement of household debt,
hours of work, and the stock of household durables in the U.S. Intuition suggests that collateral constraints should induce a positive comovement of debt,
hours worked and durable goods of debtor households, and our analysis below verifies this. Observing the aggregate comovements can indicate whether
this mechanism is strong enough to be potentially relevant for macroeconomic
fluctuations. We consider three aspects of the data: Cyclical comovements,
long-run trends, and changes of cyclical comovements over time. Household
debt is measured by mortgage debt plus consumer credit, and ‘household
durables’ are measured as real estate plus the stock of durable consumer
goods. Both are nominal values deflated by the consumption price index,
and the three variables are expressed in per-capita terms.2 Figure 1 plots
the three HP-filtered variables, showing strong positive comovement of both
the stocks of household debt and durables with hours worked. The correlation coefficients are 0.79 and 0.60, respectively. As Figure 1 suggests, the
correlation between the stocks of debt and durables is also large and positive,
0.79.
The strong positive comovement of household debt with hours worked at
the aggregate level suggests that studying the behavior of debtor households
can add to the understanding of labor fluctuations. In particular, the positive correlations of the durables’ stock with both hours worked and household
debt suggests that collateral constraints, which link debt with durable purchases and labor supply–to earn the downpayment and the repayment of the
debt–may be an important factor in generating the aggregate fluctuations
2

The debt and durables are end of quarter values, and the deflator is the total consumption chain price index from NIPA, base 1996. The per-capita values are computed
using the civilian noninstitutional population.

4

of these variables.
The unfiltered observations of these variables are shown in Figure 2.
Household debt and the stock of durable goods are scaled by the wage–
the nominal per-capita stocks divided by an index of the nominal hourly
wage–and logged. A particular interesting year for this figure is 1983, when
the debt starts to increase at a rapid rate. As it can be seen from the logarithmic scale on the left, the debt, relative to the wage, increased by 60%
from 1983:1 to the end of the sample (19 years). In comparison, from 1954:1
to 1982:4 (29 years) the rate of change is only 33%. The expansion of the
debt from the early 1980s onwards is consistent with a relaxation of borrowing constraints around this time, which induced borrowing against a large
fraction of the durable stock.
An additional feature of Figure 2 is that the trend of per-capita hours
changes from negative to positive around the same time the debt starts to
expand faster. The present model suggests a link between relaxing the borrowing constraint and higher levels of debt and hours worked. Using steadystate comparative statics, it follows from the model that improving the terms
of collateralized borrowing increases the attractiveness of durable goods relative to leisure and nondurables. This leads to a lower level of leisure and
higher levels of the stocks of durables and debt relative to the wage.
A relaxation of collateral requirements raises the possibility that the cyclical comovement of hours worked with debt and durables as well as the volatility of hours decline–as the mechanism stressed in this paper is weakened.
The statistics for the periods prior and after 1983:1 are consistent with this
consideration. In the first part of the sample, the correlations of debt and
durables with hours are 0.86 and 0.71, respectively, and in the second part
of the sample they decline to 0.53 and 0.40, respectively. Additionally, as it
can be seen from Figure 1, the volatility of hours is lower in the second part
of the sample: The standard deviation of hours before 1983:1 is 1.9 percent,
and from 1983:2 onwards it is 1.4 percent.

3

The Model

The model characterizes the choices of a single infinitely lived household
that faces a collateral constraint on its borrowing. This constraint has three
features that are typical of consumer loan contracts in the United States.
First, debt collaterlized by homes and vehicles is almost 90% of total house5

hold debt.3 Here, we make the assumption that all debt is collateralized by
durable goods. Second, the majority of new home mortgages in the United
States have loan to value ratios at or below 80%, and a typical loan to value
ratio for a new car purchase is 90%.4 Accordingly, we assume that borrowing
requires a minimum downpayment that exceeds the user cost. Third, typical loan contracts are for a fixed term that is much less than the useful life
of the durable good, so we assume that loan contracts require accelerated
repayment.5
Given that our goal is to analyze the implications of collateral constraints for a representative debtor’s choices, the collateral constraint is the
model’s only nonstandard feature. A possible extension of our moel would include nonconvex costs of durable goods adjustment–which would be relevant
for housing and automobiles–and aggregation over heterogeneous debtor
dhouseholds. King and Thomas (2003) examine models of lumpy labor demand by heterogeneous employers and find that they are observationally
similar to models of smooth adjustment by a representative employer. This
suggests to us that the interaction of debt and labor supply through the budget constraint that we consider here would manifest itself in such an extension
of the present paper.6

3.1

Preferences, Prices and Trade

The model’s household values three goods: leisure, nondurable consumption,
and the service-flow from durable goods. Each period the household has one
3

Using data from the 2002 Survey of Consumer Finances, Aizcorbe, Kennickell, and
Moore (2003) report that borrowing collateralized by residential property account for
81.5% of households’ debt in 2001 (Table 10), and installment loans, which include both
collateralized vehicle loans and unbacked education and other loans, amounts to an additional 12.3%. Credit card balances and other forms of debt account for the remainder.
The reported uses of borrowed funds (Table 12) indicate that vehicle debt represents 7.8%
of total household debt, and, hence, collateralized debt (by homes and vehicles) is almost
90% of total household debt.
4
Evidence on the terms of mortgages comes from Federal Housing Finance Board’s
Monthly Interest Rate Survey. Federal Reserve Statistical Release G.19 reports the terms
of new automobile loans.
5
We refer below to the possibility of refinancing, which practically amounts to extending
the horizon of repayment.
6
Single automobiles and homes are not divisible, but divisibility of the durable goods
stock will arise if the consumer can make supplemental expenditures to improve the current
durable good’s quality (home improvement).

6

unit of time which it splits between leisure and work. Let Nt , Ct , and St
denote the household’s hours worked, nondurable consumption, and durable
goods stock in period t.7 A time-separable expected utility function with
a constant elasticity of intertemporal substitution expresses the household’s
preferences over stochastic sequences of these goods.
i
hP
¡ θ 1−θ ¢1−σ
∞
1−λ
−ρt
(1 − Nt )
/ (1 − σ)
if σ 6= 1,
St Ct
E
t=0 e
P∞
−ρt
E [ t=0 e (θ ln St + (1 − θ) ln Ct + ln (1 − Nt ))] if σ = 1.

(1)

In (1), ρ is the household’s rate of time preference. We restrict (1 − σ) (1 − λ)
to be positive if σ 6= 1 to ensure that the utility function is concave. The
assumption that preferences are additively separable in consumption and
leisure when the intertemporal elasticity of substitution equals one guarantees that they satisfy the balanced growth restrictions of King, Plosser, and
Rebelo (1988).
The household funds its expenditures from the labor and the credit markets. The corresponding stochastic prices are Wt –the real wage–and Rt –
the (gross) real interest rate on borrowing at time t. Both prices are taken
as given by the household. The mean growth rate of the real wage is µ, and
interest rate is stationary with mean R, where R > µ.
The impatient nature of the household is characterized by the assumption
that eρ µσ > R. As in Becker (1980) and Krusell and Smith (1998), this inequality can arise if the source of borrowed funds is a more patient household,
whose rate of time preference determines the market rate of interest.
Denoting the household’s debt issued at t and repayable in t + 1 with Bt
and the depreciation rate of durables with δ, the budget constraint is
Rt−1 Bt−1 + Ct + (St − (1 − δ) St−1 ) ≤ Wt Nt + Bt .

(2)

We now turn to the specification of the constraint on the household’s
borrowing. All debt must be backed by collateral, and the durable good’s
collateral value is generally less than its replacement cost. The collateralizable value of the household’s current durable goods stock is
Vt = (1 − π)
7

∞
X
j=0

(1 − φ)j (St−j − (1 − δ) St−j−1 ) .

(3)

In this model, the flow of services from the durable goods stock is proportional to
that stock. We normalize this constant of proportionality to equal one and use the same
notation for the durable goods stock and its service flow.

7

Here, π represents the fraction of a newly purchased good’s value that cannot
serve as collateral. Hence, this is the downpayment required for the purchase
of a new durable good. The rate at which the durable good’s collateral value
depreciates is φ ≥ δ. If this rate of collateral depreciation exceeds the rate
of physical depreciation, then the household must accumulate equity in its
previous durable goods purchases. With this specification for the evolution
of collateral value, the household’s borrowing constraint is simply8
Bt ≤ Vt .

(4)

We have excluded holdings of financial assets or productive capital from
the right-hand side of (4). This is without loss of generality, because no
household will choose to hold such assets if (4) constrains its intertemporal
substitution.

3.2

Household Choices

We now develop the solution to the household’s problem given the assumption that (4) always binds. Replacing Vt with Bt in (3), the constraint can
rewritten as
Bt = (1 − φ) Bt−1 + (1 − π) (St − (1 − δ) St−1 ) .

(5)

Given its initial stocks of debt and durable goods, B−1 , and S−1 , the borrower
chooses state-contingent sequences of Ct , St , Nt , and Bt to maximize the
utility function in (1) subject to the sequences of budget and borrowing
constraints in (2) and (5).
Denote the appropriately discounted Lagrange multiplier on (2) with Ψt ,
which will always be positive. If we express the discounted Lagrange multiplier on (5) as Ξt Ψt , then Ξt measures the value (in units of either consumption good) of relaxing the borrowing constraint. In addition to the two
binding constraints, the first order conditions for this utility maximization
8

Note that accumulated equity is by assumption not collaterizable. Hence, the implied
assumption is that transaction costs in the appropriation and liquidation of a used durable
good generate a wedge between the market value, St , and the net value for the lending
institution, Vt .

8

problem are
µ ¶
¡ θ 1−θ ¢−σ St θ
Ψt = (1 − θ) St Ct
(1 − Nt )1−λ ,
(6)
Ct
·
¸
µ ¶
θ
Ψt+1
Ct
−ρ
= (1 − Ξt (1 − π)) − e (1 − δ) E
(1 − Ξt+1 (1 − π)) ,
1 − θ St
Ψt
(7)
Ct
1−λ
Wt =
,
(8)
(1 − σ) (1 − θ) 1 − Nt
¸
¸
·
·
Ψt+1
Ψt+1
−ρ
−ρ
Rt + e (1 − φ) E
(9)
Ξt = 1 − e E
Ξt+1 .
Ψt
Ψt
A state-contingent sequence of (Ct , St , Nt , Bt , Ψt , Ξt ) that satisfies these first
order conditions, the two constraints, and the transversality conditions
lim E [Ψt ] = lim E [Ψt Ξt ] = 0

t→∞

t→∞

(10)

is a solution to the household’s utility maximization problem.
The interpretation of (6) and (8) is familiar. The first equates Ψt to the
marginal utility of current nondurable consumption, and the second is a labor supply condition. Equations (7) and (9) arise from differentiating the
Lagrangian with respect to St and Bt . In the absence of a binding constraint
on Bt , the household would equate the marginal utility of durable consumption with the good’s purchase price minus its discounted and depreciated
expected resale value. This is the condition that arises if we artificially set
Ξt and Ξt+1 to zero in (7). If we define 1 − Ξt (1 − π) to be the net purchase
price of a durable good–the actual price less the benefit from relaxing the
borrowing constraint by purchasing one more unit–then (7) has a similar
interpretation when Ξt and Ξt+1 are positive.
Similarly, setting Ξt and Ξt+1 to zero reduces equation (9) to the familiar
condition that the consumer equates the marginal rate of intertemporal substitution with the real interest rate. In general, (9) determines the marginal
value of debt, Ξt , to equal the expected wedge between the real interest rate
and the marginal rate of intertemporal substitution plus its discounted and
depreciated expected future value.

9

3.3

The Balanced Growth Path

Here we consider the deterministic balance growth path with Rt = R and
log(Wt+1 /Wt ) = µ, and derive the long-run comparative statics to changes
in the parameters of the collateral constraint, π and φ. On the balanced
growth path, Nt and Ξt equal constants, N and Ξ, all remaining quantities
grow at the rate µ (e.g. Bt = Bµt , Ct = Cµt , . . . ), and the marginal utility
of nondurable consumption shrinks at the rate µσ .
We begin with (9), which immediately implies that
Ξ=

1 − e−ρ µ−σ R
.
1 − e−ρ µ−σ (1 − φ)

(11)

Given the assumption that e−ρ µ−σ R < 1, Ξ is positive and the associated
constraint always binds on the balanced growth path. From (11), we can
interpret Ξ as the present discounted value of the violation of the standard
Euler equation.
From (7), the ratio of durable to nondurable consumption is
S
θ
1
=
.
C
1 − θ (1 − Ξ (1 − π)) [1 − e−ρ µ−σ (1 − δ)]

(12)

B
(1 − π) (1 − (1 − δ) /µ)
=
.
S
1 − (1 − φ) /µ

(13)

This is a usual expression for the ratio of durable goods to nondurable consumption, involving the ratio of the two expenditure shares and their relative
price. In this case, the relative price is the net cost of durables, discussed
above.9 Using (11) to replace Ξ, S/C can be expressed as function of the
model’s primitive parameters.
The steady state borrowing constraint immediately implies that

This is the household’s leverage ratio. If φ exceeds δ, then B/S is less than
(1 − π), the fraction of new durable goods purchases financed by debt. Thus,
the household’s equity share in the most recent purchases is lower than the
corresponding share in the entire durable goods stock.
9

Note that (12) implies that a household facing a binding constraint on borrowing will
direct its consumption more heavily towards durable goods than a household without such
constraint, given that the net purchase price of durables is lower than 1 when Ξ > 0.

10

The derivation of the steady-state growth path is complete after expressing C in terms of W, R and the exogenous parameters. To do so, we substitute
(8) into (2) and use (12) and (13) to get
C=

1+

S
C

W
.
1−λ
(1 − (1 − δ) /µ) + (1−σ)(1−θ)
+ (R/µ − 1) BS CS

(14)

The balanced growth solution can be used to derive the long-run implications of changes in the collateral requirements, represented by the values
of π and φ. Increasing the downpayment rate, π, directly reduces S/C and
B/S in (12) and (13). Hence, C/W increases and N declines according to
(8). From the budget constraint and (13), both B/W and S/W decline as
well. Intuitively, increasing the downpayment rate increases the net cost of
durable goods by decreasing the debt associated with their purchase (equation (12)), and this induces the household to shift expenditures away from
durable goods towards both leisure and nondurable consumption. Increasing
the rate of debt repayment, φ, has the same qualitative implications on S/C.
In this case, however, the effect on the net cost of durables works through Ξ.
These comparative statics can be used to interpret the observations shown
in Figure 2. In particular, the figure shows an acceleration in the debt/wage
ratio and an increase in hours worked starting around 1983. This is a useful
reference point because it can be associated with the acceleration of financial
innovation in the United States in the early 1980s, which facilitated refinancing of home loans. Increasing the frequency of loan refinancing makes
possible a reduction of the borrower’s rate of equity accumulation, and hence
it can be captured in our model by reducing φ. The model predicts that a
reduction in φ permanently increases B/W , S/W , and N.

4

Results from a Simple Case

To develop intuition for the household’s choices in a stochastic environment
under the present borrowing constraint, we consider in this section a version
of the model with two simplifying assumptions. The first is that φ = δ, so
that there is no accelerated repayment requirement, but there is still a downpayment requirement. The second assumption is that σ = λ = 1, so that
the household’s preferences are additively separable in the two consumption
goods and leisure.

11

With φ = δ, it follows from the borrowing constraint in (5) that if the
household starts off with no assets and no durables, that is B−1 = S−1 = 0,
then Bt = (1 − π) St for all t ≥ 0. Replacing Bt and Bt−1 with (1 − π) St and
(1 − π) St−1 , the budget constraint (2) can be expressed as
µ
¶
Rt−1 − 1 + δ
St−1 .
Ct + πSt ≤ Wt Nt + Rt−1 π −
(15)
Rt−1

The household’s sources of funds in period t are its labor income, Wt Nt ,
and the value of the depreciated durable goods net of debt repayment,
(1 − δ) St−1 − Rt−1 (1 − π) Bt−1 –which, when φ = δ, can be written as
Rt−1 (π − (Rt−1 − 1 + δ) /Rt−1 ) St−1 . Nondurable consumption and downpayments on its current durable goods stock are the uses of funds. Also, the
first-order conditions can be combined to yield
´
 ³
Rt −1+δ
R
π
−
t
Rt
θ
π
.
− e−ρ E 
=
(16)
St
Wt (1 − Nt )
Wt+1 (1 − Nt+1 )

Here, the marginal utility of durable goods consumption is equated with the
utility cost of working to acquire the downpayment less the expected utility
in the following period from the leisure equivalent of accumulated equity.
We begin by using the budget constraint to illustrate a key implication
of the present framework for household’s choices. In (15), the current decision variables Nt , Ct , and St appear along with predetermined variables and
Wt . From the first order condition for labor in (8), Ct and Wt (1 − Nt ) are
proportional. Hence, if the household decides to expand its durable goods’
stock because of an interest rate cut or expected future changes in wages or
interest rates, (15) dictates that hours of work must go up. This positive comovement between hours worked and durable consumption contrasts sharply
with the results from a model of a financially unconstrained household, as
in Barro and King (1984). In such a model, when the current wage is held
constant, a change in hours worked should be accompanied by changes in all
forms of consumption in the opposite direction.10
A key term in both equations is π − (Rt−1 − 1 + δ) /Rt−1 –the difference
between the downpayment rate and the conventionally defined user cost of
durable goods in period t − 1. When the downpayment is higher than the
10

The model in Barro and King has no durable consumption goods, but their result
carries over to a model with durables if the household faces frictionless rental markets.

12

user cost, the borrowing constraint forces the household to acquire some
ownership of its durable goods stock. We focus next on two cases regarding
this term.

4.1

Full Collateral

A benchmark case consists of setting π = (R − 1 + δ) /R, where R is the
mean of Rt , and φ = δ, as assumed earlier in this section: The downpayment
covers only the average user cost and there is no accelerated repayment requirement. We call this the case of full collateral, because the values of the
outstanding debt and the depreciated durable goods stock are equal at the
average interest rate.
Consider the effects of changes in Wt , holding constant Rt at R. Because
the last terms in both (15) and (16) are now equal to zero, these equations
and the first order condition for Nt are satisfied only by an immediate and
full adjustment of Ct and St to the wage change, while Nt is unchanged.
If the wage change is permanent, then these choices correspond exactly to
those of a household facing no borrowing constraints. Here, however, this
result holds regardless as to whether the change is permanent or transitory.
An unconstrained household borrows to finance leisure when the wage falls
temporarily, but this option is unavailable to the the present household because borrowing must be backed by purchases of durables goods. Therefore,
full collateral eliminates completely the variation of hours worked following
wage changes.
In contrast to the lack of response to wage changes, hours worked do
respond to interest rates changes. The interest rate affects durable purchases
through the user cost, and then, as discussed above, hours worked comove
with durables so as to satisfy the budget constraint.

4.2

Partial Collateral

When π > (R − 1 + δ) /R, the borrowing constraint forces the household
to accumulate equity on its durable goods stock. Correspondingly, only a
fraction of the durable stock can serve as collateral, and thus we label this
case as one of partial collateral.
Here, when Wt changes permanently, the choice of immediate proportional adjustment in Ct and St leaving Nt unchanged violates the budget
constraint. Hence, the adjustment of Ct and St to their new long-run levels is
13

gradual. The optimal labor supply condition (8) and the gradual adjustment
of Ct /Wt imply that Nt exceeds its long-run level during the adjustment.
In the present context, a temporary wage change is less interesting because
the household’s response is qualitatively similar to the case of unconstrained
borrowing.11
The value of π has opposite implications for the responses of hours worked
to wage and to interest rate changes. A higher π increases the sensitivity
of N to the real wage, but lowers its sensitivity to the interest rate. The
first implication was already stressed as the result of moving from full to
partial collateral. The second implication follows from the fact that a higher
downpayment reduces the amount of allowed borrowing, and hence it makes
the household less sensitive to changes in R. The extreme case is of course
π = 1, when the interest rate becomes irrelevant.
The main results from considering this simple version of the model can
be summarized as follows. First, the lack of access to uncollateralized credit
overturns the standard model’s prediction that labor comoves negatively with
all consumption goods when the interest rate or expectations about the future
change. Second, with full collateral, the borrowing constrained household
does not change its hours worked in response to wage variation, permanent
or temporary, while under the realistic assumption of partial collateral, labor
does respond positively to wages even when the changes are permanent.
Third, with partial collateral the household’s adjustment of the durable goods
stock is gradual.12
11

Both with unconstrained borrowing and here, Ct and Nt comove positively from (8).
An alternative causality story which also leads to positive comovement of hours, debt
and durable goods is a Keynesian-type one, where N changes exogenously, given a shifting
demand for labor under excess supply. Because of the collateral constraint, the household
durable purchases–and thus downpayments and the debt–comove positively with N and
thus with labor income. In contrast, if this household had perfect access to the capital
market, the exogenous changes in N would lead to opposite changes in the net debt. The
difference between this type of causality story and the one addressed in this paper has to do
with nondurable consumption. In this alternative setup, nondurable consumption moves
positively with labor, given that the labor supply condition ceases to be relevant, while in
the present model this condition dictates opposite movements of nondurable consumption
and hours worked.
12

14

5

Results from a Calibrated Model

For the general case, we follow the procedure of calibrating the model’s parameters and then calculating the household’s optimal responses. We first
describe the parameter choices and then the household’s optimal responses
to wage and interest rate shocks. We also present the results of experiments
designed to illustrate the role of the borrowing constraint in generating the
results.

5.1

Calibration

The assignment of vallues to the model’s parameters is straightforward. We
set µ = 1.0047, the average quarterly growth rate of the real hourly compensation in the business sector from 1954 through 2001. The depreciation
rate δ is equated to its empirical analogue constructed from the Bureau of
Economic Analysis’ Fixed Tangible Reproducible Wealth. The estimate of δ
is 0.0115, the average of 0.0018 for residential structures and 0.034 for other
durable goods, with weights 0.7 and 0.3, respectively. We set R = 1.01.
The impatience parameter is harder to calibrate. We set ρ = 0.015,
i.e., half of a percentage point higher than the interest rate. This degree of
impatience is similar in magnitude to that used by Krusell and Smith (1998).
Using a model with 3 levels of time preference, they calibrate the differences
between each type as 0.36%; or 0.72% between the two extremes. We have
experimented with various values for this parameter with almost identical
results to those reported below.
For π, we use 0.15, which is between 0.20, a typical downpayment fraction for home loans, and 0.1, a typical value for car loans. The rate of
repayment φ is computed using the average term of home loans of 104 quarters and the average term of car loans of 12 quarters during the 1952-2002
sample. The corresponding linear repayment rates are 0.0096 and 0.083.
During that sample, the average shares of the two types of loans are 0.78 and
0.28, respectively, and thus φ is set equal to the weighted average repayment
rate of 0.03.13 The parameter θ is set so as to match the ratio of quarterly
durable consumption expenditures to nondurable consumption expenditures,
((1 − (1 − δ)/µ) S/C), which is 0.25 during the 1952-2002 sample. The resulting value is 0.26. We adopt σ = 2 as our baseline case, and experimented
13

See footnote 4 for the sources of our observations of loan terms for automobile installment loans and residential mortgages.

15

also with σ = 1. Finally, the value of λ is computed using the balanced
growth version of the condition for N in (8) and the other parameters which
determine the C/W ratio, so that N = 0.3.

5.2

Baseline Results

The solution procedure is standard, we log-linearize the first order conditions (6), (7), (8) and (9) as well the constraints (2) and (5) around the
balanced-growth path and solve the resulting log-linear system for particular
stochastic processes of the wage and the interest rate. The assumption that
the borrowing constraint always binds seems reasonable because only small
deviations from the balanced growth path are considered. We assume that
log Wt follows a random walk with drift µ and that log Rt follows a first-order
autoregression with autoregressive coefficient 0.95.
Figure 3 plots the household’s responses–expressed as percentage deviations from the initial balanced growth path–to an unexpected 1% permanent
increase in Wt . Immediately after the wage shock, the stock of durables increases 0.25%, nondurable consumption 0.91% and hours worked 0.22%. As
described in Section 4, this partial adjustment of the two consumption goods
and the increase in hours worked stand in contrast with the behavior of an
unconstrained household–which would adjust the two consumption goods
immediately by 1% and leave hours worked unchanged.
After three years, the accumulation of the stock of durables to its new
level is close to be complete, while the debt behaves quite differently. The
debt overshoots its long-run increase of 1%, and after two years it is more
than 1.5% higher than its initial level. Then, the debt falls very slowly
towards its long-run level. The source of the marked difference between the
behavior of the debt and the behavior of the stock of durable goods is the
accelerated repayment of the debt. Because φ exceeds δ, the collateral value
of a durable good as a fraction of its current value declines with its age–
newer goods can support more borrowing.14 For the parameter values used,
B/S = 0.4, so that a new purchase can collateralize more than twice as
much debt as the average durable good in the household’s stock. Hence, the
surge in the durable goods stock allows the consumer to borrow more. Even
after the durable goods stock has approached its long-run level, its average
age remains below the long-run average age. The long decline of the debt
14

t
For goods of age t, this fraction equals ( 1−φ
1−δ ) .

16

after its initial surge reflects the gradual aging of the household’s durable
goods stock. The slow convergence of the household’s debt accounts for the
sluggish behavior of its nondurable consumption and hours worked. While
debt remains high, the consumer reduces consumption and leisure to finance
its repayment. Thus, accelerated repayment causes both higher volatility of
the debt–because of the overshooting behavior–and persistent variation in
nondurable consumption and hours worked.
We now turn to consider the household’s response to a transitory but persistent interest rate shock–as the autoregression coefficient is 0.95. Figure
4 plots the responses following a 0.25% decrease in the interest rate. The
household’s stocks of the durable good and debt are extremely interest sensitive. In the period of the interest rate reduction, the stock of the durable
good rises about 1% and the debt rises by more than 2%. The strong positive
response of hours worked, 1.4%, reflects the point stressed in Section 4 about
the positive comovement of hours worked and durables when the interest rate
or expected future variables change. It is this link that leads to the nonstandard negative effect of the interest rate on labor supply. This argument also
implies a negative response of nondurable consumption, whose initial change
is −0.6%. The movements in hours worked and in nondurable consumption
persist for several quarters. The stock of durables and the debt peak after
six quarters. The durable goods stock peaks at 3.7% above its value before
the shock, while the stock of debt rises over 7% before beginning to fall.15
The negative comovement of the durable goods stock and nondurable
consumption expenditure following an interest rate change raises the question
of how does total consumption of this household comove with hours worked.
We calculated the responses of a total consumption measure that adds the
user cost of the durable goods stock to nondurable consumption.16 Initially,
this consumption measure falls slightly, −0.2%, following the interest rate
shock. Thereafter, it rises above its long-run level and achieves a peak of
0.7% after eight quarters, before slowly reverting to its steady state value.
Thus, the household’s hours worked covary positively with this particular
measure of its total consumption.
Whether the household faces variation in the wage, the interest rate, or
both; the calibrated model’s simulations indicate that its debt and labor
15

When σ = 1 is used, the responses are larger in magnitude and with a similar degree
of persistence.
16
We calculated the service flow from the durable goods stock using the user cost of
durable goods, (R − 1 + δ) /R, with a constant interest rate.

17

supply comove positively. This behavior is qualitatively consistent with the
observations presented in Section 2.

5.3

Relaxing the Borrowing Constraint

We now turn to consider two versions of the model in which the borrowing
constraint requires either a slower repayment or a lower downpayment. Figures 5 and 6 plot the household’s responses to the wage and interest rate
shocks when φ is lowered to the level of δ, so that there is no accelerated
repayment, while holding all the other parameters constant.
The contrast between the responses to the 1% permanent wage increase
in Figure 5 and those in Figure 3 illustrates the implications of accelerated
repayment. Hours worked become less volatile and much less persistent. The
response of the durable goods stock is very similar in the two cases, but here
it converges faster. By construction, debt perfectly tracks the durable goods
stock when φ = δ, so that also the debt converges very quickly to its long-run
level. Given that debt does not overshoot, it becomes less volatile following
wage changes.
Figure 6 plots the household’s responses to the interest rate decline examined earlier. The most important difference between these responses and
those from the baseline version is that they are now weaker and less persistent. This is particularly the case for hours worked.
We also lowered π from 0.15 to 0.1. These responses, not shown, are
quantitatively very similar to those from the calibrated model. As discussed
in Section 4, the effects of changing the downpayment rate on the responses
of hours worked are mixed. Hours worked become less sensitive to the wage,
but more sensitive to the interest rate. Also, the easier access to credit
amplifies the responses of debt and durables.
Typical downpayment rates on automobile loans and home mortgages
changed little over the sample used in Figures 1 and 2, but the frequency of
refinancing fixed term mortgages, or borrowing against accumulated home
equity, went up dramatically during this period.17 This is reflected in the
sharp increase in household debt since around 1983. In the model, φ − δ
represents the rate at which a debtor builds equity in durable goods. Lowering this rate increases the levels of a representative debtor’s debt and hours
17

Brady, Canner, and Maki (2000) report that 8% of homeowners in 1977 had refinanced
the first mortgage on their current home. The analogous statistics in 1989, 1994, and 1999
were 20%, 45%, and 47%.

18

worked, and at the same time it decreases their volatility because debt comoves more closely with the durable goods stock. Hence, the present model
of a representative debtor suggests a consistent interpretation of the salient
features of both levels and variances of household debt and hours worked
shown in Figures 1 and 2. Around the same time, about 1983, when the
trends of debt and hours increase, the volatilities of both variables decline.

5.4

Disaggregation: Housing and Automobiles

In reality, two distinct goods account for most of a typical household’s durable
goods: houses and automobiles. Houses depreciate much slower than automobiles, both physically and for collateral purposes, and downpayments are
typically higher than for automobiles. Additionally, the two goods may have
different utility parameters.
To examine the robustness of our results to disaggregating the durable
goods stock, we extended the model to include two durable goods, each with a
separate collateral constraint. The downpayment and repayment rates were
calibrated for houses and automobiles separately using the values referred
to in Section 5. The utility parameters were inferred from the expenditure
shares of residential investment (owner occupied) and automobiles: 0.05 and
0.14, respectively.
The main conclusion from this extension is that the results–not reported
in the paper–are very similar to those reported above. Here, we also obtain
the behavior of the two stocks and two types of debt. For example, when the
wage increases permanently, the immediate percentage increase in car debt
is much higher than for home debt–given the lower downpayment rate–
and converges faster to the long-run one-percent increase–given the higher
repayment rate.
This extension of our model could be used to address relative price changes
of the two durable goods. We do not pursue this avenue because the mechanism triggered by relative price changes in the present model is not essentially
different than in the standard household framework. In both, a relative price
decline will shift demand toward that good, and away from the other goods,
including leisure.

19

6

Concluding Remarks

Limiting a household’s borrowing by imposing collateral constraints fundamentally alters its intertemporal choices. In models of unconstrained intertemporal substitution with preferences consistent with balanced growth,
a permanent wage gain increases immediately both consumption and net indebtedness while leaving hours worked unchanged. In the same models, a
reduction in interest rates induces the household to work less and consume
more in the present at the expense of the future. Collateral constraints render
these choices infeasible. A household facing collateral constraints responds
to both of these shocks by working more to accumulate downpayments for
durable goods purchases and later to repay the principal. The result is a
gradual accumulation of durable goods. Surprisingly, forcing the household
to repay its debts at a rate faster than durable goods’ depreciation rate amplifies fluctuations in the debt. This arises from new durable goods having
higher value as collateral.
Because our household is a net debtor, it obviously cannot be considered
a representative household for the analysis of aggregate data. Nevertheless,
we find the positive comovement of household debt with hours worked and
the stock of durable goods in the aggregate U.S. data to be striking enough
to merit the investigation of a representative debtor’s choices. The natural
generalization of the present framework adds a more patient household to
serve as a source of funds. Because the patient representative creditor holds
all of the economy’s tangible wealth minus the debtor’s equity in his durable
goods stock, its supply of labor is likely to be smaller than that of the borrower. On these grounds, if the wealth differential is large enough, total
hours worked in such an economy will reflect primarily the debtor’s choices.
One of the issues suggested by the present analysis for a general equilibrium analysis of this type is the implications of financial innovation for
aggregate fluctuations. The current analysis suggests that financial innovation may reduce the volatility of business cycles, given that the positive
comovement between household durable purchases, debt and hours worked
is weakened. The analysis of a general equilibrium setup of this type is the
subject of our next research project.

20

References
[1] Aizcorbe, Ana M., Arthur B. Kennickell, and Kevin B. Moore, 2003,
“Recent Changes in U.S. Family Finances: Evidence from the 1998 and
2001 Survey of Consumer Finances,” Federal Reserve Bulletin, 89, pp.
1-32.
[2] Barro, Robert J., and Robert G. King, 1984, “Time-Separable Preferences and Intertemporal-Substitution Models of Business Cycles,” The
Quarterly Journal of Economics, 99, pp. 817-839.
[3] Becker, Robert A., 1980, “On the Long-Run Steady State in a Simple
Dynamic Model of Equilibrium with Heterogeneous Households,” The
Quarterly Journal of Economics, 95, pp. 375-382.
[4] Brady, Peter J., Canner, Glenn B. and Dean M. Maki , 2000, “The Effects
of Recent Mortgage Refinancing,” Federal Reserve Bulletin, July, pp. 441450.
[5] Bureau of Economic Analysis, 1999, Fixed Tangible Reproducible Wealth,
Washington, D.C.
[6] Del Boca, Daniela and Annamaria Lusardi, 2003, “Credit Market Constraints and Labor Market Decisions,” Labour Economics, Forthcoming.
[7] Federal Housing Finance Board, Monthly Interest Rate Survey, Various
Issues.
[8] Federal Reserve Board, Consumer Credit: Statistical Release G.19, Various Issues.
[9] Fortin, Nicole, M., 1995, “Allocation Inflexibilities, Female Labor Supply, and Housing Assets Accumulation: Are Women Working to Pay the
Mortgage?,” Journal of Labor Economics, 13, pp. 524-557
[10] Greenwood, Jeremy and Zvi Hercowitz, 1991, The Allocation of Capital
and Time over the Business Cycle,” Journal of Political Economy, 99,
pp. 1188-1214.
[11] King, Robert G., Charles I. Plosser and Sergio T. Rebelo, 1988, “Production, Growth and Business Cycles I,” Journal of Monetary Economics,
21, pp. 195-232.
21

[12] King, Robert G. and Julia K. Thomas, 2003, Partial Adjustment without Apology, NBER Working Paper 9946.
[13] Krusell, Per, and Anthony A. Smith Jr.,1998, “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of Political Economy, 106, pp.
867-896.
[14] McConnell and Quiros, 2001, “Output Fluctuations in the United
States: What has Changed Since the Early 1980’s?” American Economic
Review, 90, pp. 1464-1476.
[15] Rupert, Peter, Richard Rogerson and Randall Wright, 2000, “Homework
in Labor Economics: Household Production and Intertemporal Substitution,” Journal of Monetary Economics, 46, pp. 557-579.
[16] Stock James and Mark W. Watson, August 2003, “Has the Business
Cycle Changed? Evidence and Explanations.” Unpublished Manuscript,
Princeton University.
[17] Zeldes, Stephen, P., 1989, “Consumption and Liquidity Constraints: An
Empirical Investigation,” Journal of Political Economy, 97, pp. 305-346.

22

Household Durables (Including Real Estate) and Hours Worked
HP-filtered

Household Debt and Hours Worked
HP-filtered

.06

.08

Durables

Debt

.06

.04

.04
.02

.02

.00

.00
-.02

-.02

-.04

Hours

-.04

-.06

Hours
-.06

-.08
55

60

65

70

75

80

85

90

95

00

55

60

65

70

75

80

85

90

95

00

Figure 1: Cyclical Fluctuations of Household Debt, the Durable Goods Stock,
and Hours Worked

Household Debt/Wage and Durables/Wage Ratios
and Hours Worked
.20
Hours

-0.5

.15

Durables

-1.0

.10

-1.5

.05

-2.0

.00

Debt

-2.5

Hours (logs)

Debt/Wage Durables/Wage (logs)

0.0

-.05

-3.0

-.10
55

60

65

70

75

80

85

90

95

00

Figure 2: Levels of Household Debt, the Durable Goods Stock, and Hours
Worked

23

Percentage Points

Stock of Durable Goods

Outstanding Debt

1

2

0.8

1.5

0.6

1

0.4

0.5

0.2

0

10

20

30

40

0

0

Percentage Points

Nondurable Consumption
0.25

0.98

0.2

0.96

0.15

0.94

0.1

0.92

0.05

0

10

20

30

20

30

40

Hours Worked

1

0.9

10

40

0

Quarters After Shock

0

10

20

30

40

Quarters After Shock

Figure 3: Baseline Responses to a 1% Permanent Wage Increase

24

Stock of Durable Goods

Outstanding Debt

4

8

Percentage Points

3.5
6

3
2.5

4

2
1.5

2

1
0.5

0

10

20

30

40

0

0

Nondurable Consumption

10

20

30

40

Hours Worked

0.1

1.5

Percentage Points

0
1

-0.1
-0.2

0.5

-0.3
-0.4

0

-0.5
-0.6

0

10

20

30

40

-0.5

Quarters After Shock

0

10

20

30

40

Quarters After Shock

Figure 4: Baseline Responses to a Persistent 1/4% Interest Rate Decrease

25

Percentage Points

Stock of Durable Goods

Outstanding Debt

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0

10

20

30

40

0.2

0

Percentage Points

Nondurable Consumption
0.2

1

0.15

0.98

0.1

0.96

0.05

0.94

0

0

10

20

30

20

30

40

Hours Worked

1.02

0.92

10

40

-0.05

Quarters After Shock

0

10

20

30

40

Quarters After Shock

Figure 5: Responses to a Permanent 1% Wage Increase when φ = δ

26

Percentage Points

Stock of Durable Goods

Outstanding Debt

6

6

5

5

4

4

3

3

2

2

1

1

0

0

10

20

30

0

40

0

Nondurable Consumption

10

20

30

40

Hours Worked

0.2

1.5

Percentage Points

0.1
1

0
-0.1

0.5

-0.2
-0.3

0

-0.4
-0.5

0

10

20

30

-0.5

40

Quarters After Shock

0

10

20

30

40

Quarters After Shock

Figure 6: Responses to a Persistent 1/4% Interest Rate Decrease when φ = δ

27

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Gadi Barlevy

WP-03-18

Banking Market Conditions and Deposit Interest Rates
Richard J. Rosen

WP-03-19

Creating a National State Rainy Day Fund: A Modest Proposal to Improve Future
State Fiscal Performance
Richard Mattoon

WP-03-20

Managerial Incentive and Financial Contagion
Sujit Chakravorti, Anna Llyina and Subir Lall

WP-03-21

Women and the Phillips Curve: Do Women’s and Men’s Labor Market Outcomes
Differentially Affect Real Wage Growth and Inflation?
Katharine Anderson, Lisa Barrow and Kristin F. Butcher

WP-03-22

Evaluating the Calvo Model of Sticky Prices
Martin Eichenbaum and Jonas D.M. Fisher

WP-03-23

5

Working Paper Series (continued)

The Growing Importance of Family and Community: An Analysis of Changes in the
Sibling Correlation in Earnings
Bhashkar Mazumder and David I. Levine

WP-03-24

Should We Teach Old Dogs New Tricks? The Impact of Community College Retraining
on Older Displaced Workers
Louis Jacobson, Robert J. LaLonde and Daniel Sullivan

WP-03-25

Trade Deflection and Trade Depression
Chad P. Brown and Meredith A. Crowley

WP-03-26

China and Emerging Asia: Comrades or Competitors?
Alan G. Ahearne, John G. Fernald, Prakash Loungani and John W. Schindler

WP-03-27

International Business Cycles Under Fixed and Flexible Exchange Rate Regimes
Michael A. Kouparitsas

WP-03-28

Firing Costs and Business Cycle Fluctuations
Marcelo Veracierto

WP-03-29

Spatial Organization of Firms
Yukako Ono

WP-03-30

Government Equity and Money: John Law’s System in 1720 France
François R. Velde

WP-03-31

Deregulation and the Relationship Between Bank CEO
Compensation and Risk-Taking
Elijah Brewer III, William Curt Hunter and William E. Jackson III

WP-03-32

Compatibility and Pricing with Indirect Network Effects: Evidence from ATMs
Christopher R. Knittel and Victor Stango

WP-03-33

Self-Employment as an Alternative to Unemployment
Ellen R. Rissman

WP-03-34

Where the Headquarters are – Evidence from Large Public Companies 1990-2000
Tyler Diacon and Thomas H. Klier

WP-03-35

Standing Facilities and Interbank Borrowing: Evidence from the Federal Reserve’s
New Discount Window
Craig Furfine

WP-04-01

Netting, Financial Contracts, and Banks: The Economic Implications
William J. Bergman, Robert R. Bliss, Christian A. Johnson and George G. Kaufman

WP-04-02

Real Effects of Bank Competition
Nicola Cetorelli

WP-04-03

Finance as a Barrier To Entry: Bank Competition and Industry Structure in
Local U.S. Markets?
Nicola Cetorelli and Philip E. Strahan

WP-04-04

Working Paper Series (continued)
6

The Dynamics of Work and Debt
Jeffrey R. Campbell and Zvi Hercowitz`

WP-04-05

7