# Elasticity Market Demand & Supply

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 12

## Today's Agenda

Part 1: Elasticity

Part 2: Market Demand and Supply

• Why elasticity?
• Degrees of elasticity
• Calculating elasticity
• Demand elasticities
• Supply elasticities
• Aggregating demand and supply
• How market demand aggregates preferences

# Notation

\epsilon_{Y,X} = \frac{\% \Delta Y}{\% \Delta X}
\text{Elasticity }(\epsilon) = \frac{\% \text{ change in endogenous variable}}{\% \text{ change in exogenous variable}}

"X elasticity of Y"
or "Elasticity of Y with respect to X"

Examples:
"Price elasticity of demand"
"Income elasticity of demand"
"Cross-price elasticity of demand"
"Price elasticity of supply"

Perfectly Inelastic

Inelastic

Unit Elastic

Elastic

Perfectly Elastic

|\epsilon| = 0
|\epsilon| < 1
|\epsilon| = 1
|\epsilon| > 1
|\epsilon| = \infty

Doesn't change

Changes by less than the change in X

Changes proportionally to the change in X

Changes by more than the change in X

Changes "infinitely" (usually: to/from zero)

How does the endogenous variable Y respond to a
change in the exogenous variable X?

\text{Elasticity }(\epsilon) = \frac{\% \text{ change in endogenous variable}}{\% \text{ change in exogenous variable}}

(note: all of these refer to the ratio of the perentage change, not absolute change)

## Using Elasticities

• Suppose the price elasticity of demand is -2.
• This means that each % increase in the price
leads to approximately a 2% decrease in the quantity demanded
• Example 1: a 3% increase in price would lead to a ~6% decrease in quantity
• Example 2: a 0.5% decrease in price would lead to a ~1% increase in quantity
• These are approximations in the same way as if $$dy/dx = -2$$ along a function, increasing $$x$$ by 3 would cause $$y$$ to decrease by approximately 6.
\epsilon_{Y,X}
= \frac{\% \Delta Y}{\% \Delta X}
= \frac{\Delta Y / Y}{\Delta X / X}
= \frac{\Delta Y}{\Delta X}\times \frac{X}{Y}
\text{Suppose }Y \text{ is a function of }X.

General formula:

Linear relationship:

Using calculus:

Multiplicative relationship:

Y = mX + b
Y = f(X)
Y = aX^b
\text{General formula: }\epsilon_{Y,X} = \frac{\Delta Y}{\Delta X}\times \frac{X}{Y}
\text{If }Y = a + bX \text{, then }\frac{\Delta Y}{\Delta X} =
b
\epsilon_{Y,X} = b\times \frac{X}{a + bX}

Note: the slope of the relationship is $$b$$.

Elasticity is related to, but not the same thing as, slope.

= \frac{bX}{a + bX}
\epsilon_{y,x}
= \frac{\% \Delta y}{\% \Delta x}
= \frac{\Delta y / y}{\Delta x / x}
= \frac{\Delta y}{\Delta x}\times \frac{x}{y}
\text{Suppose }y \text{ is a function of }x.
\text{In the limit, as }\Delta x \rightarrow 0:
\epsilon_{y,x} = \frac{dy}{dx}\times \frac{x}{y}
\text{General formula: }\epsilon_{y,x} = \frac{dy}{dx}\times \frac{x}{y}
\text{If }y = ax^b \text{, then }\frac{dy}{dx} =
abx^{b-1}
\epsilon_{y,x} = abx^{b-1} \times \frac{x}{ax^b}
= b

This is related to logs, in a way that you can explore in the homework.

This is a super useful trick and one that comes up on midterms all the time!

# Demand Elasticities

x_1^*(p_1,p_2,m)

How much of a good a consumer wants to buy, as a function of:

• the price of that good
• the price of other goods
• their income

We can ask: how much does the amount of this good change, when one of those determinants changes?

# Own-Price Elasticity

What is the effect of a 1% change
in the price of good 1 $$(p_1)$$ on the quantity demanded of good 1 $$(x_1^*)$$?

no change

perfectly inelastic

less than 1%

inelastic

exactly 1%

unit elastic

more than 1%

elastic

1. How does $$x_1^*$$ change with $$p_1$$?
• ​​Own-price elasticity
• Elastic vs. inelastic
x_1^*(p_1,p_2,m)

# Three Relationships

## Price Elasticity of Demand or Supply: $$\epsilon = \frac{\%\Delta Q}{\%\Delta P}$$

\text{Perfectly elastic}
|\epsilon| = \infty
\text{Elastic}
|\epsilon| >1
\text{Unit Elastic}
|\epsilon| = 1
\text{Inelastic}
|\epsilon| < 1
\text{Perfectly Inelastic}
|\epsilon| = 0

[poll question coming up...]

pollev.com/chrismakler

# Cross-Price Elasticity

What is the effect of an increase
in the price of good 2 $$(p_2)$$ on the quantity demanded of good 1 $$(x_1^*)$$?

no change

independent

decrease

complements

increase

substitutes

1. How does $$x_1^*$$ change with $$p_1$$?
• ​​Own-price elasticity
• Elastic vs. inelastic
2. How does $$x_1^*$$ change with $$p_2$$?
• ​​Cross-price elasticity
• Complements vs. substitutes
x_1^*(p_1,p_2,m)

# Income Elasticity

What is the effect of an increase
in income $$(m)$$ on the quantity demanded of good 1 $$(x_1^*)$$?

decrease

good 1 is inferior

increase

good 1 is normal

1. How does $$x_1^*$$ change with $$p_1$$?
• ​​Own-price elasticity
• Elastic vs. inelastic
2. How does $$x_1^*$$ change with $$p_2$$?
• ​​Cross-price elasticity
• Complements vs. substitutes
3. How does $$x_1^*$$ change with $$m$$?
• ​​Income elasticity
• Normal vs. inferior goods
x_1^*(p_1,p_2,m)

# Supply Elasticities

Think about all the things we calculated for the function $$f(L,K)=L^{1 \over 4}K^{1 \over 4}$$

LONG RUN

L^c(w,r,y) = w^{-{1 \over 2}}r^{1 \over 2}y^2
K^c(w,r,y) = w^{1 \over 2}r^{-{1 \over 2}}y^2
y^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

L^c_s(y|\overline K) = {1 \over \overline K}y^4
y_s^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

LONG RUN

L^c(w,r,y) = w^{-{1 \over 2}}r^{1 \over 2}y^2

SHORT RUN

L^c_s(y|\overline K) = {1 \over \overline K}y^4

What is the output elasticity of conditional labor demand in the short run and long run?

Intuitively, why this difference?

\epsilon_{L^c,y}=
\epsilon_{L^c_s,y}=
2
4

In the long run, the firm uses
both labor and capital to increase output;
in the short run, it only increases labor.

# Supply Elasticities

Think about all the things we calculated for the function $$f(L,K)=L^{1 \over 4}K^{1 \over 4}$$

LONG RUN

L^c(w,r,y) = w^{-{1 \over 2}}r^{1 \over 2}y^2
K^c(w,r,y) = w^{1 \over 2}r^{-{1 \over 2}}y^2
y^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

L^c_s(y|\overline K) = {1 \over \overline K}y^4
y_s^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

LONG RUN

y^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

y_s^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

What is the price elasticity of supply
in the long run and short run?

Intuitively, why this difference?

\epsilon_{y,p}=
\epsilon_{y_s,p}=
1
{1 \over 3}

In the long run, the firm can adjust its capital to keep its costs down; so its marginal cost rises less steeply, and its supply curve is flatter.

# Supply Elasticities

Think about all the things we calculated for the function $$f(L,K)=L^{1 \over 4}K^{1 \over 4}$$

LONG RUN

L^c(w,r,y) = w^{-{1 \over 2}}r^{1 \over 2}y^2
K^c(w,r,y) = w^{1 \over 2}r^{-{1 \over 2}}y^2
y^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

SHORT RUN

L^c_s(y|\overline K) = {1 \over \overline K}y^4
y_s^*(w,r,p) = \left({\overline K \over 4}\right)^{1 \over 3}w^{-{1 \over 3}} p^\frac{1}{3}

Work with a partner:
How would the firm respond to a
6% increase in the wage rate in the long run?

pollev.com/chrismakler

L^c(w,r,y) = w^{-{1 \over 2}}r^{1 \over 2}y^2
K^c(w,r,y) = w^{1 \over 2}r^{-{1 \over 2}}y^2
y^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

How would the firm respond to a
6% increase in the wage rate in the long run?

A 6% increase in w decreases y by 3%.

and decreases by 6% (due to the -3% change in y),
for a total decrease of 9%.

K increases by 3% (due to the +6% change in w)

\epsilon_{L^c,w}=-{1 \over 2}
\epsilon_{L^c,y}=+2
\epsilon_{y,w}=-{1 \over 2}

L decreases by 3% (due to the +6% change in w)

\epsilon_{K^c,w}=+{1 \over 2}
\epsilon_{K^c,y}=+2

and decreases by 6% (due to the -3% change in y),
for a total decrease of 3%.

L^c(w,r,y) = w^{-{1 \over 2}}r^{1 \over 2}y^2
y^*(w,r,p) = {1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p

How would the firm respond to a
6% increase in the wage rate in the long run?

A 6% increase in w decreases y by 3%.

and decreases by 6% (due to the -3% change in y),
for a total decrease of 9%.

L decreases by 3% (due to the +6% change in w)

L^*(w,r,p) = w^{-{1 \over 2}}r^{1 \over 2}[y^*(w,r,p)]^2
= w^{-{1 \over 2}}r^{1 \over 2}\left[{1 \over 4}w^{-{1 \over 2}}r^{-{1 \over 2}}p\right]^2
= {1 \over 16}w^{-{3 \over 2}}r^{-{1 \over 2}}p^2

Note: we can calculate the LR profit-maximizing demand for labor:

## What did we just show?

• If there is a direct causality $$X \rightarrow Y$$, elasticity measures how Y responds to X.
• If there is a chain of causality $$X \rightarrow Y \rightarrow Z$$, the elasticity composes just like a function does (like the chain rule for elasticity)

# Individual and Market Demand

Individual demand curve, $$d^i(p)$$: quantity demanded by consumer $$i$$ at each possible price

Market demand sums across all consumers:

\displaystyle D(p) = N_Cd(p)
\displaystyle D(p) = \sum_{i=1}^{N_C}{d^i(p)}

If all of those consumers are identical and demand the same amount $$d(p)$$:

There are $$N_C$$ consumers, indexed with superscript $$i \in \{1, 2, 3, ..., N_C\}$$.

Market demand curve, $$D(p)$$: quantity demanded by all consumers at each possible price

Market demand sums across all consumers:

\displaystyle D(p) = \sum_{i=1}^{N_C}{d^i(p)}

# Individual and Market Supply

Firm supply curve, $$s^j(p)$$: quantity supplied by firm $$j$$ at each possible price

Market supply sums across all firms:

\displaystyle S(p) = N_Fs(p)
\displaystyle S(p) = \sum_{j=1}^{N_F}{s^j(p)}

If all of those firms are identical and supply the same amount $$s(p)$$:

There are $$N_F$$ competitive firms, indexed with superscript $$j \in \{1, 2, 3, ..., N_F\}$$.

Market supply curve, $$S(p)$$: quantity supplied by all firms at each possible price

# How Demand Aggregates Preferences

\Sigma
NOTATION AHEAD

STAY FOCUSED ON
ACTUAL ECONOMICS

## Special Case: Cobb-Douglas

u(x_1,x_2,...,x_n) = \alpha_1 \ln x_1 + \alpha_2 \ln x_2 + \cdots + \alpha_n \ln x_n

Suppose each consumer has the utility function

where the $$\alpha$$'s all sum to 1.

x_{k,i}^*(p_1,p_2,...,p_n) = \frac{\alpha_{k,i}\times m_i}{p_k}

We've shown before that if consumer $$i$$'s income is $$m$$, their demand for good $$k$$ is

quantity demanded of good $$k$$ by consumer $$i$$

consumer $$i$$'s preference weighting of good $$k$$

consumer $$i$$'s income

price of good $$k$$

There are 200 people, and they each have $$\alpha = \frac{1}{2}, m = 30$$

u_i(x_1,x_2) = \alpha_i \ln x_1 + (1-\alpha_i) \ln x_2

Suppose there are only two goods, and each consumer has the utility function

x_{1,i}^*(p_1) = \frac{\alpha_{i}\times m_i}{p_1}

So consumer $$i$$ will spend fraction $$\alpha_i$$ of their income $$m_i$$ on good 1:

x_1^*(p_1) = \frac{15}{p_1}
D_1(p_1) = 200 \times \frac{15}{p_1}

Market demand:

= \frac{3000}{p_1}

number of consumers

quantity demanded by each consumer

= \frac{3000}{p_1}

Note: total income is $$200 \times 30 = 6000$$, so this means the demand is the same "as if" there were one "representative agent" with $$\alpha = \frac{1}{2}, m = 6000$$

Individual demand:

Now suppose there are two types of consumers:

u_i(x_1,x_2) = \alpha_i \ln x_1 + (1-\alpha_i) \ln x_2

Again there are only two goods, and each consumer has the utility function

x_{1,i}^*(p_1) = \frac{\alpha_{i}\times m_i}{p_1}
x_{1,L}^*(p_1) = \frac{5}{p_1}
x_{1,H}^*(p_1) = \frac{30}{p_1}
D_1(p_1) = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +

100 low-income consumers who don't like this good: $$\alpha_L = \frac{1}{4}, m_L = 20$$

100 high-income consumers who do like this good:$$\alpha_H = \frac{3}{4}, m_H = 40$$

100\times \frac{30}{p_1}
= \frac{3500}{p_1}
100\times \frac{5}{p_1}

(demand from
low-income)

(demand from
high-income)

\Rightarrow

Market demand:

Individual demand:

Note: total income is $$100 \times 20 + 100 \times 40 = 6000$$, so this means the demand is the same "as if" there were one "representative agent" with $$\alpha = \frac{7}{12}, m = 6000$$

# Conundrum

In both cases, average income was 30 and average preference parameter $$\alpha$$ was $$\frac{1}{2}$$.

When everyone was identical, it was "as if"
there was a representative agent with all the money
with preference parameter $$\alpha = \frac{1}{2}$$.

When rich people had a higher $$\alpha$$, it was "as if"
there was a representative agent with all the money
with preference parameter $$\alpha = \frac{7}{12} > \frac{1}{2}$$.

Feel free to tune out the intermediate steps, but hang on to the econ...

## How market demand aggregates preferences

x_{k,i}^*(p_1,p_2,p_3,...,p_n) = \frac{\alpha_{k,i} m_i}{p_k}

If consumer $$i$$'s demand for good $$k$$ is

then the market demand for good $$k$$ is

\displaystyle = \sum_{i=1}^{N_C}\frac{\alpha_{k,i} m_i}{p_k}
\displaystyle = \frac{\alpha_k M}{p_k}
\displaystyle D_k(p_k) = \sum_{i=1}^{N_C}x_{k,i}^*(p_k)

where $$M = \sum m_i$$ is the total income of all consumers
and $$\alpha_k$$ is an "aggregate preference" parameter.

Conclusion: we can model demand from $$N_C$$ consumers with Cobb-Douglas preferences
"as if" they were a single consumer with "average" Cobb-Douglas preferences.

so what is $$\alpha_K$$?

\displaystyle \text{We want to get from }D(p) = \sum_{i=1}^{N_C}\frac{\alpha_{k,i} m_i}{p_k}\text{ to }D(p) = \frac {\alpha_k M}{p_k}

If everyone has the same income ($$m_i = \overline m$$ for all $$i$$), then demand simply aggregates preferences:

Let $$\overline m = M/N_C$$ be the average income. Then we can rewrite market demand as:

\displaystyle D_k(p_k) = \sum_{i=1}^{N_C}\frac{\alpha_{k,i} m_i}{p_k} \times \frac{M}{N_C \overline m}
\displaystyle = \frac{1}{N_C}\sum_{i=1}^{N_C}\left(\alpha_{k,i} \times \frac{ m_i}{\overline m}\right) \times \frac{M}{p}

$$\alpha_K$$

\displaystyle \alpha_k = \frac{1}{N_C}\sum_{i=1}^{N_C}\alpha_{k,i}

But if there is income inequality, $$\alpha_k$$ gives more weight to the prefs of those with higher income.

$$=1$$

Example: consider an economy in which rich consumers like a good more:

100 low-income people with $$\alpha_L = \frac{1}{4}, m_L = 20$$,

100 high-income people with $$\alpha_H = \frac{3}{4}, m_H = 40$$

\displaystyle \alpha = \frac{1}{N_C}\sum_{i=1}^{N_C}\left(\alpha_{i} \times \frac{ m_i}{\overline m}\right)
\displaystyle = \frac{1}{200}\left[100 \times \left(\frac{1}{4} \times \frac{20}{30}\right) + 100 \times \left(\frac{3}{4} \times \frac{40}{30}\right)\right]
\displaystyle = \frac{7}{12}

Average income is $$\overline m = 30$$, total income is $$M = 6000$$

So, market demand is

\displaystyle D(p) = \alpha \times \frac{M}{p} = \frac{7}{12} \times \frac{6000}{p} = \frac{3500}{p}

closer to $$\alpha_H$$ than $$\alpha_L$$

## Example, revisited

Source: trulia.com, 5/12/22

By Chris Makler

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