Simple equilibrium in monopoly question

Question

Given an inverse demand function for hangars: $$P = 3 - \frac{Q}{16,000}$$ and constant marginal cost of $1$, what is the equilibrium price and quantity of hangars in a monopoly?

In a perfectly competitive market we need $P = MC$, which gives us an equilibrium price of $1$ and quantity of $32,000$. However, in a monopoly we need $MC = MR$, and in this case that equates simply to price = $1$. Then, the demand gives us, again, an equilibrium quantity of $32,000$. Am I making some mistake somewhere, or are the equilibrium prices and quantities in a monopoly and competitive market in this example equal simply by coincidence?

Answer 1

The monopolist solves,

$\max \pi=R(Q)-C(Q)=P(Q)Q-C(Q)$

The general solution is (if the problem is globally concave)

$MR(Q^*)=P'(Q^*)Q^*+P(Q^*)=C'(Q^*)$.

Since $P'(Q)<0$, notice that $MR(Q)<P(Q)$ and, therefore, $Q^*<Q^{C}$.

In this example, we have $P(Q)=3-\frac{Q}{16000}$ and $C(Q)=Q$. Hence,

$MR(Q)=-\frac{1}{160000}Q+(3-\frac{Q}{16000})=1=MC(Q)$.

Rewritting, we get $3-\frac{Q}{8000}=1$ or $Q=16000$. To compute the price simply substitute in the demand function and get $P(16000)=3-1=2>1$.