<p>I suggest that you ask yourself the following questions:</p>

<ul>
<li>If $\tilde{z}(y;w)$ is the input demand from cost minimization, can there be any difference between $\tilde{z}\big(y(p);w\big) \equiv \hat{z} (p;w)$ and $z(p;w)$ from Hotelling's lemma and profit maximization?</li>
<li>Even if $\tilde{z}(y;w)$ is a correspondance rather than a function, is there any difference between $ p \hat{z} (p;w) = p \tilde{z}\big(y(p);w\big) = c(y;w) $ and $p z(p;w)$ (where the later are dot products).</li>
</ul>

<p>Hope this helps,</p>

CC BY-SA 3.0
2014-12-01T15:51:20.703Z
2014-12-01T15:51:20.703Z
Martin Van der Linden
<p>If you have recovered the supply function <span class="math-container">$y_s(\bullet)$</span>, from the equation <span class="math-container">$$Q=y_s(p,\mathbf{r})$$</span> hopefully you could solve for p, e.g. <span class="math-container">$p=f\left(Q,\mathbf{r}\right)$</span>. If the firm is maximizing profits, the equation <span class="math-container">$p=f\left(Q,\mathbf{r}\right)$</span> is equivalent to the condition <span class="math-container">$p=C'\left(Q,\mathbf{r}\right)$</span>.
So, one recovers <span class="math-container">$$C\left(Q,\mathbf{r}\right)=\int {f\left(Q,\mathbf{r}\right)dQ}$$</span></p>

CC BY-SA 4.0
2021-06-03T16:07:35.793Z
2021-06-03T16:07:35.793Z
Maximiliano Gabriel Miranda Za
<p>If you know quantity as a function of prices, then you know revenue as a function of prices.  If you also know profits, then cost are straightforwardly the difference.</p>

CC BY-SA 3.0
2014-12-02T01:35:11.727Z
It would help if you elaborated a little (and addressed the specific question mentioned in the original post).
Steve S
CC BY-SA 3.0
2014-12-01T20:51:14.480Z
2014-12-01T20:51:14.480Z
ssladler
<p>How do I recover the cost function from the profit function?</p>

<p>Suppose I have $$ \pi(p, w_1, w_2) = p^3 \cdot w_1 \cdot w_2 $$.</p>

<p>How do I get the cost function?</p>

<p>Using Hotelling's lemma I get the supply function: $$ y_s(p) = \frac{\partial  \pi(p, w_1, w_2)}{\partial p} = 3 p^2 \cdot w_1 \cdot w_2$$ and the input demands:
$$z_1(p, w)= - \frac{\partial  \pi(p, w_1, w_2)}{\partial w_l} = - p^3 w_2 $$ and $$z_2(p, w)= - \frac{\partial  \pi(p, w_1, w_2)}{\partial w_2} = - p^3 w_1 $$but what is the next step?</p>

<p>Note that the functional forms are hypothetical and highly probable to be invalid.</p>

<microeconomics>
Recover cost function from profit function
Konstantinos
# Recover cost function from profit function

(Asked by: username_0 on 2014-11-30)

How do I recover the cost function from the profit function?

Suppose I have $$ \pi(p, w\_1, w\_2) = p^3 \cdot w\_1 \cdot w\_2 $$.

How do I get the cost function?

Using Hotelling's lemma I get the supply function: $$ y\_s(p) = \frac{\partial \pi(p, w\_1, w\_2)}{\partial p} = 3 p^2 \cdot w\_1 \cdot w\_2$$ and the input demands:
$$z\_1(p, w)= - \frac{\partial \pi(p, w\_1, w\_2)}{\partial w\_l} = - p^3 w\_2 $$ and $$z\_2(p, w)= - \frac{\partial \pi(p, w\_1, w\_2)}{\partial w\_2} = - p^3 w\_1 $$but what is the next step?

Note that the functional forms are hypothetical and highly probable to be invalid.

**Tags:** microeconomics


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## 3 answers

### Answer by username_1 on 2014-12-01. Score: 5

I suggest that you ask yourself the following questions:

* If $\tilde{z}(y;w)$ is the input demand from cost minimization, can there be any difference between $\tilde{z}\big(y(p);w\big) \equiv \hat{z} (p;w)$ and $z(p;w)$ from Hotelling's lemma and profit maximization?
* Even if $\tilde{z}(y;w)$ is a correspondance rather than a function, is there any difference between $ p \hat{z} (p;w) = p \tilde{z}\big(y(p);w\big) = c(y;w) $ and $p z(p;w)$ (where the later are dot products).

Hope this helps,


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### Answer by username_2 on 2021-06-03. Score: 1

If you have recovered the supply function $y\_s(\bullet)$, from the equation $$Q=y\_s(p,\mathbf{r})$$ hopefully you could solve for p, e.g. $p=f\left(Q,\mathbf{r}\right)$. If the firm is maximizing profits, the equation $p=f\left(Q,\mathbf{r}\right)$ is equivalent to the condition $p=C'\left(Q,\mathbf{r}\right)$.
So, one recovers $$C\left(Q,\mathbf{r}\right)=\int {f\left(Q,\mathbf{r}\right)dQ}$$


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### Answer by username_3 on 2014-12-01. Score: 0

If you know quantity as a function of prices, then you know revenue as a function of prices. If you also know profits, then cost are straightforwardly the difference.

> **Comments:**
> **username_4**: It would help if you elaborated a little (and addressed the specific question mentioned in the original post).  

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