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

Last Two Classes: What is the optimal bundle for a given budget line?

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

Specific Prices & Income

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}

Specific Prices & Income

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...

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.

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?

Normal Goods

Inferior Goods

When your income goes up,
demand for the good increases.

When your income goes up,
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

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}