Question
Let T = 24 be total available hours(note that T=24, not T=12),L be hours of leisure, and n be hours of work. Let w >0 be the hourly wage and p >0 be the price of the consumption good. Finally, let C be consumption. Treat each part of this question as a separate and independent scenario. Assume the utility function is U(L, C) = LC.
Suppose that the law forces the agent to work no less than 3 hours and no more than 8 hours, i.e. 𝑛 ∈ [3, 8]. Find (L*,C*) and illustrate graphically your solution
Suppose that the law forces the agent to work no less than 3 hours and no more than 8 hours, i.e. 𝑛 ∈ [3, 8]. Find (L*,C*) and illustrate graphically your solution
Answers
There are no human answers yet.
Answered by
GPT 3.5
To find the optimal values of L and C, we need to maximize the utility function U(L, C) = LC, subject to the constraints.
First, let's analyze the constraints. The agent is forced to work no less than 3 hours and no more than 8 hours. Therefore, the feasible range for n is 𝑛 ∈ [3, 8]. Considering that T = 24, we can express the leisure hours (L) in terms of work hours (n): L = T - n = 24 - n.
Now, let's substitute this expression for L in the utility function: U(n) = LC = (24 - n)n = 24n - n^2.
To maximize U(n), we need to find the value of n that maximizes this quadratic function. The maximum value occurs at the vertex of the quadratic, which is given by n = -b/2a, where a = -1 and b = 24.
n = -24/(2*(-1)) = 24/2 = 12.
Therefore, the optimal value for work hours is n* = 12. Substituting this value back into the expression for L, we find L* = 24 - n* = 24 - 12 = 12.
So, the optimal values are L* = 12 and n* = 12.
To illustrate graphically, you can plot the utility function U(n) = 24n - n^2 as a quadratic curve with n on the x-axis and U(n) on the y-axis. Mark the point (12, U(12)) as the maximum point on the curve.
First, let's analyze the constraints. The agent is forced to work no less than 3 hours and no more than 8 hours. Therefore, the feasible range for n is 𝑛 ∈ [3, 8]. Considering that T = 24, we can express the leisure hours (L) in terms of work hours (n): L = T - n = 24 - n.
Now, let's substitute this expression for L in the utility function: U(n) = LC = (24 - n)n = 24n - n^2.
To maximize U(n), we need to find the value of n that maximizes this quadratic function. The maximum value occurs at the vertex of the quadratic, which is given by n = -b/2a, where a = -1 and b = 24.
n = -24/(2*(-1)) = 24/2 = 12.
Therefore, the optimal value for work hours is n* = 12. Substituting this value back into the expression for L, we find L* = 24 - n* = 24 - 12 = 12.
So, the optimal values are L* = 12 and n* = 12.
To illustrate graphically, you can plot the utility function U(n) = 24n - n^2 as a quadratic curve with n on the x-axis and U(n) on the y-axis. Mark the point (12, U(12)) as the maximum point on the curve.
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.