Shifts in Demand Curves; Indirect Utility Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 10

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What does this imply about complements and substitutes for marijuana?

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)

x_1

x_2

p_1

DEMAND CURVE FOR GOOD 1

BL_{p_1 = 2}

BL_{p_1 = 3}

BL_{p_1 = 4}

"Good 1 - Good 2 Space"

"Quantity-Price Space for Good 1"

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: \(p_2 \uparrow \Rightarrow x_1^* \uparrow\)

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Goods are complements if which of the following would cause a RIGHTWARD shift in the demand curve for good 1?

An increase in the price of good 2

A decrease in the price of good 2

An increase in income

A decrease in income

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

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: \(m \uparrow \Rightarrow x_1^* \downarrow\)

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The "rule" for Cobb-Douglas is that you spend a certain fraction of your income on each good, regardless of prices or income.

What does this make the two goods?

Complements

Substitutes

Normal

Inferior

CES Utility

= \begin{cases}\infty & \text{ if } x_1 < x_2 \\ 0 & \text{ if } x_1 > x_2 \end{cases}

-\infty

MRS = \left(x_2 \over x_1\right)^\infty

MRS = {x_2 \over x_1}

MRS = 1

u(x_1,x_2) = \min\{x_1,x_2\}

u(x_1,x_2) = x_1x_2

u(x_1,x_2) = x_1 + x_2

PERFECT
SUBSTITUTES

PERFECT
COMPLEMENTS

INDEPENDENT

PERFECT
SUBSTITUTES

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}

[50Q only]

u(x_1,x_2) = (x_1^r+x_2^r)^{1 \over r}

Constant Elasticity of Substitution (CES) Utility

MRS = \left(x_2 \over x_1\right)^{1-r}

-\infty

COMPLEMENTS: \(r < 0\)

SUBSTITUTES: \(r > 0\)

-1

MRS = \left(x_2 \over x_1\right)^2

{1 \over 2}

MRS = \left(x_2 \over x_1\right)^{1 \over 2}

u(x_1,x_2) = (x_1^{-1}+x_2^{-1})^{-1}

u(x_1,x_2) = \left(x_1^{1 \over 2} + x_2^{1 \over 2}\right)^2

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

Why are goods complements or substitutes?

Substitution Effect

Effect of change in relative prices

Effect of change in real income

Income Effect

Decomposition Bundle

Suppose that, after a price change,
we adjusted the consumer's income
so that they had exactly the amount of money needed to afford their initial bundle \(A\)?

Approach

TOTAL EFFECT

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME EFFECT

THIS SLIDE NOT ON HOMEWORK OR TESTS

How does this explain complements and substitutes?

Complements

Substitutes

When the price of good 1 goes up...

Net effect: buy less of both goods

Net effect: buy less good 1 and more good 2

Substitution effect: buy less of good 1 and more of good 2

Income effect (if both goods normal): buy less of both goods

Substitution effect dominates

Income effect dominates

THIS SLIDE NOT ON HOMEWORK OR TESTS

u(x_1,x_2) = x_1x_2

x_1^*(p_1,p_2,m) = {m \over 2p_1}

x_2^*(p_1,p_2,m) = {m \over 2p_2}

What is your utility when you are
maximizing your utility?

V(p_1,p_2,m)=u(x_1^*(p_1,p_2,m),x_2^*(p_1,p_2,m))

= x_1^*(p_1,p_2,m) \times x_2^*(p_1,p_2,m)

= \displaystyle{{m \over 2p_1} \times {m \over 2p_2}}

= \displaystyle{{m^2 \over 4p_1p_2}}

INDIRECT UTILITY FUNCTION

What is the marginal utility
of another dollar?

\displaystyle{{\partial V(p_1,p_2,m) \over \partial m} = {2m \over 4p_1p_2}}

\displaystyle{= {m \over 2p_1p_2}}

Write these down, and let's look at Lagrange one last time.

Indirect Utility Functions and the Utility of Money

UTILITY FUNCTION

DEMAND FUNCTIONS

\max

The Lagrange Method: Utility Maximization

x_1,x_2

\text{s.t.}

m - p_1x_1 - p_2x_2

Income left over

x_1x_2

Utility

\ge 0

The Lagrange Method: Utility Maximization

m - p_1x_1 - p_2x_2

Income left over

\mathcal{L}(x_1,x_2,\lambda)=

\lambda

x_1x_2

(

)

Utility

(utils)

(dollars)

utils/dollar

\frac{\partial \mathcal{L}}{\partial x_1} = x_2 - \lambda p_1

First Order Conditions

\frac{\partial \mathcal{L}}{\partial x_2} = x_1 - \lambda p_2

\frac{\partial \mathcal{L}}{\partial \lambda} = m - p_1x_1 - p_2x_2

= 0 \Rightarrow \lambda = \frac{x_2}{p_1}

= 0 \Rightarrow \lambda = \frac{x_1}{p_2}

The Lagrange Method: Utility Maximization

m - p_1x_1 - p_2x_2

\mathcal{L}(x_1,x_2,\lambda)=

\lambda

(

)

x_1x_2

x_1^*(p_1,p_2,m) = {m \over 2p_1}

x_2^*(p_1,p_2,m) = {m \over 2p_2}

Solutions:

What's the value of \(\lambda\) at the optimal bundle?

= \frac{m/2p_2}{p_1}

= \frac{m/2p_1}{p_2}

= \frac{m}{2p_1p_2}

\displaystyle{{\partial V(p_1,p_2,m) \over \partial m} = {m \over 2p_1p_2}}

Next Week

Foundations of Finance:
Applications of consumer theory

MONDAY

WEDNESDAY

FRIDAY

Intertemporal Choice

Uncertainty and Risk

Applications & Review

We will assume there is some
indirect utility function over consumption \(v(c)\),
and look at decisions about shifting consumption
across time and across states of the world.