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

Landon Kelley · Accepted Answer

Final Answer : C, Explanation :  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)RWe can now set up the problem as follows:Maximize U(sqrt(L), R) = sqrt(L)RSubject to L + R = 48We can use the method of Lagrange multipliers to solve for the optimal value of L:L' = R' = λU(sqrt(L), R) = sqrt(L)RL + R = 48Solving 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)λTo 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) = 0Multiplying both sides by 2 (48-λ)^(1/2), we get:λ = (48-λ)^(1/2)Squaring both sides, we get:λ^2 = 48-λRearranging, we get:λ^2 + λ - 48 = 0Solving for λ, we get:λ = 4 or λ = -12Since 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.

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