# How is the translog cost function derived?

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?

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).