Hardcore elasticity of substitution (bad results)

Question

I have a following function and would like to find the elasticity of substitution between pairs:

$$U = \left( x_1^\delta + x_2^\delta + x_3^\gamma + x_4^\gamma \right)^{\frac{1}{\delta + \gamma}}$$

With elasticity of substitution formula being:

$$ \sigma_{ij} = \frac{\frac{\partial (x_j/x_i)}{x_j/x_i}}{\frac{\partial MRS_{ij}}{MRS_{ij}}} = \frac{1}{\frac{\partial MRS_{ij}}{\partial (x_j/x_i)}} \frac{MRS_{ij}}{x_j/x_i}$$

Where $MRS_{ij} = MU_i/MU_j$.

I can divide the elasticity of substitution into two groups based on sameness of parameters. What concerns the results, $\sigma_{1,2}$ and $\sigma_{3,4}$ is easy to compute... It gives the same result as CES function.

HOWEVER, the case of $\sigma_{1,3}$ and similar seems rather hard to find analythically. See what the $MRS_{13}$ is equal to:

$$MRS_{13} = \frac{\frac{\delta}{\delta + \gamma}}{\frac{\gamma}{\delta + \gamma}} \frac{x_1^{\delta-1}}{x_3^{\gamma-1}} = \frac{\delta}{\gamma} \frac{x_1^{\delta-1}}{x_3^{\gamma - 1}} $$

The difficulty comes because of the next step when we should differentiate it by fraction of $x_3/x_1$ and math SE told me this is usually solved through substitution $z= x_3/x_1$, which means I should rewrite the variables as: $x_1 (z) = x_3/z$ and $x_3 (z) = z x_1$.

So why is this approach not correct:

$$\frac{\partial MRS_{13}}{\partial \frac{x_3}{x_1}} = \frac{\partial \left[\frac{\delta}{\gamma} \frac{x_1^{\delta-1}}{x_3^{\gamma - 1}} \right]}{\partial \frac{x_3}{x_1}} = \frac{\delta}{\gamma} \frac{\partial \left[ (x_1 (z) )^{\delta-1} \cdot (x_3 (z))^{-(\gamma-1)} \right]}{\partial z} = \frac{\delta}{\gamma} \left[ \frac{\partial (x_1 (z) )^{\delta-1}}{\partial z} (x_3 (z))^{-(\gamma-1)} + (x_1 (z) )^{\delta-1} \frac{\partial (x_3 (z))^{-(\gamma-1)}}{\partial z} \right] = \frac{\delta}{\gamma} \left[ (\delta-1) (x_1 (z) )^{\delta-2} \frac{\partial x_1 (z)}{ \partial z} (x_3 (z))^{-\gamma+1} + (x_1 (z) )^{\delta-1} (-\gamma +1) (x_3 (z))^{-\gamma} \frac{\partial x_3 (z)}{\partial z} \right]$$

Now substituing back for $x_1(z)$, $x_3(z)$ and $z$, we get:

$$\frac{\partial MRS_{13}}{\partial \frac{x_3}{x_1}} = \frac{\delta}{\gamma} \left[ (\delta-1) \left( \frac{x_3}{\frac{x_3}{x_1}} \right)^{\delta-2} \left( x_1 \frac{x_3}{x_1}\right)^{-\gamma +1} \frac{\partial x_1 (z)}{ \partial z} + \left( \frac{x_3}{\frac{x_3}{x_1}} \right)^{\delta-1} (-\gamma + 1) \left( x_1 \frac{x_3}{x_1} \right)^{-\gamma} \frac{\partial x_3 (z)}{\partial z} \right]= \frac{\delta}{\gamma} \left[ (\delta-1) ( x_1)^{\delta-2} ( x_3)^{-\gamma +1} \frac{\partial x_1 (z)}{ \partial z} + (x_1)^{\delta-1} (-\gamma + 1) (x_3)^{-\gamma} \frac{\partial x_3 (z)}{\partial z} \right]$$

Now solving the interior derivatives since I know that $\partial x_3(z) / \partial z = \partial (x_1 \cdot z )/ \partial z = x_1$ etc... I should get:

$$\frac{\partial MRS_{13}}{\partial \frac{x_3}{x_1}} = \frac{\delta}{\gamma} \left[(\delta-1) ( x_1)^{\delta-2} ( x_3)^{-\gamma +1} (-1) \frac{x_3}{\left( \frac{x_3}{x_1} \right)^2} + (x_1)^{\delta-1} (-\gamma + 1) (x_3)^{-\gamma} x_1 \right] = \frac{\delta}{\gamma} \left[ (-\delta +1) x_1^\delta x_3^{-\gamma} + x_1^\delta x_3^{-\gamma} (-\gamma + 1) \right] = \frac{\delta}{\gamma} \frac{x_1^\delta}{x_3^\gamma} \left[ 2 - \gamma - \delta \right]$$

I know it is not correct because I have simulated the results

Answer 1

I think that making the substitutions you overlooked some dependence on $z=x_3/x_1$. I mean, if you look at your calculations, you will find some $x_3/x_1$ which is not substituted by $z$.$^1$

By the way, in general, your method is correct, but cumbersome.

It is formally correct to write the substitution as $x_1 (z) = x_3/z$ and $x_3 (z) = z x_1$, but if you substitute them in the $MRS_{13}$ as such, and then you differentiate using the chain rule for composite functions, you make your life harder when calculating the derivative.

Instead, just plug into the formula of $MRS_{13}$ $x_1 = x_3/z$ and $x_3 = z x_1$, and treat $x_1$ and $x_3$, after the substitution, as constants. But one must be careful to not forget some dependence on $z$.

The following are my calculations for the derivative of $MRS_{13}$ with respect to $z=x_3/x_1$.

$MRS_{13}$ is:

$$MRS_{13} = \frac{\delta}{\gamma} \frac{x_1^{\delta-1}}{x_3^{\gamma - 1}} \qquad (1)$$

In order to 'isolate' $z$, I write $(1)$ as:

$$MRS_{13} = \frac{\delta}{\gamma} \frac{x_1^{\delta-1+\gamma-1-\gamma+1}}{x_3^{\gamma - 1}}== \frac{\delta}{\gamma} \frac{x_1^{\gamma-1} }{x_3^{\gamma - 1}} x_1^{\delta -1-\gamma +1}=\frac{\delta}{\gamma}z^{1-\gamma} x_1^{\delta-\gamma}\qquad (2)$$

Differentiating with respect to $z$ and substituting back $x_3/x_1$ for $z$ I have:

$$\frac{\partial MRS_{13}}{\partial z} = \frac{\delta}{\gamma} (1-\gamma) z^{-\gamma} x_1 ^{\delta-\gamma}=\frac{\delta}{\gamma} (1-\gamma) (\frac {x_1}{x_3})^{\gamma} x_1 ^{\delta-\gamma} =\frac{\delta}{\gamma}(1-\gamma)\frac {x_1^\delta}{x_3^\gamma}.\qquad (3)$$

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$^1$ ^{I tried to follow your method making the substitutions, I had the same result as you, but I realized that some dependence on $z$ had been forgotten, because in the formula remained some $x_3/x_1$ that was not substituted by $z$. This mistake was indicated to you also in the post you linked to Math.SE. I think that here there is a similar mistake during the substitutions.}