PS2Solution - Economics 100A Spring 2011 Problem Set 2...

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Economics 100A Spring 2011 Problem Set 2 Suggested Solutions 1. A. Since I only value ice cream and bananas in the form of banana splits, the two goods are perfect complements and my preferences for them can be expressed using the utility representation min { 1 3 x 1 , x 2 } , where x 1 denotes scoops of ice cream and x 2 denotes bananas. Notice that this function also outputs the number of banana splits I can make with the available ice cream and bananas. The Lagrangean would then be: L = min { 1 3 x 1 , x 2 } + λ (5 - . 5 x 1 - . 25 x 2 ) . B. The utility function is not differentiable everywhere - its derivative at the kink points is undefined. The Lagrange method requires differentiability, however (intuitively, it helps us find tangencies). C. Please see graph on page 4. D. Since I derive utility only from the number of banana splits I consume, I want to maximize that number. Since each banana split is made up of three ice cream scoops (each costing $.5) and one banana (each costing $.25), each banana split costs $1.75. I have $5, therefore I can consume at most 5 / 1 . 75 = 20 7 banana splits. This means that the optimal consumption of the two goods is: x 1 = 3 * 20 7 = 60 7 and x 2 = 20 7 . 2. A. The MRS of the utility function is: - ∂u ( x 1 , x 2 ) /∂x 1 ∂u ( x 1 , x 2 ) /∂x 2 = - 2 3 x - 1 3 1 1 = - 2 3 x - 1 3 1 The slope of the budget line (i.e. the OC) is - p 1 p 2 . Thus the tangency condition is: - 2 3 x - 1 3 1 = - p 1 p 2 .
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