Recover cost function from profit function

Question

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.

Martin Van der Linden · Answer

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 $tilde{z}big(y(p);wbig) equiv hat{z} (p;w)$ 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);wbig) = c(y;w) $ and $p z(p;w)$ (where the later are dot products).

Hope this helps,

ssladler · Answer

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.

Maximiliano Gabriel Miranda Za · Answer

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. $p=fleft(Q,mathbf{r}right)$. If the firm is maximizing profits, the equation $p=fleft(Q,mathbf{r}right)$ is equivalent to the condition $p=C'left(Q,mathbf{r}right)$.
So, one recovers $$Cleft(Q,mathbf{r}right)=int {fleft(Q,mathbf{r}right)dQ}$$

Recover cost function from profit function

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