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Utility maximization subject to an income constraint is equivalent to expenditure
minimization subject to a utility constraint.
1. In the utility maximization method for finding the optimal consumption bundle, we
choose the bundle that allows us to reach the highest level of utility U subject to an
2. Likewise, the consumer can choose a level of consumption of X and Y that minimizes
expenditure on the two goods subject to a constraint that utility must be at least as high
as the a certain utility level, say U=U2.
Exogenous: U=U2, Px, Py
Endogenous: X, Y, and Expenditure (I)
Think of shifting the budget constraint inward to lower income levels until the expenditure
line is just tangent to the given utility curve U=U*. This is the optimal consumption bundle. Examples of optimization with different budget constraints
*Be able to draw and label the budget constraints in these examples.
When analyzing public policy it is often easiest to think of the consumer consuming the good of
interest and a composite good H with a price of 1 dollar.
Composite good: represents expenditure on all other goods and has a price set at $1.00. If
the composite good is on the vertical axis, then the y-intercept of the budget constraint
represents both the quantity of the composite good purchased and total income.
a. Coupons versus Cash subsidies
Composite good, no housing subsidy
NOTE: with the composite good y on the y axis