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4. Suppose preferences are represented by the Cobb-Douglas utility function, u(x1x2) = xx-. a) Show that...
Suppose Bill has preferences over chocolate,x, and ice cream,y, that are represented by the Cobb-Douglas utility...
Suppose Bill has preferences over chocolate,x, and ice cream,y, that are represented by the Cobb-Douglas utility function u(x, y) =x^2 y. 1. Write down two other Cobb-Douglas utility functions, besides the one above, that represent Bill’s preferences. 2. Write down two more Cobb-Douglas utility functions that do NOT represent Bill’s prefer- ences. 3. Draw 3 indifference curves that represents Bill’s preferences at 3 different levels of satsifaction. 4. What is Bill’s marginal rate of substitution between chocolate and ice cream?...
Problem 1 (10 marks) Answer the following questions regarding a Cobb-Douglas utility function U(X,Y)= X0.3 0.7...
Problem 1 (10 marks) Answer the following questions regarding a Cobb-Douglas utility function U(X,Y)= X0.3 0.7 (a) Does this utility function exhibit diminishing marginal utility in X? Show why or why not. (b) Does this utility function exhibit diminishing marginal rate of substitution? Mathematically show and verbally explain why it has (or doesn't have) such property. Problem 2 (10 marks) Consider the following utility function U(X,Y)= X14734 Suppose that prices and income are given as following Px= 1 Py =...
2. (25%) Consider a consumer with preferences represented by the utility function: u(x1, x2) = min...
2. (25%) Consider a consumer with preferences represented by the utility function: u(x1, x2) = min {axı, bx2} If the income of the consumer is w > 0 and the prices are p1 > 0 and P2 > 0. (a) Derive the Marshallian demands. Be sure to show all your work. (b) Derive the indirect utility function. (c) Does the utility function: û(x1, x2) = axı + bx2 represent the same preferences?
5. Suppose that each consumer has the Cobb-Douglas utility function u:(X1i, X2i) X11 X21-4. In addition...
5. Suppose that each consumer has the Cobb-Douglas utility function u:(X1i, X2i) X11 X21-4. In addition the endowments are wi=(1,2) and w2=(2,1). What should be the vector of prices (pı", p2') in order to achieve equilibrium (supply-demand). [Note use an increasing transformation of the utility functions given by a In Xii+(1-a) In X2i] . . following utility functions:
Suppose Alex’s preferences are represented by u(x1,x2) = x1x32. The marginal utilities for this utility function...
Suppose Alex’s preferences are represented by u(x1,x2) = x1x32. The marginal utilities for this utility function are MU1 = x23 and MU2 = 3x1x22. (a) Show that Alex’s utility function belongs to a class of functions that are known to be well-behaved and strictly convex. (b) Find the MRS. [Note: find the MRS for the original utility function, not some monotonic transformation of it.] (c) Write down the tangency condition needed to find an optimal consumption bundle for well-behaved preferences....
4. Suppose you have the following Cobb-Douglas Utility Function: And $200 to spend. a. Use the...
4. Suppose you have the following Cobb-Douglas Utility Function: And $200 to spend. a. Use the method of Lagrangian Multipliers, to maximize this consumer's utility and derive demand equations for both goods. Sketch their respective demand curves. Show all work. (5 pts) b. If Px = Py = $1, how much utility will the consumer enjoy? Show work/explain. (2.5 pts) c. Does this allocation satisfy the rule of equal marginal utility per dollar spent? Explain/show work. (2.5 pts)
A consumer's preferences are given by the following utility function: u(x,y) = xy Assume Pold =...
A consumer's preferences are given by the following utility function: u(x,y) = xy Assume Pold = 1, Py = 1, and I = 8. a. Solve for the Marshallian demand functions of x and y (your answer should have numbers, not variables. You should round your answers to three decimal places): * old 4 y = 4 b. What is the utility associated with these demands, prices, and income? u = 16 c. Suppose the price of x rises to...
Instructor-created question A consumer's preferences are given by the following Cobb-Douglas utility function: Assume Px >...
Instructor-created question A consumer's preferences are given by the following Cobb-Douglas utility function: Assume Px > 0, P, > 0, and I > 0 a. In the limit, what is the marginal utility of xas x goes to zero and what is the marginal utility of y as y goes to zero? lim MUY ra3 lim MU, -
4. Anne's utility function is given by u(xi,x2) Michael's utility function is given by u(xı, x2)' = x1x2. S...
4. Anne's utility function is given by u(xi,x2) Michael's utility function is given by u(xı, x2)' = x1x2. She has two friends, Michael and Peter 1 and Peter's is given by u(x1, X2) = = 1042 1 Τ) -x12. Who has the same preferences as Anne? And who has indifference curves with similar shape as Anne's indifference curves? Explain.
4. Anne's utility function is given by u(xi,x2) Michael's utility function is given by u(xı, x2)' = x1x2. She has two...
Income and substitution, Compensating Variation: Show your work in the steps below. Consider the utility function...
Income and substitution, Compensating Variation: Show your work in the steps below. Consider the utility function u(x,y)-x"y a. Derive an expression for the Marshallian Demand functions. b. Demonstrate that the income elasticity of demand for either good is unitary 1. Explain how this relates to the fact that individuals with Cobb-Douglas preferences will always spend constant fraction α of their income on good x. Derive the indirect utility function v(pxPod) by substituting the Marshallian demands into the utility function C....
Problem 1 (10 marks) Answer the following questions regarding a Cobb-Douglas utility function U(X,Y)= X0.3 0.7 (a) Does this utility function exhibit diminishing marginal utility in X? Show why or why not. (b) Does this utility function exhibit diminishing marginal rate of substitution? Mathematically show and verbally explain why it has (or doesn't have) such property. Problem 2 (10 marks) Consider the following utility function U(X,Y)= X14734 Suppose that prices and income are given as following Px= 1 Py =...
2. (25%) Consider a consumer with preferences represented by the utility function: u(x1, x2) = min {axı, bx2} If the income of the consumer is w > 0 and the prices are p1 > 0 and P2 > 0. (a) Derive the Marshallian demands. Be sure to show all your work. (b) Derive the indirect utility function. (c) Does the utility function: û(x1, x2) = axı + bx2 represent the same preferences?
5. Suppose that each consumer has the Cobb-Douglas utility function u:(X1i, X2i) X11 X21-4. In addition the endowments are wi=(1,2) and w2=(2,1). What should be the vector of prices (pı", p2') in order to achieve equilibrium (supply-demand). [Note use an increasing transformation of the utility functions given by a In Xii+(1-a) In X2i] . . following utility functions:
4. Suppose you have the following Cobb-Douglas Utility Function: And $200 to spend. a. Use the method of Lagrangian Multipliers, to maximize this consumer's utility and derive demand equations for both goods. Sketch their respective demand curves. Show all work. (5 pts) b. If Px = Py = $1, how much utility will the consumer enjoy? Show work/explain. (2.5 pts) c. Does this allocation satisfy the rule of equal marginal utility per dollar spent? Explain/show work. (2.5 pts)
A consumer's preferences are given by the following utility function: u(x,y) = xy Assume Pold = 1, Py = 1, and I = 8. a. Solve for the Marshallian demand functions of x and y (your answer should have numbers, not variables. You should round your answers to three decimal places): * old 4 y = 4 b. What is the utility associated with these demands, prices, and income? u = 16 c. Suppose the price of x rises to...
Instructor-created question A consumer's preferences are given by the following Cobb-Douglas utility function: Assume Px > 0, P, > 0, and I > 0 a. In the limit, what is the marginal utility of xas x goes to zero and what is the marginal utility of y as y goes to zero? lim MUY ra3 lim MU, -
4. Anne's utility function is given by u(xi,x2) Michael's utility function is given by u(xı, x2)' = x1x2. She has two friends, Michael and Peter 1 and Peter's is given by u(x1, X2) = = 1042 1 Τ) -x12. Who has the same preferences as Anne? And who has indifference curves with similar shape as Anne's indifference curves? Explain.
4. Anne's utility function is given by u(xi,x2) Michael's utility function is given by u(xı, x2)' = x1x2. She has two...
Income and substitution, Compensating Variation: Show your work in the steps below. Consider the utility function u(x,y)-x"y a. Derive an expression for the Marshallian Demand functions. b. Demonstrate that the income elasticity of demand for either good is unitary 1. Explain how this relates to the fact that individuals with Cobb-Douglas preferences will always spend constant fraction α of their income on good x. Derive the indirect utility function v(pxPod) by substituting the Marshallian demands into the utility function C....
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