2. [This question describes a class of preferences that combine aspects of perfectcomplements and perfect substitutes.]Suppose u(x) = min{arı + ßr2, yz₁+z2}, for some parameters a, ß, 7, 6 > 0.Assume that > 7.(a) Will this consumer will spend all their money (i.e. choose z so thatp·x* = m)? Why or why not?(b) If r¡ > 0 and î; > 0, what is z*? In this case, what is u(z)?(c) Name an example for (a, ß, 7, 8, P₁, P2, m) such that the consumer will onlybuy good 1. Justify your answer.(d) Name an example for (a, 6, 7, 8, P₁, P₂, m) such that the consumer will onlybuy good 2. Justify your answer.(e) Name an example for (a, ß, y, d, P1, P2, m) such that the consumer will buyboth goods. Justify your answer.

Answered: Suppose u(x) = min{ax₁ + ßx2, 7x₁+x2},…

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Suppose u(x) = min{ax₁ + ßx2, 7x₁+x2}, for some parameters a, ß, y,d > 0. Assume that >.

Principles of Economics 2e

2nd Edition

ISBN:9781947172364

Author:Steven A. Greenlaw; David Shapiro

Publisher:Steven A. Greenlaw; David Shapiro

Chapter6: Consumer Choices

Section: Chapter Questions

Problem 10RQ: What is the rule relating the ratio of marginal utility to prices of two goods at the optimal...

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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 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 r E 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 MU₂(2)). What is it? goods (MU(z) P1 P2 (d) What is the consumer's optimal bundle from her budget set? =
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)
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?
. (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?
4. [In addition to giving you some practice thinking about utility representations, this question will give us a bit more intuition for the idea that utility is an “ordinal” concept. It will also show how some of our intuitions about utility might apply when X is finite but not when X is infinite.] Let X be a set, let be a preference relation on X, and let u be a utility representation for . (a) Suppose X is finite. Explain why some (large enough) number M > 0 exists such that −M ≤ u(x) ≤ M for every à € X. That is, the utility is bounded. (b) Show that, even if X is infinite, some alternative utility representation ũ and some (large enough) number M > 0 exists such that –M ≤ û(x) ≤ M for every r € X. [Hint: The function y : R → R given by y(t) = 1+4 is strictly increasing.] (c) Suppose X is finite. Show that some (small enough) number m> 0 exists such that u(x) − u(y) > m for any ï ➤ y. (d) Suppose X = Z+, and is the preference relation defined in question 1(e). i. Explain why any…
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.
(Table) In the table, assume that a consumer has a budget of $75. Movies cost $15 per movie (with popcorn), and dinner costs $15 per dinner. Utility will be maximized by buying Hint: You will need to create a few extra columns to solve Price Movie = $15.00 Price Dinner = $15.00 Movies per Total Utility month Eating out Total Utility per month 5 135 50 4 125 2 90 110 125 2 90 4 155 1 55 175 O A. 2 movies /3 dinners O B. 1 movie / 4 dinners O C.4 movies / 1 dinners. O D.3 movies /2 dinners O E. there is no combination of goods that maximize utility Moving to another question will save this response. Question 1 of 75 Priv Terr MacBook Pro 吕0 DII DD esc F4 F5 F6 F7 F8 F9 F10 F11 F12 F1 F2 F3 %23 24 & 1 2 4 5 7 8. { Q W E R Y tab D F G つ K caps lock P # 3
(a.) Consider a consumer who consumes two good- X and Y - where X is an inferior (but NOT Giffen good. Consider an initial equilibrium where with an income of 100 and individual consumes 10 units of X and 12 units of Y . Now, suppose the price of the good X increases, keeping everything else unchanged. Draw a clearly labelled diagram and indicate the range of the new equilibrium due to price change. [Note: only a clean graph is required and the information that are NOT given are NOT necessary.] (b.) A monopolist- who sells microprocessors in an entire Indian market- charges INR 1200 per unitand sells 1.213 million units in the country. The firm is a recent entry- so the economists are not sure about the demand curve for the Indian market, except that it is a linear curve. Can you comment anything about the elasticity at the equilibrium? Specifically, does the equilibrium lie in the elastic or the inelastic part of the demand curve? (c.) Consider a monopolist who serves a market given…
(Figure: Optimal Selection of Cola (c) and Sushi Rolls (r)) Use Figure: Optimal Selection of Cola (c) and Sushi Rolls (r). If the consumer is at point C, what should the consumer do to maximize utility? Marginal utility per dollar (utils/$) 10 6 5 st 4 w 2 1 1. O 10 8 B 3 2 Quantity of sushi rolls 6 Quantity of cola (bottles) Consuming fewer rolls and more cola. Consume fewer rolls and more cola. Nothing, since utility is already maximized. Consume more rolls and less cola. 2 MUC/PC MU+/Pt 5
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.
5. [This question gives you some practice with a type of preference called "Cobb- Douglas preferences" which shows up in many applied economic models because they are easy to do calculus with.] Let X = R2, let u: X → R be given by u(x) = r2r3, and let be the preference relation on X given by utility representation u. Show that: (a) The function u is not concave. [Hint: Take a second partial derivative with respect to one of the coordinates.] (b) The preference relation is convex. [Hint: You may use the fact that the function f : X → R given by f(x) = x/²x₂ 1/2/2 is concave. If you feel like doing some math, try to show that f is concave, but you don't need to do this.]
5) Which of the following statements is not true about utility? ο ο ο We can compare utility across consumers. Utility depends on individual's tastes and preference. Higher utility is always preferred by a consumer. Utility can only be used to rank alternative combinations of goods and services.