Show that if in R² are continuous, highly monotonic preferences then indifference curves (i.e. classes of abstraction of relations ~) are graphs of continuous, decreasing functions from the interval [a, b] to [c, d] (the boundaries of the intervals can also be infinite). Give an example illustrating that continuity is essential.

Show that if in R² are continuous, highly monotonic preferences then indifference curves (i.e. classes of abstraction of relations ~) are graphs of continuous, decreasing functions from the interval [a, b] to [c, d] (the boundaries of the intervals can also be infinite). Give an example illustrating that continuity is essential.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.13P

See similar textbooks

Similar questions

Consider the following utility functions: (1) u(x₁, x₂) = x₁ + 2x2. (2) u(x1, 1₂) = 2125 12.75 (3) u(x₁,1₂)=-1²-1₂. (4) u(x1, 1₂) min{x1, 2x2). For utility functions (1) to (3), give the equations of the indifference curves correspond- ing to utility level k, where 22 is expressed as a function of 2₁ and k. For utility functions (1) to (4), draw two indifference curves for each function. That is, draw four graphs (one for each utility function) and two indifference curves on each graph.
Suppose that one of your friends does not care if she consumes spaghetti (S) or noodle (N). She wants toconsume either two-dish spaghetti or two-dish noodle or any combination of both which adds up to twodishes.(i) Write the utility function of your friend to represent his preferences.(ii) Draw couple of indifference curves of your friend on graph with appropriate labels.(iii) Denote prices as Pn, Ps; income as M. If Pn < Ps, then determine the demand function of your friend foreach product as a function of prices and income.
2. Now consider u(x1, 22) = 2 ln xỉ + ln x2 (a) Find expressions for MU1 and MU2. (b) Find an expression for MRS12. (c) Are the preferences represented by this function convex? Justify your answer using the MRS or a graph of some indifference curves.
We claimed in the text that if preferences were monotonic, then a diagonal line through the origin would intersect each indifference curve exactlyonce. Can you prove this rigorously? (Hint: what would happen if itintersected some indifference curve twice?)
Que Robert's utility function is U(x, y) = 2/T + y. In every graph for this question, set y in the vertical axis and z in the horizontal axis. (a) Obtain the marginal rate of substitution MRS. Obtain the equation of the indifference curves for Ū = 2 and Ū = 4 (solving for y as a function of 1). Draw these indifference curves, identifying the intersections with each of the axes. (b) Suppose p, = 2 and Robert's income is M = 6. In a new graph, show the total, income and substitution effects (on the demand of r) of a drop in the price of z from p, = 4 to Pz = 2. (c) Now, the store where Robert goes for shopping has changed the prices to p, = 2 and py = 6, however, it also has a new special offer: for any amount of Y that Robert buys, he gets the same amount of X for free. Robert's income is M = 6. Draw Robert's budget line. Write down the equation of this line. Find Robert's utility-maximising bundle. (d)(d.1) Arthur's income is M = 6 and he goes to the same store as Robert. His utility…
Question 4: Consider a general utility function U(x₁, x₂). Let's now solve for the optimal bundle generally using the Lagrangian Method. 1. Write down the objective function and constraint in math. 2. Set up the Lagrangian Equation. 3. Fnd the first derivatives. 4. Find the first order conditions. What's the interpretation for >? 5. Rearrange them to get the tangency condition.
A utility function is called separable if it can be written as U(x, y) = U (x) + U,(y), where U > 0, U" < 0, and U1, U, need not be the same function. a. What does separability assume about the cross-partial derivative U„? Give an intuitive discussion of what word this condition means and in what situations it might be plausible. b. Show that if utility is separable then neither good can be inferior. c. Does the assumption of separability allow you to con- clude definitively whether x and y are gross substitutes or gross complements? Explain. d. Use the Cobb-Douglas utility function to show that separability is not invariant with respect to monotonic transformations. Note: Separable functions are examined in more detail in the Extensions to this chapter.
b) Define indifference curve and discuss the characteristics of an indifferencecurve.
Suppose an individual has the following utility function: U(x, y) = -4(x – 5.5)? – 2(y - 3.5)? Further assume that the price of good x, Pr individual has an income m = $65 $6, the price of good y, Py = $8 and the %3D %3! a) enčes. Please label the graph properly including values for x and y. Draw an indifference curve (one IC is enough) that represents this person's efer- b) and y* for the above utility function. Explain your answer. Intuitively, and without formally solving, can you guess the maximized values r* Derive the optimal values z* and y* by formally solving the above utility function c) subject to the above constraint. You can use any of the utility maximization techniques we learned in class. d) is given in this problem as well as the budget constraint, can you explain the differences between the answers you found in parts b) and c) (if any). Be as detailed as possible with your explanation. Compare your answers in parts b) and c). Based on the utility function that
ASAP 1. Consider a consumer with utility u(x1,x2) = .5lnx1 + .5lnx2 (b) Now, consider an equivalent representation of the above utility functionx51 x52 Does this utility function have the "increasing difference property"? Howabout the "strict increasing difference" property?
What is the logarithmic transform of the utility function U=xα1xβ2xγ3 given the budget constraint px1x1+px2x2+px3x3=M . Select one: a. ln U=ln xα1+ln xβ2+ln xγ3 b. ln U=ln xγ1+ln xβ2+ln xα3 c. ln U=α ln x1−β ln x2−γ ln x3 d. ln U=α ln x1+β ln x2+γ ln x3
Suppose that consumer has the following utility function: U(X, Y) = X@Y where 1 > a > 0 and 1 > b>0 are constants. Which of the following is correct? Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Utility holds. Preferences are convex and indifference curves are bowed outward from the origin since Lavw of Diminishing Marginal Utility fails to hold. Preferences are concave and indifference curves are bowed outward from the origin since Law of Diminishing Marginal Rate of Substitution fails to hold. O Preferences are convex and indifference curves are bowed inward towards the origin since Law of Diminishing Marginal Rate of Substitution holds.