4 4. (Based on Renshaw (2016)) Consider the utility function u = f(x₁, x₂) = 0.75 ln x₁ +0.25 ln x₂.Using the implicit-function rule, find the marginal rate of substitution of x₂ for x₁ (i.e., finddx2/dx₁). Is the slope of any indifference curve associated with this item positive or negative?(Note for the slope: Assume that we have x₂ on the vertical axis and x₁ on the horizontal axis.)

Utility is a measure of satisfaction or happiness that an individual derives from consuming a…

4. (Based on Renshaw (2016)) Consider the utility function u = f(x₁,x2) = 0.75 ln x₁ + 0.25 ln x2. Using the implicit-function rule, find the marginal rate of substitution of x2 for x₁ (i.e., find dx2/dx₁). Is the slope of any indifference curve associated with this item positive or negative? (Note for the slope: Assume that we have x₂ on the vertical axis and x₁ on the horizontal axis.)

4. (Based on Renshaw (2016)) Consider the utility function u = f(x₁,x2) = 0.75 ln x₁ + 0.25 ln x2. Using the implicit-function rule, find the marginal rate of substitution of x2 for x₁ (i.e., find dx2/dx₁). Is the slope of any indifference curve associated with this item positive or negative? (Note for the slope: Assume that we have x₂ on the vertical axis and x₁ on the horizontal axis.)

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.2P

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4. If we have the utility function, U(x)=(x−7)^2, and we know that there is an indifference set around x = 4 and an upper contour set of x = 6. How to graphically represent this function? I'm confused about how to draw the upper contour set on the graph? I know the concept that upper contour is the area over the convex line, but don't know how to put it on the graph. Could you draw a simple graph to help me identify the upper contour?
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1.5) For any two goods (things you would like more of): Select one or more: Indifference curves are downward-sloping. Indifference curves can sometimes cross. Indifference curves can have straight line sections. Indifference curves cannot have kinks.
Find the marginal utility (MU) and marginal rate of substitution (MRS) of following utility functions. 1. U(x1,x2)=ln(x1)+x2,findMUx1,MUx2,andMRS(x1,x2) 2. U(x1,x2)=(2x21+x32)1/2,findMUx1,MUx2,andMRS(x1,x2)
1. Calculate the marginal rate of substitution for the following utility functions: (a) u(x1, x2) = ax₁ + bx2 (b) u(x1, x2) = 2√√x₁ + x₂ (c) u(x1, x2) = ln(x₁) + x2 (d) u(x1, x2) = x1x2 (e) u(x1, x2) = xx h (f) u(x₁, x₂) = x + x² 2. Marie Curie has a utility function U(x, y) = max{x, 2y}. Note that the max{} function es- sentially says that the value of the function takes the highest of the values in the set (the set is denoted as the things inside the curly braces). (a) Draw a graph, plotting the indifference curve along which U(x, y) = 10. (b) Are Marie's preferences convex?
4 Suppose the preferences of an individual are represented by a quasilinear utility function: U(x, y) = ln(x) + 3y a) Initially, px=1, py=6 and I=102. Then, the price of x increases to 2 (px=2). Calculate the changes in the demand for x. Please also calculate the substitution and income effects of the change in px on x. (Hint: since the change in price is not small, you cannot use the Slutsky equation. You need to have numbers instead of functions as the answer.) b) Please also calculate the substitution and income effects of the change in pa on y. c) Instead of doubling to 2, suppose px is only increased by a small amount. Use the Slutsky equation to find the substitution and income effects of the change in the price of x on x. Compare your result to (a). Explain why there's no income effect of the change in px on x. Show your result on an indifference curve. d) Use the Slutsky equation to find the substitution and income effect of the change in Px on y. Compare your result to (b).
Questions 1. Will's utility from vacations (91) and meals (92) is given by the function U(V, M) = 91 x 92. Last year, the price of vacations was $200 and the price of meals was $50. This year, the price of meals rose to $75, while the price of vacations remained the same. Both years, Will had an income of $1500. (a) What is the compensating variation for the price change in meals? (b) What is the equivalent variation for the price change in meals?
Henry's utility function is u(x,y)=max{x+3y,3x+y}. (a) Suppose Henry's current consumption bundle is (2,1.5), what is his current utility level? If he consumes the bundle (1.5,2), will his utility change from the current utility level?
4. If an individual's preferences are described by the utility function U(X1, X2 ) = Xx² + Xz?, a) graph the indifference curve for U = 20 and U= 40. b) Find the optimal consumption quantities if P1 = US$2.50 ; P2 = US$ 7.50; and M = US$ 60.
10. which of the utility function U=5x+3y^2. the indifference curves for this utility function. @) will have a constant marginal rate of substitution of x for y as x increases. B) will have an increasing marginal rate of substitution of x for y as x increases. please explain
12. [G] To find the Income/Substitution effects, graphically we find the point on the original indifference curve that is tangent to a line having the slope of the new budget line. We call this point (xub, x2ubs) Analytically, we can find this point as well. Specifically, the solution to what two equations will give us (x1ubs x2subs) ? Hint: It's the two equations that are equivalent to the graphical approach of finding this point. U, (xubs xubs) Pew U2 (xub xubsy ppew ubs xubs) Uz(xubs xubsy U, (xubs xgubsy A. 1-Pnewxubs - Pnewx2" subs 03; В. U(x1oris,x2ori) = U(x1ubs xubs); Pnew Pnew С. I- P,origxsubs– P,origx,subs= 0: Phew Uz (x{ubs xubs) Pnew D. U(x1"e",x2 mew) = U(x1ubs x ubs). U;(x}ubs x3ubs)_ Pporig Uz (xubs xubs) Porig E. None of the above system of equations will correctly solve for (x1ubs