1. Suppose that an individual's direct utility function is represented implicitly by:u = ΣΦ, logx-2Φi=1,2where i=α + Bu1+uYi, Oi, and Bi are parameters with Σ a; = B₁ = 1.a) Derive the Hicksian demand and expenditure functions.b) Prove the homogeneity property.c) Use the Shephard's lemma to derive the Hicksian demand functions.

Step-by-Step Solutiona) Derive the Hicksian Demand and Expenditure Functions1. Define the…

Answered: 1. Suppose that an individual's direct…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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1. For each of the following scaling functions for a von Neumann-Morgenstern utility func- tion, determine the Marginal Rate of Substitution between X1 and X2 and the equation for an indifference curve through the consumption bundle (100,100) (solve for X2 on the left hand side of the equation). State 1 is the bad outcome that occurs with probability 0.2 and State 2 the good outcome that occurs with probability 0.8. Graph these indif- ference curves and comment on what you see. Is a consumer with these preferences risk averse, risk neutral, or risk loving? (a) V(X) = InX (b) V(X)= VX (c) V(X)= X (d) V(X)= X²
5.6. Bunde's preferences are given by the utility function u(x1, x2) = x1 + X2. For each of the following cases, decompose the price effects into the substitution and income effects using either the Hicks-Allen or Slutsky decompositions. - (a) Suppose m = 120, p₁ = 12, and p2 P1 5, keeping p2 and m fixed. (b) Suppose m = 100, P1 = 20. The price p₁ then falls to = 20, and p2 = 10. The price p₁ then falls to 5, keeping p2 and m fixed.
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…
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3- Assuming that the equation F(U, x, X2 = f(x1, x2,, ... Xn): ******* **** ,xn) = 0 implicitly defines a utilityfunction U 073 a) Find the expressions for 60, 6U and 6x4 " 3 6x2 6xn 6x2 6xn b) Interpret their respective economic meanings. c) Now. assume the utility function is U(x, y) = y√x. Does the consumer believe that more is better for each good? Do the consumer's preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y diminishing?
A. Consider a consumer whose preferences can be represented by Cobb-Douglas utility function u(x₁, x2) = x³x² where x₁ and ⁄2 are the quantities of good 1 and good 2 she consumes. Let p₁ and p2 be the prices of good 1 and good 2 and let m denote her income. Derive the consumer's Marshallian demand functions. Derive the consumer's Hicksian demand functions. Derive the consumer's expenditure function. Let m = = 120, p₁ = 2, and p2 = 1. Suppose that the price of good 1 drops to p₁ = 1. Find the following Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV. Let m = 120, P₁ P₁ = 2. Find the following = 1, and p2 = 1. Suppose that the price of good 1 increases to Compensating variation (CV) Equivalent variation (EV) Change in consumer surplus (ACS) Compare CV, ACS, and EV.
K should be used to TL? 3. A consumer's preferences are represented by the utility function, U(X,Y)= Xay. Budget constraint is M = PxX+ P,Y. Based on this information; E a) Derive the Marshallian demand function. b) Calculate zero degree homogeneity for Marshallian demand for X. c) Derive the Hicksian demand function.
2) For the following utility functions, using the budget constraint M = Pxx & Pyy, find the compensated (Hicksian) demand functions. Note: For a, minimize expenditure; For b, maximize utility and use the Shepard's Lemma. a) U(x,y) = x0.5 + 2y0.5 b) U (x, y) = min{2x, 5y}
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4. Maximize utility U = xy subject to 4x+3y=60, where x and y are the two consumption goods, 4 is the price of good x, 3 is the price of good y, and 60 is the budget. a) Set up the Lagrangian objective function. How much is the utility-maximizing consumption bundle? b) Show that the second-order condition/s is/are met (Use |Ħ₂ to identify whether it is a maximum or a minimum). c) What is the maximum utility?
Consider a consumer with indirect utility function v(p, w) = w - B1p1 - B2p2 √P1P2 where 3₁ and 32 are nonnegative constants. (a) Find the consumer's expenditure function. (b) Find the Hicksian and Marshallian demands for Good 1. (c) Find expressions for the substitution and income effects on Good 1 associated with a marginal increase in the price of Good 2. (d) Suppose that the price of good 1 changes from pi to kp₁ where 0 < k < 1. Find the compensating variation of this price change.

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