5. [This question demonstrates how blindly using calculus can lead us astray.]Suppose the consumer's utility is given by u(x) x² + (1 + 2)², and theirincome and prices are given by (m, p1, p2) (10, 2, 3). In what follows, youmay take it for granted that the consumer has some optimal solution to theconsumer's problem.=(a) Given r E X, write a formula for the marginal utilities MU₁(x) andMU₂(x).(b) Show that they will optimally spend all their money.(c) A unique bundle r exists such that the consumer spends all their money(pr = m) and their per-price marginal utility is the same for the twoMU₂(2)). What is it?goods (MU(z)P1P2(d) What is the consumer's optimal bundle from her budget set?=

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Suppose the consumer's utility is given by u(x) = x² + (1 + x2)², and their income and prices are given by (m, p₁, P2) = (10, 2, 3). In what follows, you may take it for granted that the consumer has some optimal solution to the consumer's problem. (a) Given € X, write a formula for the marginal utilities MU₁(x) and MU₂(x). (b) Show that they will optimally spend all their money. (c) A unique bundle r exists such that the consumer spends all their money (pr m) and their per-price marginal utility is the same for the two goods (MU₁(¹) — MU₂(¹)). What is it? P2

Suppose the consumer's utility is given by u(x) = x² + (1 + x2)², and their income and prices are given by (m, p₁, P2) = (10, 2, 3). In what follows, you may take it for granted that the consumer has some optimal solution to the consumer's problem. (a) Given € X, write a formula for the marginal utilities MU₁(x) and MU₂(x). (b) Show that they will optimally spend all their money. (c) A unique bundle r exists such that the consumer spends all their money (pr m) and their per-price marginal utility is the same for the two goods (MU₁(¹) — MU₂(¹)). What is it? P2

ECON MICRO

5th Edition

ISBN:9781337000536

Author:William A. McEachern

Publisher:William A. McEachern

Chapter6: Consumer Choice And Demand

Section: Chapter Questions

Problem 2.5P

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2. [This question shows that our marginal utility calculations for consumer behav- ior essentially do not depend on which utility we use to represent preferences, even though the marginal utilities themselves do.] Suppose u: X→ R is differentiable at some r E X, with strictly positive marginal utilities. Let f: R→ R be a function with strictly positive derivative; let u fou. (In particular, u and u represent the same preferences because f is strictly increasing.) = (a) Show by example that the marginal utility for good 1 could be different under u and under u. (b) Show that the ratio MU() is the same for utility u as it is for utility u. MU₂ (1)
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4. [This question gives us some practice using calculus to think about a utility's properties, and to solve the consumer's problem.] - Suppose the consumer's utility is given by u(x) = −(x₁ − 1)² – (x2 − 1)², and their income and prices are given by (m, p1, p2) = (3,2,2). In what follows, you may take it for granted that the consumer has some optimal solution to the consumer's problem. (a) Given re X, write a formula for the marginal utilities MU₁(r) and MU₂(x). (b) Show that these preferences are not increasing on X(p, m). (c) Show that the conusmer will optimally spend all their money. [Hint: The previous question's first part may be useful.] (d) Explain why the consumer's preferences are convex. (e) What is the consumer's optimal bundle from their budget set?
Title Suppose that Lynn enjoys coconut oil in her coffee. She has very particular preferences, and she mus Description Suppose that Lynn enjoys coconut oil in her coffee. She has very particular preferences, and she must have exactly two spoonfuls of coconut oil for each cup of coffee. Let C be the number of cups of coffee, and O be the number of spoonfuls of coconut oil. Also, let PC be the price of a cup of coffee. Suppose Lynn has $12 to spend on coffee and coconut oil. Also, the price of coconut oil is $.50 per spoonful.a) Graph Lynnâs Price Consumption Curve for prices, PC = $1, PC = $2, and PC = $3. Please put the number of cups of coffee on the horizontal axis, and the number of spoonfuls of coconut oil on the vertical axis. Be sure to label your graph carefully and accurately.b) Graph Lynnâs demand curve for coffee. You may assume that both coconut oil and coffee are continuous variables so she can consume any amount of coffee and coconut oil that she could afford.…
3. [In class, we noted that the consumer will spend all their money if they have in- creasing preferences. Here, we give a calculus-based condition for the consumer to spend all their money, that doesn't assume more is always better. The con- dition is a stronger version of what is sometimes called "local non-satiation".] Let E X, and let MU, denote the marginal utility for good i € {1,2} at r. (a) Suppose some good i has MU, #0 (it could be positive or negative), and suppose the consumer is not spending all her money at bundle x (that is pr 0 if MU; <0. Argue that the consumer would not optimally choose r. x (b) Suppose MU₁(x) = MU₂(x) = 0, the good is affordable (p ≤ m), and u is concave. Explain why r is optimal.
. (b) Suppose there are two Bohemian roommates with identical preferences who derive utility from the number of paintings hung on their hotel's walls (X) and the number of granola bars (Y) that they eat. The specific utility function underlying their preferences is given by: U₁ (X,Y)= X1¹³ Y₁23 (for i=1.2) mes Given that each roommate has $300 to spend and P=$100, Py = $ 0.20, explore the consequences of various expenditure allocations. In particular provide a detailed analysis of the possibility leading to free ridership Will that solution be efficient? If so or if not, why? Calculate the efficient allocation. What arrangement for cost sharing would be Pareto superior?
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4 Assume that a person's utility over two goods is given by U(g1; g2) = g1 + ln g2. The price of good g1 is equal to p1 and the price of good g2 is p2. The total income of the individual is given by I. The marginal rate of substitution between g1 and g2 is given by 1/(1/g2). Then, the expressions for this person's (1) budget constraint, (2) budget line's slope (assume that, graphically, g1 is on the horizontal axis and g2 on the vertical axis), and (3) the person's demand function for g2 (that is, g2 as a function of price ratio) are respectively:
2. (All right, this one IS tricky. Use the hints and do the best you can!) Suppose there are 2 brothers - Cain and Abel. Cain has a utility function U(L,R) 2min {L,R}, where L and R are left and right shoes respectively. Abel's utility function for shoes is U(L,R)-3L+3R. Further, suppose Cain's endowment is (4 left, 8 right) while Abel's endowment is (7 left, 2 right). a. b. C. d. e. Carefully construct the Edgeworth Box for this exchange economy being careful to label everything and to indicate the box's dimensions. Label the endowment point as well as the initial indifference curves for both Cain and Abel. Label the Contract Curve. (Hint: they cannot trade ½ a shoe.) What are the possible equilibria in this economy? Describe how those equilibria would occur.
48 aces How big would that budget have to be before he would spend a dollar buying a first cup of coffee? Instructions: Enter your answer as a whole number. here to search Suppose that Omar's marginal utility for each additional cup of coffee is 2.5 utils per cup no matter how many cups he drinks. On the other hand, his marginal utility per doughnut is 10 for the first doughnut he eats, 9 for the second he eats, 8 for the third he eats, and so on (that is, declining by 1 util per additional doughnut). In addition, suppose that coffee costs $1 per cup, doughnuts cost $1 each, and Omar has a budget that he can spend only on doughnuts, coffee, or both. O Saved II Help 4 Saw
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