# 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

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

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)

# 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

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)

# Three Relationships

### 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.

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)

# 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

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)

# Three Relationships

### Inferior Goods

demand for the good increases.

demand for the good decreases.

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)

# Three Relationships

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}