Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 12
Part 1: Elasticity
Part 2: Market Demand and Supply
"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
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?
(note: all of these refer to the ratio of the perentage change, not absolute change)
General formula:
Linear relationship:
Using calculus:
Multiplicative relationship:
Note: the slope of the relationship is \(b\).
Elasticity is related to, but not the same thing as, slope.
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!
How much of a good a consumer wants to buy, as a function of:
We can ask: how much does the amount of this good change, when one of those determinants changes?
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
[poll question coming up...]
pollev.com/chrismakler
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
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
Think about all the things we calculated for the function \(f(L,K)=L^{1 \over 4}K^{1 \over 4}\)
LONG RUN
SHORT RUN
LONG RUN
SHORT RUN
What is the output elasticity of conditional labor demand in the short run and long run?
Intuitively, why this difference?
In the long run, the firm uses
both labor and capital to increase output;
in the short run, it only increases labor.
Think about all the things we calculated for the function \(f(L,K)=L^{1 \over 4}K^{1 \over 4}\)
LONG RUN
SHORT RUN
LONG RUN
SHORT RUN
What is the price elasticity of supply
in the long run and short run?
Intuitively, why this difference?
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.
Think about all the things we calculated for the function \(f(L,K)=L^{1 \over 4}K^{1 \over 4}\)
LONG RUN
SHORT RUN
Work with a partner:
How would the firm respond to a
6% increase in the wage rate in the long run?
pollev.com/chrismakler
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)
L decreases by 3% (due to the +6% change in w)
and decreases by 6% (due to the -3% change in y),
for a total decrease of 3%.
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)
Note: we can calculate the LR profit-maximizing demand for labor:
Individual demand curve, \(d^i(p)\): quantity demanded by consumer \(i\) at each possible price
Market demand sums across all consumers:
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:
Firm supply curve, \(s^j(p)\): quantity supplied by firm \(j\) at each possible price
Market supply sums across all firms:
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
NOTATION AHEAD
STAY FOCUSED ON
ACTUAL ECONOMICS
Suppose each consumer has the utility function
where the \(\alpha\)'s all sum to 1.
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 \(\alpha = \frac{1}{2}, m = 30\)
Suppose there are only two goods, and each consumer has the utility function
So consumer \(i\) will spend fraction \(\alpha_i\) of their income \(m_i\) on good 1:
Market demand:
number of consumers
quantity demanded by each consumer
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:
Again there are only two goods, and each consumer has the utility function
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:\(\alpha_H = \frac{3}{4}, m_H = 40\)
(demand from
low-income)
(demand from
high-income)
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 \(\alpha = \frac{7}{12}, m = 6000\)
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...
If consumer \(i\)'s demand for good \(k\) is
then the market demand for good \(k\) is
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 \(\alpha_K\)?
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:
\(\alpha_K\)
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 \(\alpha_H = \frac{3}{4}, m_H = 40\)
Average income is \(\overline m = 30\), total income is \(M = 6000\)
So, market demand is
closer to \(\alpha_H\) than \(\alpha_L\)
Source: trulia.com, 5/12/22