4.1 — Modeling Firms With Market Power — Appendix

Monopolists Only Produce Where Demand is Elastic: Proof

Let’s first show the relationship between $MR(q)$ and price elasticity of demand, ${ϵ}_D$.

$$\begin{array}{rlrl}MR(q)& =p+(\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q})q& & \text{Definition of }MR(q)\\ \frac{MR(q)}{p}& =\frac{p}{p}+(\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q})\frac{q}{p}& & \text{Dividing both sides by }p\\ \frac{MR(q)}{p}& =1+\underset{\frac{1}{ϵ}}{\underbrace{(\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q}\times \frac{q}{p})}}& & \text{Simplifying}\\ \frac{MR(q)}{p}& =1+\frac{1}{{ϵ}_D}& & \text{Recognize price elasticity }{ϵ}_D=\frac{\mathrm{\Delta }q}{\mathrm{\Delta }p}\times \frac{p}{q}\\ MR(q)& =p(1+\frac{1}{{ϵ}_D})& & \text{Multiplying both sides by }p\end{array}$$

Remember, we’ve simplified ${ϵ}_D=\frac{1}{slope}\times \frac{p}{q}$, where $\frac{1}{slope}=\frac{\mathrm{\Delta }q}{\mathrm{\Delta }p}$ because on a demand curve, $slope=\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q}$.

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

$$\begin{array}{rl}MR(q)& =MC(q)\\ p(1+\frac{1}{{ϵ}_D})& =MC(q)\\ p(1-\frac{1}{|{ϵ}_D|})& =MC(q)\end{array}$$

I rearrange the last line only to remind us that ${ϵ}_D$ is always negative!

Now note the following:

• If $|{ϵ}_D|<1$, 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 $|{ϵ}_D|=1$, then $MR(q)$ is 0. Only if $MC(q)$ is also 0 is this profit-maximizing.
• If $|{ϵ}_D|>1$, 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 33.

Derivation of the Lerner Index

Marginal revenue is strongly related to the price elasticity of demand, which is $E_D=\frac{\mathrm{\Delta }q}{\mathrm{\Delta }p}\times \frac{p}{q}$¹

We derived marginal revenue (in the slides) as:

$$MR(q)=p+\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q}q$$

Firms will always maximize profits where:

$$\begin{array}{rlrl}MR(q)& =MC(q)& & \text{Profit-max output condition}\\ p+(\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q})q& =MC(q)& & \text{Definition of }MR(q)\\ p+(\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q})q\times \frac{p}{p}& =MC(q)& & \text{Multiplying the left by }\frac{p}{p}\text{ (i.e. 1)}\\ p+\underset{\frac{1}{ϵ}}{\underbrace{(\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q}\times \frac{q}{p})}}\times p& =MC(q)& & \text{Rearranging the left}\\ p+\frac{1}{ϵ}\times p& =MC(q)& & \text{Recognize price elasticity }ϵ=\frac{\mathrm{\Delta }q}{\mathrm{\Delta }p}\times \frac{p}{q}\\ p& =MC(q)-\frac{1}{ϵ}p& & \text{Subtract }\frac{1}{ϵ}p\text{ from both sides}\\ p-MC(q)& =-\frac{1}{ϵ}p& & \text{Subtract }MC(q)\text{ from both sides}\\ \frac{p-MC(q)}{p}& =-\frac{1}{ϵ}& & \text{Divide both sides by }p\end{array}$$

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.

Firms With Market Power vs. Competitive Firms’ Responses to Market Changes

Consider a firm in a competitive market (left) and a firm with market power (right):

An Increase in Firms’ Marginal Cost

A competitive firm responds by only changing its output $q^{\star }$ (since it cannot control price), whereas the firm with market power changes both its $p^{\star }$ and $q^{\star }$.

A Shift in Market Demand

Both firms change $p^{\star }$ and $q^{\star }$, but there is a much smaller change in $q^{\star }$ for the monopolist.

A Change in Price Elasticity of Demand

For the competitive market on the left, there is no change in $q^{\star }$ or $p^{\star }$ for the industry! On the right, the monopolist will lower (raise) $p^{\star }$ and raise (lower) $q^{\star }$ as demand becomes more (less) elastic!

Footnotes

1. 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 $\frac{\mathrm{\Delta }p}{\mathrm{\Delta }q}=\frac{rise}{run}$.