4.1 — Modeling Firms With Market Power — Class Content

Contents

• Section 1: Monday, April 19, 2021
• Section 2: Tuesday, April 27, 2021Section 2, because of no class on Thursday Feb 18 and Tuesday Mar 9, and no class on Wednesday April 21, is 1 class behind Section 1.
  
  

Overview

Today we begin our look at “imperfect competition,” where firms have market power, meaning they can charge p>MC and search for the profit-maximizing quantity and price. Today is merely about how do change the model to understand how a firm with market power behaves. To assist us, we begin with an extreme case of a single seller, i.e. a monopoly. For now we only assume that there is a single firm, and see how it behaves differently than if it were in a competitive market. Next class we will begin to explore what could cause a market to have only a single seller, and what are some of the social consequences of market power.

Readings

• Ch. 9-2-9.5 in @textbook

Slides

• Download as PDF

Practice

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

Assignments

Problem Set 5 Due Sun Apr 25

Problem set 5 (on 3.1-3.5) is due by 11:59 PM Sunday April 25 (both sections) by PDF upload to Blackboard Assignments. This will be your final graded problem set this semester.

Appendix

Monopolists Only Produce Where Demand is Elastic: Proof

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

MR(q)=p+(ΔpΔq)qDefinition of MR(q)MR(q)p=pp+(ΔpΔq)qpDividing both sides by pMR(q)p=1+(ΔpΔq×qp)⏟1ϵSimplifyingMR(q)p=1+1ϵDRecognize price elasticity ϵD=ΔqΔp×pqMR(q)=p(1+1ϵD)Multiplying both sides by p

Remember, we’ve simplified ϵD=1slope×pq, where 1slope=ΔqΔp because on a demand curve, slope=ΔpΔq.

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

MR(q)=MC(q)p(1+1ϵD)=MC(q)p(1−1|ϵD|)=MC(q)

I rearrange the last line only to remind us that ϵD is always negative!

Now note the following:

• If |ϵD|<1, then MR(q) is negative. Since MC(q) is assumed to be positive, it cannot equal a negative MR(q), hence this is not profit-maximizing.
• If |ϵD|=1, then MR(q) is 0. Only if MC(q) is also 0 is this profit-maximizing.
• If |ϵD|>1, then MR(q) is positive. It can equal a positive MC(q) to be profit-maximizing.

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 31.

Derivation of the Lerner Index

Marginal revenue is strongly related to the price elasticity of demand, which is ED=ΔqΔp×pqI sometimes simplify it as ED=1slope×pq, where “slope” is the slope of the inverse demand curve (graph), since the slope is ΔpΔq=riserun.

We derived marginal revenue (in the slides) as: MR(q)=p+ΔpΔqq

Firms will always maximize profits where:

MR(q)=MC(q)Profit-max outputp+(ΔpΔq)q=MC(q)Definition of MR(q)p+(ΔpΔq)q×pp=MC(q)Multiplying the left by pp (i.e. 1)p+(ΔpΔq×qp)⏟1ϵ×p=MC(q)Rearranging the leftp+1ϵ×p=MC(q)Recognize price elasticity ϵ=ΔqΔp×pqp=MC(q)−1ϵpSubtract 1ϵp from both sidesp−MC(q)=−1ϵpSubtract MC(q) from both sidesp−MC(q)p=−1ϵDivide both sides by p

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