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If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem, $$ x c'(x) = k c(x).$$ That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is ...


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One of the simplest specification I can think of (and for which the first order condition can be solved analytically in $L$) is: $$ y=\left\{ \begin{array}{ccc} L^{\alpha} & & L\leq L_{e} \\ L_{e}^{\alpha}+g\left( L-L_{e}\right) & & L>L_{e}% \end{array}% \right. $$ with $g\left( L-L_{e}\right) =(L-L_{e})^\beta$ and $\alpha\geq1$ and ...


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I'm not sure but it seems to me that the logistic function $\frac{e^{x}}{1+e^{x}}$ could serve your purpose. You may need to scale it as its output falls between 0 and 1, but it does have an analytical derivative that you can then use to solve for the labour demand function.


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