# 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.

# 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

A
C
B

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

INCOME  EFFECT

# 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

A
C
B

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

## Cost Minimization

\max \ u(x_1,x_2)
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }u(x_1,x_2) = U

Solution functions:
"Ordinary" Demand functions

x_1^*(p_1,p_2,m)
x_2^*(p_1,p_2,m)

Solution functions:
"Compensated" Demand functions

x_1^c(p_1,p_2,U)
x_2^c(p_1,p_2,U)

## Cost Minimization

\text{Objective function: } x_1x_2
\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }p_1x_1 + p_2x_2 = m
\text{Constraint: }x_1x_2 = U

Plug tangency condition back into constraint:

\displaystyle{\text{Tangency condition: } x_2 = {p_1 \over p_2}x_1}

## When Lagrange Doesn't Work: Perfect Complements

\text{Objective function: } p_1x_1 + p_2x_2
\text{Constraint: }\min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\} = U

## 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 $$p_1/p_2$$:

(x_1^*,x_2^*): \text{intersection of IOC, BL}
(x_1^c,x_2^c): \text{intersection of IOC, IC}

# 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

\max \ u(x_1,x_2)
\min \ p_1^\prime x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }u(x_1,x_2) = U(A)

Decomposition Bundle (B):
Solves
cost minimization
problem

\max \ u(x_1,x_2)
\text{s.t. }p_1^\prime x_1 + p_2x_2 = m

Part II:

Solve a cost minimization problem

Calculate the coordinates of A, B, C

Calculate the income and substitution effects