Unemployment and income‐distribution effects of economic growth: A minimum‐wage analysis with optimal saving

• DOI: http://doi.org/10.1111/ijet.12197
• Author: Richard A. Brecher,Till Gross
• Date: 01 September 2020
• Published date: 01 September 2020

Unemployment and income-distribution effects of economic

growth: A minimum-wage analysis with optimal saving

Richard A. Brecher



and Till Gross



Theoretically and numerically, we analyze the unemployment and income-distribution effects of

economic growth, in a model with optimal saving (investment) and a minimum wage for

unskilled labor. Within this three-factor model (including skilled labor), an exogenous rise in the

growth rate increases unemployment if capital and unskilled labor are complements (versus

substitutes), implying a tradeoff between (faster) growth and (lower) unemployment. We also

show how the growth rate affects the skill premium and factor shares of national income,

providing little support for Piketty’s controversial thesis that capital’s share is higher when

growth is slower.

Key words optimal growth, minimum wage, unskilled unemployment, income distribution

JEL classification E24, O41

Accepted 14 April 2018

1 Introduction

The specter of a slowdown in economic growth has recently caught the attention of economists and

policy-makers alike. Possible reasons for such a slowdown are plentiful. For example, Summers

(2015) revives the idea of secular stagnation, whereby growth slows because of insufficient demand.

On the other hand, Gordon (2012) questions whether productivity-enhancing innovations can

continue on a scale observed in the past, and he identifies a host of other issues (such as

demographics) that may further decrease the growth rate in the United States (and elsewhere). In any

case, the potential ramifications of slower growth are wide-ranging and important. For instance,

unemployment may rise, as predicted by Okun’s law, and the distribution of income may become

severely skewed, as argued by Piketty (2014).

In light of these concerns, the present paper analyzes how the level of unemployment and the

distribution of income respond to changes in the rate of economic growth. To abstract from the

underlying determinants of this rate, we specify (and change) it exogenously. Our analysis occurs

within a one-good model, under perfect competition and constant returns to scale. This model also

includes physical capital arising from optimal savings, as well as fixed endowments of skilled and

unskilled labor.

Within our analytic framework, some unskilled labor is unemployed because of a minimum real

wage for this particular factor of production. Although minimum wages hold a long-standing place

of prominence in the history of economic thought dating back at least as far as Mill (1848)

1

they



Department of Economics, Carleton University, Ottawa, Ontario, Canada. Email: richard.brecher@carleton.ca

We gratefully acknowledge helpful comments and suggestions from Thomas Fischer, workshop participants at Carleton

University, and an anonymous referee.

doi: 10.1111/ijet.12197

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appear to be scarce in the theoretical literature on optimal growth. This apparent scarcity might well

result from an inherent problem of overdetermination, which is explained (and solved) below.

Although we assume that the minimum wage arises simply from government legislation, one

could also interpret it as the result of some other institutional arrangement, such as social custom or

labor unions. Alternatively, workers may refuse to accept any job that pays less than some type of

unemployment benefit (financed by lump-sum taxes), which could thus be viewed as analytically

equivalent to a minimum wage. In any case, within our model, the wage for unskilled labor is

constrained to exceed the level required for full employment.

There are two main reasons for assuming a thirdfactor in the form of skilled labor, which remains

fullyemployed becauseits wage is perfectly flexible.First, it is realisticto recognize that a minimumwage

usually applies to only part ofthe labor force. Second, in an optimally savingeconomy with exogenous

growth and constant returns to scale,a binding minimum wage wouldoverdetermine the steady-state

equilibriumif there were only two factors(capital and labor).

2

Our three-inputspecificationalso allows

us to consider the implications of factor substitutes versus complements (as defined below), and

discover an additional determinant of the wage differential between the two types of labor.

Within our model, an exogenous rise in the rate of growth increases (decreases) the

unemployment rate when capital and unskilled labor are complements (substitutes), in the sense that

the marginal product of each of these two inputs depends positively (negatively) on the quantity of

the other input.

3

In other words, if and only if capital and unskilled labor are complements, there is a

tradeoff between (faster) growth and (lower) unemployment.

Such a growth–unemployment tradeoff has been studied previously under alternative

assumptions about the labor market. For example, Aghion and Howitt (1994) and Pissarides

(2000, chap. 3) use search-and-matching models of frictional unemployment, whereas Brecher et al.

(2002) assume that unemployment arises for efficiency-wage reasons. The present paper contributes

to this literature by analyzing the simpler but classic case of unemployment due to a minimum-wage

constraint. This case sheds new light on the relationship between growth and unemployment, by

featuring the important role of factor substitutes/complements.

We also address the recent controversy over Piketty’s (2014, especially p. 233) thesis that a fall in

the rate of growth implies a rise in capital’s share of national income.

4

Although our minimum-wage

analysis does not generally support his thesis, some support is provided under certain assumptions

about depreciation of capital and elasticities of factor substitution (between capital and both types of

labor).

Section 2 sets up our basic model of optimal growth with a minimum wage. Using this model,

Section 3 explores the relationship between the rates of growth and unemployment. Section 4

analyzes how a change in the growth rate affects the distribution of income among the three factors of

production. To estimate the magnitude of our theoretical results, Section 5 numerically simulates the

1

See his critique ‘‘Of PopularRemedies for Low Wages’’ (the title of his chap. XII on pp. 424–438 in bk. II of vol. I), as well as

Leonard’s (2000) section on the ‘‘History of Minimum-Wage Legislation and Its Economics.’’

2

The growth rate determines the rate of return on capital (via the household’s Euler equation), thereby pinning down the

wage rate (in the two-factor case), which thus cannot be fixed also by the minimum wage. For alternative (two-factor)

solutions to this overdetermination problem, see Brecher et al. (2013) and Brecher and Gross (in press).

3

Although various empirical studies suggest that capital is more complementary with skilled than with unskilled labor (as

discussed by Violante, 2008), there appears to be no consensus on whether capital and unskilled labor are in fact

complements rather than substitutes (as defined here). We thus consider both of these alternative possibilities.

4

For a detailed critique of this book and of some related work, see Rognlie (2015). See also Fischer’s (2017) critique within

an optimal-growth model without a minimum wage.

Unemployment and economic growth Richard A. Brecher and Till Gross

2International Journal of Economic Theory xxx (2018) 1–17 ©IAET

International Journal of Economic Theory 16 (2020) 243–259 © IAET

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