4.1 — Modeling Firms With Market Power — Class Content



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.




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


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.


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 31.

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.