Non‐separable utilities and aggregate instability

• Author: Shun‐Fa Lee,Been‐Lon Chen,Xavier Raurich
• Published date: 01 June 2020
• Date: 01 June 2020
• DOI: http://doi.org/10.1111/ijet.12188

Non-separable utilities and aggregate instability



Been-Lon Chen



, Shun-Fa Lee

y

and Xavier Raurich

z

This paper studies an infini te-horizon two-sector gr owth model with sector-sp ecific

externalities and prefer ences that are non-separab le between consumption and leisure. We

find two main results. First, a larger income effect on the lab or supply increases the possibility

of macroeconomic instabi lity. Second, a larger elast icity of the labor supply may increase or

decrease the possibility of ag gregate instability, dep ending on the intensity of the inc ome

effect.

Key words indeterminacy, non-separable preference, income effect, labor supply elasticity

JEL classification E3, O41

Accepted 30 January 2018

1 Introduction

A strand of the growth litera ture has shown that equili brium indeterminacy can explain

macroeconomic instab ility in the infinite-horizon growth model w hen sector-specific externalities

are considered. This lite rature has analyzed how th e existence of aggregate in stability depends

both on the intensity of the income ef fect on the labor supply an d on the value of the wage

elasticity of the labor su pply. It has obtained differ ent results in the one-secto r and two-sector

growth models.

In a one-sector growth model, Jaimovich (2008) studied aggregate instability or indeterminacy

with a non-separable utility function. The utility features different intensities of the income effect on

the labor supply and nests, as special cases, the two classes of utility functions most widely used in the

business cycle literature, those characterized in King et al. (1988) and in Greenwood et al. (1988).

1

Jaimovich found that, regardless of the degree of increasing returns to scale, the one-sector growth

model does not generate aggregate instability when there is no income effect on the labor supply.

Similarly, in a one-sector growth model, Meng and Yip (2008) found that the presence of labor-

supply income effects is required for indeterminacy to occur. These results are different from the

existence of indeterminacy in one-sector growth models with sufficiently strong increasing returns

proved by Benhabib and Farmer (1994) and Farmer and Guo (1994). Those latter authors use a



Institute of Economics, Academia Sinica, Taipei, Taiwan. Email: bchen@econ.sinica.edu.tw

y

Department of Industrial Economics, Tamkang University, New Taipei City, Taiwan.

z

Department of Economics, University of Barcelona, Barcelona, Spain.



Early versions have benefited from comments and suggestions made by Leonor Modesto, Thomas Seegmuller, Alain

Venditti and an anonymous reviewer.

1

Jaimovich and Rebelo (2009) also consider this class of utility functions to show that labor-supply income effects lie at the

heart of the ability to generate aggregate and sectoral co-movement, which is a central feature of business cycles. In

particular, they can generate aggregate co-movement in the presence of moderate labor-supply income effects, and low

labor-supply income effects are essential to generate sectoral co-movement.

doi: 10.1111/ijet.12188

International Journal of Economic Theory xxx (2018) 1–16 ©IAET 1

International Journal of Economic Theory 16 (2020) 222–237 © 2018 IAET

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International Journal of Economic Theory

utility function that is characterized by the presence of income effects on the labor supply. Moreover,

they showed that a higher elasticity of the labor supply increases the possibility of aggregate

instability.

2

In a two-sector growth mode l with no income effect on the labo r supply, Guo and Harrison

(2010) obtained a very di fferent result, as they proved the existen ce of equilibrium indeterminacy.

Specifically, th ey studied a cla ss of tw o-sector mode ls 

alaBenhabib and Farmer (1 996) and

Harrison (2001), where t he technologies in the tw o sectors are differenti ated by the introduction

of sector-specific exte rnalities and the two secto rs are identified as the inv estment and

consumption sectors. Th ese authors adopted the no n-separable utility fun ction used in

Greenwood et al. (1988). The lab or supply obtained from this ut ility function does not exh ibit

income effects. They s howed that equilibrium i ndeterminacy emerge s when sufficiently stron g

investment externali ties are present. They a lso numerically showe d for a logarithmic uti lity

function that indetermi nacy is more likely for lo w values of the labor suppl y elasticity (LSE).

Dufourt et al. (2015) proved this result a nalytically under a mo re general specificatio n of no-

income effect preferen ces. These results are very di fferent from the finding th at indeterminacy is

more likely for high valu es of LSE, which is obtaine d in models with income eff ects on the labor

supply (e.g. Benhabib an d Farmer 1996).

The aforementioned literature has obtained very different results concerning the effect on

aggregate instability of the intensity of the income effect on the labor supply and of the LSE. The

purpose of this paper is to contribute to this literature by clarifying these effects. To this end, we

consider a two-sector growth model with sector-specific externalities 

alaBenhabib and Farmer

(1996) and a non-separable utility function that introduces a varying degree of income effects on the

labor supply.

3

At a limiting case, when the income effect on the labor supply is zero, this utility

function encompasses the preferences considered by Guo and Harrison (2010). In another limiting

case, when the income effect is the largest, the income elasticity of the labor supply equals the LSE. In

non-limiting cases, the income elasticity is smaller than the LSE.

We obtain two main findings. First, the presence of the income effect on the labor supply

increases the possibility of indeterminacy, as it reduces the minimum value of the intensity of the

investment externality needed for indeterminacy. Second, the relationship between the LSE and

indeterminacy depends on the intensity of the income effect. If the income effect is small, a decrease

in the LSE reduces the minimum value of the intensity of the investment externality needed for

indeterminacy and, hence, increases the possibility of indeterminacy. In contrast, if the income effect

is large, a decrease in the LSE decreases the possibility of indeterminacy. Thus, the relationship in

Guo and Harrison (2010) arises only when the income effect is small. This second result is

particularly interesting as evidence on the value of the intensity of the income effect is not

conclusive.

4

The intuition on these two results is based on expectations on a higher interest rate that cause a

reduction in today’s consumption and an increase in tomorrow’s consumption. We show that these

changes increase the marginal rate of substitution (MRS) between today’s and tomorrow’s

consumption. As in the consumption Euler equation the MRS equals the marginal product of capital

2

See also Lloyd-Braga et al. (2006) and Coury and Wen (2009).

3

The utility function is simpler than the one used in Jaimovich (2008) and Jaimovich and Rebelo (2009) which, in the case

with income effects on the labor supply, involves a consumption habit that accumulates from past consumption. We do

not consider consumption habits in our analysis.

4

The business cycle literature has considered that preferences with no income effects provide a better fit to explain business

cycle facts (Greenwood et al. 1988). However, other authors provide estimates of large positive income effects (Khan and

Tsoukalas 2011).

Non-separable utilities and instability Been-Lon Chen et al.

2International Journal of Economic Theory xxx (2018) 1–16 ©IAET

International Journal of Economic Theory 16 (2020) 222–237 © 2018 IAET 223