Consider an agent whose preferences over any couple (r1, r2), where ri E R+ and r2 E R4, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u : R → R, which represent these preferences.
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- 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?Suppose you have $10, which you can wager in a game called "double or quits." In this game, you roll a fair die. If you roll a 4, 5, or 6, you win the game, getting back your $10 plus $10 more for a total of $20. If you roll a 1, 2, or 3, you lose the game and your $10. Your utility function is shown on the following graph Utility 10 20 Income According to this graph of your utility function, would you be considered risk-averse or a risk-taker? Why? Risk averse because of increasing marginal utility of income Risk taker because of increasing marginal utility of income Risk taker because of diminishing marginal utility of income O Risk averse because of diminishing marginal utility of incomeI am unsure the direction the utility functions would go in , with this specific scenario
- A consumer with income I=120 facing prices pX = 4 and pY = 8 for two goods X and Y (for each good she prefers more to less) chooses optimally to consume 12 units of X. If the prices change and now pX = 6 and pY = 4, what is the possible range for her optimal X consumption? (like, x >/ 7 or 10 >/ x >/7…etc. Use indifference curve analysis on a graph to reason about the possible locations of the new optimal bundle.)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.A consumer's preferences are represented by the following utility function: u(x, y) = 2 T+ Vỹ +7 1. Recall that for any two bundles A = (TA, YA) and B (TB, YB) the following equivalence then holds AEB u(A)> u (B) Which of the two bundles (x1,Yı) = (9,4) or (x2, Y2) = (4,9) does the consumer prefer? 2. Take as given for now that this utility function represents a consumer with strictly convex preferences. Also remember that preferences are strictly convex when for any three bundles A, B, and C, if B E A and C E A then aB +(1 – a)C > A for any a E (0, 1). Use this information and your answer to (1) to determine which of the two bundles (03, Y3) = (7,6) or (x2, Y2) = (4, 9) the consumer prefers. Verify your answer. 3. Derive an expression for the marginal rate of substitution. Interpret the MRS. What is the MRS at the point (x3, Y3) = (7,6)? 4. Derive an equation for the indifference curve through the bundle (x2,Y2) = (4,9). With an equation it is meant: y as a function of x, y(x), such…
- You are maxiimizing utility when :A)(MU of X)/(P of X) < (MU of Y)/(P of Y). B)(MU of X)/(P of X) > (MU of Y)/(P of Y). C)(MU of X)/(P of X) = (MU of Y)/(P of Y). D)(MU of X)/(P of Y) = (MU of Y)/(P of X).Question 1 There are four utility functions defined as follows: U(r, y) 5 ln r +6 ln y U2(x, y) = r+y U3(x, y) r'y U(a, y) = r/y +9 1. Show that the utility functions U1(), Us() and Us() represent the same preference ordering. 2. Show that utility functions U2() and Ua(-) do not represent the same preference ordering. 3. Draw the indifference curve of each utility function at threshold 10 and shade in the upper contour set. Based on your plots, state whether each utility function is quasi-concave or not.Linda loves buying shoes and going out to dance. Her utility function for pairs of shoes, S, and the number of times she goes dancing per month, T, is U(S,T) = 2ST, so MUs = 2T and MUT = 2S. It costs Linda $50 to buy a new pair of shoes or to spend an evening out dancing. Assume that she has $500 to spend on shoes and dancing. A. What is the equation for her budget line? Draw it (with T on the vertical axis) and label the slope and intercepts. B. What is Linda's marginal rate of substitution? Explain. C. Use math to solve for her optimal bundle. Show how to determine this bundle in a diagram using indifference curves and a budget line.
- Rafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples and bananas are p. = 7 and pb - 7. Which of the following utility functions over quantities of apples (a) and bananas (b) could represent Rafe's preferences? = u(a, b)-a4/5 61/5 Ou(a, b) = a¹/5 64/5 Ou(a, b)-a2/3 b1/3 Ou(a, b)-a¹/3 2/3Donna and Jim are two consumers purchasing strawberries and chocolate. Jim’s utility function is ?(?,?) = ?? and Donna’s utility function is ?(?,?) = ?2? where x is strawberries and y is chocolate. Jim’s marginal utility functions are MUx=y and MUy=x while Donna’s are MUx=2xy and MUy=x2. Jim’s income is $100, andDonna’s income is $150. What is the optimal bundle for Donna if the price of strawberries is $2 and the price of chocolate is $4?Sam's extended family spends $3,200 per month on wine and beer. Their utility function is given by U = 200WB, where W represents the number of bottles of wine that they buy, and B represents the number of cases of beer that they buy. Wine costs $25 per bottle and beer costs $32 per case. Sam's family wants to maximize their utility. Calculate how many bottles of wine and how many cases of beer they should buy. Show your calculation(s).