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A consumer has preferences for two goods that are represented by the utility function u(x1, x2) =

A consumer has preferences for two goods that are represented by the utility function u(x1, x2) =

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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A consumer has preferences for two goods that are represented by the utility function u(x₁, x₂) = 2√₁ + √√₂ if x₁ ≤. For 2₁ > T, he gets no additional benefit from consuming more of good 1; that is, if #₁ > then he is indifferent between (21, 2) and (T, T2). (a) Show that this consumer's preferences are monotone and convex. Solution: If a' »r, then 2√√₁+√√√₂ > 2√₁ + √₂ since √ is strictly increasing. This implies that preferences are monotone for 1 ≤. For ₁>, once can replace x₁ with T. For convexity, note that, for x₁ ≤, indifference curves consist of solutions to 2√√₁ + √₂ = k for some k. Along the indifference curve, for x₁ < we have d >0 dx and for ₁> we have 2 = 0. Since preferences are continuous and monotone, it dz follows that they are (weakly) convex. (b) Find the consumer's Hicksian demands for each good. Solution: Solving the expenditure minimization problem gives x1 = 2up2 P1+ 4p2, h(p, u) = 2 (( | (T, (u - 2 √T) ²) and 2up2 P₁+4P2 x₂ = up1 P1+ 4p2, provided that ₁ ≤, i.e. provided that u ≤ (2+p₁/2p₂) √ (note that part (a) implies that the solution to the FOCs is indeed a minimum). Otherwise, the solution is x₁ = T and x2 = (u - 2√7)². Putting these together, we have 2 2 +)², (P²) ²) if u ≤ (2 + p₁/2²p2)√E, P1+4P2 otherwise. =
Transcribed Image Text:A consumer has preferences for two goods that are represented by the utility function u(x₁, x₂) = 2√₁ + √√₂ if x₁ ≤. For 2₁ > T, he gets no additional benefit from consuming more of good 1; that is, if #₁ > then he is indifferent between (21, 2) and (T, T2). (a) Show that this consumer's preferences are monotone and convex. Solution: If a' »r, then 2√√₁+√√√₂ > 2√₁ + √₂ since √ is strictly increasing. This implies that preferences are monotone for 1 ≤. For ₁>, once can replace x₁ with T. For convexity, note that, for x₁ ≤, indifference curves consist of solutions to 2√√₁ + √₂ = k for some k. Along the indifference curve, for x₁ < we have d >0 dx and for ₁> we have 2 = 0. Since preferences are continuous and monotone, it dz follows that they are (weakly) convex. (b) Find the consumer's Hicksian demands for each good. Solution: Solving the expenditure minimization problem gives x1 = 2up2 P1+ 4p2, h(p, u) = 2 (( | (T, (u - 2 √T) ²) and 2up2 P₁+4P2 x₂ = up1 P1+ 4p2, provided that ₁ ≤, i.e. provided that u ≤ (2+p₁/2p₂) √ (note that part (a) implies that the solution to the FOCs is indeed a minimum). Otherwise, the solution is x₁ = T and x2 = (u - 2√7)². Putting these together, we have 2 2 +)², (P²) ²) if u ≤ (2 + p₁/2²p2)√E, P1+4P2 otherwise. =
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