Oscar consumes only two goods, X and Y. Assume that Oscar is not at a corner solution, but he is maximizing utility. Which of the following is NOT necessarily true? (a) MRSxy = Px/Py. (b) MUx/MUy = Px/Py. (c) Px/Py = Income. (d) Px/Py = slope of the indifference curve at the optimal choice. (e) MUx/Px = MUy/Py.

Oscar consumes only two goods, X and Y. Assume that Oscar is not at a corner solution, but he is maximizing utility. Which of the following is NOT necessarily true? (a) MRSxy = Px/Py. (b) MUx/MUy = Px/Py. (c) Px/Py = Income. (d) Px/Py = slope of the indifference curve at the optimal choice. (e) MUx/Px = MUy/Py.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter3: Preferences And Utility

Section: Chapter Questions

Problem 3.4P

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1. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are px and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. = (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px = 10, Py = 5 and m = 100. Suppose that px increases to p how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y > 0). = 15,
2. Sally consumes two goods, X and Y. Her utility function is given by the expression U = 3XY². The current market price for X is £10, while the market price for Y is £5. Sally's current income is £500. (a) Sketch a set of two indifference curves for Sally in her consumption of X and Y. (b) Write the expression for Sally's budget constraint. Graph the budget constraint and determine its slope. (c) Determine the X, Y combination which maximises Sally's utility, given her budget constraint. (d) Calculate the impact on Sally's optimum market basket of an increase in the price of X to $15. What will happen to her utility as a result of the price increase?
(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.
(a) Suppose we have preferences U(X, Y) = min [X, 3Y]. Graph/sketch the indifference curve through the bundle (X = 30, Y = 30). What is the utility of (30, 30) and explain why the indifference curves look the way they do. (b) What does the Marginal Rate of Substitution tell us about preferences? < (c) Why is the Marginal Rate of Substitution not applicable in this example? < (d) What do we mean by a composite good? What does this composite good look like with these preferences? Show and explain.< (e) State the consumer's maximization problem and express this in words.< (f) Now let Px = 10, Px= 20 and income M = 2000. Find optimal X*, Y*, and the resulting Utility (U*). Show your work. < (g) Now let Py = 15. How does optimal consumption (X*.Y*) and utility (U*) change relative to (e)? Explain in simple terms and show in a diagram.<
21. A consumer has a utility function defined over two goods X and Y. Let the quantity of Good X be x ≥ 0 and the quantity of Good Y be y ≥ 0. The utility function is given below: u(x, y) = xy + 2y. Assume that the consumer has income m and that prices are på and py. (a) Explain whether the preferences underlying this utility function satisfy completeness and transitivity. (b) Explain whether the preferences underlying this utility function satisfy monotonicity and convexity. (c) Find the consumer's Marshallian demands for Good X and Good Y at prices px > 0 and Py > 0. (d) Show that goods X and Y are normal goods and explain whether either good is a substitute for the other. (e) Assume that px 10, Py = 5 and m = 100. Suppose that px increases to px = 15, how much of the change in demand for Good X is via the substitution effect and how much is via the income effect? Note: You may assume an interior solution (i.e. x > 0 and y> 0). =
4. Assume the prices of r and y are 1 and income is 10. For the following two utility functions, find the quantities of r and y consumed at the con- sumer's optimum. (Here, it may be helpful to solve by drawing out some indifference curves.) (a) U(r, y) = Min(x, 2y) (b) U(x, y) = 2x + y (c) Suppose the price of a rises to 3. What are the new optima in each case. What are the substitution effects?
1. For each of the three utility functions below, find the substitution effect, the income effect, and the total effect that result when prices change from p = (2,1) to p' = (2,4). Assume the consumer has income I = 20. (a) Before doing any calculation, make an educated guess about the relative magnitude of the three substitution effects and the three income effects to be found below. (b) u(x1, x2) = x1 + x2. (c) u(x1, x2) = x1x2. (d) u(x1,x2) = min {x1,x2). (e) Rank the substitution effects and the income effects found above by their magnitude. To what extent do they conform to your guess?
13) An indifference curve is related to which of the following? A) Choices and preferences of consumer B) Prices of goods X and Y C) Consumer’s income D) Total utility from goods X and Y
(a) Suppose we have preferences U(X, Y) = 10X²/³ Y¹/3, Create a table and graph/sketch the indifference curve through the bundle X = 30 and Y = 30.< (b) The Marginal Rate of Substitution is MRSxy=-2Y/X. For the bundle (X= 30, Y = 30), calculate and then interpret what the value of the MRS means.< (c) Cobb-Douglas preferences are strictly convex. What does this imply about the MRS as we move along the indifference curve? Explain/discuss (you may want to draw a picture). < (d) What are the two conditions (equations) that identify the optimum given these preferences and the consumer's budget constraint? Sketch this in a figure and explain.< (e)_From (d) we can show that optimal demands are: X=½ M/PX and Y = ½ M/Px. (you do not have to derive these, just use the equations I have given you.) Calculate optimal demands (X*, Y*) and utility if Px = 10, Px= 5 and income M = 1200. < (f)_Suppose Px falls to Px = 8 but Py and M are unchanged (Px = 5 and M = 1200). Calculate the new optimal demands…
1. If an individual consumer faced the following utility function: U(X, Y) = 2X² + Y³, where X, Y > 0, which of the following assumptions would not hold? (a) Completeness (b) Monotonicity (c) Transitivity (d) Convexity (e) Continuity
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16. Utility functions are simply mathematical representations of preferences.(a) True(b) False