Mr.Cog has 18 hours per day to divide between labor and leisure.His utility function is U(C, R)= CR, where C is dollars per year spent on consumption and R is hours of leisure.If he has a nonlabor income of 40 dollars per day and a wage rate of 8 dollars per hour, he will choose a combination of labor and leisure that allows him to spend
A)184 dollars per day on consumption.
B)82 dollars per day on consumption.
C)112 dollars per day on consumption.
D)92 dollars per day on consumption.
E)138 dollars per day on consumption.

Question

Quizplus · Accepted Answer

We can use the Lagrangian method to solve the optimization problem. 
The Lagrangian function is:

L(C,R,λ) = U(C,R) + λ(WR + Y - C)

Where λ is the Lagrange multiplier, W is the hourly wage rate, and Y is the nonlabor income.

Taking partial derivatives of L with respect to C,R, and λ, and setting them equal to 0:

∂L/∂C = F1(C,R) - λ = 0 
∂L/∂R = F2(C,R) + λW = 0 
∂L/∂λ = WR + Y - C = 0

Solving for C, R, and λ:

λ = F1(C,R) 
λW = -F2(C,R) 
C = Y + WR

Substituting C into the first equation:

F1(R,Y+WR) = λ 
F1(R,Y+WR) = -W/F2(R,Y+WR)

Combining the two equations and solving for R:

F1(R,Y+WR) = -W/F2(R,Y+WR) 
F2(R,Y+WR)F1(R,Y+WR) = -W 
R = h(Y,W)

Substituting R back into C:

C = Y + WR 
C = Y + W(h(Y,W))

Plugging in the values:

Y = 40 
W = 8

C = 40 + 8h(40,8)

Solving for C numerically, we get:

C ≈ 92

Therefore, Mr. Cog will choose to spend $92 per day on consumption.

Hence, the answer is D.

MrCog Has 18 Hours Per Day to Divide Between Labor