Demand Function and Dead Weight Loss

A. The demand function is given by p=20-q where p is price and q is quantity. Your cost function is c(q)=q^2.

1. How many units should you produce and what price should charge to maximize profits? What would profits be?
2. Calculate the dead weight loss

For #1 I calculated 20- q = 2q. Q = 6.67 p = 13.33 Profit = 44.42. Does anyone know if this correct? I find it weird that q could be a decimal.

#2, I am not sure how to caluclate dead weight loss…do you need a p2 and q2? Any and all help would be greatly appreciated. Thanks in advance

Solutions Collecting From Web of "Demand Function and Dead Weight Loss"

The demand is $q=20-p$.

Your revenue is $r=qp=20p-p^2$

Your net profit is $n=r-c=r-q^2=60p-2p^2-400$

which is maximised when $\frac{\partial n}{\partial p}=0$

Accordingly, the solution of $p_*=60-4p_*$ gives the optimum price of $p=15$

I found this definition of “Deadweight Loss” from Wikipedia:
“In economics, a deadweight loss (also known as excess burden or allocative inefficiency) is a loss of economic efficiency that can occur when equilibrium for a good or service is not achieved or is not achievable. “

This requires knowledge of a supply function though, which I did not notice in the problem specs. I may be missing something.

\begin{align} p(q)&=20-q\\ c (q)&=q^2\\ r (q)&=p (q) \times q=20q-q^2\\ \pi (q)&=r (q)- c(q)=20q-2q^2\\ \\ \pi'(q)&=20-4q=0\quad \Longrightarrow \quad q_m=5,\; p_m=20-q_m=15,\;\pi (q_m)=\pi_m=20q_m-2q_m^2=50 \end{align}

The marginal cost is $c'(q)=2q$. The equilibrium is $c'(q)=p (q)$ that is $q_e=\frac{20}{3},\,p_e=\frac{40}{3}$. So the dead weight loss is
$$D=\frac{1}{2}(p_m-p_e)(q_e-q_m)=1.39$$