ajaz_204_2013_2014_HW_5

Solve this ump hint exploit the rules derived from

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Unformatted text preview: h the utility function {( } Assume ) . Solve this UMP. Hint: Exploit the “rules” derived from the general UMP. 2 University of Toronto, Department of Economics, ECO 204, 2013 - 2014 SOLUTIONS Question 1 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 problem: ( ) ⏟ ⏟ ) ⏟ ⏟ 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). Answer 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) we have: ⏞ ⏞ From the KT conditions we know that: [ Since we see that: ⏟ [⏟ If the consumer “likes” at least one good then...
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