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

Chapter6: Demand Relationships Among Goods

Section: Chapter Questions

Problem 6.9P

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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.
A consumer has a (direct) utility function over x1 and x2: u(x₁, x2) = x₁x2.5 The price of good 1 is p1=1; the price of good 2 is p2=2; and the consumer has m=24 in income. Indicate which of the following statements is true. Select one: O Demand for good 1 is equal to 24. O Demand for good 2 is equal to 6. O Indirect utility function is equal to 32. O The consumer will demand 0 units of good 2. O None of the statements is correct.
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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Individual that consumes two goods (X and Y) and has a CES Utility Function of the form: U = 100(X^(0.75) + Y^(0.75)). Income of 1000, the price of Good X is 10 and the Price of Good Y is 20 a) Find the Marginal Rate of Substitution as a function of the quantities consumed of Good X and Good Y. b) Write out the Lagrangian for this problem. c) Solve to find the demand for Good X, the demand for Good Y, and the highest level of utility for this individual. d) Now consider an increase in the price of Good X to 20. What is the demand for Good X and Good Y? What is the Utility of the consumer following the price change? e) Considering the change in demand for each good between parts c) and d), how much is due to the substitution effect and how much is due to the income effect? f) Show your answers on a graph.
Consider the following utility function that expresses the preferences of any consumer for good x and good U(x, y) Min {2x, y) Consider that the price of x is $4, the price of y is $2, and the income R is $100. Reply: a) What is the marginal rate of substitution for x? b) What is the marginal rate of substitution for y? c) What is the expression of demand ( demand function) of x? d) What is the demand expression (demand function) for y? e) What is the basket composition that maximizes this consumer's satisfaction given the given restriction? f) What is the utility achieved with this basket?
Utility maximization under constraint, substitution and income effect, CV and EV Josh gets utility (satisfaction) from two goods, A and B, according to the utility function U(A, B) = 4[A-4 + B-4-4 + 40. While Josh would like to consume as much as possible he is limited by his income. a. Maximize Josh's utility subject to the budget constraint using the Lagrangean method b. Suppose PA increases. Show graphically the income, substitution effect and total effect and explain c. Suppose PA increases. Show the graphs for CV and EV and explain. One graph for CV and one graph for EV
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.
1. Calculate the marginal rate of substitution for the following utility functions: (a) u(x1, x2) = ax₁ + bx2 (b) u(x1, x2) = 2√√x₁ + x₂ (c) u(x1, x2) = ln(x₁) + x2 (d) u(x1, x2) = x1x2 (e) u(x1, x2) = xx h (f) u(x₁, x₂) = x + x² 2. Marie Curie has a utility function U(x, y) = max{x, 2y}. Note that the max{} function es- sentially says that the value of the function takes the highest of the values in the set (the set is denoted as the things inside the curly braces). (a) Draw a graph, plotting the indifference curve along which U(x, y) = 10. (b) Are Marie's preferences convex?

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