Don't use Ai. Varna I was given a dislike. 1. Consider an individual with the utility function, U = √x/2 + y1/2. The prices of x and y andhis income are Px > 0, Py > 0, and I > 0.(a) Show that this utility function satisfies diminishing marginal rate of substitution.(b) Derive his Marshallian (uncompensated) demand functions for x and y.(c) Derive his indirect utility function.(d) Verify that Roy's identity holds.(e) Without solving his dual problem (i.e., minimizing expenditure subject to a given utility),derive his expenditure function.(f) Show that his expenditure function is homogenous of degree 1 in prices.

Answered: Image /qna-images/answer/9c094042-a10a-4298-904b-5413b5d0ec38.jpg

Answered: 1. Consider an individual with the…

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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Find utility functions given each of the following indifferencecurves [defined by U()=k ]: a. z=k1/x/y/ b. y=0.5x24(x2k)0.5x c. z=y44x(x2yk)2xy22x
Recent research confirms that the demand for cigarettes is not only inelastic, but it also indicates that smokers with incomes in the lower half of all incomes respond to a given price increase by reducing their purchases by amounts that are more than four times as large as the purchase reductions made by smokers in the upper half of all incomes. How can the income and substitution effects of a price change help explain this finding?
1/2 1/2 2. Cynthia has preferences represented by the utility function u(x) = x¹/² + x¹/² (a) Calculate Cynthia's marginal rate of substitution. (b) Calculate Cynthia's Marshallian Demand functions x(p, 1) and x(p,1). (c) Does Cynthia consider either of these goods to be Inferior Goods? (d) Show that Cynthia's demands generate an indirect utility function V(p,1)=√√IP₁P2 (e) Solve for Cynthia's expenditure function.
A consumer’s preferences over two goods x and y are given bythe utility function U(x, y) = xαyβ with α, β > 0. The prices of the goods are px = 2 and py = 4.The consumer has an income of I > 0.(a) For what values of α and β are these utility functions strictly monotone?(b) For what values of α and β will the consumer demand (i.e., Walrasian demand) be more x than y?(c) For what values of α and β are these goods gross substitutes? For what values of α and β are these goods gross complements? Provide a justification for your answer.
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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…
Consider a consumer who has a direct utility function given as U(x₁, x2) = (x1.5 + x2.5)² where x₁ > 0 and x₂ > 0. The consumer chooses to spend his entire income m on the consumption of two goods, where each good has a price of p₁ and p2 respectively. • Obtain the marginal utility of each good. give the domain of these quantities. • Obtain the marginal rate of substitution for goods 1 and 2 (i.e. MRS12). • Calculate the direct second-order partial derivatives for goods 1 and 2. What economic principle does this adhere to?
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