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
A
C
B
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
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
Utility Maximization
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)
Utility Maximization
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}
Same tangency condition, different constraints
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}
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
\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
Talk about complements and substitutes.
Econ 50 | 7 | Income and Substitution Effects
By Chris Makler
