a. (2, 3). b. (2,4). c. (2,5). d. (3, 3). U(x, y) = 4x + 2y With this utility function, the bundle (3,2) provides the same utility as the bundle For this utility function, the MRS a. depends on the values of x and y. b. is always 0. c. is always 2. d. is always 4.
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These questions refer to an individual whose utility function is given. Please answer showing your work. Thanks.
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- 2. Utility function: U(X,Y)= 2XY* Budget Constraint: 2X +4Y = 80 a. Calculate the utility maximizing amount of X and Y.1. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are px and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. = (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px = 10, Py = 5 and m = 100. Suppose that px increases to p how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y > 0). = 15,b. A consumer purchases two goods, the amounts of which are x and y. Her marginal utility of consuming good x is Ux , while that of consuming good ? is Uy. If the two marginal utility functions are given by Ux = y2 + 2xy and Uy = 4xy + 3x2 − 3 and the consumer is known to stay on the same utility curve, find the total utility function of this consumer buying the two goods
- 3. Suppose Mary enjoys Pepsi and Coke according to the functionU(P;C) = 4C + 5P. What does her utility function say about her MRS of Coke for Pepsi? What do her indi§erence curves look like? What type of goods are Pepsi and Coke for Mary? If Pepsi and Coke each cost $1 and Mary has $20 to spend on these products, how many units of each product should she buy in order to maximize her utility? Show this utility maximiz- ing combination combination of Pepsi and Coke on the graph. how would her consumption and utility maximizing bundle of Coke and Pepsi change if the price of Coke decreases to 50 cents. 4. Vera is an impoverished graduate student who has only $100 a month to spend on food. She has read in a government publication that she can assure an adequate diet by eating only peanut butter and carrots in the Öxed ratio of 2 pounds of peanut butter to 1 pound of carrots, so she decides to limit her diet to that regime. a) If peanut butter costs $4 per pound and carrots cost $2 per…21. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are på and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity. (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and Py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px 10, Py = 5 and m = 100. Suppose that px increases to px = 15, how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y> 0). =1. 2. Suppose Frank enjoys traveling and his utility is a function of the number of airplane trips he takes and the number of bus rides he takes. His utility function is given by U(b,p) = 3b0-4p0.8. A trip on a plane costs $400, a trip on the bus costs $50 and Frank has $3,000 to spend on travel. a. Draw Frank's budget line on a graph with bus trips on the vertical axis. b. Calculate AND INTERPRET the slope of Frank's budget line. c. Draw a representative indifference curve on your graph and provide a function that describes the slope of this indifference curve (i.e., find Frank's marginal rate of substitution). What is the INTERPRETATION of the marginal rate of substitution? d. What two conditions must be met in order for Frank to maximize his utility? e. What are the utility maximizing quantities of plane trips and bus trips for Frank? Given income of $1000, Px = 25 and Py = 40, find (1) a function for the MRS and (2) the utility maximizing consumption bundle for each of the…
- 3. Assuming a price of X (Px) and a price of Y (Py) are $ 1 each, the individual's income is $ 8 per time period and all of it is spent. The Schedule Utility Marginal for items X and Y for individuals is: Specify : a. How should this individual spend his income in order to maximize total utility? b. What is the total amount of utility received by an individual when it is in equilibrium?4. Minnie enjoys dessert by eating chocolate cake (C) and apple pie (P) according to the utility function U = Min [C, P]. In any given week, Minnie %3D spends £20 to purchase these two products. Assume that Minnie is a utility maximiser and has to pay 50p for a unit of apple pie and £2 for a unit of chocolate cake. (i) Consider the bundle C = 20 and P = 5. Is this the utility maximizing choice subject to Minnie's budget?. %3! (ii) Suppose that Minnie chooses the exact bundle that maximizes her utility. How would the utility maximizing choice of C be affected if price of P went up? (iii) Explain under which conditions the tangency condition does not work. Given that the tangency condition does not work, discuss how we can find out the optimal consumption bundle for a consumer.Lorelai's choice behavior can be represented by the utility function u(x1, x₂) = 0.9ln(x₁) + 0.1x2. The prices of both x₁ and x₂ are $10 and she has an income of $80. 1. What preference does this utility function represent? (Hint: the utility function is not linear, but at least linear in good x₂.) 2. Drawinwg indifference curves: you can copy down the graph on your paper using econgraphs. Set the preferences and parameters accordingly as given in the question. Click on "snap to optimal bundle" to see the optimal choice. Click on "show indifference curve map." Draw the lowest four indifference curves as you see from the graph. 3. Find the marginal utility of x₁ and x₂. What is the maximum number of x₁ so that MUX₁ is bigger than or equal to MUX₂ 4. Given her income of $80, how many units of x₁ can she buy? Would she buy any positive number of x₂ in light of the answer from Q5.3? Find the optimal bundle. 5. Suppose instead her income is $100. Would she buy any positive number of x2?…
- 5. What kind of preferences are represented by a utility function of the form u(x1, x2) = x1 + V*2? Is the utility function v(x1, x2) = xỉ + 2x1Vx2+x2 a monotonic transformation of u(x1, x2)? 6. Consider the utility function u(x1, x2) = Vx1x2. What kind of pref- erences does it represent? Is the function v(1, x2) transformation of u(x1, x2)? Is the function w(x1, x2) = xỉx, a monotonic transformation of u(x1, x2)? = x²x2 a monotonic1. Suppose Frank enjoys traveling and his utility is a function of the number of airplane trips he takes and the number of bus rides he takes. His utility function is given by U(b,p) = 3b0-4p0.8. A trip on a plane costs $400, a trip on the bus costs $50 and Frank has $3,000 to spend on travel. a. Draw Frank's budget line on a graph with bus trips on the vertical axis. b. Calculate AND INTERPRET the slope of Frank's budget line. c. Draw a representative indifference curve on your graph and provide a function that describes the slope of this indifference curve (i.e., find Frank's marginal rate of substitution). What is the INTERPRETATION of the marginal7. Bob receives utility from days spent traveling on vacation domestically (D) and daysspent traveling in a foreign country (F) as given by the utilityU(D; F) = DFThe price of a day spent traveling domestically is 160 pounds and in a foreigncountry 200 pounds. Bobís annual budget for traveling is 8000 pounds. (a) Find Bobís utility maximising choice of days traveling domestically and of daystravelling in a foreign country. Find also his utility level from consuming thatbundle. Suppose that the price of domestic traveling increases to 250 pounds per day.Denoting his budget for traveling x, (suppose by now that it is unknown) findthe demand for D and F under the new prices as a function of x.
