<p>Your math is not wrong.</p>
<p>To get to the correct solution, you should not treat <span class="math-container">$(w_2/w_1)$</span> as a constant. If <span class="math-container">$x_1/x_2$</span> changes then the prices will have to change to keep the factor shares optimal for a given level of output (I know that the question is not clear in this respect).</p>
<p>Use the first order conditions <span class="math-container">$\dfrac{w_2}{w_1} = \dfrac{\partial f(x_1,x_2)/\partial x_2}{\partial f(x_1, x_2)/\partial x_1}= -\dfrac{1}{TRS}$</span> to subsitute out.</p>

CC BY-SA 4.0
2021-11-08T04:57:57.333Z
Thank you. Is there any justification for treating $w_2/ w_1$ as a function of $x$? The question comes from a chapter on profit maximization, which assumes a competitive market. If the firm is a price taker, it seems strange to talk about moving (market) prices around in order to accommodate firm-based optimality.
Max
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2021-11-08T16:20:58.523Z
Maybe it is better to see it from a macro perspective (in this case, $w_1$ and $w_2$ do change). If $x_1$ is labour and $x_2$ is capital then $w_2 x_2/w_1 x_1$ gives the renumeration of capital versus labour. $x_1/x_2$ then gives the 'supply or usage' of labour versus capital. So a positive elasticity means that if the share of labour increases, the share of total renumeration that goes towards labor (vis a vis) capital in fact decreases.
tdm
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2021-11-05T12:55:02.310Z
2021-11-05T12:55:02.310Z
tdm
<p>My question concerns exercise 2.4 from Hal Varian's <em>Microeconomic Analysis</em>.</p>
<p>Please note that I am <em>not</em> requesting a solution to the exercise, and only asking how to understand what I am being asked to do here.</p>
<blockquote>
<p><strong>2.4.</strong> Let <span class="math-container">$f(x_1, x_2)$</span> be a production function with two factors and let <span class="math-container">$w_1$</span> and <span class="math-container">$w_2$</span> be their respective prices. Show that the elasticity of the factor share <span class="math-container">$(w_2 x_2 / w_1 x_1)$</span> with respect to <span class="math-container">$(x_1 / x_2)$</span> is given by <span class="math-container">$1/\sigma - 1$</span>.</p>
</blockquote>
<p>Here <span class="math-container">$\sigma$</span> is defined earlier as the elasticity of substitution <span class="math-container">$$\frac{\text{TRS}}{x_2 / x_1}\frac{d(x_2 / x_1)}{d \text{TRS}}$$</span></p>
<p>where TRS is the technical rate of substitution <span class="math-container">$\frac{-\partial f / \partial x_1}{\partial f / \partial x_2}$</span>.</p>
<p>The text has not yet defined the factor share or its elasticity yet, but in general, I interpret the phrase &quot;the elasticity of <span class="math-container">$A$</span> with respect to <span class="math-container">$B$</span>&quot; as referring to the quantity <span class="math-container">$\frac{B}{A}\frac{dA}{dB}$</span>. It is not hard to show using the chain rule that this quantity is equal to <span class="math-container">$d (\ln A) / d (\ln B)$</span>.</p>
<p>Using the first definition, I take <span class="math-container">$A = w_2 x_2 / w_1 x_1$</span> and <span class="math-container">$B = x_1 / x_2$</span>. Then <span class="math-container">$A = \frac{w_2}{w_1}\frac{1}{B}$</span>, and
<span class="math-container">$$\frac{B}{A}\frac{dA}{dB} = \frac{B}{A}\frac{w_2}{w_1} \frac{-1}{B^2}= -1$$</span></p>
<p>Using the second definition, I get the same result. We know
<span class="math-container">$$\ln A = \ln (w_2 x_2 / w_1 x_1 ) = \ln (w_2 / w_1) - \ln (x_1 / x_2) = \text{const.} - \ln B
$$</span>
Differentiating both sides with respect to the symbol <span class="math-container">$\ln B$</span> yields  <span class="math-container">$d (\ln A) / d (\ln B) = -1$</span>.</p>
<p>Is my math wrong, or have I misunderstood the question?</p>

<elasticity>
Elasticity of the factor share
Max
# Elasticity of the factor share

(Asked by: username_0 on 2021-11-05)

My question concerns exercise 2.4 from Hal Varian's *Microeconomic Analysis*.

Please note that I am *not* requesting a solution to the exercise, and only asking how to understand what I am being asked to do here.

> **2.4.** Let $f(x\_1, x\_2)$ be a production function with two factors and let $w\_1$ and $w\_2$ be their respective prices. Show that the elasticity of the factor share $(w\_2 x\_2 / w\_1 x\_1)$ with respect to $(x\_1 / x\_2)$ is given by $1/\sigma - 1$.

Here $\sigma$ is defined earlier as the elasticity of substitution $$\frac{\text{TRS}}{x\_2 / x\_1}\frac{d(x\_2 / x\_1)}{d \text{TRS}}$$

where TRS is the technical rate of substitution $\frac{-\partial f / \partial x\_1}{\partial f / \partial x\_2}$.

The text has not yet defined the factor share or its elasticity yet, but in general, I interpret the phrase "the elasticity of $A$ with respect to $B$" as referring to the quantity $\frac{B}{A}\frac{dA}{dB}$. It is not hard to show using the chain rule that this quantity is equal to $d (\ln A) / d (\ln B)$.

Using the first definition, I take $A = w\_2 x\_2 / w\_1 x\_1$ and $B = x\_1 / x\_2$. Then $A = \frac{w\_2}{w\_1}\frac{1}{B}$, and
$$\frac{B}{A}\frac{dA}{dB} = \frac{B}{A}\frac{w\_2}{w\_1} \frac{-1}{B^2}= -1$$

Using the second definition, I get the same result. We know
$$\ln A = \ln (w\_2 x\_2 / w\_1 x\_1 ) = \ln (w\_2 / w\_1) - \ln (x\_1 / x\_2) = \text{const.} - \ln B
$$
Differentiating both sides with respect to the symbol $\ln B$ yields $d (\ln A) / d (\ln B) = -1$.

Is my math wrong, or have I misunderstood the question?

**Tags:** elasticity


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## 1 answers

### Answer by username_1 on 2021-11-05. Score: 3 | Accepted answer

Your math is not wrong.

To get to the correct solution, you should not treat $(w\_2/w\_1)$ as a constant. If $x\_1/x\_2$ changes then the prices will have to change to keep the factor shares optimal for a given level of output (I know that the question is not clear in this respect).

Use the first order conditions $\dfrac{w\_2}{w\_1} = \dfrac{\partial f(x\_1,x\_2)/\partial x\_2}{\partial f(x\_1, x\_2)/\partial x\_1}= -\dfrac{1}{TRS}$ to subsitute out.

> **Comments:**
> **username_0**: Thank you. Is there any justification for treating $w_2/ w_1$ as a function of $x$? The question comes from a chapter on profit maximization, which assumes a competitive market. If the firm is a price taker, it seems strange to talk about moving (market) prices around in order to accommodate firm-based optimality.  
> **username_1**: Maybe it is better to see it from a macro perspective (in this case, $w_1$ and $w_2$ do change). If $x_1$ is labour and $x_2$ is capital then $w_2 x_2/w_1 x_1$ gives the renumeration of capital versus labour. $x_1/x_2$ then gives the 'supply or usage' of labour versus capital. So a positive elasticity means that if the share of labour increases, the share of total renumeration that goes towards labor (vis a vis) capital in fact decreases.  

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