<p>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?</p>
<p>For completeness, one can say a production function with variable inputs <span class="math-container">$M_{it}$</span>, <span class="math-container">$L_{it}$</span> and fixed input <span class="math-container">$K_{it}$</span> is weakly homothetically separable if it is of the form,</p>
<p><span class="math-container">\begin{equation}
    q_{it} = F\left( K_{it}, h( K_{it}, L_{it}, M_{it} ) \right),
\end{equation}</span></p>
<p>with <span class="math-container">$h( K_{it}, \cdot, \cdot )$</span> homogeneous of arbitrary degree for all <span class="math-container">$K_{it}$</span>.</p>
<p><em>References</em></p>
<p>Shephard, Ronald W. COST AND PRODUCTION FUNCTIONS. PRINCETON UNIV NJ, 1953.</p>

CC BY-SA 4.0
2021-05-19T11:41:43.887Z
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.
Bertrand
CC BY-SA 4.0
2021-05-19T16:33:16.403Z
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.
hrrrrrr5602
<production-function><cost-functions><duality>
Self-dual production functions that do not satisfy weak homothetic separability
hrrrrrr5602
# Self-dual production functions that do not satisfy weak homothetic separability

(Asked by: username_0 on 2021-05-18)

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,

\begin{equation}
q\_{it} = F\left( K\_{it}, h( K\_{it}, L\_{it}, M\_{it} ) \right),
\end{equation}

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.

**Tags:** production-function, cost-functions, duality

> **Comments:**
> **username_1**: 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.  
> **username_0**: 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.  

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## 0 answers

