please only do: if you can teach explain steps of how to know of each cases?of the x 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 ofgood 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 x₁with 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>0dxand for ₁> we have 2 = 0. Since preferences are continuous and monotone, itdzfollows that they are (weakly) convex.(b) Find the consumer's Hicksian demands for each good.Solution: Solving the expenditure minimization problem givesx1 =2up2P1+ 4p2,h(p, u) =2((| (T, (u - 2 √T) ²)and2up2P₁+4P2x₂ =up1P1+ 4p2,provided that ₁ ≤, i.e. provided that u ≤ (2+p₁/2p₂) √ (note that part (a) impliesthat the solution to the FOCs is indeed a minimum). Otherwise, the solution is x₁ = Tand x2 = (u - 2√7)². Putting these together, we have22+)², (P²) ²) if u ≤ (2 + p₁/2²p2)√E,P1+4P2otherwise.=

Utility function : U = 2(x1)0.5 + x20.5 Price of good 1 : P1 , Price of good 2 : P2 Monotonic…

Answered: A consumer has preferences for two…

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

ENGR.ECONOMIC ANALYSIS

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Concept explainers

Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

Question

please only do: if you can teach explain steps of how to know of each cases?

of the x

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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