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
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.
To find the optimal values of leisure (L*) and consumption (C*) in this scenario, we need to maximize the utility function U(L, C) = LC subject to the constraint that the agent works between 3 and 8 hours.
Let's start by graphing the feasible region based on the given constraint:
- On the x-axis, we represent the number of hours of work (n).
- On the y-axis, we represent the number of hours of leisure (L).
Since T = 24 and the agent is required to work between 3 and 8 hours, the feasible region is a vertical rectangle with the following vertices: (3, 0), (3, T), (8, T), and (8, 0).
Now, let's find the optimal values (L*, C*) within this feasible region that maximize the utility function.
Since the utility function is U(L, C) = LC, we can rewrite it as U(n, C):
U(n, C) = n*(T - n)*C
To find the maximum utility, we need to find the values of n and C that maximize U(n, C).
One approach to finding the maximum is to calculate the utility for each point within the feasible region and find the maximum value.
However, the utility function is increasing with respect to n and decreasing with respect to T - n, so the maximum utility will occur at one of the corners of the feasible region.
Let's calculate the utility at each corner of the feasible region:
For (3, 0):
U(3, 0) = 3*(24 - 3)*C = 63C
For (3, T):
U(3, T) = 3*(24 - 3)*C = 63C
For (8, T):
U(8, T) = 8*(24 - 8)*C = 128C
For (8, 0):
U(8, 0) = 8*(24 - 8)*C = 128C
Comparing these utilities, we can see that U(8, T) = U(8, 0) gives the maximum utility, which is 128C.
Therefore, the optimal values are:
L* = T = 24 hours
C* = 128C
Graphically, the optimal point (L*, C*) lies at the top-right corner of the feasible region.
To find the solution (L*, C*), we need to maximize the utility function U(L, C) = LC subject to the constraint n ∈ [3, 8].
To start, let's write the budget constraint for this scenario. The agent's total available hours can be divided into leisure hours (L) and working hours (n). Since T = 24 and n must be between 3 and 8, we can write the equation:
T = L + n
24 = L + n
Now, we can rewrite this equation as n = 24 - L and substitute it into the utility function:
U(L, C) = LC
U(L, C) = L(24 - L) [Substituting n with 24 - L]
To maximize this function, we need to take the derivative with respect to L and set it equal to zero:
dU/dL = 24 - 2L = 0
Solving this equation, we find L = 12. Plugging this value back into our equation for n, we get:
n = 24 - L
n = 24 - 12
n = 12
So, the optimal values for L* and n* are L* = 12 and n* = 12. Now, we can plug these values back into the utility function to find C*:
U(L*, C*) = L*C
U(12, C*) = 12*C
Since we don't have a specific equation or constraint for C, we cannot determine its exact value. However, we know it will be dependent on L and n and needs to be consistent with the budget constraint.
To illustrate this solution graphically, we can plot a graph where the x-axis represents leisure hours (L), the y-axis represents consumption (C), and the budget constraint n = 24 - L is shown as a line segment with n ranging from 3 to 8.
We plot the utility function U(L, C) = L*C as a contour plot or a 3D surface. The maximum point on this plot, where L = 12 and C is undetermined, represents the solution (L*, C*) within the given constraints.