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Journal ArticleDOI

Reconciling Jaimovich–Rebello preferences, habit in consumption and labor supply

25 Apr 2018-Economics Letters (Elsevier)-Vol. 168, pp 132-137

AbstractThis note studies a form of a utility function of consumption with habit and leisure that (a) is compatible with long-run balanced growth, (b) hits a steady state observed target for hours worked and (c) is consistent with micro-econometric ev- idence for the inter-temporal elasticity of substitution and the Frisch elasticity of labor supply. Employing Jaimovich-Rebello preferences our results highlight a con- straint on the preference parameter needed to target the steady-state Frisch elastic- ity. This leads to a lower bound for the latter that cannot be reconciled empirically with external habit, but the introduction of a labor wedge solves the problem. We also propose a dynamic Frisch inverse elasticity measure and examine its business cycle properties.

Summary (2 min read)

1 Introduction

  • In particular, RBC-DSGE models in which a consumer’s utility level not only depends on her consumption level but also how that level compares to a standard set either by her own past consumption levels (internal habit-formation) or the levels of those in her peerage (catching-up with the Joneses’ or external habit) are now ubiquitous in the literature.
  • At the same time to achieve co-movement of output, hours, consumption and investment modellers turn to preferences proposed by Jaimovich and Rebello (2008) (henceforth JR) that control short-run wealth effects.

2 The Frisch Elasticity of Labor Supply

  • Bilbiie (2011) shows that the constantmarginal-utility inverse elasticity of labor supply is given by δ = ULCH UL ( ULC UCC − ULL ULC ) and it is straightforward to show that g′′ >.
  • Microeconomic and macroeconomic estimates of the Frisch elasticity differ significantly, the former typically ranging from 0 to 0.5 and the latter from 2 to 4 (Peterman, 2016).
  • Estimations of the elasticity of labor supply found using microeconomic data depend on factors such as gender, age, marital status and dependants.
  • Reichling and Whalen (2017) give a thorough review of the estimates found in the literature based on microeconomic data, finding that estimates typically range from 0 to over 1.

3 The Household Problem

  • Households choose between work and leisure and therefore how much labor they supply.
  • They also own the capital stock which is rented to firms at a rental rate rKt and choose an optimal investment path.
  • The single-period utility is given by JR preferences (3) and (4).
  • Tt where All variables are expressed in real terms relative to the price of output.

3.4 Numeral Illustration

  • The authors can now assess the empirical plausibility of JR preferences with habit in consumption.
  • From their numerical results for the lower bound δJR(ψ), this rules out external habit if the authors are to choose JR preferences that allow for only weak wealth effects (γ very small).
  • But even without habit, or with internal habit, it is difficult to reconcile the extreme choice of γ almost zero with a Frisch elasticity within this empirical range.

4 A Resolution: Generalized JR Utility and Labor Wedge

  • Again for a given H this pins down % given the remaining parameters.
  • The second modification is to introduce a labor wedge into the household problem (as in Shimer (2009).
  • Figure 1 shows that a combination of their generalized JR utility and an empirically supported wedge, the authors can calibrate parameters to achieve the desired empirical value for δJR.

5 Conclusions

  • This note has reviewed a utility function commonly used in RBC-DSGE models that is nonseparable in habit-adjusted consumption and leisure, compatible with balanced growth and elim- 7 inates counterfactual wealth effects highlighted by Jaimovich and Rebello (2008).
  • The authors main contributions are first, Theorems 1 and 2 that highlight a constraint on the preference parameter ψ needed to target the Frisch elasticity.
  • This leads to a lower bound for the latter that cannot be reconciled empirically with external habit.
  • Even with internal or no habit, the range of possible values of the Frisch elasticity lie outside empirical results unless the authors allow for a modest wealth effect.

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Discussion Papers in Economics
DP 10/17
School of Economics
University of Surrey
Guildford
Surrey GU2 7XH, UK
Telephone +44 (0)1483 689380
Facsimile +44 (0)1483 689548
Web www.surrey.ac.uk/economics
ISSN: 1749-5075
RECONCILING JAIMOVICH-REBELLO PREFERENCES,
HABIT IN CONSUMPTION AND LABOR SUPPLY
By
Tom Holden,
(University of Surrey)
Paul Levine,
(University of Surrey)
&
Jonathan Swarbrick
(Bank of Canada)
DP 05/15

Reconciling Jaimovich-Rebello Preferences,
Habit in Consumption and Labor Supply
Tom Holden
University of Surrey
t.holden@surrey.ac.uk
Paul Levine
University of Surrey
p.levine@surrey.ac.uk
Jonathan Swarbrick
Bank of Canada
jswarbrick@bankofcanada.ca
November 20, 2017
Abstract
This note studies two forms of a utility function of consumption with habit and leisure that
are (a) compatible with long-run balanced growth, (b) hit a steady state observed target for
hours worked and (c) are consistent with micro-econometric evidence for the inter-temporal
elasticity of substitution and the Frisch elasticity of labor supply. For Jaimovich-Rebello pref-
erences our Theorems 1 and 2 highlight a constraint on the preference parameter needed to
target the Frisch elasticity leading to a lower bound for the latter that cannot be reconciled
empirically with external habit. Even with internal or no habit, the range of possible values
of the Frisch elasticity lie outside empirical results unless we allow for a modest wealth effect.
In Theorem 3 we propose a generalized JR utility function that in conjunction with a labor
wedge solves the problem.
JEL Classification: E21, E24.
Keywords: Jaimovich-Rebello Preferences, Habit in Consumption, Labor Supply, Frisch Elas-
ticity, Labor Wedge
School of Economics, University of Surrey, Guildford GU2 7XH, UK. Email: p.levine@surrey.ac.uk

Contents
1 Introduction 1
2 The Frisch Elasticity of Labor Supply 1
3 The Household Problem 2
3.1 Solution of the Household Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 JR Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.3 External versus Internal Habit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.4 Numeral Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 A Resolution: Generalized JR Utility and Labor Wedge 6
5 Conclusions 7

1 Introduction
Whether it is in the context of the equity-premium puzzle (see for example Abel (1999)), the
savings-growth relation (Carroll and Weil (2000) ) or monetary policy - business cycle analysis
(Christiano et al. (2005)), researchers have used the concept of relative preferences to advance
their various agendas. In particular, RBC-DSGE models in which a consumer’s utility level not
only depends on her consumption level but also how that level compares to a standard set either
by her own past consumption levels (internal habit-formation) or the levels of those in her peerage
(catching-up with the Joneses’ or external habit) are now ubiquitous in the literature.
At the same time to achieve co-movement of output, hours, consumption and investment mod-
ellers turn to preferences proposed by Jaimovich and Rebello (2008) (henceforth JR) that control
short-run wealth effects. This note discusses two forms of this utility function, U (C, L), where
C is consumption modified by habit and L = 1 H is leisure, as the proportion of the day, H
being hours. The objective is to choose a form (a) compatible with long-run balanced growth, (b)
hits a steady state observed target for H (c) is consistent with micro-econometric evidence for the
inter-temporal elasticity of substitution and the Frisch elasticity of labor supply.
2 The Frisch Elasticity of Labor Supply
We examine a broad class of utility functions that satisfy criterion (a) above proposed by King
et al. (1988) henceforth KPR:
U(C, L) =
C
1σ
1 σ
exp((σ 1)g(L)); σ > 1, g
0
< 0, g
00
> 0 (1)
Note that (1) is a CRRA function in f (L) = exp(g(L)). Bilbiie (2011) shows that the constant-
marginal-utility inverse (Frisch) elasticity of labor supply is given by
δ =
U
LC
H
U
L
U
LC
U
CC
U
LL
U
LC
and it is straightforward to show that g
00
> 0 is sufficient for leisure to be a normal good. Differ-
entiating (1) we can substitute for partial derivatives U
C
, U
CC
, U
LC
, U
L
and U
LL
to arrive at the
Frisch elasticity for the KPR class of utility functions:
δ =
H(g
00
(L) +
σ1
σ
(g
0
(L))
2
g
0
(L)
(2)
1

Jaimovich and Rebello (2008) propose a utility function of the KPR form in its balanced growth
steady state:
U
t
= U(C
t
, L
t
, X
t
) =
(C
t
%(1 L
t
)
1+ψ
X
t
)
1σ
1 σ
; ψ > 0 (3)
X
t
= C
γ
t
X
1γ
t1
; γ [0, 1] (4)
In a zero growth steady state equilibrium with γ > 0 we have X = C where for the household
decision (as we shall see below) C is a function of γ and
U(C, L) =
C
1σ
1 σ
(1 %(1 L)
1+ψ
)
1σ
=
C
1σ
1 σ
exp
(σ 1) log(1 %(1 L)
1+ψ
)
which is of KPR form (1) where g(L) = log(1 %(1 L)
1+ψ
). Differentiating g(L) and using
(2), for JR preferences the Frisch elasticity is then given by
δ
JR
= ψ +
(2σ 1)(1 + ψ)%H
1+ψ
σ(1 %H
1+ψ
)
(5)
Note that in equilibrium H = 1 L, and therefore δ
JR
, are dependent on γ (see Section 3.2).
Microeconomic and macroeconomic estimates of the Frisch elasticity differ significantly, the
former typically ranging from 0 to 0.5 and the latter from 2 to 4 (Peterman, 2016). Estimations of
the elasticity of labor supply found using microeconomic data depend on factors such as gender,
age, marital status and dependants. Keane (2011) offers a survey of labor supply, restricting the
sample to men, finding a range of between 0 to 0.7 with an average of 0.31. Reichling and Whalen
(2017) give a thorough review of the estimates found in the literature based on microeconomic
data, finding that estimates typically range from 0 to over 1. The higher estimates corresponding
to married women with children, whereas the labor supply of men is far lower. Combining the
results, Reichling and Whalen (2017) propose a range of between 0.27 and 0.53, with a central
point estimate of 0.4. This corresponds to a Frisch coefficient, δ, between 1.89 and 3.7, with a
point estimate of 2.5.
3 The Household Problem
Households choose between work and leisure and therefore how much labor they supply. They also
own the capital stock which is rented to firms at a rental rate r
K
t
and choose an optimal investment
path. The single-period utility is given by JR preferences (3) and (4). In a stochastic environment,
2

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Frequently Asked Questions (1)
Q1. What are the contributions in "Reconciling jaimovich-rebello preferences, habit in consumption and labor supply" ?

This note studies two forms of a utility function of consumption with habit and leisure that are ( a ) compatible with long-run balanced growth, ( b ) hit a steady state observed target for hours worked and ( c ) are consistent with micro-econometric evidence for the inter-temporal elasticity of substitution and the Frisch elasticity of labor supply. In Theorem 3 the authors propose a generalized JR utility function that in conjunction with a labor wedge solves the problem.