1.5 Monopoly I: Behavior


Thursday, January 30, 2020 and Thursday, February 6, 2020


We begin looking at monopoly by discussing how a monopolist (as compared to a price-taking firm) chooses both its output and its price to maximize profits.



Price Elasticity of Demand Refresher

Price elasticity of demand measures how much (in %) quantity demanded changes in response to a (1%) change in price.

\[\begin{align*} \epsilon_{q,p} &= \frac{\% \Delta q}{\% \Delta p}\\ &= \cfrac{\left(\frac{\Delta q}{q}\right)}{\left(\frac{\Delta p}{p}\right)}\\ &= \frac{\Delta q}{\Delta p} \times \frac{p}{q}\\ &= \frac{1}{slope} \times \frac{p}{q}\\ \end{align*}\]

“Elastic” “Unit Elastic” “Inelastic”
Intuitively: Large response Proportionate response Little response
Mathematically: \(\vert \epsilon_{q_D,p}\vert > 1\) \(\vert \epsilon_{q_D,p}\vert = 1\) \(\vert \epsilon_{q_D,p} \vert < 1\)
Numerator \(>\) Denominator Numerator \(=\) Denominator Numerator \(<\) Denominator
A 1% change in \(p\) More than 1% change in \(q_D\) 1% change in \(q_D\) Less than 1% change in \(q_D\)

Price elasticity of demand changes along the demand curve:

Determinants of Price Elasticity

What determines how responsive your buying behavior is to a price change?

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:

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

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.