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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

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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