1.7 — Price Elasticity

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com

Outline

Price Elasticity of Demand and Revenues

Price Elasticity of Demand

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

$ϵ_{q_{D}, p} = \frac{% Δ q_{D}}{% Δ p}$

Price Elasticity of Demand: Elastic vs. Inelastic

$ϵ_{q_{D}, p} = \frac{% Δ q_{D}}{% Δ p}$

	“Elastic”	“Unit Elastic”	“Inelastic”
Intuitively:	Large response	Proportionate response	Little response
Mathematically:	$\| ϵ_{q_{D}, p} \| > 1$	$\| ϵ_{q_{D}, p} \| = 1$	$\| ϵ_{q_{D}, p} \| < 1$
	Numerator $>$ Denominator	Numerator $=$ Denominator	Numerator $<$ Denominator
1% change in $p$ causes	More than 1% change in $q_{D}$	Exactly 1% change in $q_{D}$	Less than 1% change in $q_{D}$

Visualizing Price Elasticity of Demand

An identical 50% price cut on an:

“Inelastic” Demand Curve

“Elastic” Demand Curve

Price Elasticity of Demand Formula

$ϵ_{q_{D}, p} = \frac{% Δ q_{D}}{% Δ p}$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{% Δ q}{% Δ p} = \frac{(\frac{Δ q}{q})}{(\frac{Δ p}{p})}$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{% Δ q}{% Δ p} = \frac{(\frac{Δ q}{q})}{(\frac{Δ p}{p})} = \frac{Δ q}{Δ p} \times \frac{p}{q}$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{Δ q}{Δ p} \times \frac{p}{q}$

First term: direction of the effect
- This is the price effect!
- Always negative!
Second term: magnitude of the effect
- Will change depending on $p$ and $q$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{Δ q}{Δ p} \times \frac{p}{q}$

You’ve learned “arc”-price elasticity using the “midpoint formula’ between 2 points
Here is a more general formula: the elasticity at any single point!
We can actually simplify this even more...does the first term remind you of anything?

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{1}{s l o p e} \times \frac{p}{q}$

First term is actually the inverse of the slope of the inverse demand curve (that we graph)!
To find the elasticity at any point, we need 3 things:
1. The price
2. The associated quantity demanded
3. The slope of (inverse) demand

Example

Example: The demand for movie tickets in a small town is given by:

$q = 1000 - 50 p$

Find the inverse demand function.
What is the price elasticity of demand at a price of $5.00?
What is the price elasticity of demand at a price of $12.00?
At what price is demand unit elastic (i.e. $ϵ_{q, p} = - 1)$ ?

Price Elasticity Changes Along the Demand Curve

$ϵ_{q, p} = \frac{1}{s l o p e} \times \frac{p}{q}$

Elasticity $\neq$ slope (but they are related)!
Price elasticity changes along the demand curve
Gets less elastic as $↓$ price $(↑$ quantity )
- $\frac{1}{s l o p e}$ is constant
- $\frac{p}{q}$ gets smaller as $↓ p$ and $↑ q$

Determinants of Price Elasticity of Demand

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

More (fewer) substitutes $⟹$ 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 $⟹$ more (less) elastic

Price Elasticity of Demand and Revenues

Price elasticity of demand is closely related to Revenues $(R)$ ^†

$R (q) = p q$

^† From the buyer’s side, this is total expenditures.

Price Elasticity of Demand and Revenues

Price elasticity of demand is closely related to Revenues $(R)$ ^†

$R (q) = p q$

Demand is	$Δ R$ and $Δ p$
Elastic $\| ϵ \| > 1$	p & R change opposite
Unit Elastic $\| ϵ \| = 1$	R maximized
Inelastic $\| ϵ \| < 1$	p & R change together

^† From the buyer’s side, this is total expenditures.

Price Elasticity of Demand and Revenues

Price elasticity of demand is closely related to Revenues $(R)$ ^†

$R (q) = p q$

Demand is	$Δ R$ and $Δ p$
Elastic $\| ϵ \| > 1$	p & R change opposite
Unit Elastic $\| ϵ \| = 1$	R maximized
Inelastic $\| ϵ \| < 1$	p & R change together

Revenues: Example I

Revenues: Example II

Visualizing Price Elasticity of Demand and Revenues

“Inelastic” Demand Curve

(Agricultural Products)

“Elastic” Demand Curve

(Computer Chips)

Price Elasticity and Revenues

$R (q) = p q$

$q$	$p$	$R (q)$
0	10	0
1	9	9
2	8	16
3	7	21
4	6	24
5	5	25
6	4	24
7	3	21
8	2	16
9	1	9
10	0	0

Revenue max'ed at price where $ϵ = - 1$

Price Elasticity and Revenues: Example I

Source: CNN (July 2, 2018)

"Build-A-Bear announced its Pay Your Age event earlier this week. Customers who show up to the stores can pay their current age for the popular stuffed animals. On Wednesday, the retailer wrote on its Facebook page that it was 'anticipating potential of long lines and wait times.'"

Price Elasticity and Revenues: Example II

Source: Wall Street Journal (Feb 3, 2019)

"While leaguewide average attendance dropped .43% this season to its lowest level since 2010, Atlanta’s attendance rose for the second season. Mercedes-Benz Stadium and the Falcons have become the model for drawing fans and keeping them happy."

"Instead of charging elevated sums—a long-held industry practice that fans despised—the Falcons would price most of its food at what it sold for on the street...Prices plunged 50%. Fans rejoiced. Although the team made less money on each $2 hot dog it sold, it made more overall. Average fan spending per game rose 16%. Atlanta’s food services, which ranked 18th in the NFL in the 2016 annual league survey, shot up to No. 1 in 2017 in every metric—and by a wide margin."

Price Elasticity and Revenues: Example III

Cowen & Tabarrok (2014: p.75)

Price Elasticity and Revenues: Example IV

Summing Up Unit 1

Models of Individual Choice I

"All models are lies. The art is telling useful lies." - George Box

Remember, we're not modelling the process by which people actually choose
We're predicting consequences (in people's choices) when parameters change

Models of Individual Choice II

Constrained optimization models are the main workhorse model in economics
All constrained optimization models have three moving parts:

Choose: < some alternative >
In order to maximize: < some objective >
Subject to: < some constraints >

Models of Individual Choice III

Applications of Consumer Theory

See today's class notes page for some applications of consumer theory:

Uncertainty: risky outcomes & insurance
Exchange: two individuals trading their endowments, general equilibrium, & Pareto efficiency
Taxes: Which is better for consumers, a consumption tax or a (revenue-equivalent) income tax?
Intertemporal choice: saving, borrowing, lending, & interest

1.7 — Price Elasticity

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com

Price Elasticity of Demand

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

$ϵ_{q_{D}, p} = \frac{% Δ q_{D}}{% Δ p}$

Price Elasticity of Demand: Elastic vs. Inelastic

$ϵ_{q_{D}, p} = \frac{% Δ q_{D}}{% Δ p}$

	“Elastic”	“Unit Elastic”	“Inelastic”
Intuitively:	Large response	Proportionate response	Little response
Mathematically:	$\| ϵ_{q_{D}, p} \| > 1$	$\| ϵ_{q_{D}, p} \| = 1$	$\| ϵ_{q_{D}, p} \| < 1$
	Numerator $>$ Denominator	Numerator $=$ Denominator	Numerator $<$ Denominator
1% change in $p$ causes	More than 1% change in $q_{D}$	Exactly 1% change in $q_{D}$	Less than 1% change in $q_{D}$

Visualizing Price Elasticity of Demand

An identical 50% price cut on an:

“Inelastic” Demand Curve

“Elastic” Demand Curve

Price Elasticity of Demand Formula

$ϵ_{q_{D}, p} = \frac{% Δ q_{D}}{% Δ p}$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{% Δ q}{% Δ p} = \frac{(\frac{Δ q}{q})}{(\frac{Δ p}{p})}$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{% Δ q}{% Δ p} = \frac{(\frac{Δ q}{q})}{(\frac{Δ p}{p})} = \frac{Δ q}{Δ p} \times \frac{p}{q}$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{Δ q}{Δ p} \times \frac{p}{q}$

First term: direction of the effect
- This is the price effect!
- Always negative!
Second term: magnitude of the effect
- Will change depending on $p$ and $q$

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{Δ q}{Δ p} \times \frac{p}{q}$

You’ve learned “arc”-price elasticity using the “midpoint formula’ between 2 points
Here is a more general formula: the elasticity at any single point!
We can actually simplify this even more...does the first term remind you of anything?

Price Elasticity of Demand Formula

$ϵ_{q, p} = \frac{1}{s l o p e} \times \frac{p}{q}$

First term is actually the inverse of the slope of the inverse demand curve (that we graph)!
To find the elasticity at any point, we need 3 things:
1. The price
2. The associated quantity demanded
3. The slope of (inverse) demand

Example

Example: The demand for movie tickets in a small town is given by:

$q = 1000 - 50 p$

Find the inverse demand function.
What is the price elasticity of demand at a price of $5.00?
What is the price elasticity of demand at a price of $12.00?
At what price is demand unit elastic (i.e. $ϵ_{q, p} = - 1)$ ?

Price Elasticity Changes Along the Demand Curve

$ϵ_{q, p} = \frac{1}{s l o p e} \times \frac{p}{q}$

Elasticity $\neq$ slope (but they are related)!
Price elasticity changes along the demand curve
Gets less elastic as $↓$ price $(↑$ quantity )
- $\frac{1}{s l o p e}$ is constant
- $\frac{p}{q}$ gets smaller as $↓ p$ and $↑ q$

Determinants of Price Elasticity of Demand

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

More (fewer) substitutes $⟹$ 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 $⟹$ more (less) elastic

Price Elasticity of Demand and Revenues

Price elasticity of demand is closely related to Revenues $(R)$ ^†

$R (q) = p q$

^† From the buyer’s side, this is total expenditures.

Price Elasticity of Demand and Revenues

Price elasticity of demand is closely related to Revenues $(R)$ ^†

$R (q) = p q$

Demand is	$Δ R$ and $Δ p$
Elastic $\| ϵ \| > 1$	p & R change opposite
Unit Elastic $\| ϵ \| = 1$	R maximized
Inelastic $\| ϵ \| < 1$	p & R change together

^† From the buyer’s side, this is total expenditures.

Price Elasticity of Demand and Revenues

Price elasticity of demand is closely related to Revenues $(R)$ ^†

$R (q) = p q$

Demand is	$Δ R$ and $Δ p$
Elastic $\| ϵ \| > 1$	p & R change opposite
Unit Elastic $\| ϵ \| = 1$	R maximized
Inelastic $\| ϵ \| < 1$	p & R change together

Revenues: Example I

Revenues: Example II

Visualizing Price Elasticity of Demand and Revenues

“Inelastic” Demand Curve

(Agricultural Products)

“Elastic” Demand Curve

(Computer Chips)

Price Elasticity and Revenues

$R (q) = p q$

$q$	$p$	$R (q)$
0	10	0
1	9	9
2	8	16
3	7	21
4	6	24
5	5	25
6	4	24
7	3	21
8	2	16
9	1	9
10	0	0

Revenue max'ed at price where $ϵ = - 1$

Price Elasticity and Revenues: Example I

Source: CNN (July 2, 2018)

"Build-A-Bear announced its Pay Your Age event earlier this week. Customers who show up to the stores can pay their current age for the popular stuffed animals. On Wednesday, the retailer wrote on its Facebook page that it was 'anticipating potential of long lines and wait times.'"

Price Elasticity and Revenues: Example II

Source: Wall Street Journal (Feb 3, 2019)

Price Elasticity and Revenues: Example III

Cowen & Tabarrok (2014: p.75)

Price Elasticity and Revenues: Example IV

Summing Up Unit 1

Models of Individual Choice I

"All models are lies. The art is telling useful lies." - George Box

Remember, we're not modelling the process by which people actually choose
We're predicting consequences (in people's choices) when parameters change

Models of Individual Choice II

Constrained optimization models are the main workhorse model in economics
All constrained optimization models have three moving parts:

Choose: < some alternative >
In order to maximize: < some objective >
Subject to: < some constraints >

Models of Individual Choice III

Applications of Consumer Theory

See today's class notes page for some applications of consumer theory:

Uncertainty: risky outcomes & insurance
Exchange: two individuals trading their endowments, general equilibrium, & Pareto efficiency
Taxes: Which is better for consumers, a consumption tax or a (revenue-equivalent) income tax?
Intertemporal choice: saving, borrowing, lending, & interest

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1.7 — Price Elasticity

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner Assistant Professor of Economics safner@hood.edu ryansafner/microS22 microS22.classes.ryansafner.com

Outline

Price Elasticity of Demand

Price Elasticity of Demand

Price Elasticity of Demand: Elastic vs. Inelastic

Visualizing Price Elasticity of Demand

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Example

Price Elasticity Changes Along the Demand Curve

Determinants of Price Elasticity of Demand

Price Elasticity of Demand and Revenues

Price Elasticity of Demand and Revenues

Price Elasticity of Demand and Revenues

Price Elasticity of Demand and Revenues

Revenues: Example I

Revenues: Example I

Revenues: Example II

Revenues: Example II

Visualizing Price Elasticity of Demand and Revenues

Price Elasticity and Revenues

Price Elasticity and Revenues: Example I

Price Elasticity and Revenues: Example II

Price Elasticity and Revenues: Example III

Price Elasticity and Revenues: Example IV

Summing Up Unit 1

Models of Individual Choice I

Models of Individual Choice II

Models of Individual Choice III

Applications of Consumer Theory

Outline

Help

1.7 — Price Elasticity

1.7 — Price Elasticity

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner Assistant Professor of Economics safner@hood.edu ryansafner/microS22 microS22.classes.ryansafner.com

Outline

Price Elasticity of Demand

Price Elasticity of Demand

Price Elasticity of Demand: Elastic vs. Inelastic

Visualizing Price Elasticity of Demand

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Price Elasticity of Demand Formula

Example

Price Elasticity Changes Along the Demand Curve

Determinants of Price Elasticity of Demand

Price Elasticity of Demand and Revenues

Price Elasticity of Demand and Revenues

Price Elasticity of Demand and Revenues

Price Elasticity of Demand and Revenues

Revenues: Example I

Revenues: Example I

Revenues: Example II

Revenues: Example II

Visualizing Price Elasticity of Demand and Revenues

Price Elasticity and Revenues

Price Elasticity and Revenues: Example I

Price Elasticity and Revenues: Example II

Price Elasticity and Revenues: Example III

Price Elasticity and Revenues: Example IV

Summing Up Unit 1

Models of Individual Choice I

Models of Individual Choice II

Models of Individual Choice III

Applications of Consumer Theory

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com