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Attendance trivia question:

The name of what company is formed from a portmanteau of Danish words that translate to English as "play well"?

Demand Shifters I: Changes in the Prices of Complements and Substitutes

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 14

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

Income and Substitution Effects

Demand function: how does an optimal bundle change when prices or income changes?

If we want to know how best to implement a policy, we want to know why it changes.

For example: we could be interested in how far a cannonball travels, so we can aim it at a target.

To do this, a physicist would decompose its velocity
into the horizontal portion and vertical portion:

Two Effects

Substitution Effect

Effect of change in relative prices, holding utility constant.

Effect of change in real income,
holding relative prices constant.

Income Effect

Decomposition Bundle

Suppose that, after a price change,
we compensated the consumer
just enough to afford some bundle
that would give the same utility
as they had before the price change?

The Hicks decomposition bundle
is the lowest-cost bundle
that satisfies this condition.

Approach

TOTAL EFFECT

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME EFFECT

Must the compensated budget line lie below A?

Income Effects with Inferior Goods

Let's think about an increase in the price of good 1.

\(BL_1\)

\(BL_2\)

We start out at bundle A.

The price of good 1 increases, pivoting the budget line in.

Suppose that if we found the decomposition bundle, it was this one, bundle B.

COMPENSATED BUDGET LINE

...so our compensated budget line looks like this.

Income Effects with Inferior Goods

\(BL_2\)

COMPENSATED BUDGET LINE

The shift from bundle B to the final bundle is the income effect; it represents the loss in purchasing power due to the price increase.

If both goods are normal goods,
then the income effect results in the consumer buying less of both goods.

Income Effects with Inferior Goods

\(BL_2\)

COMPENSATED BUDGET LINE

The shift from bundle B to the final bundle is the income effect; it represents the loss in purchasing power due to the price increase.

If good 1 is an inferior good, then the
income effect results in the consumer buying
less of good 2 and more of good 1.

Income Effects with Inferior Goods

\(BL_2\)

COMPENSATED BUDGET LINE

The shift from bundle B to the final bundle is the income effect; it represents the loss in purchasing power due to the price increase.

If good 1 is an extremely inferior good, then the income effect results in the consumer buying so much more of good 1 that they're buying more than at point A!!!

\(BL_1\)

Giffen Good

p_1 \uparrow \Rightarrow x_1 \uparrow

Next time...

How do we calculate the Hicks decomposition bundle?

New optimization problem:
cost minimization.