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:

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?

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)

4. The variable (A) in the utility function represents the:A. investor's return requirement.B. investor's aversion to risk.C. certainty-equivalent rate of the portfolio.D. minimum required utility of the portfolio.E. none of the above.1Downloaded by My H?u Ng?c ([email protected])lOMoARcPSD|13409944

The concept of utility can be measured objectivelyTrueFalse

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