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Given $ Y=f(K,L;\sigma) $, the effect on labor from a change in the price of capital can be gauged through a substitution effect and a scale effect: \begin{align*} \frac{\partial L}{\partial r} & =\underset{Sub}{\underbrace{\left. \frac{\partial L^{c}}{\partial r}\right\vert _{Y}}}% +\underset{Scale}{\underbrace{\frac{\partial L^{c}}{\partial Y}\frac{\partial Y}{\partial r}}}\\ \\ & =SH_{_{Labor}}\left( \sigma+\eta\right) \end{align*} where $L^c$ is the conditional demand (the smooth movement along the isoquant).

This can then - as far as I can see - be also expressed as the sum of the factor substitution elasticity and the firm's elasticity of demand, weighted by the labor cost share.

But are $\eta$ and $\sigma$ independent of one another? If a firm can readily substitute factors in response to factor price changes, this will presumably impact its price elasticity of demand. Any thoughts?

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