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Suppose that all you do in a day is work hard, play nard, sleep hard. Let Xị per day you spend playing, X2 number of hours you spend sleeping, and x3 is the number of hours you spend working. Suppose that sleeping is free, but playing costs you $19 an hour. Furthermore, you spend all the money you earn working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: U = x{' x , where a + az = 1 NOTE: By construction, x2 can represent the hours you spend consuming anything that is free, and x1 can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting. , and let your hourly wage be w. Find the number of hours you should work a day (x; ) in order to maximize your utility as a function of w. Let aj x; = If w = 38: %3D Use Lagrange multipliers. You can simplify the algebra a lot by noticing that maximizing In U instead of U will give the same answer.

Suppose that all you do in a day is work hard, play nard, sleep hard. Let Xị per day you spend playing, X2 number of hours you spend sleeping, and x3 is the number of hours you spend working. Suppose that sleeping is free, but playing costs you $19 an hour. Furthermore, you spend all the money you earn working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: U = x{' x , where a + az = 1 NOTE: By construction, x2 can represent the hours you spend consuming anything that is free, and x1 can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting. , and let your hourly wage be w. Find the number of hours you should work a day (x; ) in order to maximize your utility as a function of w. Let aj x; = If w = 38: %3D Use Lagrange multipliers. You can simplify the algebra a lot by noticing that maximizing In U instead of U will give the same answer.

Algebra and Trigonometry (MindTap Course List)
Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter14: Counting And Probability
Section14.CT: Chapter Test
Problem 5CT: A commuter must travel from Ajax to Barrie and back every day. Four roads join the two cities. The...
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Suppose that all you do in a day is work hard, play hard, and sleep hard. Let x1 be the number of hours per day you spend playing, x2 is the number of hours you spend sleeping, and x3 is the number of hours you spend working. Suppose that sleeping is free, but playing costs you $19 an hour. Furthermore, you spend all the money you earn working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: U = x" x, where aj + az = 1 NOTE: By construction, x, can represent the hours you spend consuming anything that is free, and x can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting. Let aj = 2, and let your hourly wage be w. Find the number of hours you should work a day (x) in order to maximize your utility as a function of w. x = If w = 38: x = Use Lagrange multipliers. You can simplify the algebra a lot by noticing that maximizing In U instead of U will give the same answer.
Transcribed Image Text:Suppose that all you do in a day is work hard, play hard, and sleep hard. Let x1 be the number of hours per day you spend playing, x2 is the number of hours you spend sleeping, and x3 is the number of hours you spend working. Suppose that sleeping is free, but playing costs you $19 an hour. Furthermore, you spend all the money you earn working on playing. The utility you get from sleeping and playing is given by a Cobb-Douglas utility function: U = x" x, where aj + az = 1 NOTE: By construction, x, can represent the hours you spend consuming anything that is free, and x can be the number of hours consuming goods you have to pay for. This does not change the question, it is just interesting that this set-up can be applied to a more general setting. Let aj = 2, and let your hourly wage be w. Find the number of hours you should work a day (x) in order to maximize your utility as a function of w. x = If w = 38: x = Use Lagrange multipliers. You can simplify the algebra a lot by noticing that maximizing In U instead of U will give the same answer.
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