# Income and Substitution Effects of a Price Change

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 12

### Lecture 11: 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 12: 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.

# Approach

TOTAL EFFECT

A
C
B

INITIAL BUNDLE

FINAL BUNDLE

DECOMPOSITION BUNDLE

SUBSTITUTION EFFECT

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.

# Today's Agenda

Mathematics of cost minimization

Finding the decomposition bundle

Analyzing the income and substitution effects

# Cost Minimization

## Cost Minimization

\text{Utility Maximization}
\text{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

Set up the Lagrangian for each of these problems.

Examine the units.

Find the first-order conditions.

Solve for the Lagrange multiplier. What does it represent?

Set the values of the Lagrange multiplier equal. What's the same in both cases? What's different?

\text{Utility Maximization}
\text{Cost Minimization}
\max \ x_1x_2
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }x_1x_2 = U

Set up the Lagrangian for each of these problems.

Find the first-order conditions.

Solve for the optimal bundle.

\text{Utility Maximization}
\text{Cost Minimization}
\max \ \min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\}
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }\min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\} = U

Solve for the optimal bundle.

How does it change when prices change?

\text{Utility Maximization}
\text{Cost Minimization}
\max \ x_1x_2
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }x_1x_2 = U
x_1^*(p_1,p_2,m) = \frac{m}{2p_1}
x_2^*(p_1,p_2,m) = \frac{m}{2p_2}
x_1^*(p_1,p_2,U) = \sqrt{\frac{p_2U}{p_1}}
x_2^*(p_1,p_2,U) = \sqrt{\frac{p_1U}{p_2}}

Objective function:

Constraint:

Optimized values:

Objective function evaluated at optimized values:

V(p_1,p_2,m) = u(x_1^*,x_2^*)
E(x_1^*,x_2^*) = p_1x_1^* + p_2x_2^*

utility from utility-maximizing choice (in utils)

cost of cost-minimizing bundle (in $) ## Constant Elasticity of Substitution (CES) Utility \text{Utility Maximization} \text{Cost Minimization} \max \ x_1x_2 \min \ p_1x_1 + p_2x_2 \text{s.t. }p_1x_1 + p_2x_2 = m \text{s.t. }x_1x_2 = U x_1^*(p_1,p_2,m) = \frac{m}{2p_1} x_2^*(p_1,p_2,m) = \frac{m}{2p_2} x_1^*(p_1,p_2,U) = \sqrt{\frac{p_2U}{p_1}} x_2^*(p_1,p_2,U) = \sqrt{\frac{p_1U}{p_2}} Objective function: Constraint: Optimized values: Objective function evaluated at optimized values: V(p_1,p_2,m) = u(x_1^*,x_2^*) E(x_1^*,x_2^*) = p_1x_1^* + p_2x_2^* utility from utility-maximizing choice (in utils) cost of cost-minimizing bundle (in$)

\text{Utility Maximization}
\text{Cost Minimization}
\max \ \min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\}
\min \ p_1x_1 + p_2x_2
\text{s.t. }p_1x_1 + p_2x_2 = m
\text{s.t. }\min\left\{\frac{x_1}{2},\frac{x_2}{3}\right\} = U
x_1^*(p_1,p_2,m) = 2 \times \frac{m}{2p_1 + 3p_2}
x_2^*(p_1,p_2,m) = 3 \times \frac{m}{2p_1 + 3p_2}
x_1^*(p_1,p_2,U) = 2U
x_2^*(p_1,p_2,U) = 3U

Objective function:

Constraint:

Optimized values:

Objective function evaluated at optimized values:

V(p_1,p_2,m) = u(x_1^*,x_2^*)
E(x_1^*,x_2^*) = p_1x_1^* + p_2x_2^*

utility from utility-maximizing choice (in utils)

cost of cost-minimizing bundle (in \$)

By Chris Makler

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