A monotone preference relation on R is homothetic if the following is true for any a 2 0: if xvy, axvay. Prove that a continuous preference relation is homothetic if and only if it admits a utility function that is homogeneous of degree 1.
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- Give examples of preference relation that is monotone but not continuous and is representable by a utility function.T/F Explain If preference is strongly convex, the utility function will exhibit dimin- ishing marginal utility (MU).1. For each of the following scaling functions for a von Neumann-Morgenstern utility func- tion, determine the Marginal Rate of Substitution between X1 and X2 and the equation for an indifference curve through the consumption bundle (100,100) (solve for X2 on the left hand side of the equation). State 1 is the bad outcome that occurs with probability 0.2 and State 2 the good outcome that occurs with probability 0.8. Graph these indif- ference curves and comment on what you see. Is a consumer with these preferences risk averse, risk neutral, or risk loving? (a) V(X) = InX (b) V(X)= VX (c) V(X)= X (d) V(X)= X²
- Consider the following utility functions: a) U= xy b) U=(xy)l3 c) U=min(x,y/2) d) U=2x + 3y e) U=x?y? + xy 1. Check if any of utility functions a) – e) are homogeneous. If so, state the degree of homogeneity of each.We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactlyonce. Can you prove this rigorously? (Hint: what would happen if itintersected some indifference curve twice?)I am unsure the direction the utility functions would go in , with this specific scenario
- 2 Utility representations Recall that for a finite outome set X, the rank-score function U# (x) = #({x' = X : x ≥ x'}) is a utility function representation for any preference relation. In this problem we'll consider the case of a countably infinite outcome set X = {x¹, x², x³, ...}. 1. Write down a preference relation over X for which the rank-score function is a utility representation. Write down another preference relation for which it is not a utility representation. 2. Consider the preference relation over X defined by x¹ ≥ x¹ iff i ≤ j. Write down a utility representation which assigns non-positive utility to every outcome. Write down another utility representation which assigns non-negative utility to every outcome.Suppose that consumer has the following utility function: U(X, Y) = X@Y where 1 > a > 0 and 1 > b>0 are constants. Which of the following is correct? Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Utility holds. Preferences are convex and indifference curves are bowed outward from the origin since Lavw of Diminishing Marginal Utility fails to hold. Preferences are concave and indifference curves are bowed outward from the origin since Law of Diminishing Marginal Rate of Substitution fails to hold. O Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Rate of Substitution holds.Which form of utility is expressed on ranking the bundles According to preferences
- Q12 please help quick Consider a utility function for apples and bananas. You are given the following three bundles: (12, 4), (4, 12), and (8, 8). Under the standard assumptions of the well-behaved preference relationship. you should strictly prefer (8, 8) to (12, 4) and (4, 12). Briefly explain why?The price of X is $20 and the price of Y is $40. Y 30 Uz 24 40 Units of good X based on the graph, at point B, Multiple Choiİce < Prev B. 14 Units of good YNo written by hand solution Assume Fred has the following preference relation on [0, 1]: x ≿ y if and only if x ≤ y for all x ̸= 1 and y ̸= 1; and 1 ≻ z for all z ∈ (0, 1), 1 ∼ 1 and 0 ≻ 1. Does there exists a utility representation for this preference relation? If yes, provide a utility function. If no, explain. Show your work.