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The share of total income obtained by workers rather than capital owners is for obvious reasons of interest. Assuming that the economy can be desribed by an aggregate production function

$$Y = F(K,L)$$

where $K$ is capital input and $L$ is labor input the total expenditure on labor and capital is given as

$$pY = rK + wL,$$

where $p$ is output price, $r$ is user cost of capital and $w$ is the wage (perfect competition is assumed on all markets). Labors share of total revenue or expenditure is therefore given as

$$s_L := \frac{wL}{pY} = \frac{wL}{rK + wL},$$

depending only on $w,r,L,K$.

The elasticity of substitution is defined as

$$\sigma := - \frac{d\log(K/L)}{d\log(r/w)},$$

describing how the capital to labor ratio changes given a change in the ratio of user cost of capital to wage.

Since both labor share of expenditure and elasticity of substitution depend on the variables $w,r,L,K$ this suggest the question:

How does changes in labors share of income $s_L$ as a result of changes in relative price $(r/w)$ relate to the elasticity of substitution $\sigma$?

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From the shares equation, we obtain: $$ 1 + \frac{rK}{wL} = \frac{1}{s_L} \to tk = \frac{1 - s_L}{s_L} $$ where I defined $t = \frac{r}{w}$ and $k = \frac{K}{L}$.

then taking logs gives: $$ \ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L) $$ Then take the derivative of this expression with respect to $\ln(t)$: $$ \begin{align*} -\sigma &= -1 - \dfrac{\dfrac{\partial s_L}{\partial \ln(t)}}{1 - s_L} - \gamma,\\ &=-1 - \gamma - \frac{s_L}{1 - s_L} \gamma = -\left(\frac{\gamma}{1 - s_L} +1\right) \end{align*} $$ Where we defined $\gamma$ to be the elasticity of the labour revenue share with respect to the $r/w$ ratio.

Solving for $\gamma$ gives: $$ \gamma = \left(\sigma-1\right)(1 - s_L) $$ which means that $\gamma > 0$ if $\sigma < 1$. In other words, if the elasticity of subsititution is smaller than one, then the labour share increases when $r/w$ increases.

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  • $\begingroup$ Thx, nice derivation. Had not seen it done like this before. $\endgroup$ May 29 at 21:10

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