# 1.5 Monopoly I: Behavior

## Contents

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

## Overview

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.

## Slides

## Appendix

### 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?

- More (fewer)
**substitutes**\(\implies\) more (less) elastic- Larger categories of products (less elastic) vs. specific brand (more elastic)
**Necessities**(less elastic) vs.**luxuries**(more elastic)- Large (more elastic) vs. small (less elastic)
**portion of budget**

- More (less)
**time to adjust**\(\implies\) more (less) elastic

### 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.