<p>First, this only works in perfect competition so it is not a general result. Second, the reason why it works is that, first derivative wrt quantity is the condition for maximizing profit. Profit maximizing firm would want to supply quantity to a market such that its profit is maximized. So firm supply will be given by q that satisfies this foc for given price.</p>

CC BY-SA 4.0
2023-03-20T12:52:50.277Z
In this exercise, we found the FOC in regards to price. Does it make any difference?
aliosha karamazov
CC BY-SA 4.0
2023-03-20T12:59:32.717Z
@alioshakaramazov no, in that case q will still by implied by optimum p
1muflon1
CC BY-SA 4.0
2023-03-20T12:42:13.603Z
2023-03-20T12:42:13.603Z
1muflon1
<p>Hotelling’s Lemma states that for the indirect profit function <span class="math-container">$\Pi^\star(p,w,r)$</span>, we have that the firm’s output supply is given by:</p>
<p><span class="math-container">$q^s = \frac{d \Pi^\star}{dp}$</span></p>
<p>Here is its proof:</p>
<p>The optimized profit function is given by</p>
<p><span class="math-container">$\Pi^\star(p,w,r) = \Pi(p,L^\star(p),K^\star(p))$</span></p>
<p><span class="math-container">$ = p \cdot q(L^\star(p), K^\star(p)) - w L^\star(p) - r K^\star(p)$</span></p>
<p>Recall the optimality conditions for profit maximization are given by</p>
<p><span class="math-container">$\frac{\partial \Pi}{\partial L}|_{L = L^\star} = 0$</span>,</p>
<p><span class="math-container">$\frac{\partial \Pi}{\partial K}|_{K = K^\star} = 0$</span>.</p>
<p>Taking the total derivative of the optimized profit <span class="math-container">$\Pi^\star$</span> with respect to <span class="math-container">$p$</span> at the optimal inputs,</p>
<p><span class="math-container">$\frac{d\Pi^\star}{dp} = \frac{\partial \Pi}{\partial L}|_{L=L^\star(p)} \frac{\partial L}{\partial w} + \frac{\partial \Pi}{\partial K}|_{K=K^\star(p)} \frac{\partial K}{\partial w} + \frac{\partial \Pi}{\partial p} = 0 + 0 + \frac{\partial \Pi}{\partial p} = \frac{\partial \Pi}{\partial p}$</span></p>
<p><span class="math-container">$= q(L^\star(p),K^\star(p)) = q^s(p)$</span>.</p>

CC BY-SA 4.0
2023-03-20T14:34:37.130Z
2023-03-20T14:34:37.130Z
Nicolas Torres
<p>We have the profit function of the firm   profit = <span class="math-container">$p^2 -2p -399$</span>.</p>
<p>We take derivative of it we say that the output supply function is =<span class="math-container">$2p-2$</span></p>
<p>I understand that Profit = q*p - TC
But why do we say the output supply function is equal to it's first derivative? We are not maximizing here. Simply finding the derivative.</p>

CC BY-SA 4.0
2023-03-20T12:52:10.037Z
Did you mean to state the profit function only in terms of the output price $p$?
Nicolas Torres
<microeconomics><supply-and-demand><long-run>
Why is the supply function the first derivative of the profit function in the long run?
aliosha karamazov
# Why is the supply function the first derivative of the profit function in the long run?

(Asked by: username_0 on 2023-03-20)

We have the profit function of the firm profit = $p^2 -2p -399$.

We take derivative of it we say that the output supply function is =$2p-2$

I understand that Profit = q\*p - TC
But why do we say the output supply function is equal to it's first derivative? We are not maximizing here. Simply finding the derivative.

**Tags:** microeconomics, supply-and-demand, long-run
> **Comments:**
> **username_1**: Did you mean to state the profit function only in terms of the output price $p$?  

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## 2 answers

### Answer by username_2 on 2023-03-20. Score: 3

First, this only works in perfect competition so it is not a general result. Second, the reason why it works is that, first derivative wrt quantity is the condition for maximizing profit. Profit maximizing firm would want to supply quantity to a market such that its profit is maximized. So firm supply will be given by q that satisfies this foc for given price.

> **Comments:**
> **username_0**: In this exercise, we found the FOC in regards to price. Does it make any difference?  
> **username_2**: @alioshakaramazov no, in that case q will still by implied by optimum p  

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### Answer by username_1 on 2023-03-20. Score: 1

Hotelling’s Lemma states that for the indirect profit function $\Pi^\star(p,w,r)$, we have that the firm’s output supply is given by:

$q^s = \frac{d \Pi^\star}{dp}$

Here is its proof:

The optimized profit function is given by

$\Pi^\star(p,w,r) = \Pi(p,L^\star(p),K^\star(p))$

$ = p \cdot q(L^\star(p), K^\star(p)) - w L^\star(p) - r K^\star(p)$

Recall the optimality conditions for profit maximization are given by

$\frac{\partial \Pi}{\partial L}|\_{L = L^\star} = 0$,

$\frac{\partial \Pi}{\partial K}|\_{K = K^\star} = 0$.

Taking the total derivative of the optimized profit $\Pi^\star$ with respect to $p$ at the optimal inputs,

$\frac{d\Pi^\star}{dp} = \frac{\partial \Pi}{\partial L}|\_{L=L^\star(p)} \frac{\partial L}{\partial w} + \frac{\partial \Pi}{\partial K}|\_{K=K^\star(p)} \frac{\partial K}{\partial w} + \frac{\partial \Pi}{\partial p} = 0 + 0 + \frac{\partial \Pi}{\partial p} = \frac{\partial \Pi}{\partial p}$

$= q(L^\star(p),K^\star(p)) = q^s(p)$.


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