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 r e X, write a formula for the marginal utilities MU₁(x) 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?

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 r e X, write a formula for the marginal utilities MU₁(x) 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?

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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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
3 Question 11 Suppose a consumer's utility is given by: u(x1, x2) = axı + bx₂ Where a= 63, and b = 22 If the price of good 1 is P₁ =$57/unit, the price of good 2 is P2 -$37/unit, and income is I = $423, what is this consumer's optimal consumption level of good 1? (Note: The answer may not be a whole number, so round to the nearest hundredth) (Note: The numbers may change between questions, so read carefully) Question 12
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5) Consider indifference curves for the consumption of milk and chocolates (you may assume that both are 'goods'). The indifference curves are drawn with the number of chocolate bars on the horizontal axis and pints of milk on the vertical axis. Suppose that at every point, consumer A's indifference curve is flatter than consumer B's. What can you conclude about consumers A and B tastes and preferences?
Questions 4 (Choice & Demand). Suppose a consumer's utility function is given by: U(xi, x2 ) = x" a) What is the consumer's demand for x, as a function of income and prices? What is the consumer's demand for x; as a function of income and prices? Use Lagrange method or your intuition. b) What percentage of income does the consumer spend on x;? What percentage of income does the consumer spend on x2?
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.
Is it reasonable to assume that people seek to equate the marginal utility/price ratios of the goods that they purchase, if they have never heard of ‘utility’, let alone ‘marginal utility’, and marginal utility cannot be measured in any absolute way?
ADVANCED ANALYSIS Let: MUA/PA=ZA = 10 - 1 Instructions: Enter your answers as a whole number. How is the $10 best allocated between A and B? MUB/PB=ZB=21-2y, where z is marginal utility per dollar measured in utils, x is the amount spent on product A, and y is the amount spent on product B. Assume that the consumer has $10 to spend on A and B-that is, x+y=10. Note: Px = $1 and Py = $1. LA on A and $ on B How much utility will the marginal dollar yield? utils Prev 1 of 7 Exit Submit Next > Check my work
25 Columns 1 through 4 in the following table show the marginal utility, measured in utils, that Ricardo would get by purchasing various amounts of products A, B, C, and D. Column 5 shows the marginal utility Ricardo gets from saving. Assume that the prices of A, B, C, and D are, respectively, $9, $3, $2, and $6 and that Ricardo has an income of $71. Column 1 Column 2 Column 3 Column 4 Column 5 Units of Good A Units of Units of Units of MU MU MU MU Good B Good C Good D Number of Dollars Saved MU 1 72 1 24 1 15 1 36 1 5 2 54 2 15 2 12 2 30 2 4 3 45 3 12 3 8 3 24 3 3 4 36 4 9 4 7 4 18 4 2 5 27 5 7 5 5 5 13 5 1 6 18 6 5 6 4 6 7 6 0.50 7 15 7 2 7 3.5 7 4 7 0.25 8 12 8 1 8 3 8 2 8 0.13 Instructions: Enter your answers as a whole number. a. What quantities of A, B, C, and D will Ricardo purchase in maximizing his utility? 4 units of A 3 units of B units of C units of D b. How many dollars will Ricardo choose to save?
3. [30+10] A consumer who consumes two goods and has the following utility function: U(x₁, x₂) = log(x₁ + 1) + log(x2 + 1), where ₁ and ₂ denote the monthly consumption of two goods. The consumer maximizes her utility subject to the following constraints: P₁x1 + P₂x2 ≤m, x₁ ≥ 0, X₂ ≥ 0, where p₁ > 0, p2 > 0 and m > 0. [Hint: Note that log f(x) = f(df (2).] 1_df f(x) dx (a) Write down the Kuhn-Tucker conditions for a point (x₁, x2) to solve the problem, and find all points that satisfy the conditions. Do any of those points solve the problem? (b) Show the above results by drawing indifference curves and budget set on the x₁x2 plane.
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