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
Elasticity
Why Elasticity?
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
-
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
-
How does \(x_1^*\) change with \(p_1\)?
- Own-price elasticity
- Elastic vs. inelastic
-
How does \(x_1^*\) change with \(p_2\)?
- Cross-price elasticity
- Complements vs. substitutes
x_1^*(p_1,p_2,m)
Three Relationships
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
-
How does \(x_1^*\) change with \(p_1\)?
- Own-price elasticity
- Elastic vs. inelastic
-
How does \(x_1^*\) change with \(p_2\)?
- Cross-price elasticity
- Complements vs. substitutes
-
How does \(x_1^*\) change with \(m\)?
- Income elasticity
- Normal vs. inferior goods
x_1^*(p_1,p_2,m)
Three Relationships
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)
Marked Demand and Supply
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
Conclusion: Aggregating Preferences
You can model market demand as reflecting the preferences of a single representative agent.
But...know that you're weighting the preferences of richer people more.
Econ 50 | Fall 2022 | 12 | Market Demand
By Chris Makler
