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Unformatted text preview: she’ll expend her entire income (budget).
(c) As you’ve shown in part (b), assuming that at least one good (say good 1) the consumer’s UMP becomes:
( ) ⏟ ⏟ Use the envelope theorem to interpret the Lagrange multipliers (i.e. after you solve the UMP, what will the Lagrange
multipliers tell you?).
The UMP is:
( ) ⏟ ⏟ Here, the consumer must consume at least zero units of goods 1 and 2. More generally, the consumer might have to
consume at least
units of good 1 and at least
of good 2. As such, we can re-state the problem as:
4 University of Toronto, Department of Economics, ECO 204, 2013 - 2014 ( ( ) ⏟ ) ( ) The Lagrangian equation would be:
( ) [ [ [ The FOCs and KT conditions would be: ⏟ ⏟ [ [ [ From the FOCs and KT conditions we see that at the optimal solution the optimal value of the Lagrangian equation will
be (make sure you know why): ( ) [⏟ ( ⏟[ ⏟[ ) The optimal value of the Lagrangian equals the optimal value of the utility function. Thus, a chang...
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