<p>First of all, I think that 'linear and homogeneous' is a typo of 'linearly homogeneous.' Indeed, it can be shown that if the production function <span class="math-container">$V$</span> is linearly homogeneous, Allen Elasticity of Substitution between <span class="math-container">$K$</span> and <span class="math-container">$L$</span> can be expressed as
<span class="math-container">$$
\sigma = \frac{V_K V_L}{V \cdot V_{KL}}
$$</span>
by using the fact that <span class="math-container">$V_K$</span> and <span class="math-container">$V_L$</span> are homogeneous of degree 0, resulting in
<span class="math-container">$$
V_{LL}L + V_{LK}K=0
\\
V_{KL}L + V_{KK}K=0
$$</span>
.</p>

<p>Moreover, by the definitions in the paper (Lu, 1967), note that
<span class="math-container">$$
X \equiv \frac{K}{L}
\\
Y \equiv \frac{V}{L} = F(\frac{K}{L}, 1) \equiv f(X)
\\
$$</span></p>

<p>Now, by substituting the results given by the author, we have
<span class="math-container">$$
\sigma = \frac{\bigg[\frac{1}{K}(V-V_LL)\bigg]  \bigg[\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}\bigg]} {V \cdot \frac{\alpha}{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )}
$$</span>
, or
<span class="math-container">$$
\sigma = \frac{b}{V} \cdot \frac{L}{K} \cdot \frac{(V-V_LL)XY}{X\frac{dY}{dX}-cY}
\\
= \frac{b}{V} \cdot \frac{(V-V_LL)Y}{X\frac{dY}{dX}-cY}
\\
= \frac{b}{V} \cdot \frac{V-V_LL}{\frac{X}{Y} \frac{dY}{dX}-c}
$$</span></p>

<p>Note that, since <span class="math-container">$V$</span> is linearly homogeneous,
<span class="math-container">$$
V-V_LL = V_KK = \frac{dY}{dX}K
$$</span>
where the second equality holds for (3.18) in the paper.</p>

<p>Therefore,
<span class="math-container">$$
\sigma = \frac{b}{V} \cdot \frac{\frac{dY}{dX}K}{\frac{X}{Y} \frac{dY}{dX}-c}
\\
= b \cdot \frac{\frac{X}{Y} \frac{dY}{dX}}{\frac{X}{Y} \frac{dY}{dX}-c}
\\
= \frac{b}{1-c(\frac{X}{Y} \frac{dY}{dX})^{-1}}
$$</span></p>

<p>Finally, note that
<span class="math-container">$$
\frac{X}{Y} \frac{dY}{dX} = \frac{X}{f(X)} \frac{dY}{dX} = \frac{X}{f(X)}f'(X)
$$</span>
, and thus we have
<span class="math-container">$$
\sigma = \frac{b}{1-\frac{cf}{Xf'}}
$$</span></p>

<p>(You have a typo on this expression)</p>

CC BY-SA 4.0
2018-11-23T06:54:34.613Z
2018-11-23T06:54:34.613Z
K. Ban
<p>Following the work of <em>Lu (1967)</em> (Full text available <a href="https://lib.dr.iastate.edu/cgi/viewcontent.cgi?referer=https://www.google.es/&amp;httpsredir=1&amp;article=4405&amp;context=rtd" rel="nofollow noreferrer" title="link to the paper">here</a>!) I got stuck trying to derive the elasticity of substitution between factors. He use the formula developed by Allen, that when the production function is linear and homogeneous is the following:</p>

<p><span class="math-container">$$\sigma =\frac{\frac{\partial V}{\partial K}\frac{\partial V}{\partial L}}{V\frac{\partial^2 V}{\partial K\partial L}}$$</span></p>

<p>The partial derivatives of L and K and the cross second-order partial derivative are the following: </p>

<p><span class="math-container">$$(1)\frac{\partial V}{\partial K}=\frac{dY}{dX}=\frac{1}{X}\left ( Y-\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}} \right )=\frac{1}{K}\left ( V-\frac{\partial V}{\partial L}\cdot L \right )$$</span>
<span class="math-container">$$(2)\frac{\partial V}{\partial L}=Y-X\frac{dY}{dX}=\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}$$</span>
<span class="math-container">$$(3)\frac{\partial^2 V}{\partial K \partial L}=\frac{\alpha}{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$</span></p>

<p>And the expected result (the one the author gets) is:</p>

<p><span class="math-container">$$\sigma =\frac{b}{1-\frac{cf}{cf'}}$$</span></p>

<p>Is there anyone who can help me with this? Thanks.</p>

<microeconomics><mathematical-economics><linear-algebra>
Calculating the elasticity of substitution between factors of production
Alessandro
# Calculating the elasticity of substitution between factors of production

(Asked by: username_0 on 2018-11-21)

Following the work of *Lu (1967)* (Full text available [here](https://lib.dr.iastate.edu/cgi/viewcontent.cgi?referer=https://www.google.es/&httpsredir=1&article=4405&context=rtd "link to the paper")!) I got stuck trying to derive the elasticity of substitution between factors. He use the formula developed by Allen, that when the production function is linear and homogeneous is the following:

$$\sigma =\frac{\frac{\partial V}{\partial K}\frac{\partial V}{\partial L}}{V\frac{\partial^2 V}{\partial K\partial L}}$$

The partial derivatives of L and K and the cross second-order partial derivative are the following:

$$(1)\frac{\partial V}{\partial K}=\frac{dY}{dX}=\frac{1}{X}\left ( Y-\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}} \right )=\frac{1}{K}\left ( V-\frac{\partial V}{\partial L}\cdot L \right )$$
$$(2)\frac{\partial V}{\partial L}=Y-X\frac{dY}{dX}=\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}$$
$$(3)\frac{\partial^2 V}{\partial K \partial L}=\frac{\alpha}{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )$$

And the expected result (the one the author gets) is:

$$\sigma =\frac{b}{1-\frac{cf}{cf'}}$$

Is there anyone who can help me with this? Thanks.

**Tags:** microeconomics, mathematical-economics, linear-algebra


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## 1 answers

### Answer by username_1 on 2018-11-23. Score: 1

First of all, I think that 'linear and homogeneous' is a typo of 'linearly homogeneous.' Indeed, it can be shown that if the production function $V$ is linearly homogeneous, Allen Elasticity of Substitution between $K$ and $L$ can be expressed as
$$
\sigma = \frac{V\_K V\_L}{V \cdot V\_{KL}}
$$
by using the fact that $V\_K$ and $V\_L$ are homogeneous of degree 0, resulting in
$$
V\_{LL}L + V\_{LK}K=0
\\
V\_{KL}L + V\_{KK}K=0
$$
.

Moreover, by the definitions in the paper (Lu, 1967), note that
$$
X \equiv \frac{K}{L}
\\
Y \equiv \frac{V}{L} = F(\frac{K}{L}, 1) \equiv f(X)
\\
$$

Now, by substituting the results given by the author, we have
$$
\sigma = \frac{\bigg[\frac{1}{K}(V-V\_LL)\bigg] \bigg[\alpha X^{-\frac{c}{b}}Y^{\frac{1}{b}}\bigg]} {V \cdot \frac{\alpha}{bL} X^{-\frac{c}{b}-1}Y^{\frac{1}{b}-1}\left ( X\frac{dY}{dX}-cY \right )}
$$
, or
$$
\sigma = \frac{b}{V} \cdot \frac{L}{K} \cdot \frac{(V-V\_LL)XY}{X\frac{dY}{dX}-cY}
\\
= \frac{b}{V} \cdot \frac{(V-V\_LL)Y}{X\frac{dY}{dX}-cY}
\\
= \frac{b}{V} \cdot \frac{V-V\_LL}{\frac{X}{Y} \frac{dY}{dX}-c}
$$

Note that, since $V$ is linearly homogeneous,
$$
V-V\_LL = V\_KK = \frac{dY}{dX}K
$$
where the second equality holds for (3.18) in the paper.

Therefore,
$$
\sigma = \frac{b}{V} \cdot \frac{\frac{dY}{dX}K}{\frac{X}{Y} \frac{dY}{dX}-c}
\\
= b \cdot \frac{\frac{X}{Y} \frac{dY}{dX}}{\frac{X}{Y} \frac{dY}{dX}-c}
\\
= \frac{b}{1-c(\frac{X}{Y} \frac{dY}{dX})^{-1}}
$$

Finally, note that
$$
\frac{X}{Y} \frac{dY}{dX} = \frac{X}{f(X)} \frac{dY}{dX} = \frac{X}{f(X)}f'(X)
$$
, and thus we have
$$
\sigma = \frac{b}{1-\frac{cf}{Xf'}}
$$

(You have a typo on this expression)


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