Labor Share, Capital Share, and Rents: A Macrohistorical Perspective

Abstract

This article reviews the literature on value added distribution in a long-run perspective. In so doing, it discusses: (i) the measure of the labor and capital shares; (ii) the set of explanation behind the evolution of those ratios, whether this is due to labor and capital substitution effect, the automation and creation of new tasks, and the evolution of firms market power and markup; (iii) the consequences on inequalities, growth, and welfare. In all cases, historical series are used to enlighten the issue while historical events are proposed to document-specific mechanisms at work.

Appendix

Production Function and Elasticity of Substitution Value

Recall (4) and set $\rho =\frac{\sigma -1}{\sigma }$. This gives:

$$ Y={\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho}\right]}^{\frac{1}{\rho }} $$

(21)

The Cobb-Douglas case: ρ → 0

Applying a monotonic (log) transformation to our production function gives:

$$ \ln (Y)=\frac{\ln \left[\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho}\right]}{\rho } $$

(22)

Remembering that, in a Cobb-Douglas function, the elasticity of substitution is 1, it can be approached by the limit of our CES when ρ tends toward 0:

$$ \underset{\rho \to 0}{\lim}\ln (Y)=\underset{\rho \to 0}{\lim}\frac{\ln \left[\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho}\right]}{\rho } $$

Which leads us to the undetermined form $\frac{0}{0}$. By L’Hôpital’s rule:

$$ \underset{\rho \to 0}{\lim}\ln (Y)=\underset{\rho \to 0}{\lim}\frac{\left(1-\alpha \right){\left({A}_LL\right)}^{\rho}\ln {\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho}\ln \left({A}_KK\right)}{\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho }} $$

$$ \underset{\rho \to 0}{\lim}\ln (Y)=\frac{\left(1-\alpha \right)\ln \left({A}_LL\right)+\alpha \ln \left({A}_KK\right)}{1-\alpha +\alpha } $$

Applying the inverse transformation operated in (21), we get the standard Cobb-Douglas production function:

$$ \underset{\rho \to 0}{\lim }Y={\left({A}_LL\right)}^{1-\alpha }{\left({A}_KK\right)}^{\alpha } $$

(23)

The Leontieff case: ρ → −∞

Starting from (21), and remembering that the elasticity of substitution in a Leontieff function is 0, the limit of our CES, when ρ tends to −∞ gives:

$$ \underset{\rho \to -\infty }{\lim}\ln (Y)=\underset{\rho \to -\infty }{\lim}\frac{\ln \left[\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho}\right]}{\rho } $$

Which is again an undetermined form $\frac{\infty }{\infty }$. Then by L’hôpital’s rule:

$$ {\displaystyle \begin{array}{c}\underset{\rho \to -\infty }{\lim}\ln (Y)=\underset{\rho \to -\infty }{\lim}\frac{\left(1-\alpha \right){\left({A}_LL\right)}^{\rho}\ln \left({A}_LL\right)+\alpha {\left({A}_KK\right)}^{\rho}\ln \left({A}_KK\right)}{\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho }}\\ {}\underset{\rho \to -\infty }{\lim}\ln (Y)=\underset{\rho \to -\infty }{\lim}\left\{\frac{\ln \left({A}_KK\right)\left[\alpha {\left({A}_KK\right)}^{\rho }+\left(1-\alpha \right){\left({A}_LL\right)}^{\rho}\right]}{\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho }}\right.\kern1.2em \\ {}\kern6em \left.+\frac{\left(1-\alpha \right)\ln \left({A}_LL\right){\left({A}_LL\right)}^{\rho }-\left(1-\alpha \right)\ln \left({A}_KK\right){\left({A}_LL\right)}^{\rho }}{\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho }}\right\}\\ {}\underset{\rho \to -\infty }{\lim}\ln (Y)=\underset{\rho \to \infty }{\lim}\left\{\ln \left({A}_KK\right)+\left(1-\alpha \right)\frac{\ln \left({A}_LL\right){\left({A}_LL\right)}^{\rho }-\ln \left({A}_KK\right){\left({A}_LL\right)}^{\rho }}{\left(1-\alpha \right){\left({A}_LL\right)}^{\rho }+\alpha {\left({A}_KK\right)}^{\rho }}\right\}\\ {}\underset{\rho \to -\infty }{\lim}\ln (Y)=\underset{\rho \to -\infty }{\lim}\left\{\ln \left({A}_KK\right)+\left(1-\alpha \right)\frac{\ln \left({A}_LL\right)-\ln \left({A}_KK\right)}{\left(1-\alpha \right)+\alpha {\left(\frac{A_KK}{A_LL}\right)}^{\rho }}\right\}\end{array}} $$

When A_KK > A_LL, $\underset{\rho \to -\infty }{\lim }{\left(\frac{A_KK}{A_LL}\right)}^{\rho }=0$, and $\underset{\rho \to -\infty }{\lim}\ln (Y)=\ln \left({A}_LL\right)$. On the contrary, if A_KK < A_LL, $\underset{\rho \to -\infty }{\lim }{\left(\frac{A_KK}{A_LL}\right)}^{\rho }=+\infty$, and $\underset{\rho \to -\infty }{\lim}\;\ln (Y)=\ln \left({A}_KK\right)$. In a nutshell:

$$ \underset{\rho \to -\infty }{\lim }Y=\min \left\{{A}_LL,{A}_KK\right\} $$

(24)

Cost-minimization Problem

Assuming the aggregate production function is (4), and taking a potential markup in the economy, the cost-minimization problem is:

$$ {\displaystyle \begin{array}{l}\underset{L,K}{\mathbf{\min}}\kern1em \left\{C(Y)= wL+ rK\right\}\\ {}\ \mathrm{s}.\mathrm{t}.\kern1em Y={\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{\frac{\sigma }{\sigma -1}}\end{array}} $$

Where C(Y) is the aggregate total cost in the economy, and the price P is the numeraire, such that PY = Y. By defining the following Lagrangian function:

$$ \mathrm{\mathcal{L}}\left(L,K,\uplambda \right)=\left( wN+ rK\right)-\uplambda \left[{\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{\frac{\sigma }{\sigma -1}}-Y\right] $$

where λ is the Lagrange multiplier, the first-order conditions lead to:

$$ \frac{\mathrm{\partial \mathcal{L}}\left(L,K,\uplambda \right)}{\partial N}=0\iff \kern1em w=\uplambda \left[\left(1-\alpha \right){A}_L^{\frac{\sigma -1}{\sigma }}{L}^{\frac{-1}{\sigma }}{\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{\frac{1}{\sigma -1}}\right] $$

(25)

$$ \frac{\mathrm{\partial \mathcal{L}}\left(L,K,\uplambda \right)}{\partial K}=0\iff \kern1em r=\uplambda \left[\alpha {A}_K^{\frac{\sigma -1}{\sigma }}{K}^{\frac{-1}{\sigma }}{\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{\frac{1}{\sigma -1}}\right] $$

(26)

$$ \frac{\mathrm{\partial \mathcal{L}}\left(L,K,\uplambda \right)}{\mathrm{\partial \uplambda }}=0\iff \kern1em Y={\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{\frac{\sigma }{\sigma -1}} $$

(27)

By the Envelope theorem, we have:

$$ {\left.\frac{\mathrm{\partial \mathcal{L}}}{\partial Y}\right|}_{\begin{array}{c}L={L}^{\ast }(Y)\\ {}\phantom{\rule{0ex}{0.3em}}K={K}^{\ast }(Y)\\ {}\phantom{\rule{0ex}{0.3em}}\uplambda ={\uplambda}^{\ast }(Y)\end{array}}={\uplambda}^{\ast }(Y) $$

(28)

Equation (28) shows that the Lagrange multiplier can be interpreted as the rate at which the optimal value of total costs (C(Y)) changes with respect to changes in the quantities produced (Y). In other words, λ are the marginal costs. Recording that the markup of price over marginal costs is the price divided by the marginal cost:

$$ \mu =\frac{P}{C^{\prime }(Y)} $$

(29)

and that P = 1, there is a direct link between the Lagrange multiplier (λ) and the markup (μ):

$$ \uplambda =\frac{1}{\mu } $$

(30)

Accordingly, we can rewrite Eqs. (25) and (26) as (7) and (8).

Factor Shares Derivation

For the labor share:

$$ {\displaystyle \begin{array}{c}{S}_L=\frac{1}{\mu}\left(1-\alpha \right){A}_L^{\frac{\sigma -1}{\sigma }}{\left(\frac{L}{Y}\right)}^{\frac{\sigma -1}{\sigma }}\\ {}=\frac{1}{\mu}\left(1-\alpha \right){A}_L^{\frac{\sigma -1}{\sigma }}{L}^{\frac{\sigma -1}{\sigma }}{\left[{\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+{\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{\frac{\sigma }{\sigma -1}}\right]}^{\frac{-\sigma -1}{\sigma }}\\ {}=\frac{1}{\mu}\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}{\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+{\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{-1}\\ {}=\frac{1}{\mu}\frac{\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}}{\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}}\\ {}=\frac{1}{\mu}\frac{1}{\frac{\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}}{\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}}}\\ {}=\frac{1}{\mu}\frac{1}{1+\left(\frac{\alpha }{1-\alpha}\right){\left(\frac{A_KK}{A_LL}\right)}^{\frac{\sigma -1}{\sigma }}}\end{array}} $$

For the capital share:

$$ {\displaystyle \begin{array}{c}{S}_K=\frac{1}{\mu}\left(1-\alpha \right){A}_K^{\frac{\sigma -1}{\sigma }}{\left(\frac{K}{Y}\right)}^{\frac{\sigma -1}{\sigma }}\\ {}=\frac{1}{\mu}\left(1-\alpha \right){A}_K^{\frac{\sigma -1}{\sigma }}{K}^{\frac{\sigma -1}{\sigma }}{\left[{\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+{\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{\frac{\sigma }{\sigma -1}}\right]}^{\frac{-\sigma -1}{\sigma }}\\ {}=\frac{1}{\mu}\left(1-\alpha \right){\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}{\left[\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+{\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}\right]}^{-1}\\ {}=\frac{1}{\mu}\frac{\left(1-\alpha \right){\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}}{\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}}\\ {}=\frac{1}{\mu}\frac{1}{\frac{\left(1-\alpha \right){\left({A}_LL\right)}^{\frac{\sigma -1}{\sigma }}+\alpha {\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}}{\left(1-\alpha \right){\left({A}_KK\right)}^{\frac{\sigma -1}{\sigma }}}}\\ {}=\frac{1}{\mu}\frac{1}{1+\left(\frac{\alpha }{1-\alpha}\right){\left(\frac{A_LL}{A_KK}\right)}^{\frac{\sigma -1}{\sigma }}}\end{array}} $$