<p>The monopolist solves,</p>

<p>$\max \pi=R(Q)-C(Q)=P(Q)Q-C(Q)$</p>

<p>The general solution is (if the problem is globally concave)</p>

<p>$MR(Q^*)=P'(Q^*)Q^*+P(Q^*)=C'(Q^*)$. </p>

<p>Since $P'(Q)&lt;0$, notice that $MR(Q)&lt;P(Q)$ and, therefore, $Q^*&lt;Q^{C}$. </p>

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

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

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

CC BY-SA 3.0
2017-09-15T16:09:27.377Z
Great! If you delete the comments, even better! :)
luchonacho
CC BY-SA 3.0
2017-09-15T16:02:43.653Z
2017-09-15T16:02:43.653Z
Fato
<blockquote>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? </blockquote>

<p>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?</p>

<monopoly><competitive-equilibrium>
Simple equilibrium in monopoly question
mizichael
# Simple equilibrium in monopoly question

(Asked by: username_0 on 2017-09-14)

> 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?

**Tags:** monopoly, competitive-equilibrium

---

## 1 answers

### Answer by username_1 on 2017-09-15. Score: 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)

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 $P(16000)=3-1=2>1$.

> **Comments:**
> **username_2**: Great! If you delete the comments, even better! :)  

---

