4.1: Monopoly - Class Notes

Contents

Overview

Today we begin our look at market power, when firms can charge \(p>MC\) and can search for the profit-maximizing quantity and price. We begin with our first extreme case of a single firm, a monopoly. We look at how market power changes how a firm chooses its optimal production.

Class Livestream/Videos

Slides

Practice Problems

Today we will be working on practice problems. Answers will be posted later.

Exam 2 Corrections

I am still grading your Exam 2, I hope to finish for next class. Sorry it’s been a crazy semester!

Problem Set 5 Due 11:59PM Sunday April 19

Homework 5 on classes 3.1-3.3 is due by email (as a PDF) at 11:59PM Sunday April 19.

This will be your final graded homework of the semester. A reminder, the homework grade (20% of course grade) is the average of your homework grades. Again, see the grade calculator or the syllabus to estimate how this impacts your overall grade.

I will give a “homework” (with answers, of course) on Unit 4 which we are currently covering, so that you can practice for the Final exam, but I will not collect or grade it.

Appendix

Monopolists Only Produce Where Demand is Elastic: Proof

Let’s first show the relationship between \(MR(q)\) and price elasticity of demand, \(\epsilon_D\).

\[\begin{align*} MR(q) &= p+\bigg(\frac{\Delta p}{\Delta q}\bigg)q && \text{Definition of } MR(q)\\ \frac{MR(q)}{p} &= \frac{p}{p}+\bigg(\frac{\Delta p}{\Delta q}\bigg) \frac{q}{p} && \text{Dividing both sides by } p\\ \frac{MR(q)}{p} &= 1+\underbrace{\bigg(\frac{\Delta p}{\Delta q}\times \frac{q}{p} \bigg)}_{\frac{1}{\epsilon}} && \text{Simplifying}\\ \frac{MR(q)}{p} &= 1+\frac{1}{\epsilon_D} && \text{Recognize price elasticity } \epsilon_D=\frac{\Delta q}{\Delta p} \times \frac{p}{q}\\ MR(q) &= p\bigg(1+\frac{1}{\epsilon_D}\bigg) && \text{Multiplying both sides by }p\\ \end{align*}\]

Remember, we’ve simplified \(\epsilon_D = \frac{1}{slope} \times \frac{p}{q}\), where \(\frac{1}{slope} = \frac{\Delta q}{\Delta p}\) because on a demand curve, \(slope = \frac{\Delta p}{\Delta q}\).

Now that we have this alternate expression for \(MR(q)\), lets assume \(MC(q) \geq 0\) and set them equal to one another to maximize profits:

\[\begin{align*} MR(q) &= MC(q)\\ p(1+\frac{1}{\epsilon_D}) & = MC(q)\\ p(1-\frac{1}{|\epsilon_D|}) & = MC(q)\\ \end{align*}\]

I rearrange the last line only to remind us that \(\epsilon_D\) is always negative!

Now note the following:

Hence, a monopolist will never produce in the inelastic region of the demand curve (where \(MR(q)<0)\), and will only produce at the unit elastic part of the demand curve (where \(MR(q)=0)\) if \(MC(q)=0\). Thus, it generally produces in the elastic region where \(MR(q)>0\).

See the graphs on slide 22.

Derivation of the Lerner Index

Marginal revenue is strongly related to the price elasticity of demand, which is \(E_{D}=\frac{\Delta q}{\Delta p} \times \frac{p}{q}\)I sometimes simplify it as \(E_{D}=\frac{1}{slope} \times \frac{p}{q}\), where “slope” is the slope of the inverse demand curve (graph), since the slope is \(\frac{\Delta p}{\Delta q} = \frac{rise}{run}\).

We derived marginal revenue (in the slides) as: \[MR(q)=p+\frac{\Delta p}{\Delta q}q\]

Firms will always maximize profits where:

\[\begin{align*} MR(q)&=MC(q) && \text{Profit-max output}\\ p+\bigg(\frac{\Delta p}{\Delta q}\bigg)q&=MC(q) && \text{Definition of } MR(q)\\ p+\bigg(\frac{\Delta p}{\Delta q}\bigg) q \times \frac{p}{p}&=MC(q) && \text{Multiplying the left by } \frac{p}{p} \text{ (i.e. 1)}\\ p+\underbrace{\bigg(\frac{\Delta p}{\Delta q}\times \frac{q}{p} \bigg)}_{\frac{1}{\epsilon}} \times p &=MC(q) && \text{Rearranging the left}\\ p+\frac{1}{\epsilon} \times p &=MC(q) && \text{Recognize price elasticity } \epsilon=\frac{\Delta q}{\Delta p} \times \frac{p}{q}\\ p &=MC(q) - \frac{1}{\epsilon} p && \text{Subtract }\frac{1}{\epsilon}p \text{ from both sides}\\ p-MC(q) &= -\frac{1}{\epsilon} p && \text{Subtract }MC(q) \text{ from both sides}\\ \frac{p-MC(q)}{p} &= -\frac{1}{\epsilon} && \text{Divide both sides by }p\\ \end{align*}\]

The left side gives us the fraction of price that is markup \(\left(\frac{p-MC(q)}{p}\right)\), and the right side shows this is inversely related to price elasticity of demand.