Working Paper Series

2008-07

Congressional Budget Office

Washington, D.C.

September 2008

Labor Supply Response in Macroeconomic Models

Assessing the Empirical Validity of the Intertemporal Labor Supply

Response from a Stochastic Overlapping Generations Model with

Incomplete Markets

Juan M. Contreras and Sven H. Sinclair

Congressional Budget Office

Authors: juan.contreras@cbo.gov, sven.sinclair@cbo.gov. CBO’s Working and Technical Papers are preliminary and are circulated to stimulate discussion and critical comment. The

papers are not subject to CBO’s formal review and editing process. The analysis and conclusions expressed are those of the authors and should not be interpreted as those of the

Congressional Budget Office. References in publications should be cleared with the authors.

Papers in this series can be obtained from www.cbo.gov/publications.
Labor Supply Response in Macroeconomic Models

Assessing the Empirical Validity of the Intertemporal Labor Supply

Response from a Stochastic Overlapping Generations Model with

Incomplete Markets

Juan M. Contreras and Sven H. Sinclair1

Congressional Budget Office

September 2008

Abstract

We evaluate the labor supply response in a stochastic overlapping generations

model with incomplete markets and a non separable utility function in labor and

consumption. Using a simulated panel from the model, we calculate the labor supply response to anticipated changes in wages (holding the marginal utility of wealth

constant-that is, the Frisch elasticity) and to unanticipated change in wages (which

describes the effect of uncertainty in labor supply responses). The model’s Frisch

elasticity estimate is 0.33, which is slightly higher than the empirical estimates in

the earlier literature but somewhat lower than more recent estimates. The paper

also shows that the borrowing constraints in the model reduce substantially the

estimates of the Frisch elasticity. The labor supply response to an unanticipated

change in wages is small because of large wealth effects. Having all the variables

required and no measurement error, we calculate the omitted variable bias of not

controlling for the level and variance (risk) of the unexpected changes in wages.

Omitting both variables biases the estimates of the Frisch elasticity downward by

a factor of 8; omitting measures of wage risk alone biases it by a factor of 1.4.

JEL CODES: J22, D91, D58

Keywords: labor supply, intertemporal substitution, computable GE models.

1We thank Doug Hamilton for very useful comments and for important insights about the relevance

of the wealth effect in the labor supply response, and Bob Dennis and Thomas Woodward for comments

on earlier drafts.

2
1 Introduction

The validity of dynamic macroeconomic models for analyzing policy experiments depends on key behavioral parameters that determine the agents’ responses to changes

in policy variables. The intertemporal elasticity of labor supply, which determines the

labor supply response, is among the most important of those parameters.

In this paper, we analyze the empirical validity of the labor supply response in a

stochastic overlapping generations model with incomplete markets.2 We estimate the

labor supply response in the model to anticipated changes in wages, holding constant

the marginal utility of wealth (Frisch elasticity), and also to unanticipated changes in

wages.3 We calculate the Frisch elasticity on the intensive margin (that is, we account for

the hours decision and not for the participation decision given) because that approach is

consistent with most of the empirical studies that estimate the wage elasticity of labor

supply.

This paper makes three main contributions. First, it reconciles the Frisch elasticity

estimates from the empirical labor literature with the values used in the macroeconomic

literature, using a methodology that is consistent across fields. Second, it emphasizes the

importance and quantifies the magnitude of the effect of wage uncertainty in the labor

supply response in macroeconomic models. Third, it quantifies the omitted variable bias

of not controlling for the level and variance (risk) of the unexpected changes in wages.

Many macroeconomic models focus on the Frisch elasticity values because their goal

is to analyze real business cycles. Equally important but less studied is the life cycle

2This is the model that CBO regularly uses in its analysis of the President’s budget.

3Economists usually consider four basic elasticities of labor supply (Blundell and MaCurdy [2008]):

compensated and uncompensated elasticity of labor supply, Frisch elasticity of labor supply, and a

life cycle elasticity that considers unanticipated (parametric) changes in wage changes. Because the

compensated and uncompensated elasticities of labor supply usually are derived and estimated from

static models, they are unsuitable for application in dynamic macroeconomic models. The Frisch

elasticity of labor supply is defined as the percentage change in labor supply resulting from a one

percent increase in the expected wage rate, holding the marginal utility of wealth constant. The Frisch

elasticity measures the response of wages to anticipated wage changes, as opposed to the other life cycle

elasticity, which measures the response of work hours to unanticipated wage changes or shifts in the

wage profile.

1
elasticity that considers labor supply responses to unanticipated changes in wages. This

elasticity is also very important for the study and the analysis of tax and benefit reforms,

which almost always cause unanticipated shifts in wages today and in the future (see

Blundell and MaCurdy 1999 and 2008). Of course, the relative importance of the two

elasticities depends on the type of tax policy implemented and on the mix of wealth and

intertemporal substitution effects it triggers.

The estimates of the Frisch elasticity of labor supply that come from the model are

consistent with estimates from the empirical literature, once borrowing constraints and

the level and variance of unexpected changes in wages are accounted for. The Frisch

elasticity from macroeconomic models often differs from that estimated in microeconomic

studies.4 Previous work has posited that those differences could be accounted for in two

ways. The first way is considering the entry to and exit from the labor market—the

extensive margin at which labor supply is adjusted.5 Although the elasticity at the

intensive margin is generally small, adjustments at the extensive margin are larger.

The second way to account for the differences between the Frisch elasticity coming

from macroeconomic models and from the empirical studies is the consideration of borrowing constraints.6 Commonly used utility functions, including the constant relative

risk aversion (CRRA) utility, can be shown analytically to have high Frisch elasticities

when there is no uncertainty or binding constraints. Most empirical studies find much

lower elasticities. However, borrowing constraints play a central role in our model; indeed, we also show in this paper that once we account for borrowing constraints, the

Frisch elasticity that comes from the model using the CRRA utility is reduced by a

factor of nearly 4. Borrowing constraints are important because without them house4Blundell and MaCurdy (2008) argue that the aggregation of individual agents into a representative

agent to describe the aggregate labor supply elasticity requires stringent assumptions. That criticism

does not apply to the class of models used in this paper, because the agents are ex-post heterogeneous

in wealth, and we can simulate a longitudinal panel of agents to estimate the elasticity of labor supply

with microeconomic techniques.

5The elasticities of labor supply can refer to the extensive margin (labor force enter or exit), or to the

intensive margin (change in the number of hours worked). Rogerson and Wallenius (2007) propose an

elegant model that reconciles the differences between the relatively small micro estimates of the Frisch

elasticity with the relatively large macro responses.

6See Domeij and Floden (2006).

2
holds respond to a bad wage shock by reducing labor and increasing borrowing so as to

smooth consumption over time. With borrowing constraints, however, households can’t

completely smooth consumption, so they don’t reduce labor supply as much as they

otherwise would. That dampens the labor supply response in the model.

We analyze a third factor that affects the difference between the Frisch elasticity

from empirical estimates and the Frisch elasticity from macroeconomic models: the

omitted variable bias that is present in many empirical studies. If we do not control

for the variance of the unexpected changes in future wages as a measure of wage risk or

for both the variance and the level of unexpected changes in future wages, the Frisch

elasticity estimates are biased downward. If we omit measures of level and variance of

unexpected changes in wages, the point estimate of the Frisch elasticity is reduced by a

factor of 8 (0.33 vs. 0.04). If we omit measures of the variance of unexpected changes

in wages (wage risk) the point estimate is 1.4 times smaller (0.33 compared with 0.24).

The intuition behind the omitted variable bias is related to the presence of the wealth

and the intertemporal substitution effect. If the econometrician does not control for

unexpected wage changes, the error term would be correlated with the coefficient on

wages that measures the intertemporal substitution (Frisch) effect.

Our estimates from the model of the labor supply response to unanticipated wage

changes are small because wealth effects in the model largely offset the intertemporal

substitution effect. That result comes from important wealth effects in our OLG model.

Unexpected wage changes shift the wage profile, generating a labor supply response from

a wealth effect and from an intertemporal substitution (Frisch) effect. A wage increase

generates a wealth effect and a consequent labor supply reduction, but the same wage

increase generates an intertemporal substitution effect that tends to increase the labor

supply. Depending on which effect is bigger, the labor supply response to unanticipated

wage changes would be negative or positive. In our model, wealth effects have almost

the same magnitude as intertemporal substitution effects.

Our analysis used a stochastic OLG model with incomplete markets that was similar to those of Nishiyama (2002), Nishiyama and Smetters (2005) and CBO (2008).

3
Households in the model have a CRRA utility function that is non separable between

consumption and leisure. First, we use that model to show analytically how borrowing

constraints affect the Frisch elasticity measures. We then generate a simulated panel

from the model and use the Pistaferri (2003) econometric method to estimate the Frisch

elasticity of labor supply. Next we compare the estimates with estimates from the empirical literature and quantify the omitted variable bias that is induced by the lack of

measures of the level and variance of unexpected changes in future wages. We can calculate that bias because the variables from the simulated panel are known exactly and

thus have no measurement error.

In the next section we review the literature that estimates the Frisch elasticity of

labor supply and put the different estimates in perspective. In section 3, we discuss

methodological and empirical issues regarding the measurement of the Frisch elasticity,

present our estimates of various measures of intertemporal labor supply elasticities,

and quantify the omitted variable bias from the empirical estimates. We present our

conclusions in section 4.

2 Literature Review

Most empirical studies of the intertemporal elasticity of labor supply focus on the labor

supply response to anticipated wage changes or Frisch elasticity (with the additional

qualification of holding the marginal utility of wealth constant); just a few, among them

MaCurdy (1981) and Pistaferri (2003), estimate the intertemporal elasticity of labor

supply in response to unanticipated wage changes. MaCurdy estimates a male labor

supply response to unanticipated wage changes of 0.08 in the median range; Pistaferri’s

estimate is 0.5.7

7Early studies showed that the female intertemporal response is much larger. Heckman and MaCurdy

(1980, 1982) estimated a Frisch labor supply elasticity for women of 1.6. Blundell, Meghir, and Neves

(1993) estimated a Frisch elasticity between 0.8 and 1.2 for married women with children and an

elasticity of 0.6 for childless women. The high elasticities obtained for women in part reflect the

importance of considering the decisions of entry and exit from the labor market not present in the

studies that consider only the decision of hours worked for males. Heckman (1979, 1993) and Blundell

and MaCurdy (1999) argued that most of the labor supply response stems from choices at the extensive

4
Empirical measures of the intertemporal Frisch elasticity of labor supply present a

huge variation. Estimates range from close to zero to about 3, although most have values

of less than 1. The variation arises from differences in methodology, in the samples used,

and in the type of population and identification strategies.

MaCurdy (1981) and Altonji (1986) were among the first to identify the intertemporal

response of hours worked to changes in wages for a sample of males in the United States.

MaCurdy’s estimates of the Frisch elasticity of labor supply are between 0.1 and 0.45.

Altonji’s estimates are between 0 and 0.35. Pencavel (1986) surveys this early literature

and presents a mean value of 0.2 and a range of 0 to 0.45.

Among the difficulties associated with early work estimating the Frisch elasticity of

labor supply are measurement error, lack of data about expected wages and difficulty

in accounting for wealth effects through the life cycle.8 Angrist (1991) addressed some

of the problems associated with measurement error and estimated Frisch labor supply

elasticities of 0.6 to 0.8. French (2004) also tried to address measurement error problems

finding a Frisch elasticity close to zero with a standard error of 0.25.

Moreover, Lee (2001) addresses the finite sample bias in estimates of intertemporal

labor supply. He finds that the estimates of the Frisch elasticity of many studies in the

field are biased downward because of small samples and, when that bias is corrected,

the Frisch labor supply elasticities for men rise from 0–0.2 to about 0.5.

Domeij and Floden (2006) argue that when the econometrician ignores borrowing

constraints, the estimated intertemporal Frisch elasticities have a downward bias. They

also identify an additional downward bias in many studies from the approximation errors

of log-linearizing the Euler equation.

Most recent studies tend to find higher values than was the case earlier. Estimates

generated by Ziliak and Kneisner (2005) are around 0.5. Pistaferri (2003) estimates a

(entry-exit) margin. In his Nobel lecture, Prescott (2006) emphasized this point; he argued for a Frisch

elasticity of 3.

8See Card (1994) for a critical review of the basic intertemporal model. He points out the difficulty of

identifying the parameters that govern the intertemporal substitution of labor supply and the ambiguity

in the literature that is attributable to the presence of wealth effects.

5
Frisch elasticity of 0.75 in a paper that improves on previous studies by avoiding the use

of weak instruments to estimate expected wage rates. The author instead relies on a

unique data set from the Bank of Italy that contains data both on an individual’s hours

and on their expectations of future earnings (but not wages).9

Researchers who take a more structural approach and estimate directly the intertemporal elasticity of labor supply from the life cycle model also report much higher values.

Gourinchas and Parker (2002) present estimates that range from 0.7 to almost 2. Imai

and Keane (2004) report that accounting for human capital accumulation can lead to

huge Frisch elasticities of labor supply (more than 3). Laitner and Silverman (2005)

present estimates that are around 0.9.

Our paper is related to Chang and Kim (2006), who use an infinite horizon model

with incomplete markets and assume that consumption and leisure are separable in the

utility function. In contrast, we use an OLG model and a CRRA utility function nonseparable between leisure and consumption. Chang and Kim’s utility function implies

a direct mapping between the intertemporal substitution elasticity of leisure and the

Frisch elasticity; in our model it does not. Another difference is that we can compare

our results directly with previous results from the empirical literature because we apply

standard econometric techniques used in previous microeconomic studies for comparison. Finally, Chang and Kim do not estimate the sources of bias present in the empirical

studies nor the labor supply response to unexpected changes in wages

This paper also is similar to the work of Domeij and Floden (2006) in that it considers the effects of uncertainty and borrowing constraints on the estimates of the Frisch

elasticity of labor supply. It differs from Domeij and Floden (2006) in three major ways:

First, instead of using a model with infinitely lived agents, we use a life cycle model to

compare estimates from a simulated panel with the estimates that do not consider borrowing constraints and uncertainty. Second, we focus on a different type of bias present

9Fehr and G ̈otte (2007) estimate an intertemporal elasticity of substitution of about 1.1 by using

a unique data set that measures individual workers’ responses to a preannounced change in wage for

a specific piece rate (service provided by bicycle messenger) service. Because they use high-frequency

data and estimate an elasticity that does not hold the marginal utility of wealth constant, their results

are not relevant for the model used in this paper or for the calculation of the Frisch elasticity.

6
in the Frisch elasticity estimates: the omitted variable bias. Finally, we use a different

method to measure the Frisch elasticity of labor supply. We borrow Pistaferri’s (2003)

methodology, which accounts for the labor supply response to unexpected changes in

wages.

3 Quantifying the Intertemporal Labor Supply

Response Using a Stochastic OLG Model with

Incomplete Markets

In this section we lay down a theoretical framework to analyze two types of intertemporal

elasticity of labor supply. The first measures the response of hours of work to anticipated wage changes, holding the marginal utility of wealth constant (that is, the Frisch

elasticity). The second measures the response of hours of work to unanticipated wage

changes. Both elasticities are relevant to the analysis of tax and benefit reforms, and

their relative importance depends on whether the reforms favor wealth or intertemporal

substitution effects differently. We discuss the empirical challenges faced in estimating

the elasticities and the issues that arise when we want to map the results from the model

to the empirical estimates. We quantify the Frisch elasticity directly from the model,

not accounting for borrowing constraints, and we quantify the same elasticity accounting

for the borrowing constraints using data from a simulated panel. We estimate from the

model the labor supply response to unanticipated wage changes. Finally, we quantify

the omitted variable biases of previous estimates of the Frisch elasticity of labor supply.

3.1 OLG Model Used to Generate Synthetic Data

This paper uses an OLG growth model with uninsurable idiosyncratic working ability

shocks and uncertain life span.10 The unit of analysis is married households, but we

10The base model is similar to those used Aiyagari (1994), Huggett (1996), and many others, although

Aiyagari assumed infinitely-lived agents. The model is an extension of that used by Nishiyama (2002)

7
treat them as a single decision unit.11 Accordingly, we calibrate the number of hours

worked in the model as the sum of two individuals’ total hours. The consumer side is

the one relevant to the determination of the intertemporal elasticity of labor supply.

The production side, which features a perfectly competitive representative firm, and the

government sector with exogenously determined policies, only determine the interest

rate and the wage rate that feed into the consumer’s problem. For this reason, we just

briefly analyze here the consumer part of the model. For more detailed information on

the model, see Nishiyama (2002) and Nishiyama and Smetters (2005). Table 1 lists the

parameters used in the calibration.

Table 1. Parameters for the model

Time preference parameter β 0.986

Share parameter for consumption α 0.449

Coefficient of relative risk aversion γ 2.0

Capital share of output θ 0.300

Depreciation rate of capital stock δ 0.047

Long-term real growth rate μ 0.018

Long-term population growth rate ν 0.010

Total factor productivity A 0.949

Subject to a budget constraint, households choose consumption and leisure. They

face uninsurable idiosyncratic working ability shocks and mortality shocks and can hold

only one type of assets. Formally, the household problem is expressed as follows:

max

c,h

Et

X

T

t=0



1

1 + ρ

t

u(ct, ht, Zt) (1)

s.t. at+1 = (1 + rt)at + wtht − ct and at ≥ a ̄ (2)

where ρ is the discount rate, c is the consumption level, h is the number of hours worked,

Z is a vector of preference shifts, a is the asset level, r is the interest rate,  ̄a is the lower

limit of assets a household can have, T is the total number of periods a household lives

and is very similar to the model presented by Nishiyama and Smetters (2005).

11Collective decision models within the households have been studied theoretically by Chiappori

(1988, 1992) and Blundell et al (2007). We abstract from this issue in our analysis and treat the

households as if it is a single decisionmaker.

8
and w is the stochastic wage rate. The optimality conditions for this problem are

∂u(ct, ht, Zt)

∂ct

= λt (3)

−

∂u(ct, ht, Zt)

∂ht

= λtwt (4)

Et



1 + rt+1

1 + ρ



λt+1 + μt = λt (5)

λt represents both the marginal utility of wealth in period t and the Lagrange multiplier associated with the budget constraint in equations (3), (4), and (5). μt represents

the Lagrange multiplier associated with the borrowing constraint. This multiplier would

be equal to zero when the household does not face liquidity constraints and different from

zero otherwise.

From equations (3) and (4), we can derive the Frisch elasticity of labor supply, or

how the number of hours responds intertemporally to changes in the wage rate, holding

the marginal utility of wealth constant, if the borrowing constraint in equation (2) does

not bind. Dropping the time subscript, the expression for the Frisch elasticity without

considering borrowing constraints is as follows:

dh

dw

w

h







no constraint

λ

=

uhucc

uhhucc − u

2

hc

1

h

(6)

If the borrowing constraint binds, the Lagrange multiplier μ will be different from

zero. In that case, we can use equation (5) in addition to equations (3) and (4) to

express the Frisch elasticity of labor supply, as follows:

dh

dw

w

h







borrowing constraint

λ

=

uhucc

uhhucc − u

2

hc

1

h

∗

1

Et



1+r

0

1+ρ

 ∗



1 +

wμ

uh



(7)

Note that the additional term in equation (7) is less than 1 because the marginal

utility of labor is negative.12 This implies that if borrowing constraints are binding, the

12The parameters’ choice in this economy make 

1+r

0

1+ρ



≤ 1.

9
Frisch elasticity of labor supply is lower than it is when borrowing constraints are not

binding.

We can use equation (6) to calculate directly from the model’s parameters the Frisch

elasticity of labor supply . However, that calculation is more complicated if the borrowing constraint binds (equation [7]) because we don’t know the value of the Lagrange

multiplier.

Equation (6) implies also that if there is no change in consumption when the number

of hours changes (uch = 0), the Frisch elasticity is given by (uh/uhh)(1/h). That happens,

for example, when the utility function is separable in labor and consumption, as was the

case in many previous studies.

The assumption that labor and consumption are additively separable was very important to many previous empirical estimates, because it implies that the Frisch elasticity

can be calculated directly using the parameters of the utility function. In particular,

Altonji (1986), Heckman and MaCurdy (1982), MaCurdy (1981), and others use the

following functional form:

u(c, h) = Bc

1 + Bc

VckC

1+ 1

Bc

k − VnkN

1+ 1

Bn

k

(8)

where C is consumption, Vck and Vnk are preference parameters, N is labor supply,

Bc is the elasticity of intertemporal substitution of consumption (EIS consumption). In

equation (8), the Frisch elasticity of labor supply is given directly by Bn, which represents

at the same time the EIS of labor.13 That condition does not hold in utility functions

for which labor and consumption are nonseparable, as will be clear in the next section.

13This is a particular case: The difference between the EIS and the Frisch elasticity is that the EIS

does not hold the marginal utility of wealth constant.

10
3.2 Quantifying the Frisch Elasticity, Not Accounting for

Borrowing Constraints and Uncertainty

To quantify the Frisch elasticity without considering borrowing constraints or uncertainty, we use equation (6) and assume in the model the following utility function:

u(ct, ht) =

nh

1 + nt

2

−ς

ct

iα

[h

max

t − ht

]

1−α

o1−γ

1 − γ

(9)

here γ is the coefficient of relative risk aversion, α is the share parameter for consumption,

ntis the number of dependent children, ς is the consumption adjustment parameter for

children, and h

max

t

is the maximum number of working hours.

Using the model’s utility function in equation (9) and applying the general theoretical

formula for the Frisch elasticity described in equation (6) we arrive at an expression

for the Frisch elasticity of labor supply without considering borrowing constraints or

uncertainty:

dh

dw

w

h







λ

=

h

max

t − ht

ht



1 − α(1 − γ)

γ



(10)

To calculate the Frisch elasticity in (equation 10), we need measures of the maximum

number of hours and parameters for the share of consumption in the utility function

and for the coefficient of relative risk aversion. Those parameters are chosen such that

the average hours per household match the data for the United States, as well as the

capital-to-labor ratio. Following Nishiyama and Smetters (2005), we use h

average = 3414

and h

max = 8760, representing 2 persons working 12 hours each, γ = 2.0, and α = 0.465.

We calibrate the value of α (the share of consumption in the utility function) to get

consistency among the value of the risk aversion γ and the value of maximum and average

hours. The intertemporal Frisch elasticity of labor supply obtained in this fashion has a

value of 1.14.

The elasticity in equation (10) does not consider borrowing constraints or uncertainty

as they appear in the household problem (equations [1] and [2]). But uninsurable wage

11
risk and borrowing constraints are crucial features of the stochastic OLG model used in

this paper. The borrowing constraints reduce the response of hours of work to a change

in wages, and the presence of uncertainty introduces a different type of intertemporal

labor supply response, which we measure later as the response of hours of work to

unanticipated changes in wages.14

The uncertainty is implemented as idiosyncratic shocks to earnings. Bad wage shocks

can slam households onto borrowing constraints that effectively break the intertemporal links across periods in the model. When the constraints bind, households in the

model effectively become static, and the lifetime period is broken up into a series of

shorter periods. Therefore, the estimate using equation (10) would not correspond to

observable quantities in conditions of uncertainty and borrowing constraints, so we need

to use a different estimating procedure. We do so in the next section by generating a

synthetic panel and using established empirical procedures to estimate the two types of

intertemporal elasticity (response to anticipated and to unanticipated changes in wages).

3.3 Empirical Estimates of the Labor Supply Responses to

Anticipated (Frisch) and Unanticipated Changes in

Wages from a Simulated Panel

There is no reason for the Frisch elasticity estimated in a simulated dynamic economy

to be equal to the values analytically derived from the utility function. In the model

we use in this paper, agents live through many periods and face borrowing constraints,

uncertain time of death, and idiosyncratic shocks to working ability. Those factors are

expected to attenuate the labor supply response to policy changes, but quantifying that

attenuation is not easy. The implied Frisch elasticity cannot be calculated analytically

from the model’s parameters and aggregate results.

To overcome those difficulties, we use a general method that can be summarized in

two steps. First, we generate a simulated panel with the OLG model (The method is

14This type of intertemporal labor supply elasticity exists only if uncertainty is present, because

idiosyncratic shocks cannot be anticipated by the households, and they trigger this specific response.

12
described in the appendix.) Second, once the panel is generated, we estimate two types

of intertemporal elasticities using an empirical methodology that has already been used

with real data. The elasticities we estimate are the labor supply response to anticipated

changes in wages, holding the marginal utility of wealth constant (Frisch elasticity),

and the labor supply response to unanticipated changes in wages. We use Pistaferri

(2003) approach because it accounts directly for variances and shocks, and it controls

for changes in the marginal utility of wealth. Given that we can generate exact data and

calculate expectations from the model, our estimate is free of measurement errors that

arise when people misjudge their expected income or consumption or misreport current

values.

As is clear in equation (7), it is impossible to calculate directly the Frisch elasticity

accounting for borrowing constraints and uncertainty, so we need to obtain an equation

that is feasible to estimate. To obtain such equation, we follow Pistaferri (2003) and

obtain from equation (6) a log-linear specification:

ln hit = Z

0

itα + η ln wit + (η + φ) ln λit (11)

where η represents the Frisch elasticity of labor supply, φ is the elasticity between

consumption and labor, and (η+φ) is the elasticity of hours with respect to the marginal

utility of wealth. Empirical estimates of the Frisch elasticity obtained using equation (11)

will account for borrowing constraints and uncertainty, independent of the econometric

problems present in the methodology; calculations of the Frisch elasticity of labor supply

using equation (6) will not.

Equation (11) can be used to analyze the effect of borrowing constraints in the Frisch

elasticity of labor supply, in the same spirit as Domeij and Floden (2006). We can

express that equation in a differenced version and use equation (5) to arrive at an

approximated expression for the change in the marginal utility of wealth: ∆ ln λit =

(ρ−Et−1rt) + ln[1−(μ/λ)]. This way we arrive at the following new version of equation

13
(11):

∆ ln hit = ∆Zitα + ηEt−1∆ ln wit + (η + φ)[(ρ − Et−1rt) + ln(1 − (μ/λit))] + εt (12)

Through the term ln(1−(μ/λit)), equation (12) shows the negative effect of borrowing

constraints μ in the Frisch elasticity of labor supply η. It can be argued that the expected

change in wages is negatively correlated with the marginal utility of wealth because the

marginal utility of wealth decreases by the effect of the concavity of the utility function.

Then, if the marginal utility of wealth is not controlled directly, we would have a negative

correlation between the terms Et−1∆ ln wit and ln(1 − (μ/λit)), inducing a negative bias

in the parameter η, the Frisch elasticity of labor supply.

Researchers face three important problems when they try to estimate equations like

equation (12): First, neither the marginal utility of wealth nor the Lagrange multiplier

associated with the borrowing constraints is observed by the econometrician. Second,

the change in expected log of wages is not observed. Third, the change in expected log of

wages is likely to be correlated with the marginal utility of wealth or with the change in

tastes for leisure, Zit. A widely applied option is to use instrumental variables techniques

for that term. The problem, as Keane (2006) points out, is that it is difficult to develop

a good instrument to predict the expected change in log wages, and the commonly used

variables, such as age and education, could be associated with preference shifts or with

elements present in the error term, again introducing bias in the estimates.

From here, we need to make some assumptions to arrive at closed-form solutions and

calculate the intertemporal elasticities of labor supply. The Pistaferri (2003) method

uses the usual consumer problem, as described in equation (3), with the following assumptions:

1. The wage process is parameterized as

ln wit = ∆X

0

itσ + ln wit−1 + ζit (13)

14
where X is a vector of deterministic variables affecting wages (age, education) and

ζ is the innovation in the wage rate ζ = ln wit − Et−1 ln wit−1.

2. The log of the marginal utility of wealth (λ) is parameterized as

(η + φ) ln λit = γa +

X

T

τ=0

γτEt(ln wit+τ ) + νi (14)

where η is the Frisch elasticity, φ is the response of consumption to changes in

labor, and (η + φ) is the elasticity of hours with respect to the marginal utility

of wealth. From here, we can observe that Γ = PT

τ=0 γτ is the wealth effect of a

parametric permanent shift in the wage profile.

Using assumptions (13) and (14) and the general theoretical formula for the Frisch

elasticity described in equation (6) we can develop the following equation:

∆ ln hit = ∆Zitα + ηEt−1∆ ln wit − (η + φ)(Et−1rt − ρ) + υV art−1(ζit) + (η + Γ)ζit (15)

η represents the value of the Frisch elasticity, which measures the change in labor supply

in reaction to expected changes in wages, holding the marginal utility of wealth constant.

The other parameters of interest are υ, which measures the change in labor supply in

response to wage risk, and (η + Γ), which measures the change in labor supply to

unexpected changes in wages.15 The variables generated using the model are hours (h),

expected change in the log of wages (Et−1∆ ln wit), and innovation in the wage rate

ζ = ln wit − Et−1 ln wit−1. In the simulated panel, the interest rate and the discount

rate are the same for all households, and the vector of preference shifts does not change.

All those terms are reflected in the constant term. Because our data were generated by

simulation from a life-cycle model, we have the advantage over most empirical studies

of having the precise variables we need to estimate the parameters of interest and no

15The literature refers to anticipated changes in wages as “evolutionary,” and to unanticipated changes

as “parametric.”

15
measurement error problems to deal with. The equation to be estimated is the following:

∆ ln hit = α

0 + ηEt−1∆ ln wit + υV art−1(ζit) + (η + Γ)ζit + εt (16)

Note that in equation (16) the presence of uncertainty about future wages is independent from the coefficient of the Frisch elasticity, but failing to account for this uncertainty

in the estimation introduces bias to the results (omitted variable bias). Accounting for

this uncertainty allows one to identify the estimate of the labor supply response to

unanticipated changes in wages.

Table 2 shows the results of the estimation after applying equation (16) to the simulated data (and using a fixed-effect specification). We arrive at an estimate of the Frisch

elasticity of labor supply of 0.327 (standard deviation = 0.003), about one-fourth of

the value when not considering borrowing constraints and wage uncertainty. Our Frisch

elasticity estimate is within the range of 0.1-0.45 given by MaCurdy (1981), close to the

central value of 0.2 in Pencavel’s 1986 survey, within the range of 0.24-0.79 given by

Angrist (2005), but smaller than many recent estimates.

Our Frisch elasticity estimate is just over half of Pistaferri’s (2003) estimate. The

difference could have any one of several causes: First, because we generate data on

wages and expected wages directly, we isolate the estimates from unobserved taste shifts

and we can estimate the Frisch elasticity using wages instead of earnings. Second, our

data do not have measurement error. And third, we can control the macroeconomic

conditions, where unemployment and other demand-side variables can alter the labor

supply responses.

The labor supply response in the model to unanticipated wage changes is very small

(0.0007). That value is below previous empirical estimates. For example, MaCurdy

(1981) has 0.08 and Pistaferri (2003) has 0.51. At the same time, our estimate implies a

wealth effect of -0.3264, which is similar to the Frisch elasticity estimate and significantly

ameliorates the intertemporal substitution effect in response to wage changes. The

wealth effect estimate in our model is somewhat higher than Pistaferri’s estimate of -0.2

16
Table 2. Frisch elasticity estimates using the simulated panel

Parameter Estimate 95% Confidence Interval

(standard error)

0.3271

η (Frisch) (0.0029) [0.3213, 0.3328]

0.0017

υ (Wage risk) (0.0000) [0.0017, 0.0018]

η + Γ 0.0007

(Unexpected change in wages) (0.0002) [0.0003, 0.0010]

-0.2225

α

0

(0.0052) [-0.2326, -0.2124]

and significantly higher than MaCurdy’s estimate of -0.07. However, our estimate is

within 1.5 standard errors of Pistaferri estimate.

3.4 Quantifying the Omitted Variable Bias in the Frisch

Elasticity Estimation

If the econometrician does not include measures of wage risk or unexpected changes in

wages-as is the case for many empirical studies-the Frisch elasticity estimates exhibit a

significant downward bias. To quantify the omitted variable bias in the Frisch elasticity

of labor supply, we can run the specification in equation (16) without controlling for

wage risk or for unexpected changes in wages.

If measures of wage risk are not included in the regression, the estimate of the Frisch

elasticity drops from 0.33 to 0.24 (see Table 3). That estimate suggests an omitted

variable bias of around 30 percent. By contrast, the Frisch elasticity estimate does not

change much when unexpected change in wages alone is not included (Table 4).

However, if both the wage risk measure and the measure of unexpected change in

wages is excluded, the estimate of the Frisch elasticity drops to 0.04 (Table 5). That

result suggest that the omitted variable bias in regressions that exclude those control

variables could be substantial and may explain why several early studies found small

estimates of the Frisch elasticity.

17
Table 3. Frisch elasticity estimates using the simulated panel

not controlling for wage risk measures

Parameter Estimate 95% Confidence Interval

(standard error)

0.2376

η (Frisch) (0.0025) [0.2328, 0.2425]

η + Γ 0.0020

(Unexpected change in wages) (0.0002) [0.0016, 0.0023]

-0.0297

α

0

(0.0021) [-0.0255, -0.0339]

Table 4. Frisch elasticity estimates using the simulated panel

not controlling for unexpected changes in wages

Parameter Estimate 95% Confidence Interval

(standard error)

0.3269

η (Frisch) (0.0029) [0.3211, 0.3326]

0.0018

υ (Wage risk) (0.0000) [0.0017, 0.0018]

-0.2241

α

0

(0.0051) [-0.2341, -0.2140]

Indeed, most empirical studies do not include measures of wage risk or unexpected

changes in wages, because expectations are hard to measure. The exception is the

study by Pistaferri (2003), who extracted individual expectation of earnings growth from

the Bank of Italy Survey of Households’ Income and Wealth. He modifies further the

specification in equation (16) to adjust for the fact that expectations consider earnings

rather than wages.

Several important strengths of our approach enable us to measure the omitted variable

bias. First, we generate simulated data on wages and expected wages, thus isolating the

estimates from unobserved taste shifts. Second, our simulated data have no measurement

error. Third, we can control the macroeconomic conditions under which unemployment

and other demand-side variables can alter the labor supply responses. In addition, we

can run several experiments that introduce different levels of transitory wage innovations

18
Table 5. Frisch elasticity estimates using the simulated panel

not controlling for wage risk measures or

unexpected changes in wages

Parameter Estimate 95% Confidence Interval

(standard error)

0.0411

η (Frisch) (0.0009) [0.0393, 0.0429]

-0.0096

α

0

(0.0022) [-0.0052, -0.0140]

to arrive at the same point estimate for the Frisch elasticity.

4 Conclusions

In this paper, we use a stochastic OLG model with incomplete markets and Pistaferri’s

(2003) empirical method to estimate the labor supply response to anticipated changes

in wages, holding the marginal utility of wealth constant (Frisch elasticity), and to

unanticipated change in wages. We conduct an experiment by using a simulated panel

from the model, in which we can exactly observe expected wages and control for demandside variables and uncertainty, but which does not present issues with measurement error.

Once borrowing constraints and uncertainty about future wages are considered, our

Frisch elasticity estimate (0.33) is within the range of what appears in the microeconomic

empirical literature. If we exclude those factors, the estimate is almost 4 times larger

(1.17). The rationale is that, without borrowing constraints, households respond to a

bad wage shock by reducing labor and increasing borrowing so as to smooth consumption

over time.

The labor supply response to unanticipated wage changes, which also is relevant for

the analysis of tax and benefit reforms, is very small (0.0007) because of large wealth

effects in the OLG model. In principle, both intertemporal substitution and wealth

effects are relevant for determining the labor supply response to tax or benefit changes.

A wage increase generates an intertemporal substitution effect which tends to increase

the labor supply, but the same wage increase generates a wealth effect and a labor

19
supply reduction. Because reforms often differ at the extent to which they alter these

two effects, it is important to explicitly consider both of them in the analysis.

Finally, we quantify the bias-attributed to omitted variables-that occurs in many

empirical estimates because of lack of data about wage risk and unexpected changes in

wages. That bias is 88% if both variables are absent and 30% if wage risk measures are

excluded.

20
Appendix. Generation of the Simulated Panel

The simulated panel is generated in two steps. First, we solve for the optimal solution to

the consumer problem and obtain the policy functions for each point in the discretized

state space. The policy functions define a rule that maps the space into recursive decisions, which are located in the continuous space defined by the boundaries of the state

variables (not limited to the grid points). For every combination of age, wealth, and

work ability, the policy function gives the optimal consumption and work hours. The

policy function will, in general, map a state S at time t, which corresponds to a grid

point in the discretized space, to a state S

0 at time t + 1, in the continuous-state space.

When aggregating the results, a measure of households is assigned to each grid point

by reverse interpolation: Each of the grid points surrounding state S

0

is assigned such

a portion of the surviving households from state S that S

0

could be interpolated as a

combination of solutions limited to grid points.

The second step in panel generation is to simulate a time series of the household’s

optimal decisions. In every period, those decisions are linked by the law of motion from

the policy function, so that a household makes decisions at time t + 1 depending on its

state at time t.

To generate the time series, we simulate the stochastic variables of the model (the

ability shocks) as the same stochastic process defined in the solution of the model. In

each period t, the household is at a grid point S of the state space, determined by the

random numbers generated so far. The policy function is applied to that state, and

it maps the household’s wealth to its value in the next period, resulting in a state S

0

,

which generally does not correspond to a grid point. At that stage, a process equivalent

to the assigning of measure to the surrounding grid points (described above) is applied.

The procedure assigns to the household one of the surrounding grid points, with the

respective probability of each grid point proportional to the measure assigned to the

grid point during the solution of the model. This step requires the generation of an

additional random number or, more generally, a vector of as many random numbers as

21
there are continuous dimensions in the state space. The household thus ends up in a

state S

00 given by a grid point in the discretized space, and the procedure can be repeated

for the next period.16

The method described before generates the time series of length T (T is the number of

stochastic realizations for the income process—generally equal to the maximum length

of a career) for one individual. The same procedure is repeated N times to generate N

households, and thus there is a panel of dimension N×T. We generate 10,000 households

that may live up to 109 years.

16There is another option for generating simulated households, not pursued here. That option includes

generating a policy rule for each stochastic income process point and a recursive realization of state

variables; that is, to apply the solution procedure instead of applying the obtained policy rules. In

other words, the state S

0

, resulting from the last period’s policy, would not be further shocked to fall

onto a grid point S

00, but the period t + 1 policy would be computed directly for the state S0

.

22
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