# Income and Substitution Effects of a Price Change

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 7

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:

### Lecture 7: 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?

### Lecture 8: Welfare Analysis

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

# 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

TOTAL EFFECT

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME EFFECT

Movement along POC

Shift of IOC

Movement along IOC

## Today's Agenda

Part 1: Cost Minimization

Part 2: Income & Substitution Effects

Utility maximization vs. cost minimization

Cost minimization when Lagrange works

Cost minimization when Lagrange fails

Finding the decomposition bundle

Income and substitution effects

Complements and substitutes

# Cost Minimization

## Utility Maximization

## Cost Minimization

Solution functions:

**"Ordinary" Demand functions**

Solution functions:

**"Compensated" Demand functions**

## Utility Maximization

## Cost Minimization

Plug tangency condition back into constraint:

## Same tangency condition, different constraints

## When Lagrange Doesn't Work: Perfect Complements

## Utility Maximization, Cost Minimization, and the IOC

The IOC represents all

the utility-maximizing bundles

for various levels of income.

It **also** represents all

the cost-minimizing bundles

for various levels of utility

For a given price ratio \(p_1/p_2\):

## Utility Maximization, Cost Minimization,

and the 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

Part II:

Solve a cost minimization problem

Calculate the coordinates of A, B, C

Calculate the **income** and **substitution** effects

Talk about **complements** and **substitutes**.

#### Econ 50 | 7 | Income and Substitution Effects

By Chris Makler

# Econ 50 | 7 | Income and Substitution Effects

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