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Unformatted text preview: h the utility function
. Solve this UMP. Hint: Exploit the “rules” derived from the general UMP. 2 University of Toronto, Department of Economics, ECO 204, 2013 - 2014 SOLUTIONS
Consider a (rational) consumer with an arbitrary utility function
) ( ) defined over the consumption set (a) Write down the Lagrangian equation for the following (constrained optimization) general utility maximization
( ) ⏟ ⏟ ) ⏟ ⏟ Answer
First re-arrange the inequality constraints:
( ( ) ⏟ Then write down the Lagrangian equation:
( [ )
( [ ) [ [ (b) Show that if one of the goods (say, good 1) is a “good” good then the consumer will spend her entire income (or the
budget allocated to these two goods).
The general UMP is:
( ) ( ⏟ ) ⏟ [ ⏟ 3 University of Toronto, Department of Economics, ECO 204, 2013 - 2014 ⏟ [ Let’s suppose good 1 is a “good” good so that
. Re-arrange the first FOC, isolate , and noting that all
pecuniary parameters are assumed to be strictly positive and that (from the KT conditions)
⏞ ⏞ From the KT conditions we know that:
Since we see that:
⏟ [⏟ If the consumer “likes” at least one good then...
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