# Self-dual production functions that do not satisfy weak homothetic separability

I am looking for parametric production functions that do not satisfy weak homothetic separability (as first defined in Shephard, 1953), but that do allow for an analytical expression of the dual cost function to be derived from the cost minimization problem (they are self-dual). Do any exist?

For completeness, one can say a production function with variable inputs $$M_{it}$$, $$L_{it}$$ and fixed input $$K_{it}$$ is weakly homothetically separable if it is of the form,

$$$$q_{it} = F\left( K_{it}, h( K_{it}, L_{it}, M_{it} ) \right),$$$$

with $$h( K_{it}, \cdot, \cdot )$$ homogeneous of arbitrary degree for all $$K_{it}$$.

References

Shephard, Ronald W. COST AND PRODUCTION FUNCTIONS. PRINCETON UNIV NJ, 1953.

• Are you sure that $K_{it}$ appears twice, both as an argument of $F$ and of $h$? This is not the usual definition of homothetic separability. May 19 at 11:41
• As far as I understand that's the "weak" part. Strong separability would mean $K_{it}$ doesn't appear in $h(\cdot)$. Thanks for the comment. May 19 at 16:33