Calculating the elasticity of substitution between factors of production

Question

Following the work of Lu (1967) (Full text available here!) 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.

Answer 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)