<p>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, </p>

<p>$$
C(\overrightarrow{w},y)=h(y).C(\overrightarrow{w},{1})
$$</p>

<p>Substituting in the profit function gives; </p>

<p>$$
\pi(p,w) = \mathop{max}_{\textbf{y}}\quad p.y - h(y). C(\overrightarrow{w},1)
$$</p>

<p>First order condition gives us; 
$$
p=h'(y)C(\overrightarrow{w},1) 
$$</p>

<p>Which can be written as; 
$$
h'(y)=\frac{p}{C(\overrightarrow{w},1)} 
$$</p>

<p>or</p>

<p>$$
y=(h')^{-1}\frac{p}{C(\overrightarrow{w},1)} 
$$
$$
\Rightarrow y= \gamma(p).\beta(w)
$$</p>

<p>Therefore $y(p,w)$ is separable.
Since 
$$
\pi(p,w)=\int{y(p,w)}dp
$$</p>

<p>$$
\pi(p,w)=\int{\gamma(p).\beta(w)}dp
$$</p>

<p>$$
\pi(p,w)=\beta(w)\int{\gamma(p)}dp
$$</p>

<p>$$
\pi(p,w)=\beta(w)\alpha(p)
$$</p>

<p>Thus $\pi(p,w)$ is also separable in factor prices and output prices. </p>

CC BY-SA 3.0
2015-10-24T19:10:03.053Z
Nice! This part is not perfectly clear though:
$
y =(h')^{-1}\frac{p}{C(\overrightarrow{w},1)}  \Rightarrow y= \gamma(p).\beta(w)
$
Giskard
CC BY-SA 3.0
2015-10-24T19:13:23.487Z
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 $\alpha, \beta$.
Giskard
CC BY-SA 4.0
2020-11-24T21:09:23.803Z
For $h(y) = exp(y)$ we have $y^*$ additively separable in $p,w$ as $y^*(p,w)=\ln(p)-\ln(C(w,1))$
Bertrand
CC BY-SA 4.0
2020-11-24T21:32:37.333Z
The last four equalities are also not correct.
Bertrand
CC BY-SA 3.0
2015-10-24T18:58:59.147Z
2015-10-24T18:58:59.147Z
Sher Afghan
<p>I know that homothetic production function implies that cost function is multiplicatively separable in input prices and output, and it can be written as 
<em>C(w,y)=h(y)C(w,1)</em>. Can some one help me derive the functional form of profit function in case of homothetic production functions?</p>

CC BY-SA 3.0
2015-10-24T17:33:42.227Z
What exactly do you mean by "functional form"?
Giskard
CC BY-SA 3.0
2015-10-24T17:53:53.593Z
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?
Sher Afghan
CC BY-SA 3.0
2015-10-24T17:55:35.903Z
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.
Giskard
CC BY-SA 4.0
2020-11-24T20:44:02.720Z
i want to know the meaning of h(t). thank u
Rous Bags
<production-function><producer-theory><profit-maximization>
Homothetic production function and Profit Function
Sher Afghan
# Homothetic production function and Profit Function

(Asked by: username_0 on 2015-10-24)

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?

**Tags:** production-function, producer-theory, profit-maximization

> **Comments:**
> **username_1**: What exactly do you mean by "functional form"?  
> **username_0**: 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?  
> **username_1**: 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.  
> **username_2**: i want to know the meaning of h(t). thank u  

---

## 1 answers

### Answer by username_0 on 2015-10-24. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_1**: Nice! This part is not perfectly clear though: $ y =(h')^{-1}\frac{p}{C(\overrightarrow{w},1)}  \Rightarrow y= \gamma(p).\beta(w) $  
> **username_1**: 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 $\alpha, \beta$.  
> **username_3**: For $h(y) = exp(y)$ we have $y^*$ additively separable in $p,w$ as $y^*(p,w)=\ln(p)-\ln(C(w,1))$  
> **username_3**: The last four equalities are also not correct.  

---

