A person has a quadratic utility function: u(w) = w − βw2, where β > 0 is some constant.(Assume the wealth levels are such that 1 − 2βw > 0, so that marginal utility is positive.What is the Arrow-Pratt measure of risk aversion for this person. Is this function DARA?IARA?

A utility function, U(x), measures the amount of satisfaction gained by an individual who buys "x" units of a product or service. The Arrow–Pratt coefficient of relative risk aversion is defined by:

Consider the following statements for an investor regarding a quadratic utility function of the form U= E(r)-0.5*A*(SD)^2 Where, U= Utility; E(r)= expected return on the portfolio; A= coefficient of risk-aversion; (SD)^2= Variance of returns Statement (1) When the value of A=0, it suggests that the investor is risk-neutral Statement (2) When the value of A is negative, it suggests that the investor is risk-averse Statement (3) When the value of A is positive, it suggests that the investor is risk-lover Which of these above statements is (are) CORRECT? • a. Statement (1) only. • b. Statement (1) and statement (2) only. • c. Statement (2) and statement (3) only. • d. Statement (1) and statement (3) only. O e. Statement (1), statement (2) and statement (3).

4.Consider the following quadratic utility function, U(W) = 100 + 180W - 3W2 that describes the preferences for a given investor. Which of the following statements is not correct regarding this utility function? a.This utility function is consistent with the investor "preference of more to less" assumption for all levels of wealth First derivative is only positive for W< 30, second derivative is always negative b.This utility function is consistent with the assumption of investor risk aversion for all levels of wealth c.This utility function is consistent with the assumption of investor risk aversion for levels of wealth less than W = 30 d.This utility function is consistent with the investor "preference of more to less" assumption for levels of wealth less than W = 30 e.This utility function implies does not rely on the assumption of security returns being jointly normal

Consider the following utility functions (in a world with Good X andGood Y ):I. U (x, y) = √xyII. U (x, y) = 4x + 3yIII. U (x, y) = min{x, y} (That is, U (4, 3) = 3, U (1, 1) = 1, U (2, 3) = 2, etc.)IV. U (x, y) = √x + y(a) Fill out the last 3 columns of Table 1. “MRS” stands for Marginal Rate of Substi-tution here. (You would need to copy this table into your answer.)(b) Fill out the the first 4 columns of Table 1 with a Yes/No entry for each cell. Justifyyour answers.(c) Do all utility functions display diminishing MRS? Justify your answers.Note: For the utility function U (x, y) = min{x, y}, calculus cannot be used. How-ever, think about the concept of MRS as how much of y you are willing to give upfor a bit more of x, as represented by the steepness of the indifference curve at apoint. Can you find some value for MRS in this case?(d) Sketch an indifference curve for each of the above utility functions.Table 1. Question 1: Properties of Common Utility FunctionsMonotone Strongly Monotone Convex Strictly Convex MU x MU y MRSI. No 12 x 12 y − 12II. Yes −4/3III. YesIV. Yes Yes1

Suppose that in addition to the consumer described inExercise 1 we also have an another consumer, Consumer 2, whose pref-erences are given by the utility functionu2(x1, x2) = x1x2where x1 is the amount of good 1 the consumer consumes and x2 theamount of good 2. Let ωℓ2 be Consumer 2’s initial endowment of goodℓ. [This is the utility function you analysed in Homework 1.] Similarlylet u1(x1, x2) = min{x1, x2} be the utility function for Consumer 1 thatyou discussed in the previous exercise and ωℓ1 be Consumer 1’s initialendowment of good ℓ.(1) Repeat the analysis of the previous exercise for Consumer 2,finding the demand functions x12(p1, p2, ω12, ω22) andx22(p1, p2, ω12, ω22)