Frisch elasticities in a model of indivisible labor supply with endogenous workweek length

• Published date: 01 January 2024
• Author: Minchul Yum
• Date: 01 January 2024
• DOI: http://doi.org/10.1111/sjoe.12544

Scand. J. of Economics 126(1), 194–217, 2024

DOI: 10.1111/sjoe.12544

Frisch elasticities in a model of indivisible

labor supply with endogenous workweek

length

Minchul Yum†

University of Southampton, Southampton, SO17 1BJ, UK

m.yum@soton.ac.uk

Abstract

In this paper, I provide an extension of the classical indivisible labor supply model where a

large macro Frisch elasticity is reconciled with a small micro counterpart. Households take as

given state-dependent hours per worker – shaped by a nonlinear mapping from hours worked to

labor services and employment frictions – and make intertemporal labor supply decisions. In the

standard indivisible labor supply model, aggregate fluctuations are independent of the individual

preference parameter that governs the intensive-margin elasticity. In my model, however, they

are connected through the extensive margin whose elasticity is empirically reasonable and is

shaped by the individual preference parameter.

Keywords: Indivisible labor; intensive margin; extensive margin; Frisch elasticity; labor supply

JEL classification:E32;J22

1. Introduction

Models of indivisible labor supply following Rogerson (1988) can reconcile

small micro-based individual labor supply elasticities with large aggregate

counterparts. This feature is important as it can generate the large volatility

of aggregate hours which we observe in the data, while being consistent

with smaller estimates of individuals’ willingness to substitute labor over

time (Keane and Rogerson, 2012). Despite this merit, in pure indivisible

labor supply models, aggregate fluctuations (or the macro elasticity) are

disconnected from the preference parameter governing the micro elasticity.1

In this paper, I present an extension of the classical indivisible labor

supply model that circumvents this disconnect by allowing equilibrium hours

†Also affiliated with CEPR.

1This is the case not only in stand-in household models (Hansen, 1985) but also in

heterogeneous-agent models with incomplete asset markets, pioneered by Chang and Kim (2006),

in which aggregate fluctuations depend on the individual distribution but are still independent

of the preference parameter governing the underlying individuals’ Frisch elasticity.

c

2023 The Authors. The Scandinavian Journal of Economics published by John Wiley & Sons Ltd on behalf of F¨

oreningen

f¨

or utgivande av the SJE.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution

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M. Yum 195

per worker to be state-dependent. Through the exercises in the spirit of

Rogerson and Wallenius (2009) who provide a reconciliation on the micro

versus macro steady-state (or long-run) labor supply elasticities, my model

reconciles a large macro Frisch (or short-run) labor supply elasticity with

a small micro counterpart. Importantly, this reconciliation is achieved in

a framework where the aggregate labor supply elasticity depends on the

individual preference parameter that shapes the curvature of the utility

function along hours worked. This happens through the novel feature

of the model that a higher micro Frisch intensive-margin elasticity can

raise the volatility of aggregate hours along both intensive and extensive

margins.2

In the model, the firm solves a dynamic problem in the presence of a

nonlinear labor services mapping – which captures set-up costs and fatigue

effects (Prescott et al., 2009) – and employment frictions (Hall, 2004). The

firm’s optimization problem gives rise to the state-dependent workweek

length, which is then taken as given by households. I embed this setting into

an otherwise standard real business-cycle model. I follow a standard procedure

to calibrate the model in steady state except for the degree of the employment

adjustment cost in the baseline specification, which is calibrated to match the

cyclicality of employment over the business cycle.

I first evaluate the calibrated model using a set of conventional

business-cycle statistics. My model outperforms the real business-cycle model

with rich household heterogeneity in Chang et al. (2019) in terms of the cyclical

volatility of labor markets along different margins.3Moreover, the volatility

of employment increases with the individuals’ willingness to substitute labor

supply over time in my model, in contrast to Chang et al. (2019) where this

relationship is qualitatively opposite.

I then use the model economy to quantify the relationship between

individuals’ intensive-margin elasticity and its model-implied aggregate

elasticities as in Rogerson and Wallenius (2009). I find that the estimated

extensive-margin Frisch elasticity tends to increase with the intensive

margin elasticity and is indeed quite sizable, broadly in line with the

recent empirical evidence (Fiorito and Zanella, 2012; Peterman, 2016).

Therefore, the aggregate labor supply elasticity – the sum of the

intensive-margin elasticity and the extensive-margin elasticity by definition

2For example, this is in contrast to the model in Chang et al. (2019) where the two margins

are essentially substitutes. Specifically, Table 1reproduces their business-cycle results where a

higher individual Frisch elasticity raises the cyclical volatility of hours at the intensive margin

while reducing the counterpart at the extensive margin.

3Earlier real business-cycle models with both intensive and extensive margins include

Kydland and Prescott (1991), Bils and Cho (1994), Cho and Cooley (1994), and Osuna

and R´

ıos-Rull (2003), among others.

c

2023 The Authors. The Scandinavian Journal of Economics published by John Wiley & Sons Ltd on behalf of F¨

oreningen

f¨

or utgivande av the SJE.