I know that homothetic production function implies that cost function is multiplicatively separable in input prices and output, and it can be written as C(w,y)=h(y)C(w,1). Can some one help me derive the functional form of profit function in case of homothetic production functions?
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$\begingroup$ What exactly do you mean by "functional form"? $\endgroup$– GiskardCommented Oct 24, 2015 at 17:33
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$\begingroup$ I meant to ask if we can we separate profit function (multiplicatively or additively) as a function of prices and output, the way we can do it for cost function. Your answer implies that we cannot. Can you suggest me any reference on homothetic technologies for better understanding? $\endgroup$– Sher AfghanCommented Oct 24, 2015 at 17:53
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$\begingroup$ I have seen your other question but unfortunately I cannot give any references, I thought of this using the definitions of homogeneity and optimum conditions. $\endgroup$– GiskardCommented Oct 24, 2015 at 17:55
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$\begingroup$ i want to know the meaning of h(t). thank u $\endgroup$– Rous BagsCommented Nov 24, 2020 at 20:44
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I have figured this as the answer to this question; As we know profit maximization problem is given as, $$ \pi(p,w) = \mathop{max}_{\textbf{y}}\quad p.y - C(\overrightarrow{w},y) $$ When $f(\overrightarrow{x})$ is homothetic,
$$ C(\overrightarrow{w},y)=h(y).C(\overrightarrow{w},{1}) $$
Substituting in the profit function gives;
$$ \pi(p,w) = \mathop{max}_{\textbf{y}}\quad p.y - h(y). C(\overrightarrow{w},1) $$
First order condition gives us; $$ p=h'(y)C(\overrightarrow{w},1) $$
Which can be written as; $$ h'(y)=\frac{p}{C(\overrightarrow{w},1)} $$
or
$$ y=(h')^{-1}\frac{p}{C(\overrightarrow{w},1)} $$ $$ \Rightarrow y= \gamma(p).\beta(w) $$
Therefore $y(p,w)$ is separable. Since $$ \pi(p,w)=\int{y(p,w)}dp $$
$$ \pi(p,w)=\int{\gamma(p).\beta(w)}dp $$
$$ \pi(p,w)=\beta(w)\int{\gamma(p)}dp $$
$$ \pi(p,w)=\beta(w)\alpha(p) $$
Thus $\pi(p,w)$ is also separable in factor prices and output prices.
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1$\begingroup$ Nice! This part is not perfectly clear though: $ y =(h')^{-1}\frac{p}{C(\overrightarrow{w},1)} \Rightarrow y= \gamma(p).\beta(w) $ $\endgroup$– GiskardCommented Oct 24, 2015 at 19:10
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2$\begingroup$ If the production function is homogeneous of degree $t$ then $h(y) = y^{\frac{1}{t}}$. Using this I think you can derive the exact functions $\alpha, \beta$. $\endgroup$– GiskardCommented Oct 24, 2015 at 19:13
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$\begingroup$ For $h(y) = exp(y)$ we have $y^*$ additively separable in $p,w$ as $y^*(p,w)=\ln(p)-\ln(C(w,1))$ $\endgroup$– BertrandCommented Nov 24, 2020 at 21:09
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$\begingroup$ The last four equalities are also not correct. $\endgroup$– BertrandCommented Nov 24, 2020 at 21:32