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Compute marginal utilities and marginal rate of substitution for each of the utility functions given below. a) U(X,Y) = 1- e-ax e-by b) U(X,Y)= XY + 3X + 5Y - 1 c) U(X,Y)= (0.3Xm +0.7Ym) m

Compute marginal utilities and marginal rate of substitution for each of the utility functions given below. a) U(X,Y) = 1- e-ax e-by b) U(X,Y)= XY + 3X + 5Y - 1 c) U(X,Y)= (0.3Xm +0.7Ym) m

Microeconomic Theory
Microeconomic Theory
12th Edition
ISBN:9781337517942
Author:NICHOLSON
Publisher:NICHOLSON
Chapter4: Utility Maximization And Choice
Section: Chapter Questions
Problem 4.13P
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Compute marginal utilities and marginal rate of substitution for each of the utility functions given below. -ax -by a) U(X,Y)= 1- e - e b) U(X,Y)= XY + 3X + 5Y c) U(X,Y)= (0.3xm +0.7Ym)mm X¹-a yl-b d) U(X,Y) 1-a 1-b = Consider an individual who decides to allocate time between different activities. For simplicity suppose that there are two activities: working and leisure. Denote by tw the time spent for working and to the time spent for leisure and suppose that individual gains utility denoted by U(tw, t₁). Consider that the individual has 16 hours a day in total that can be spent for working and/or leisure. a) Discuss possible preferences the individual might have and graphically illustrate some examples of indifference curves associated to these preferences. b) Graphically illustrate the time constraint the individual faces. c) Graphically illustrate the optimal time allocation of the individual. Explain why it is optimal by comparing to a sub-optimal allocation choice. d) Write the optimization problem of the individual (formally, as we wrote in lectures). e) Analytically solve the problem, obtain the necessary condition, and provide an interpretation of it (explain your own words what it means).
Transcribed Image Text:Compute marginal utilities and marginal rate of substitution for each of the utility functions given below. -ax -by a) U(X,Y)= 1- e - e b) U(X,Y)= XY + 3X + 5Y c) U(X,Y)= (0.3xm +0.7Ym)mm X¹-a yl-b d) U(X,Y) 1-a 1-b = Consider an individual who decides to allocate time between different activities. For simplicity suppose that there are two activities: working and leisure. Denote by tw the time spent for working and to the time spent for leisure and suppose that individual gains utility denoted by U(tw, t₁). Consider that the individual has 16 hours a day in total that can be spent for working and/or leisure. a) Discuss possible preferences the individual might have and graphically illustrate some examples of indifference curves associated to these preferences. b) Graphically illustrate the time constraint the individual faces. c) Graphically illustrate the optimal time allocation of the individual. Explain why it is optimal by comparing to a sub-optimal allocation choice. d) Write the optimization problem of the individual (formally, as we wrote in lectures). e) Analytically solve the problem, obtain the necessary condition, and provide an interpretation of it (explain your own words what it means).
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