Question 6 Recall that the MRS at a consumption bundle (x1, x2) is defined as negativethe slope of the indifference curve at (x₁, x₂).1. Suppose a utility function is given by u(x₁, x2) = 2√√x1 + 4x2.(a) Specify the equation of the indifference curve through (9,4), and writethe equation in the form x2 = ....(b) Now take the derivative with respect to x₁, and evaluate it at x₁ = 9. Thisis the slope of the indifference curve, and hence you can determine theMRS.(c) Suppose you know that (9, 4) is a person's optimal consumption, and thatthe price of good 1 is p₁ = 4. You know that at the optimum the MRSmust be equal to the price ratio. Use this insight to determine the priceof good 2 and the person's income.2. Do the same for the utility function u(x₁, x₂) = x1x².

MRS is the marginal rate of substitution which is the rate at which one good is substituted for…

Question 6 Recall that the MRS at a consumption bundle (x₁, x2) is defined as negative the slope of the indifference curve at (x₁, x₂). =.... 1. Suppose a utility function is given by u(x₁, x2) = 2√√√x₁ + 4x₂. (a) Specify the equation of the indifference curve through (9,4), and write the equation in the form x2 9. This (b) Now take the derivative with respect to x₁, and evaluate it at x₁ = is the slope of the indifference curve, and hence you can determine the MRS. (c) Suppose you know that (9, 4) is a person's optimal consumption, and that the price of good 1 is p₁ = 4. You know that at the optimum the MRS must be equal to the price ratio. Use this insight to determine the price of good 2 and the person's income. 2. Do the same for the utility function u(x₁, x2) = x1x².

Question 6 Recall that the MRS at a consumption bundle (x₁, x2) is defined as negative the slope of the indifference curve at (x₁, x₂). =.... 1. Suppose a utility function is given by u(x₁, x2) = 2√√√x₁ + 4x₂. (a) Specify the equation of the indifference curve through (9,4), and write the equation in the form x2 9. This (b) Now take the derivative with respect to x₁, and evaluate it at x₁ = is the slope of the indifference curve, and hence you can determine the MRS. (c) Suppose you know that (9, 4) is a person's optimal consumption, and that the price of good 1 is p₁ = 4. You know that at the optimum the MRS must be equal to the price ratio. Use this insight to determine the price of good 2 and the person's income. 2. Do the same for the utility function u(x₁, x2) = x1x².

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.13P

See similar textbooks

Similar questions

Find utility functions given each of the following indifferencecurves [defined by U()=k ]: a. z=k1/x/y/ b. y=0.5x24(x2k)0.5x c. z=y44x(x2yk)2xy22x
Q2. Amanda receives satisfaction from consumption C and leisure L. Her utility function is given by U(C, L) = L x C. a. Define indifference curve. [5 points] b. Name 2 properties of indifference curves. [5 points]
(d) What point in the graph will lead to utility maximizing with the budget line LZ and the indifference curves as shown above? Please label the point C. (e) What is the marginal rate of substitution (slope of the indifference curve) at the utility maximizing point you have shown in (d)?(f) Please explain why the consumer will not choose point A or point B.
Consider an individual whose preferences are represented by the utility function U(x1,x2) = min {3x1+x2 , x1+3x2}. For this individual, Plot the indifference curve giving a utility level of 6. Calculate her MRS at consumption bundle (x1,x2), does MRS depend on the value of (x1,x2)? Calculate her optimal consumption bundle when she has an income of 24, and P1=1, P2=2. Calculate her optimal consumption bundle when she has an income of 24, and P1=1, P2=4. Calculate the income and substitution effects resulting from the change in price of Good 2 (on the level of Good 2 consumption!) Calculate her demand function X1* (P1, P2,m) Calculate her own price, cross price, and income elasticities at X1*(1, 2, 24) and at X1*(1, 4, 24). Based on these, can you say the goods 1 and 2 are (gross, or Marshallian) complements or substitutes?
Suppose that a consumer has the indifference map shown above. The relevant budget line is LZ. The price of good X is $20.(a) What is the consumer’s income? Please show your calculations.(b) What is the price of good Y? Please show your calculations.(c) What is the slope of the budget line LZ. (d) What point in the graph will lead to utility maximizing with the budget line LZ and the indifference curves as shown above? Please label the point C. (e) What is the marginal rate of substitution (slope of the indifference curve) at the utility maximizing point you have shown in (d)?(f) Please explain why the consumer will not choose point A or point B.
Q12. Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F.Suppose as Case X, Total income is $100 and per unit prices of Food (F) and Cloth (C) are $2 and $15, respectively. a. What is the value of MRS at the optimal point and what does this value mean? b. What is the optimal consumption bundle i.e (F*,C*)? c. Plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space.
Q8. Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F. Suppose as Case 1, Total income is $100 and per unit prices of Food (F) and Cloth (C) are $2 and $10, respectively. a. What is the value of MRS at the optimal point and what does this value mean? b. What is the optimal consumption bundle i.e (F*,C*)? c. Plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space. d. Now assume a new Case 3, Pc' = $15, holding all else the same as in Case 1, do the same analysis (parts a-c) and contrast your answers to Case 1. For part c, you should draw old (Case 1) and new (Case 3) budget lines/point of optimality.
Q5. Consider a utility function: U (F,C) = FC so MU_F = C and MU_C = F. Suppose as Case A, Total income is $120 and per unit prices of Food (F) and Cloth (C) are $2 and $10, respectively. a. What is the value of MRS at the optimal point and what does this value mean? b. What is the optimal consumption bundle i.e (F*,C*)? c. Plot the budget line and clearly depict the point of optimality in the F (x-axis)-C (y-axis) space. d. Now suppose Case B, where assuming if income decreases to $100, holding all else the same, do the same analysis (parts a-c) and contrast your answers to Case A. For part c, you should draw old (Case A) and new (Case B) budget lines/point of optimality.
An individual utility function is given by U(x,y) = x·y. Derive this individual indirect utility function. Using this individual indirect utility function, compute her level of utility when I = $800, px = $20 and py = $40. It is equal to?
1. Prove if the indirect utility function is quasiconvex The indirect utility function: V(p,w) = w[P1^(p/p-1) + P2^(p/p-1)]^((1-p)/p)
Ambrose has indifference curves with the equation x2 = constant 4x^1/2, where larger constants correspond to higher indifference curves. If good 1 is drawn on the horizontal axis and good 2 on the vertical axis, what is the slope of Ambrose's indifference curve when his consumption bundle is (16, 17)?
Please solve all parts (d and e) Find the marginal rate of substitution of y for x [MRS(x,y)] for the following utility functions