Decomposing a Price Change into Income and Substitution Effects

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 10

Solving Climate Change in one Easy Step

Climate change is real and caused in large part by burning fossil fuels.
Transportation is a big part of the problem.
A higher tax on gasoline would make people drive less, and have many other benefits. See Mankiw's Pigou Club Manifesto.
Estimates vary, but $10/gallon seems about right.

but...

$10/gallon gas would hurt a lot of people, especially those with lower incomes who depend on a car to get to work

Can we give them a certain amount of money that would make them no worse off than they were before, to compensate them for the price increase?

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

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

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: $m \uparrow \Rightarrow x_1^* \downarrow$

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 purchasing power constant.

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

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 $X$?

Approach

TOTAL EFFECT

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME EFFECT

Slutsky Decomposition

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

$BL_1$

$BL_2$

We start out at bundle X.

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

The new optimal bundle is Z.

Slutsky Decomposition

$BL_1$

$BL_2$

Now suppose we had given the consumer just enough income to afford their original bundle $X$ at the new price.

DECOMPOSITION BUDGET LINE $BL_D$

This "decompensation budget line" would have the same slope as $BL_2$, but pass through bundle $X$.

We call the point the consumer would have bought with this budget line the Slutsky Decomposition Bundle $Y$

SUBSTITUTION EFFECT

INCOME EFFECT

Substitution effect: $X \rightarrow Y$

Income effect: $Y \rightarrow Z$

Slutsky Decomposition

$BL_1$

$BL_2$

One way to think about this decomposition is that the budget line "pivots" around the point $X$,
and then "shifts" inward.

DECOMPOSITION BUDGET LINE $BL_D$

The pivot captures the change in the relative price, but keeps the purchasing power the same: you can still (just) afford $X$.

The shift captures the change in real purchasing power, holding the price ratio constant (parallel shift)

Slutsky Decomposition

$BL_1$

$TC_1$

Let's think of this now in terms of the tangency condition.

Think about the Cobb-Douglas utility function

u(x_1,x_2) = x_1x_2

MRS = {p_1 \over p_2}

MRS = {x_2 \over x_1}

Tangency condition

x_2 = {p_1 \over p_2}x_1

Let's let $TC_1$ be the tangency condition at the original price ratio.

Slutsky Decomposition

$BL_1$

$BL_2$

DECOMPOSITION BUDGET LINE $BL_D$

$TC_1$

$TC_2$

Tangency condition

x_2 = {p_1 \over p_2}x_1

What happens to this tangency condition when the price of good 1 increases?

Inferior good: the income effect
is negative for one good

Giffen good: the income effect
is so negative that it outweighs the substitution effect and you buy less of a good when its price goes down!

Complements, Substitutes, Normal, and Inferior Goods

Complements vs. Substitutes: relationship between X and Z
Normal and Inferior Goods:
relationship between Y and Z

Next Time

Cost minimization
Relationship between utility and money
- Indirect utility functions
- Expenditure functions