a. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve? b. A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas of the form U(x, y) = xay³, where a and 3 are positive constants, what is the utility function for this consumer? c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function?

a. A consumer is willing to trade 3 units of x for 1 unit of y when she has 6 units of x and 5 units of y. She is also willing to trade in 6 units of x for 2 units of y when she has 12 units of x and 3 units of y. She is indifferent between bundle (6, 5) and bundle (12, 3). What is the utility function for goods x and y? Hint: What is the shape of the indifference curve? b. A consumer is willing to trade 4 units of x for 1 unit of y when she is consuming bundle (8, 1). She is also willing to trade in 1 unit of x for 2 units of y when she is consuming bundle (4, 4). She is indifferent between these two bundles. Assuming that the utility function is Cobb-Douglas of the form U(x, y) = xay³, where a and 3 are positive constants, what is the utility function for this consumer? c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Solve these consumer choice problems: a. Suppose a consumer has a utility function u(x, y) = xy + 7y facing prices p,=S1 and p,=$2 and income I = $10. Draw a diagram and find the consumer's optimal consumption of goods x and y. b. Suppose that you are in a situation where you can afford purchasing 3 units of good x and 4 units of good y. You discover that another affordable bundle of 5 units of good x and 2 units of good y causes you to reach the same level of utility. Assume that your utility function and budget line are both well behaved. Are you maximizing your utility by consuming your original bundle? Why or why not? Propose an alternative, affordable bundle that would yield a higher utility level than either of the original bundles.
4. Tim wants to maximise his utility given his utility function of U (A, B) = A² B¹. He faces the following prices and income: PA = 25; PB = 20; M = 250. However, a change in the price of good A to PA 22 will change his optimal bundle. If he is only interested in being able to consume his original bundle of goods, how much income would he need after the price change? = (a) Tim would require £54 less income (b) Tim would require £18.33 more income (c) Tim would require £10 less income (d) Tim would require £16.66 more income (e) Tim would require £16.66 less income
Jennifer likes her tea, but only if prepared with 4 teaspoons of honey for every cup of tea. The price of tea is $2, and the price of honey is $0.50. Jennifer has a $12 budget for tea and honey. a. Write down a utility function that reflects her preferences for tea (T) and honey (H). b. What bundle should Jennifer choose to maximize her utility given her budget constraint? c. Graphically illustrate Jennifer's utility maximizing bundle given her budget constraint.
5. Jonas purchases only two goods, starfruit (S) and kiwi (K). He has an income of $60 and can buy starfruit at $2 per pound and kiwi at $3 per pound. His utility function is U(S,K)=3S+4K. That is, his (constant) marginal utility for starfruit is 3 and his marginal utility for kiwi is 4. What bundle of starfruit and kiwi should he purchase to maximize his utility? Why?
Joe is currently in consumer equilibrium by consuming cheese and crackers, such that the last cracker consumed yielded 8 utils and the last piece of cheese consumed yielded 12 utils. Assume the price of crackers is two cents per cracker and the price of cheese is three cents per piece. If the price of crackers increases to four cents, Joe should his consumption of crackers and his marginal utility from crackers will and also his consumption of cheese and his marginal utility from cheese will (Assume he is in the downward sloping portion of the MU curve.) Ο Ο Ο decrease; increase; increase; decrease increase; increase; increase; increase increase; decrease; increase; decrease increase; increase; decrease; decrease
9. A consumer has the utility function u(x,y) = =x+6y and the price of x is $1 and the price of y is $5. If the consumer's income is $40, what bundle of goods will maximize her utility? x = 20, y = 4 x=40, y = 0 x = 0, y = 8 x= 10, y = 6 A a. b. c. d.
Tony is throwing a party at his Fraternity and is trying to choose what booze to buy. A bottle of vodka has three times the alcohol as a six-pack of beer. Assume that Tony only cares about the total amount of alcohol in his basket. (use vodka on the X-axis and beer measured in six-pack on the Y-axis) a) Devise a utility function to represent these preferences. b) Suppose a bottle of vodka costs $40, a six-pack of beer costs $10, and the budget is $200. Write the budget constraint. c) Solve Tony’s utility maximization problem and find the optimal combination. d) Suppose that a bottle of vodka cost has increased to $50. What will be his new optimal combination.
4. Use the following information to calculate Wren's optimal bundle and his resulting utility: IC:U(x,y) = x0.5, 0.5 Px = 1, Py = 4, m = 48
II. Your preferences are represented by the utility function U(x₁, x₂) = (x² − x²)0.5 for x₁ > 0, x₂ > 0, and x² > x².
(a)_Suppose we have preferences U(X, Y) = X² Y².. What is the utility at the bundle X = 5 and Y = 20? Create a table then graph/sketch the indifference curve through (5, 20). Interpret the MRS (with MRS = Y/X) at this bundle. < (b) _Let prices be Px = $5, Py = $5 and income M = $1000. Draw/sketch the budget constraint. Explain what the slope of the budget line means in economic terms.< (c) State the consumer's utility maximization problem and express this in words. What are the two conditions (equations) that identify the optimum? Sketch this in a figure and explain.< (d) Use the information in (a) and (c) to show that demands are: X = ½ M/PX and Y = ½ M/PY.< (e) For the income and prices in (b), what is optimal X, Y, and utility? Show your work. (f) Suppose Px falls to $4 and Py rises to $7 but income stays at $1000. Does consumer utility rise or fall? Show and explain. < (g) Calculate the Compensating Variation that ensures the consumer is no worse off nor better off with these price…
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4. Suppose that a consumer with the utility function U =X0.25 Y0.75 choses a bundle where X=5 and Y=10. If the prices of the two goods are Px= $1, and Py = $3. Calculate marginal utility per dollar spent on each good. Can this consumer increase utility without increasing expenditure and if so how?