2. The consumer has Leontief preferences for two goods r and y: u(x, y) = min{2x, 5y}. Pz > 0,Py> 0 and m> 0 is given.a) Set up the Expenditure Minimization Problem.b) Write down the optimality condition for x and y.c) Solve for the compensated demand functions for good z and y.• T² (Pr, Py, u) =•y (Pz, Py, u) =d) Graph the compensated demand function for rº(Pr. Py, u) when u = 50, Py = 1, and m = 100.(place z on the z-axis and pr on the y-axis).5025002550e) What is the reason for the extreme difference in the ordinary (Marshallian) demand functionand the compensated (Hicksian) demand function for the case of Leontief preferences?

Expenditure minimization problem: For the given prices and utility level the consumer tries to…

2. The consumer has Leontief preferences for two goods and y: u(x, y) = min{2r, 5y}. pz > 0, Py> 0 and m> 0 is given. a) Set up the Expenditure Minimization Problem. b) Write down the optimality condition for r and y. c) Solve for the compensated demand functions for good z and y. xº (Pz, Py, u)= y (Pz, Py, ū)=

2. The consumer has Leontief preferences for two goods and y: u(x, y) = min{2r, 5y}. pz > 0, Py> 0 and m> 0 is given. a) Set up the Expenditure Minimization Problem. b) Write down the optimality condition for r and y. c) Solve for the compensated demand functions for good z and y. xº (Pz, Py, u)= y (Pz, Py, ū)=

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter6: Demand Relationships Among Goods

Section: Chapter Questions

Problem 6.9P

See similar textbooks

Similar questions

The preferences of a typical Californian can be represented by the following utility function: U (x1 , x2 ) = α ln(x1) + (1 − α) ln(x2) Here, x1 and x2 are the quantities of electricity and gasoline, respectively. The consumer faces prices given by p1 and p2 and has income m. Currently, the government has decided to impose a consumption restriction so that any person in the state is allowed to consume at most 50 units of electricity (x1 ≤ 50). Call this restriction a rationing constraint. (a) If α=0.25, m=100,and p1 =p2 =1, find the optimal consumption bundle of gasoline and electricity. Is the electricity rationing constraint binding (meaning does x1∗ = 50)? (b) Suppose that α = 0.75, but the other parameters are the same. What is the optimal consumption bundle? Is the rationing constraint on electricity consumption binding? (c) Now, assume that there is no rationing constraint. Assume m = 100 and p1 = p2 = 1, but α remains as a generic parameter. Solve for the optimal quantity…
Assume the demand for cherries is elastic and that the producer of cherries increases the price of cherries. As a result? A convex indifference curve implies what type of behavior? If a consumer always wishes to consume peanut butter and jam in fixed proportions, he treats these two goods as if they are? Assume PX= $3 and PY = $6 and income = $30. What is the relative price of an additional unit of good X in terms of the amount of good Y that has to be given up? Assume there are only two goods (X and Y). Assume the relative price of good X, is 2 of good Y. If income doubles, the price of X doubles and the price of Y doubles, what will be the relative price of good Y?
Assume the budget constraint and the indifference curves are both linear. Assume the consumer is willing to tradeoff 1 of good X for 1 of good Y. If the relative price of one additional good X is giving up 1/2 of good Y, then the optimal bundle of the two goods is any combination on the budget constraint all Y 1 of X and 2 of Y all X
On what portion of the demand function does total revenue rise as price rises? The price of tea cakes goes up by 1%. As a result, the quantity purchased of tea falls by 1.5%. If income rises and consumers buy more of a particular good, that good is? Assume the budget constraint and the indifference curves are both linear. Assume the consumer is willing to tradeoff 1 of good X for 1 of good Y. If the relative price of one additional good X is giving up 1/2 of good Y, then the optimal bundle of the two goods is? What condition describes the optimal bundle of goods X and Y where utility is maximized [M=income, MU=marginal utility, P=price]
Suppose a consumer’s utility function is u = x_1^(3/2) x_2^(3/2) . She spends her budget of £27 for two goods. The prices of both goods are p1 = 6 and p2 = 6 Now suppose that instead both goods are priced as follows: There is a discount of 50% on the price of good 1 on each additional unit in excess of 3 units, and there is a discount of 50% on the price of good 2 on each additional unit in excess of 3 units. Draw the new budget constraint and derive it analytically.
Max’s preferences can be represented by the utility function u(x1, x2) = 2√ x1 + x2. Take good 2 to be the numeraire (p2 = 1). a) For any price p1 and income m, write down and solve his utility maximization problem, i.e. find the demand x1(p1, m) and x2(p1, m) using the Lagrangian method (assuming an interior solution). Illustrate why this method is equivalent to using the two conditions (1) |MRS| =price ratio and (2) Budget constraint. b) In doing so, also solve for λ ∗ . c) Given any p1 and income m, find the utility of Max corresponding to his demand. Denote this by v(p1, m). That means, find v(p1, m) = u(x1(p1, m), x2(p1, m)). d) To understand how v(p1, m) changes with m, i.e. compute ∂v(p1,m) ∂m . Compare with your answer to b).
Rakhi likes to consume only two commodities X and Y, and nothing else. Both the commodities give her positive utilities, thereby giving rise to her preferences being well- behaved and convex. However, she treats commodity X as an inferior good. Suppose Rakhi earns an income of 1000 per month and the ongoing prices of commodities X and Y are 20 and 10, respectively. (a) Show, with the help of a diagram, what will happen to her optimal consumption of commodity X, in each of the following circumstances: Situation A: Price of X falls to 10, while everything else remains the same. Situation B: Her income increases to 2000, while prices remain the same. (b) What can you say about the demand curve for commodity X that Rakhi’s choice behaviour would produce? Elaborate, with the help of diagram.
Q1) For the utility function in part a below, draw a set of indifference curves showing utility levels U = 12, U = 16, and U = 24. U = XY What is the relationship between the two goods X and Y? (as implied by shape of indifference curves) What is the marginal utility of X? Q2) John and Sam have the same income. The stadium sells hot dogs for $4 and popcorn bags for $3. They have different preferences; John likes popcorn much more then Sam. At their optimal bundles, John and Sam’s Marginal Rates of Substitution will be: A) the same B) Sam’s MRS will be greater. C) John’s MRS will be greater. D) we cannot tell.
Assuming a linear budget constraint, consider the following utility maximization problem:U (x1, x2) = 2x10.5 + 4x20.5 1. Compute the Marshallian demand functions for goods 1 and good 2.2. Find the compensated demand function.3. Derive the expenditure function and verify that h (p, u) = ∇pe (p, u)4. Derive the indirect utility function and verify Roy's Identit
Amy chooses between two goods, x and y, with prices px and py, respectively. She has an income I and her preferences are represented by the utility function U(x, y) = √x + y. 1. Assuming that an interior solution exists to the constrained utility maximization problem, derive Amy ’s Marshallian demand function for each of the two goods. Are both goodsnormal? Explain 2. Find the indirect utility function, V (px, py, I). 3. Derive Amy’s Hicksian demand function for each of the two goods and the expenditure function. Compare the Marshallian demand for good x and the Hicksian demand for good x. Arethese different functions? If so, why? If not, why not? 4. Suppose that I = 100, px = 1 and py = 2. How much of good x and good y will Amy optimally choose? 5. Now the price of good x rises to px = 2, while income (I = 100) and the price of good y, py = 1, remain unchanged. What quantities does Amy buy and what is her resulting utility?Illustrate graphically 6. Find the income and substitution…
Let t = 3 The consumer has a preference relation defined by the utility function u(x, y) = −(t + 1 − x)2 − (t + 1 − y)2. He has an income of w > 0 and faces prices px and py of goods X and Y respectively. He does not need to exhaust his entire income. The budget set of this consumer is thus given by B={(x,y)∈R2+ :pxx+pyy≤w } A. Is the optimal demand for good 1 everywhere differentiable with respect to w? informal argument is sufficient
Suppose a consumer has a budget of $200 to spend on two goods, X and Y, whose prices are $20 and $10, respectively. If the consumer is observed to buy 5 units of X and 10 units of Y, where the respective Marginal Utilities of X and Y are, 50 and 40 utils, is the consumer in equilibrium? Explain why or why not. If the consumer is not in equilibrium under conditions in d), suggest another combination that would possibly achieve equilibrium. Explain your answer.