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I realize that the translog production function is derived as a second order taylor approximation of a production function (e.g. the CES-production function), as explained in this post.

Is the translog cost function derived similarly as a taylor expansion of an arbitrary multiple product cost function? Or is it derived trough cost minimization of a translog production function? Or both?

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It is not trivial (and maybe not even possible) to find the cost function corresponding to a translog production function. And this cost function corresponding to a translog production function has not a translog functional form.
If you rewrite a cost function as follow $c(w,y)=c(\exp(\ln w),\exp(\ln y)) =: C(\ln w,\ln y),$ then a second order Taylor development of $C$ around a reference point $(\ln w_0,\ln y_0)$ yields a translog cost function (after reparameterization).

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