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Microeconomic Theory

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.13P

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Question

Consider the following utility function,
u (x1,x2) = xx¹³, where 3 € (0,1)
2
(a) [15 points] Derive the Hicksian demand functions and the expenditure function.
(b) [15 points] Derive the indirect utility functions.

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1. For each of the following utility function, compute the Hicksian demand function h(p, v), the expenditure function e(p, v) and the Slutsky (Substitution) matrix S(p, w). In your com- putation procedure, feel free to use your (or my) answers in Assignment 5. 3 a) u(*1, 12) = VI, + ¤2 b) u(x1, 12) = log(1) +3 log r2 c) u(r1, 12) = min(r1, 12) d) u(x1,r2) = max(r1, r2) %3D %3D %3D %3D
2. Consider the following utility function, u (x1, x2) = x{x", where y e (0, 1) %3D (a) Derive the Hicksian demand functions and the expenditure function. (b) perive the indirect utility functions.
2
For each of the following utility functions, find the Marshallian demand function, the indirect utility function and the expenditure function. Assume that prices of x and x2 are p₁ and p₂ respectively and income is m. i) U(x1x2) = ln(x1+ x2) ii) U(x1x2) = (x1+ x2)
= Suppose you are trying to minimize your expenditures E utility level (20.5 + x9.5)² = 175. 4x1 + 3x2 subject to the target maximum • Set up the Lagrangean function for this expenditure-minimization problem with constraint • Compute for the expenditure minimizing quantities of ₁ and ₂. x2. • What is the minimum expenditure you can incur given the level of utility you have to attain?
1. Consider the utility function given by u (x1, x2) = x1x3, and budget constraint given by P1x1 + P2x2 = w. (a) Solve the EMP to find the Hicksian demand function, h (p, u). (b) Find the expenditure function e (p, u). (c) Recover h (p, u) from e (p, u). (d) Noting that the indirect utility function corresponding to the UMP for this form is 4w3 v (P1, P2, w): 27p1p3' demonstrate that v (.) and e (.) are inverses of each other. (e) Using only the indirect utility function above, recover the optimal consumption bundle of the UMP (i.e., Walrasian demand function). (f) Using your answer to (e), identify the substitution effect on the quantity demanded of good 1 of a change in the price of good 1. 1; and consider the Hicksian demand curve for good 1 corresponding to , and Walrasian demand curve for good 1 corresponding to w = 1. At what price and quantity for good 1 do they intersect? Which is steeper at this point of intersection? (g) Assume P2 What does that signify? (h) Using your…
Find the expenditure function corresponding to utility function u(x₁,x₂) = ln(x₁ +a)+ln(x₂+b). a.e(p₁u) = 2√P₁P₂)eu/2 b.e(p,u)=e¹(√P₁P₂)+ap₁ + bp₂ Ⓒc. e(p₁u) = 2(√P₁P₂)eu/2-ap₁-bp₂ d.elpu)=e"√√P₁P₂
give me solution within 30-40 mins I'll give you multiple upvote.
10.
PROBLEM 2 – Uncompensated and Compensated Demand Functions, Price Elasticities of Demand, and Expenditure Functions Susi enjoys her favorite drink (D) and food (F) which she always spends her monthly income on those goods. Her average allocated income spent on drink and food is Rp 3 million, and the current prices of drink and food are Rp 30k and Rp 90k, respectively. Suppose her happiness by consuming those goods can be capture by the utility function as follows: *see image* a. What are her uncompensated (Marshallian) demand functions for drink and food, and calculate how many drinks and food that she currently consumes? Draw and describe how her demand curves for drink and food are shifted by changes in her income or the price of the other good. (hint: Use Lagrangian method). b. If the price of drink increases to Rp 50k, calculate the price elasticity of demand for drink. c. What is her the expenditure function for drink and food? Use the expenditure function to compute the…
3. Consider the following utility function, u (21, x2) = min V#1, Varz), where a > 0 Derive the Marshallian demand functions. (Explain your derivation in details.) Does the Marshallian demand increase with price? Are the two consumption goods normal goods? Show two different ways to derive the Hicksian demand functions. (b) Does the Hicksian demand increase with price?
Assume, as in Exercise 22.1, that a consumer has utility function F or fruit and chocolate. Determine the consumer's demand functions q1(P1, P2, M) and q2(P1, P2, M). Determine also It* in terms of P1, P2 and M. Find the indirect utility function and show that It* = 8Vj8M. Suppose, as before, that fruit costs $1 per unit and chocolate $2 per unit. If the income is raised from $36 to $36.5, determine the precise value of the resulting change in the indirect utility function. Show that this is approximately equal to (O.5)λ*, where λ* is evaluated at P1 = 1,P2 = 2 and M = 36. Exercise 22.1 A consumer purchases quantities of two commodities, fruit and chocolate, each month. The consumer's utility function is For a bundle (X1, X2) of X1 units of fruit and X2 units of chocolate. The consumer has a total of $49 to spend on fruit and chocolate each month. Fruit cost $1 per unit and chocolate costs $2 per unit. How many units of each should the consumer buy…

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