Income and Substitution Effects of a Price Change
Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 12
Income and Substitution Effects
Lecture 11
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.
Income Effect
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
Utility Maximization
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 $)
Econ 50 | Fall 2020 | 12 | Welfare Analysis
By Chris Makler
Econ 50 | Fall 2020 | 12 | Welfare Analysis
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