<p>The profit function of the firm is given by <span class="math-container">$\Pi = PQ-TC = PQ - VC - FC = P(40L - L^2) - 20L - 10 = 40PL-PL^2-20L-10$</span></p>
<p>So our optimization problem is</p>
<p><span class="math-container">$\max_L \Pi = 40PL-PL^2-20L-10$</span></p>
<p>This is a single variable unconstrained optimization problem with a concave objective function, so we simply set <span class="math-container">$\frac{d\Pi}{dL} = 0$</span>.</p>
<p><span class="math-container">$\frac{d\Pi}{dL} = 40P-2PL-20=0 \implies 40P - 20 = 2PL \implies L^\star = \frac{40P-20}{2P} = \frac{20P-10}{P} = 10 \cdot \frac{2P-1}{P}$</span></p>
<p>Plugging the optimal labor into the production function,</p>
<p><span class="math-container">$Q^s = 40L^\star - (L^\star)^2 = 40 \cdot 10 \cdot \frac{2P-1}{P} - (10 \cdot \frac{2P-1}{P})^2 = 400 \cdot \frac{2P-1}{P} - 100 \frac{(2P-1)^2}{P^2} = \frac{400P(2P-1)-100(2P-1)^2}{P^2} = \frac{800P^2-400P-100(4P^2-4P+1)}{P^2} = \frac{800P^2-400P-400P^2+400P-100}{P^2} = \frac{400P^2-100}{P^2}= 100 \cdot \frac{4P^2-1}{P^2}$</span></p>
<p>Therefore, the supply curve of the firm is given by</p>
<p><span class="math-container">$Q^s(P) = 100 \cdot \frac{4P^2-1}{P^2}$</span></p>
<p>Note this expression can be negative when <span class="math-container">$P&lt;\frac{1}{2}$</span>, in which case the actual supply would be <span class="math-container">$Q^s = 0$</span>.</p>
<p>Taking into account this caveat, the supply curve is given by</p>
<p><span class="math-container">$$
Q^s(P)=
\begin{cases}
100 \cdot \frac{4P^2-1}{P^2}, P \geq \frac{1}{2}\\
0, P &lt; \frac{1}{2}\\
\end{cases}
$$</span></p>

CC BY-SA 4.0
2023-03-19T14:12:44.223Z
2023-03-19T14:28:20.517Z
2023-03-19T14:28:20.517Z
Nicolas Torres
<p>For a firm with the production function
<span class="math-container">$$Q = 40L-L^2$$</span>
where <span class="math-container">$L$</span> is labor and wage <span class="math-container">$w = 20$</span> find supply function of a price-taking firm under perfect competition. Fixed costs equal <span class="math-container">$10$</span>.</p>
<p>Following similar questions on this board, I've tried the Lagrangian:
<span class="math-container">\begin{align*}
&amp;\Phi = 20L + 10 - \lambda(40L-L^2 - Q)\\
&amp;\frac{\partial\Phi}{\partial L} = 20 - \lambda(40-2L)\Rightarrow\lambda=\frac{1}{2}+\frac{10}{L}
\end{align*}</span>
but this doesn't seem to lead anywhere.</p>
<p>I know that perfect competition implies no profit and <span class="math-container">$P=MC$</span>, but how do I express <span class="math-container">$MC=\frac{\partial TC}{\partial Q}$</span> in terms of <span class="math-container">$Q$</span> if I can't get to <span class="math-container">$TC$</span>? Is there even a way to express <span class="math-container">$TC$</span> as a function of <span class="math-container">$Q$</span>?</p>

<microeconomics><supply-and-demand><production-function><lagrangian>
Supply function of a price-taking firm with a quadratic production function
honkhonk
# Supply function of a price-taking firm with a quadratic production function

(Asked by: username_0 on 2023-03-19)

For a firm with the production function
$$Q = 40L-L^2$$
where $L$ is labor and wage $w = 20$ find supply function of a price-taking firm under perfect competition. Fixed costs equal $10$.

Following similar questions on this board, I've tried the Lagrangian:
\begin{align\*}
&\Phi = 20L + 10 - \lambda(40L-L^2 - Q)\\
&\frac{\partial\Phi}{\partial L} = 20 - \lambda(40-2L)\Rightarrow\lambda=\frac{1}{2}+\frac{10}{L}
\end{align\*}
but this doesn't seem to lead anywhere.

I know that perfect competition implies no profit and $P=MC$, but how do I express $MC=\frac{\partial TC}{\partial Q}$ in terms of $Q$ if I can't get to $TC$? Is there even a way to express $TC$ as a function of $Q$?

**Tags:** microeconomics, supply-and-demand, production-function, lagrangian

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## 1 answers

### Answer by username_1 on 2023-03-19. Score: 1 | Accepted answer

The profit function of the firm is given by $\Pi = PQ-TC = PQ - VC - FC = P(40L - L^2) - 20L - 10 = 40PL-PL^2-20L-10$

So our optimization problem is

$\max\_L \Pi = 40PL-PL^2-20L-10$

This is a single variable unconstrained optimization problem with a concave objective function, so we simply set $\frac{d\Pi}{dL} = 0$.

$\frac{d\Pi}{dL} = 40P-2PL-20=0 \implies 40P - 20 = 2PL \implies L^\star = \frac{40P-20}{2P} = \frac{20P-10}{P} = 10 \cdot \frac{2P-1}{P}$

Plugging the optimal labor into the production function,

$Q^s = 40L^\star - (L^\star)^2 = 40 \cdot 10 \cdot \frac{2P-1}{P} - (10 \cdot \frac{2P-1}{P})^2 = 400 \cdot \frac{2P-1}{P} - 100 \frac{(2P-1)^2}{P^2} = \frac{400P(2P-1)-100(2P-1)^2}{P^2} = \frac{800P^2-400P-100(4P^2-4P+1)}{P^2} = \frac{800P^2-400P-400P^2+400P-100}{P^2} = \frac{400P^2-100}{P^2}= 100 \cdot \frac{4P^2-1}{P^2}$

Therefore, the supply curve of the firm is given by

$Q^s(P) = 100 \cdot \frac{4P^2-1}{P^2}$

Note this expression can be negative when $P<\frac{1}{2}$, in which case the actual supply would be $Q^s = 0$.

Taking into account this caveat, the supply curve is given by

$$
Q^s(P)=
\begin{cases}
100 \cdot \frac{4P^2-1}{P^2}, P \geq \frac{1}{2}\\
0, P < \frac{1}{2}\\
\end{cases}
$$


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