Are the preferences represented by the utility function U (x, y) = 3x+2y homothetic?

How many independent variables are there in a general utility function of the form U=h(x,y)Question 3Select one:a.3b.4c.2d.1

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

Question 1Joe’s preferences are described by the following utility functionU (x, y) = xαyβwith α > 0 and β > 0.(a) Let I denote Joe’s income, and px and py denote the prices of good x and y, respectively.Find Joe’s optimal consumption bundle.

Consider a consumer whose preferences can be represented by a utilityfunction, U : R2+ −→ R, of the form U (x1, x2) = −x21 − x22. Suppose thatthis consumer’s budget set is given byB (p1, p2, y) = {(x1, x2) ∈ R2+ : p1x1 + p2x2 6 y} ,where (p1, p2, y) ∈ R3++ are arbitrary “prices and income parameters”.1. Find this consumer’s Marshallian demand function or correspondence.Justify your answer.2. Find this consumer’s indirect utility function. Justify your answer.

Use the following figure to answer the question below. The marginal utility of the third unit of X isMultiple Choice15 utils.2 utils.4 utils.5 utils.