# Analyzing a Price Change

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 14

### Part I: Income and Substitution Effects

Break down overall effect

of a price change

into its **component parts**

How much does a price increase

hurt a consumer?

### Part II: Welfare Analysis

More broadly: what is the relationship between **money** and **utility**?

## Today's Agenda

## Today's Agenda

Part 1: Income and Substitution Effects

Part 2: Welfare Analysis

Decomposing the effects of a price change

Finding the decomposition bundle

Income and substitution effects

Complements and Substitute

Relationship between utility & money

Compensating variation

Equivalent variation

Consumer's Surplus

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

# Income Offer Curve

# Price Offer Curve

for a Good

# Review: Offer Curves

Holding the prices of both goods constant,

show how the optimal bundle changes

as the consumer's income changes.

Holding the price of the other good

and consumer's income constant,

show how the optimal bundle changes

as the price of this good changes.

Offer curves are plotted in Good 1 - Good 2 space (along with budget lines and indifference curves)

pollev.com/chrismakler

A change in the price of good 1 will result in a ______ the price offer curve for good 1

and a ______ the income offer curve.

TOTAL EFFECT

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME EFFECT

Movement along POC

Shift of IOC

Movement along IOC

# Hicks Decomposition

## Hicks Decomposition Bundle

Suppose the price of good 1 increases from \(p_1\) to \(p_1^\prime\).

The price of good 2 (\(p_2\)) and income (\(m\)) remain unchanged.

Initial Bundle (A):

Solves

utility maximization

problem

Final Bundle (C):

Solves

utility maximization

problem

Decomposition Bundle (B):

Solves

cost minimization

problem

**"Compensated Budget Line"**

The "compenated budget line" shows the budget line *as if *the consumer was given *just enough money to achieve their initial utility at the new prices.*

A consumer's compensated budget line:

**"Compensated Budget Line"**

If a consumer's preferences are well behaved, her compensated budget line

**might lie below bundle A,
or pass through A, but cannot lie above A.**

**"Compensated Budget Line"**

If a consumer's preferences are well behaved, her compensated budget line

**might lie below bundle A,
or pass through A, but cannot lie above A.**

Why? Because A *could still be* the cost-minimizing bundle, even at the new prices. But a budget line that passed *above* A

could not find the cost-minimizing bundle, because A would be cheaper!

In this example, \(p1\) and \(p_1\) started at 2 each, and then \(p_1\) rose to 8.

How would the diagram have been different if \(p_2\) had fallen to 0.5 instead?

pollev.com/chrismakler

In this example,

C was below and to the left of B.

What does that say about whether goods 1 and 2 are **normal** or **inferior**?

pollev.com/chrismakler

# Complements and Substitutes (one last time)

## 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

PERFECT

SUBSTITUTES

PERFECT

COMPLEMENTS

INDEPENDENT

PERFECT

SUBSTITUTES

## Constant Elasticity of Substitution (CES) Utility

## Constant Elasticity of Substitution (CES) Utility

COMPLEMENTS: \(r < 0\)

SUBSTITUTES: \(r > 0\)

Which of the following would be true if these goods were **substitutes** rather than **complements?**

pollev.com/chrismakler

Note: whether the goods are complements or substitutes has to do with the income and substitution effects **on the good whose price has not changed**.

(It's about how much you change your buying habits due to the change in *another good*.)

# Hicks Decomposition: Worked Example

## Hicks Decomposition Bundle

You have the utility function \(u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}\).

Suppose the price of good 1 increases from \(p_1 = 1\) to \(p_1^\prime = 4\).

The price of good 2 and income remain unchanged at \(p_2 = 1\) and \(m = 8\).

Initial Bundle (A):

Solves

utility maximization

problem

Final Bundle (C):

Solves

utility maximization

problem

Decomposition Bundle (B):

Solves

cost minimization

problem

(A) Solves utility maximization problem

(C) Solves utility maximization problem

(B) Solves cost

minimization problem

Tangency condition:

Tangency condition:

Tangency condition:

Tangency condition:

Tangency condition:

Constraint:

Constraint:

Constraint:

Bundle B:

Bundle A:

Bundle C:

## Hicks Decomposition Bundle

You have the utility function \(u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}\).

Suppose the price of good 1 increases from \(p_1 = 1\) to \(p_1^\prime = 4\).

The price of good 2 and income remain unchanged at \(p_2 = 1\) and \(m = 8\).

Initial Bundle (A):

Solves

utility maximization

problem

Final Bundle (C):

Solves

utility maximization

problem

Decomposition Bundle (B):

Solves

cost minimization

problem

Bundle B:

Bundle A:

Bundle C:

## Hicks Decomposition Bundle

You have the utility function \(u(x_1,x_2) = x_1^{1 \over 2}x_2^{1 \over 2}\).

Suppose the price of good 1 increases from \(p_1 = 1\) to \(p_1^\prime = 4\).

The price of good 2 and income remain unchanged at \(p_2 = 1\) and \(m = 8\).

Initial Bundle (A): Solves utility maximization problem with the **initial price**

Final Bundle (C): Solves

utility maximization

problem with the **new price**

Decomposition Bundle (B):

Solves cost minimization

problem with the **new price** and **initial utility**

#### Econ 50 | Lecture 14

By Chris Makler

# Econ 50 | Lecture 14

Analyzing a Price Change

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