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- (2) Consider the utility function u(x, y) = Vx+ay. Your budget constraint is P1x+p2y = М. (a) Find the marginal utilities for these goods. (b) Show that these preferences are quasilinicar. (c) Solve the consumer's optimization problem.Donald likes fishing (X1) and hanging out in his hammock (X2). His utility function for these two activities is u(x1, x2) = 3X12X24. (A) Calculate MU1, the marginal utility of fishing. (B) Calculate MU2, the marginal utility of hanging out in his hammock. (C) Calculate MRS, the rate at which he is willing to substitute hanging out in his hammock for fishing. (D)Last week, Donald fished 2 hours a day, and hung out in his hammock 4 hours a day. Using your formula for MRS from (c) find his MRS last week. (E) This week, Donald is fishing eight hours a day, and hanging out in his ham mock two hours a day. Calculate his MRS this week. Has his MRS increased or decreased? Explain why? (F) Is Donald happier or sadder this week compared to last week? Explain.A consumer consumes pizza and soda always in the fixed proportion of 3 slices of pizza (x) with 2 cups of soda (y). Any pizza or soda not consumed together with the proportional amount of the other is useless to her. (a) Write down a utility function u(x,y) that represents her preferences. (b) If she has I = $60 budget and px = 4, py = 4, what is her optimal bundle? (c) In (b), If the price of soda increases to $6, how much money should she be compensated so that she achieves the same level of utility before the price increase? (d) Returning to (b), If the price of soda increases to $6, how much should the price of pizza be reduced so that she achieves the same level of utility before the price hike? (e) In (a), what values of a and b she is indifferent between the bundle "30 slices of pizza and a cups of soda" and the bundle "b slices of pizza and 40 cups of soda". Find at least two (a,b) pairs to get full points.
- If the prices for goods A & B are $5 & $12, respectively, and a customer has $200, which basket of goods (A, B) gives her the highest satisfaction that she can afford? (10, 10) We need her indifference curves (preferences) to know. (16,10) (10, 15) (15, 10)A consuMER hAS A UTILITY FUNCTION v(x, y) = MAX (2x+y) (x+3y) he consumes(1,0) in equilibrium THEN WHICH Of the fOLLOWING MUST BE TRUE? (a) Px ≤PV Px < (3/₂) Py PX ≤ (2/3) PY Px = Py (b) (C) (d)A utility function is given by the equation U = 20xe¬0.1x, where x is the number of glasses of wine consumed. (a) Show that this utility function has a maximum value and calculate the maximum utility. (b) Describe how marginal utility changes for glasses of wine consumed after the maximum utility is reached. Do you consider this reasonable? Give an explanation.
- Q: A consumer’s preferences for food (F) and clothing (C) are given by U(F,C) = F0.2C0.8. The price of food is $4, and the price of clothing is $8. The consumer has an income of $3600. a) What is the utility maximizing choice of food and clothing? b) How would the utility maximizing choice change if price of clothing increased to $14? c) Given the answers to the previous parts plot a linear approximation to the demand function for clothing.For a consumer whose preferences over bundles (X, Y) can be represented by the utility function U(X, Y) = ln(X) +9Y½, which of these statements about the MRS is true? (a) Varies only as the value of X varies (b) Varies only as the value of Y varies (c) Varies as the values of either X or Y vary (d) Does not vary (i.e., is everywhere constant)Ice cream and cakes are perfect substitutes for a child, and 2 units of ice cream is always worth 3 units of cakes (however many ice creams or cakes she might have, she would be willing to give up 2 ice creams to get 3 more cakes to keep the same utility level) . (a) Write down a utility function u(x,y) that represents the child's preferences, where x is the number of ice creams and y is the number of cakes she has. (b) If the prices are px= 8 and py =5, and she has $140 to spend on the two goods this summer, what is her optimal bundle? (c) If the price of ice creams decreases slightly, down to px = 7, what happens to her optimal bundle in this case? Did it change "slightly" compared to (b)?
- 4 Part Question on Caesar * * PART 1 * * Caesar has preferences that are described by the utility function u = 3x 1 + 5x 2. Caesar strictly prefers the bundle (x 1, x 2) = (0, 6.6) to the bundle A. (3, 3) B. (5, 3.6) C. (0, 8.4) D. (11, 1.8) E. (8, 3.60) * * PART 2 * * What is the functional form of Caesar’s preferences, or indifference curves? A. Linear (perfect substitutes) B. Leontief (perfect complements) C. Satiation D. Concave, but not strictly concave E. Cobb-Douglas * * PART 3 * * Caesar’s preferences are A. strictly convex. B. definitely not well-behaved. C. definitely well-behaved. D. strictly concave. E. neutral. * * PART 4 * * Caesar has been consuming the bundle (x 1, x 2) = (10, 2.4). At this point, what is the marginal rate of substitution? A. – 0.25 B. –0.6 C. – 1.67 D. – 3.0 E. – 5.0.1. For each of the three utility functions below, find the substitution effect, the income effect, and the total effect that result when prices change from p = (2,1) to p' = (2,4). Assume the consumer has income I = 20. (a) Before doing any calculation, make an educated guess about the relative magnitude of the three substitution effects and the three income effects to be found below. (b) u(x1, x2) = x1 + x2. (c) u(x1, x2) = x1x2. (d) u(x1,x2) = min {x1,x2). (e) Rank the substitution effects and the income effects found above by their magnitude. To what extent do they conform to your guess?(a). What kind of preferences are represented by a utility function of the form u(x₁, x₂) = √√x₁ + x₂? What about the utility function v(x₁, x₂) = 13x₁ + 13x₂? Compute the slope of the two indifference curves. Do their slopes depend on the bundle? (b). What kind of preferences are represented by a utility function of the form u(x1, x₂) = x₁ + √√x₂ ? Compute the slope of the indifference curve. Does it depend on the bundle? In which way? (c). Consider the utility function u(x₁, x₂) = √x₁x2. What kind of preferences does it represent? Is the function v(X₁, X2) = X₁X2 a monotonic transformation of u(x₁, x₂)? Compute the slope of the indifference curve. Does it depend on the bundle? In which way?