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The motive of my post is in search of some help in order to understand the correct way to find the elasticity of substitution by means of calculating the following function.

$$f(K, L)=\frac{K^{2}L^{2}}{K+L} $$

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My guess is that the ''elasticity of substitution'' means something like$$\frac{\%\Delta K}{\% \Delta L}=\mathrm{mrs} \times\frac lk, $$ but that is dependant on what point you are at (i.e. a coordinate $(l, k)$). To get $\mathrm{mrs} $, find $\frac {\mathrm d k} {\mathrm d l} $, and even that is dependent on the value of $f$.

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Elasticity of subsitution ($\sigma$) can be found by using this formula

$$\sigma=\frac{dln(\frac{x_1}{x_2})}{dln(MRS)}$$

alternatively you can use: $$\sigma=\frac{e(x)f(x)f_1(x)f_2(x)}{x_1x_2|BH|}$$

where $x=x_1,x_2......x_n$ and $e(x)$ is elasticity of scale. You can use this information to solve your problem.

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