Let X = R2, let u: X→ R be given by u(x) = x²r3, and let be the preference relation on X given by utility representation u. Show that: (a) The function u is not concave. [Hint: Take a second partial derivative with respect to one of the coordinates.] 1/2 1/2 (b) The preference relation is convex. [Hint: You may use the fact that the function f : X → R given by f(x) = x/²² is concave. If you feel like doing some math, try to show that f is concave, but you don't need to do this.]

Let X = R2, let u: X→ R be given by u(x) = x²r3, and let be the preference relation on X given by utility representation u. Show that: (a) The function u is not concave. [Hint: Take a second partial derivative with respect to one of the coordinates.] 1/2 1/2 (b) The preference relation is convex. [Hint: You may use the fact that the function f : X → R given by f(x) = x/²² is concave. If you feel like doing some math, try to show that f is concave, but you don't need to do this.]

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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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.
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Q.3 A utility function is given by the equation U = 120 x0.7 y0.3 where x is the number of hours spent playing golf and y is the hours spent playing cards per month. Calculate the values of x and y for which utility is maximised, subject to the constraint 21x + 3y = 180
Consider an individual with preferences represented by the following utility function: U (x1, 12) = x{x; The individual faces prices: P1 = 0.5; p2 = 1 And has income: M = 250 If the price of good one increases to p1 =1 For this change in price, mark all of the following answers that are correct. (EV = equivalent variation; CV = compensating variation) O In absolute value, EV is greater than Cv O In absolute values, change in Consumer surplus is smaller than CV. O In absolute value, CV is greater than EV In absolute values, change in Consumer Surplus is smaller than EV. There is not enough information to determine an answer.
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Derive Ryan's demand function for q₁, given his utility function is where o = = (9₁) P + (9₂)P, 1 1-p The demand curve for q₁ as a function of P₁, P2, and Y is (Properly format your expression using the tools in the palette. Hover over tools to see keyboard shortcuts. E.g., a subscript can be created with the_ 9₁ = character.) U= Let the price of q₁ be p₁, let the price of q2 be p2, and let income be Y.
Gino derives utility from only two goods, carrots (Qc) and donuts (Qd). His utility function is as follows:U(Qc,Qd) = (Qc)(Qd)The marginal utility that Donald receives from carrots (MUc) and donuts (MUd) are given as follows:MUc = Qd MUd = QcGino has an income (I) of £120 and the price of carrots (Pc) and donuts (Pd) are both £1. - what is Gino's budget line and what quantities of Qc and Qd will maximise Gino's utitlity? - holding income and Pd constant, what is Gino's demand curve for carrots? - suppose a tax of 1 pound per unit is levied on donuts. how will this alter Gino's utility maximisinzing market basket of goods?