# Demand

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 11

## Today's Agenda

Part 1:
Comparative Statics

Part 2:
Functional Forms and Behavior

• Changes in price:
• Price offer curves
• Demand curves
• Complements and Substitutes
• Changes in income:
• Income offer curves
• Engel curves
• Normal, and inferior goods
• Cobb-Douglas
• Perfect Complements
• Perfect Substitutes
• Quasilinear

## Today: What happens to the optimal bundle when prices/income change?

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BL1

We will be solving for the optimal bundle
as a function of income and prices:

The solutions to this problem will be called the demand functions. We have to think about how the optimal bundle will change when $$p_1,p_2,m$$ change.

x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)

BL2

### General Prices & Income

\text{Constraint: }2 x_1 + x_2 = 12

Plug tangency condition back into constraint:

Tangency Condition: $$MRS = p_1/p_2$$

\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Objective function: } x_1^{1 \over 2}x_2^{1 \over 2}
MRS(x_1,x_2) = {x_2 \over x_1}
{x_2 \over x_1}
=
{2 \over 1}
\Rightarrow x_2 = 2x_1
{x_2 \over x_1}
=
{p_1 \over p_2}
\Rightarrow x_2 = {p_1 \over p_2}x_1
2x_1 + 2x_1 = 12
x_1^* = 3
p_1x_1 + p_2\left[{p_1 \over p_2}x_1\right] = m
x_1^*(p_1,p_2,m) = {m \over 2p_1}
4x_1 = 12
2p_1x_1 = m
x_2^* = 2x_1^* = 6
x_2^*(p_1,p_2,m) = {m \over 2p_2}

### General Prices & Income

\text{Constraint: }2 x_1 + x_2 = 12
\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Objective function: } x_1^{1 \over 2}x_2^{1 \over 2}
MRS(x_1,x_2) = {x_2 \over x_1}
x_1^* = 3
x_1^*(p_1,p_2,m) = {m \over 2p_1}
x_2^* = 2x_1^* = 6
x_2^*(p_1,p_2,m) = {m \over 2p_2}

OPTIMAL BUNDLE

DEMAND FUNCTIONS

(optimization)

(comparative statics)

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
• 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
x_1^*(p_1,p_2,m)\ \

## Three Relationships

...its own price changes?

Movement along the demand curve

...the price of another good changes?

Complements

Substitutes

Independent Goods

How does the quantity demanded of a good change when...

...income changes?

Normal goods

Inferior goods

Giffen goods

(possible) shift of the demand curve

x_1^*(p_1,p_2,m)\ \

## Three Relationships

...its own price changes?

Movement along the demand curve

How does the quantity demanded of a good change when...

The demand curve for a good

shows the quantity demanded of that good

as a function of its own price

holding all other factors constant

(ceteris paribus)

The price offer curve shows how the optimal bundle changes in good 1-good 2 space as the price of one good changes.

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

PRICE OFFER CURVE

x_1^*(p_1,p_2,m)\ \

## Three Relationships

...the price of another good changes?

How does the quantity demanded of a good change when...

### 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.

### Independent

Demand not related

x_1
x_2
x_1
x_2

Complements:    $$p_2 \uparrow \Rightarrow x_1^* \downarrow$$

What happens to the quantity of good 1 demanded when the price of good 2 increases?

Substitutes:    $$p_2 \uparrow \Rightarrow x_1^* \uparrow$$

COMPLEMENTS:

UPWARD-SLOPING

PRICE OFFER CURVE

SUBSTITUTES:

DOWNWARD-SLOPING

PRICE OFFER CURVE

x_1^*(p_1,p_2,m)\ \

## Three Relationships

How does the quantity demanded of a good change when...

...income changes?

### Inferior Goods

demand for the good increases.

demand for the good decreases.

The income offer curve shows how the optimal bundle changes in good 1-good 2 space as income changes.

x_1
x_2
x_1
x_2

Good 1 normal:    $$m \uparrow \Rightarrow x_1^* \uparrow$$

What happens to the quantity of good 1 demanded when the income increases?

Good 1 inferior:    $$m \uparrow \Rightarrow x_1^* \downarrow$$

BOTH NORMAL GOODS:

UPWARD-SLOPING

INCOME OFFER CURVE

ONE GOOD INFERIOR:

DOWNWARD-SLOPING

PRICE OFFER CURVE

• 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

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}

By Chris Makler

• 603