Christopher Makler
Stanford University Department of Economics
Econ 50: Lecture 12
Break down overall effect
of a price change
into its component parts
How much does a price increase
hurt a consumer?
More broadly: what is the relationship between money and utility?
Effect of change in relative prices, holding utility constant.
Effect of change in real income,
holding relative prices constant.
TOTAL EFFECT
INITIAL BUNDLE
FINAL BUNDLE
DECOMPOSITION BUNDLE
SUBSTITUTION EFFECT
INCOME EFFECT
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.
Mathematics of cost minimization
Finding the decomposition bundle
Analyzing the income and substitution effects
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?
Set up the Lagrangian for each of these problems.
Find the first-order conditions.
Solve for the optimal bundle.
Solve for the optimal bundle.
How does it change when prices change?
Objective function:
Constraint:
Optimized values:
Objective function evaluated at optimized values:
utility from utility-maximizing choice (in utils)
cost of cost-minimizing bundle (in $)
Objective function:
Constraint:
Optimized values:
Objective function evaluated at optimized values:
utility from utility-maximizing choice (in utils)
cost of cost-minimizing bundle (in $)
Objective function:
Constraint:
Optimized values:
Objective function evaluated at optimized values:
utility from utility-maximizing choice (in utils)
cost of cost-minimizing bundle (in $)