Supply function of a price-taking firm with a quadratic production function

Question

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 $Q$?

Answer 1

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

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