Demand Functions and Demand Curves
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 8
Today's Agenda
Part 1: Brief Overview
Part 2: Worked Examples
Review of what demand is
Demand functions
Demand curves
Complements and Substitutes
Normal and Inferior Goods
Cobb-Douglas
Perfect Complements
Perfect Substitutes
Quadratic Quasilinear
Section: Three goods!
Remember what you learned about demand and demand curves in Econ 1 / high school:
- The demand curve shows the quantity demanded of a good at different prices
- A change in the price of a good results in a movement along its demand curve
- The demand curve represents the marginal benefit of an additional unit,
or alternatively the marginal willingness to pay for another unit - A change in income or the price of other goods results in a shift of the demand curve
- If two goods are substitutes, an increase in the price of one will increase the demand for the other (shift the demand curve to the right).
- If two goods are complements, an increase in the price of one will decrease the demand for the other (shift the demand curve to the left).
- If a good is a normal good, an increase in income will increase demand for the good
- If a good is an inferior good, an increase in income will decrease demand the good
Demand Curve for Good 1
\text{Fix }p_2\text{ and }m
\text{Plot }(x_1^*(p_1),p_1)
Demand Functions
x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)
"The demand curve shows the quantity demanded of a good at different prices"
x_1
x_1
x_2
p_1
DEMAND CURVE FOR GOOD 1
BL_{p_1 = 2}
BL_{p_1 = 3}
BL_{p_1 = 4}
2
3
4
"Good 1 - Good 2 Space"
"Quantity-Price Space for Good 1"
BL
"The demand curve represents the marginal benefit of an additional unit,
or alternatively the marginal willingness to pay for another unit"
\mathcal{L}(x_1,x_2,\lambda)=
u(x_1,x_2)+
(m - p_1x_1 - p_2x_2)
\lambda
\frac{\partial \mathcal{L}}{\partial x_1} = \frac{\partial u}{\partial x_1} - \lambda p_1
Let's look at the FOC with respect to good 1:
= 0 \Rightarrow \lambda = \frac{MU_1}{p_1}
Solve for \(p_1\):
\lambda = \text{bang for your buck, in }\frac{\text{utils}}{\$}
- A change in the price of a good results in a movement along its demand curve
- A change in income or the price of other goods results in a shift of the demand curve
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 \((x_1^*)\)?
x_1^*(p_1,p_2,m)
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
-
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
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 \((x_1^*)\)?
x_1^*(p_1,p_2,m)
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
-
How does \(x_1^*\) change with \(m\)?
- Income elasticity
- Normal vs. inferior goods
x_1^*(p_1,p_2,m)
Three Relationships
Substitutes
Complements
When the price of one good goes up, demand for the other increases.
When the price of one good goes up, demand for the other decreases.
-
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
Income Elasticity
What is the effect of an increase
in income \((m)\) on the quantity demanded of good 1 \((x_1^*)\)?
x_1^*(p_1,p_2,m)
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
Normal Goods
Inferior Goods
When your income goes up,
demand for the good increases.
When your income goes up,
demand for the good decreases.
-
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
Things to Think About
Think about how the behavior described by the demand function translates into the overall shape of the demand curve:
- Are there discontinuities/cutoff prices where behavior changes?
- What happens as the price gets really high, or approaches zero?
- What fraction of income is being spent on this good?
The reason we use different utility functions is because people's relationship with prices depends on the nature of their preferences.
Note: Maximum Possible Quantity Demanded
\overline x_1 = {m \over p_1}
Quantity of Good 1 \((x_1)\)
Price of Good 1 \((p_1)\)
All demand curves must be in this region
Quantity bought at each price if you spent all your money on good 1
x_1 = {m \over p_1}
Econ 50 | Fall 2021 | 8 | Demand
By Chris Makler
