The increase in the elasticity of substitution between capital and labour: a repeated cross-country investigation

ABSTRACT

The economics literature emphasizes the importance of the elasticity of substitution between capital and labour in several economic contexts. However, analyses of the effect of the elasticity of substitution on the direction of technological change are often overlooked. Most assessments of the direction of technological change rely on a Constant Elasticity of Substitution (CES) production framework. This strand of empirical work considers the elasticity of substitution between capital and labour as a deep and fixed parameter. In this article, we show that the change in the elasticity of substitution that has occurred in recent decades might be an alternative source of change of factor income shares in addition to changes in factor-augmenting technological change. We construct a theoretical environment in which the elasticity of substitution is determined endogenously by the capital share and capital intensity. Rolling window estimates and non-linear estimation methods show that the elasticity of substitution in nine OECD economies observed between 1950 and 2017 was not constant and that, in fact, in the latter half of the 1970s, the elasticity of substitution increased, in the presence of labour-augmenting technical change.

KEYWORDS:

• Elasticity of substitution
• factor income shares
• rolling window analysis
• biased technological change

JEL:

• O33
• E23
• C19

Acknowledgements

The authors gratefully acknowledge the comments of three anonymous referees and the journal Editor on a previous version of this paper. The authors are grateful to Pierre Mohnen and Thomas Ziesemer for their valuable comments to previous versions of this paper. We thank participants at the internal PhD seminars organized by the University of Turin.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The Online Appendix is available at https://sites.google.com/carloalberto.org/guido-pialli/.

Notes

1 In particular, the de La Grandville hypothesis states that economic growth rates and income per capita levels increase as the elasticity of substitution between capital and labour increases.

2 In the rest of the paper, we use capital intensity and capital abundance interchangeably to refer to the capital-labour ratio.

3 The capital-intensive direction of technological change in capital-abundant countries is consistent, also, with the finding that richer countries are characterized by greater output elasticity of capital (Durlauf and Johnson 1995).

4 In particular, how technological change is modelled has a substantial effect on the estimates of the elasticity of substitution. For example, Antras (2004) shows that omitting biased technological change drives the estimates towards unity. In general, assuming Hicks-neutral technological change, tends to overestimate the elasticity of substitution values.

5 The reader may refer to Zuleta (2016, cap. 3, pages 66–68), for an analytical explanation of how changes in the elasticity of substitution may be biased innovations.

6 Other contributions have developed other classes of production functions based on a changing elasticity of substitution (Antony 2010; Growiec and Mućk 2020).

7 Notice that the formulation in (1) is not Hicks’s original formulation; it is an adapted version of Robinson (1933), which assumes perfect competition in factor and product markets. Hicks later proved that definition in the second edition of Theory of Wages.

8 See Arrow et al. (1961, 229–230) for the mathematical steps to obtain the CES production function from Equation (2).

9 Similar proofs are provided in Sato and Hoffman (1968) and Burmeister and Dobell (1970).

10 Notice that $\theta \equiv \left(\mathrm{\partial F}\left(K,L\right)/\mathrm{\partial K}\right)K/F\left(K,L\right)=f{}^{\mathrm{\prime }}\left(k\right)k/f\left(k\right)$ given that $f{}^{\mathrm{\prime }}\left(k\right)=\mathrm{\partial F}\left(K,L\right)/\mathrm{\partial K}$.

11 The mathematical steps to obtain Equation (6) from Equation (5) are provided in the Online Appendix, section A.

12 The first two methods of adjustment rely on mixed-income, which is capital and labour income combined. The third assumes that self-employed individuals earn the same wage rates as employees. The fourth uses agricultural value added, since most self-employed work in the agricultural sector.

13 The countries included in the analysis are Australia, Canada, Finland, France, Italy, Japan, Netherlands, Sweden and the US.

14 We can use $\sigma$ in the estimated equation, obtaining the standard errors for its estimates directly. However, given that the relationship of interest is a non-linear one and the NLLS method is based on an optimisation algorithm, to simplify the numerical method, we prefer to retain $p$. Since studying $p$ means studying $\sigma$, this choice is not crucial for our interpretation.

15 Analogous to Federici and Saltari (2016), we implement a simple Chow breakpoint test to confirm the structural break for each country. We also experimented with alternative thresholds, such as years 1978, 1980 and 1981 and the results did not change.

16 To be in line with Arrow et al.’s (1961) original formulation, Table A2 in the Online Appendix also reports estimates of the CES production function in the absence of factor-augmenting technical change. It is interesting, that, when labour-augmenting technical change is not included, the elasticity of substitution passes from a value lower than 1 in the first period to a value higher than 1 in the second period, for all the countries analysed. Similar results were obtained by De La Grandville (2009), who analysed a sample of 16 OECD countries along the period 1966–1997 and found elasticities of substitution, estimated from the relationship $y=a{w}^{\sigma }$, above unity in the second subperiod, 1982–1997, for all the countries considered. On the contrary, the inclusion of labour-augmenting technical change lowers the value of the elasticity of substitution quite dramatically. Nonetheless, as suggested by an anonymous reviewer, the inclusion of labour-augmenting technological change is a fundamental assumption in our analysis and, hence, we present the results only with the inclusion of labour-augmenting technological change in the main text.

17 Normalization of the variables is fundamental when estimating a CES production function. De La Grandville (2009) makes it clear that normalization follows directly from the derivation of the CES production function; however, while from a pure theoretical view point, every year can be chosen as the normalization year, from an econometric perspective, it is useful to consider the mean values of GDP and capital per worker. Following Leon-Ledesma, McAdam, and Willman (2010), we use the geometric mean since there is a strong time-dependency path in both the considered variables. Thus, we impose $\tilde{y}\equiv y/\overline{y}$, $\tilde{k}\equiv k/\overline{k}$, ${\tilde{\pi }}_1\equiv {\pi }_1/{\overline{\pi }}_1$ and ${\tilde{\pi }}_2\equiv {\pi }_2/{\overline{\pi }}_2$.

18 The Kmenta approximation generates the linear function:

$\log {\tilde{y}}_t={\text{β}}_0+{\text{β}}_1\log {\tilde{k}}_t+{\text{β}}_2(\log {\tilde{k}}_t{)}^2$

where $\tilde{\pi }={\text{β}}_1$ and $p=2{\text{β}}_2/\left[{\text{β}}_1\left(1-{\text{β}}_1\right)\right]$.

Problems can arise in interpreting the estimated coefficient ${\text{β}}_0$ in the above equation; Leon-Ledesma, McAdam, and Willman (2010) suggest treating it as an integration constant, i.e. ${\text{β}}_0=\log \xi$, when it is true that $\overline{y}=\mathrm{\xi f}\left(\overline{k}\right)$, with $E\left[\xi \right]=1$. The estimates of this constant are always very close to 1. Online Appendix, Table C1 presents the results of the Kmenta method and explains, in depth, its limitations in the context of our analysis. Tables C2–C10 show the results of the Kmenta regressions for each country, respectively.

19 We use the programming language R and the procedure suggested by Henningsen and Henningsen (2012). The interested reader is kindly referred to Henningsen and Henningsen (2011) for more explanations on how one gets starting values derived applying a preceding grid search.

20 In the Online Appendix, Section B, we conduct an exercise in which we simulate the pattern of a variable elasticity of substitution in function of the capital intensity. We obtained values for the elasticity of substitution that are somewhat unrealistic, and are approaching infinity as capital intensity rises. Therefore, we leave this as an explorative exercise, but one that, nonetheless, suggests that the elasticity of substitution cannot be considered only as a function of technological market factors.

Additional information

Funding

The authors gratefully acknowledge funding from the Ministero dell'Università e della Ricerca, Italy within the context of the PRIN research project 20177J2LS9, and support from Università degli studi di Torino and Collegio Carlo Alberto local research funds.