<p>From the shares equation, we obtain:
<span class="math-container">$$
1 + \frac{rK}{wL} = \frac{1}{s_L}
\to tk = \frac{1 - s_L}{s_L}
$$</span>
where I defined <span class="math-container">$t = \frac{r}{w}$</span> and <span class="math-container">$k = \frac{K}{L}$</span>.</p>
<p>then taking logs gives:
<span class="math-container">$$
\ln(k) = -\ln(t) + \ln(1-s_L) - \ln(s_L)
$$</span>
Then take the derivative of this expression with respect to <span class="math-container">$\ln(t)$</span>:
<span class="math-container">$$
\begin{align*}
-\sigma &amp;= -1 - \dfrac{\dfrac{\partial s_L}{\partial \ln(t)}}{1 - s_L} - \gamma,\\
&amp;=-1 - \gamma - \frac{s_L}{1 - s_L} \gamma = -\left(\frac{\gamma}{1 - s_L} +1\right)
\end{align*}
$$</span>
Where we defined <span class="math-container">$\gamma$</span> to be the elasticity of the labour revenue share with respect to the <span class="math-container">$r/w$</span> ratio.</p>
<p>Solving for <span class="math-container">$\gamma$</span> gives:
<span class="math-container">$$
\gamma = \left(\sigma-1\right)(1 - s_L)
$$</span>
which means that <span class="math-container">$\gamma &gt; 0$</span> if <span class="math-container">$\sigma &lt; 1$</span>. In other words, if the elasticity of subsititution is smaller than one, then the labour share increases when <span class="math-container">$r/w$</span> increases.</p>

CC BY-SA 4.0
2021-05-29T21:10:33.870Z
Thx, nice derivation. Had not seen it done like this before.
Jesper Hybel
CC BY-SA 4.0
2021-05-29T17:29:52.700Z
2021-05-29T17:29:52.700Z
tdm
<p>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</p>
<p><span class="math-container">$$Y = F(K,L)$$</span></p>
<p>where <span class="math-container">$K$</span> is capital input and <span class="math-container">$L$</span> is labor input the total expenditure on labor and capital is given as</p>
<p><span class="math-container">$$pY = rK + wL,$$</span></p>
<p>where <span class="math-container">$p$</span> is output price, <span class="math-container">$r$</span> is user cost of capital and <span class="math-container">$w$</span> is the wage (perfect competition is assumed on all markets). Labors share of total revenue or expenditure is therefore given as</p>
<p><span class="math-container">$$s_L := \frac{wL}{pY} = \frac{wL}{rK + wL},$$</span></p>
<p>depending only on <span class="math-container">$w,r,L,K$</span>.</p>
<p>The elasticity of substitution is defined as</p>
<p><span class="math-container">$$\sigma := - \frac{d\log(K/L)}{d\log(r/w)},$$</span></p>
<p>describing how the capital to labor ratio changes given a change in the ratio of user cost of capital to wage.</p>
<p>Since both labor share of expenditure and elasticity of substitution depend on the variables <span class="math-container">$w,r,L,K$</span> this suggest the question:</p>
<p><strong>How does changes in labors share of income <span class="math-container">$s_L$</span> as a result of changes in relative price <span class="math-container">$(r/w)$</span> relate to the elasticity of substitution <span class="math-container">$\sigma$</span></strong>?</p>

<macroeconomics><elasticity><elasticity-of-substitution>
When do workers versus capital owners share of income increase?
Jesper Hybel
# When do workers versus capital owners share of income increase?

(Asked by: username_0 on 2021-03-16)

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 $\sigma$**?

**Tags:** macroeconomics, elasticity, elasticity-of-substitution

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## 1 answers

### Answer by username_1 on 2021-05-29. Score: 4 | Accepted answer

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.

> **Comments:**
> **username_0**: Thx, nice derivation. Had not seen it done like this before.  

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