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

ENGR.ECONOMIC ANALYSIS

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Chapter1: Making Economics Decisions

Section: Chapter Questions

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Form Utility

The term utility in general refers to the satisfaction earned from usage of a particular product or service. Utility always plays an important role because it influences various important elements like demand & thereby the price of the good or service as well. Though utility is not easy to measure, various types of economic models can be used indirectly. There can be various types of utilities like economic utility, marginal utility, cardinal utility, form utility.

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1. Which of the following utility functions does not admit corner solutions (x = 0 or x = 0) given a positive income m > 0? a) u(x1, x2) = x, + 2x2, b) v(x1, x2) = X1X2, c) w(x1,x2) = max{x1,x2}, d) y(x1, x2) = Inx1 + x2.
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You currently have (x, y) = (8, 2) and the utility function U(x, y) = min(x, 4y ). How many units of X are you willing to give up to get another unit of Y?
Karen has perfect complement preferences over toothbrushes B and toothpaste P, with the associated utility function U(B,P) = min (3B, 2P). Suppose the price of a toothbrush is $6, the price of toothpaste is $4 a tube, and she has $72 to buy her entire year's supply of toothbrushes and toothpaste. How many toothbrushes and tubes of toothpaste will she buy?
2) Which of the following utility functions represent the same preferences? Explain. a) U (x₁, x₂) = X₁ X₂ b) W (x₁, x₂) = 5lnx₁ +5lnx₂ c) V (x₁, x₂) = x₁¹/3x₂ ¹/3 - 0.8 d) Z(x₁, x₂) = 0.5x₁ + 0.5x₂
Alice receives an allowance of 500 dollars that she spends on buying snacks (S) and tea (T). The price of each snack is 10 dollars and the price of each tea is 5 dollars. Her utility is given by: U (S, T) = 2S³/4 +T3/4 (a) Find her marginal rate of substitution (MRS) between S and T. (b) Write Alice's budget constraint. (c) Find Alice's optimal consumption and the optimal A. (d) What is her new consumption if the price of tea becomes 10 dollars? Note: numeric solutions for questions (c) and (d) are not integers.
Yam has the following utility function for Apples (X1) and Ice Cream (X2), U(X1,X2) = Min{3X1,X2}. Yam likes to get more of both Apples and Ice Cream. Suppose Yam has an income of M = $160 budgeted to be spend on these two goods and the prices of Apples and Ice Cream are p1 =$2, p2 =$1. Yam's optimal consumption of Apples is and Ice Cream is (Please put your answer in numerical values with no comma or decimal place.) Suppose a quantity tax of $1 is imposed on Ice Cream. Yam's after tax optimal consumption of Apples is and Ice Cream is (Please put your answer in numerical values with no comma or decimal place.) Suppose instead of the quantity tax, the government choose to impose a lump-sum tax of $20. After the imposition of the lump-sum tax, Yam's optimal consumption of Apples is and Ice Cream is (Please put your answer in numerical values with no comma or decimal place.) If government would have to impose one of these 2 kinds of taxes, which of these types of tax would be better…
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1 Consider a consumer who consumes only two goods, 21 and z2. His utility function is u (21, 22) = In z1 + 2. where In represents the natural logarithm. 1.a Derive the consumer's MRS of good 1 for good 2 using calculus to calculate his marginal utility from z and his marginal utility from z2. 1.b If the price of z1 is $2 per unit and the price of z2 is $4 per unit, and the consumer's income is $100, What is the equation of this consumer's budget line? 1.c What are the optimal consumption choices, z and z5, for this consumer? Show your work.
1. Suppose you are considering to make a decent holiday dinner. In this year, to be creative, you are thinking to put Copper River Salmon from Alaska (fillet) and fresh Atlantic Lobsters from Maine as entrée on the table. Use the information given, to finish the questions below. (1) The table following provides you total utility (U) and marginal (MU) from consuming Salmon and Lobster. Suppose your total utility of Salmon consumption (U1) is a function of the units (Q as pound or lbs.) of salmon consumed, as U1 =. =/400Q. Similarly, your total utility of Lobster (U2) has the form as units (Q as lbs.) of lobster consumed, which is U2 = /36Q + 40. Now fill the empty spots in the table using the information given. Total utility and marginal utility of Salmon and Lobster. Copper River Salmon Atlantic Lobster Q (Ibs.) U1 MU1 U2 MU2 1 20.00 20.00 8.72 8.72 2 3 4. 6. 7 8 10 (2) With a certain amount of budget, you are going to purchase both Salmon and Lobsters. To maximize your total…