A consumer has preferences for two goods that are represented by the utility function u(1, 2) = 2√₁ + √₂ if x₁ ≤ T. good 1; that is, if ₁ > For 2₁ > I, he gets no additional benefit from consuming more of then he is indifferent between (2₁, 2) and (T, x₂). (a) Show that this consumer's preferences are monotone and convex. Solution: If x'>x, then 2√√x₁+√√√₂ > 2√√₁+√√₂ since √ is strictly increasing. This implies that preferences are monotone for x1 <. For x1 > T, 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 < I we have > >0 and for x₁ > we have = 0. Since preferences are continuous and monotone, it follows that they are (weakly) convex. dz (b) Find the consumer's Hicksian demands for each good. Solution: Solving the expenditure minimization problem gives 21 = 2up2 P1+ 4p2/ and x2 = upi P1+ 4p2. 2 = provided that 21 ≤, 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 2up2 P1+4p₂ h(p. u) = {((²p)², (mm)²) ifu≤ (2+p₁/2p2)√VF, | (I, (u - 2 √F)²) otherwise.
A consumer has preferences for two goods that are represented by the utility function u(1, 2) = 2√₁ + √₂ if x₁ ≤ T. good 1; that is, if ₁ > For 2₁ > I, he gets no additional benefit from consuming more of then he is indifferent between (2₁, 2) and (T, x₂). (a) Show that this consumer's preferences are monotone and convex. Solution: If x'>x, then 2√√x₁+√√√₂ > 2√√₁+√√₂ since √ is strictly increasing. This implies that preferences are monotone for x1 <. For x1 > T, 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 < I we have > >0 and for x₁ > we have = 0. Since preferences are continuous and monotone, it follows that they are (weakly) convex. dz (b) Find the consumer's Hicksian demands for each good. Solution: Solving the expenditure minimization problem gives 21 = 2up2 P1+ 4p2/ and x2 = upi P1+ 4p2. 2 = provided that 21 ≤, 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 2up2 P1+4p₂ h(p. u) = {((²p)², (mm)²) ifu≤ (2+p₁/2p2)√VF, | (I, (u - 2 √F)²) otherwise.
Microeconomics A Contemporary Intro
10th Edition
ISBN:9781285635101
Author:MCEACHERN
Publisher:MCEACHERN
Chapter6: Consumer Choice And Demand
Section: Chapter Questions
Problem 6QFR
Related questions
Question
please only do: if you can teach explain steps of how to solve each part
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.
=
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Exploring Economics
Economics
ISBN:
9781544336329
Author:
Robert L. Sexton
Publisher:
SAGE Publications, Inc
Economics (MindTap Course List)
Economics
ISBN:
9781337617383
Author:
Roger A. Arnold
Publisher:
Cengage Learning
Exploring Economics
Economics
ISBN:
9781544336329
Author:
Robert L. Sexton
Publisher:
SAGE Publications, Inc
Economics (MindTap Course List)
Economics
ISBN:
9781337617383
Author:
Roger A. Arnold
Publisher:
Cengage Learning
