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.

Question

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

Answer 1

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.