Robinson Crusoe's preferences over coconut consumption, C, and leisure, R, are represented by the utility function U(C, R)= CR.There are 48 hours available for Robinson to allocate between labor and leisure.If he works L hours, he will produce the square root of L of coconuts.He will choose to work
A)8 hours.
B)12 hours.
C)16 hours.
D)20 hours.
E)24 hours.

Question

Quizplus · Accepted Answer

To maximize Robinson's utility, we need to find the combination of coconut consumption and leisure that maximizes U(C, R) subject to the constraint that Robinson has only 48 hours to allocate between labor and leisure.

Let's first find Robinson's production function for coconuts, which is the square root of L:
P(L) = sqrt(L)

Now let's find Robinson's total utility function in terms of L, the number of hours he works:
U(P(L), R) = U(sqrt(L), R) = sqrt(L)R

We can now set up the problem as follows:

Maximize U(sqrt(L), R) = sqrt(L)R
Subject to L + R = 48

We can use the method of Lagrange multipliers to solve for the optimal value of L:

L' = R' = λ

U(sqrt(L), R) = sqrt(L)R
L + R = 48

Solving the first equation for R, we get R = λ. Substituting this into the second equation, we get L + λ = 48, or L = 48 - λ.

Substituting these expressions for L and R into the utility function, we get:

U(sqrt(48-λ), λ) = sqrt(48-λ)λ

To maximize this function, we can take the derivative with respect to λ and set it equal to zero:

d/dλ [sqrt(48-λ)λ] = (1/2) (48-λ)^(-1/2) λ - (1/2) (48-λ)^(1/2) = 0

Multiplying both sides by 2 (48-λ)^(1/2), we get:

λ = (48-λ)^(1/2)

Squaring both sides, we get:

λ^2 = 48-λ

Rearranging, we get:

λ^2 + λ - 48 = 0

Solving for λ, we get:

λ = 4 or λ = -12

Since we are looking for a maximum, we choose λ = 4.

Substituting this value of λ back into L = 48 - λ, we get L = 44.

Therefore, Robinson will choose to work 44 hours, which leaves him with 48 - 44 = 4 hours of leisure.

Robinson Crusoe's Preferences Over Coconut Consumption, C, and Leisure, R