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 utilitymight 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 utilityrepresentation for .(a) Suppose X is finite. Explain why some (large enough) number M > 0exists such that −M ≤ u(x) ≤ M for every à € X. That is, the utility isbounded.(b) Show that, even if X is infinite, some alternative utility representation ũand some (large enough) number M > 0 exists such that –M ≤ û(x) ≤ Mfor every r € X. [Hint: The function y : R → R given by y(t) = 1+4 isstrictly increasing.](c) Suppose X is finite. Show that some (small enough) number m> 0 existssuch 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 whole number k ≥ 3 has u(1) < u(2) < .

Set X : A set is a collection of distinct objects or elements. In this context , X represents a set…

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 x E X. That is, the utility is bounded. (b) Show that, even if X is infinite, some alternative utility representation u and some (large enough) number M> 0 exists such that -M≤ u(x) ≤ M for every 2 E X. [Hint: The function : R → R given by y(t) = 1 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 x ➤ y. Z+, and is the preference relation defined in question 1(e). (d) Suppose X = i. Explain why any whole number k ≥ 3 has u(1) < u(2) << u(k) < u(0).

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 x E X. That is, the utility is bounded. (b) Show that, even if X is infinite, some alternative utility representation u and some (large enough) number M> 0 exists such that -M≤ u(x) ≤ M for every 2 E X. [Hint: The function : R → R given by y(t) = 1 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 x ➤ y. Z+, and is the preference relation defined in question 1(e). (d) Suppose X = i. Explain why any whole number k ≥ 3 has u(1) < u(2) << u(k) < u(0).

Principles of Microeconomics (MindTap Course List)

8th Edition

ISBN:9781305971493

Author:N. Gregory Mankiw

Publisher:N. Gregory Mankiw

Chapter21: The Theory Of Consumer Choice

Section: Chapter Questions

Problem 5CQQ

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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)
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? =
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.
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.]
(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
7) Suppose that upon solving for optimal bundles of hamburgers and french fries, we determine that when the price of hamburgers is $5 and the price of fries is $2, that we consume 2 hamburgers and 5 orders of fries. Suppose further that, upon changing the price of fries to $4, we only consume 3 orders of fries. Draw out a demand curve for fries, labeling two points. This is not a trick question, this one is supposed to be easy.
Suppose consumer consume two goods. Can both goods be inferior? Can one of the goods be inferior? Explain your reasoning clearly. 19
Question3 a) Suppose a household is faced with the choice between consuming gasoline (G) and all other goods (OG). Today the household consumes 800 liter of gasoline a year.Suppose then that a gasoline price increase is perfectly compensated by a wage increase. If the family followed the utility maximization model, how would this affect their consumption of gasoline? Explain by using a figure. b) Explain by using an example why an MRS (Marginal Rate of Substitution) between two goods must equal the ratio of the price of the goods for the consumer to achieve maximum satisfaction?
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.
9. Behavioral economics Indicate whether each of the following examples of behavior is consistent with the way the traditional economic framework suggests people should act, or if it is reserved for behavioral economists to examine. Example Some people would be willing to pay money to lower the incomes of others. Some people treat $60 they earn differently from $60 they win in a random drawing. Some people sacrifice disposable income to help their children pay for college. Some people would turn down a new, higher-paying job if it meant spending less time with their family. Consistent with the Predictions of Traditional Economic Models Reserved for Behavioral Economics
5. A consumer buys only two goods, X and Y. No other goods exist and there is no possibility of saving. The marginal utility of X is independent of the quantity of Y consumed, and the marginal utility of Y is independent of the quantity of X consumed. MUX is constant no matter how he consumes, but MUY falls as consumption increases. In the initial equilibrium he consumes one of each good. How much can you infer about the following:a.) The slope of the indifference curveb.) The curvature of the indifference curvec.) Whether the marginal utility of money is constant, rising, or falling as money income increases.d.) the income elasticity of demand for Ye.) The price elasticity of demand for X.
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.…