The HandyMandy Company uses skilled and unskilled labor to do a particular const

Question

The HandyMandy Company uses skilled and unskilled labor to do a particular construction project. The cost of doing the project depends on the number of hours of skilled labor and the number of hours of unskilled labor used, the relationship being:

C = 4 – 3X – 4Y + 2X2 + 3Y2 + XY

Where C is the cost (in thousands of dollars), X is the number of hours (in thousands) of skilled labor, and Y is the number of hours (in thousands) of unskilled labor.

a. find the number of hours of skilled labor and the number of hours of unskilled labor that minimizes the cost of doing the project. (2 pts.)

b. If HandyMandy has to purchase a license costing $2,000 to do this project (and the cost of this license is not included in C), will this alter the answer to part a? If so, how will the answer change? (1 pt.)

Explanation / Answer

Calculation of numbers of hours of skilled labors:

To get Minimizing function of Cost, first calculate the first order Condition (F.O.C). Derive the Cost function with respect to X and holding Y constant and then equate it with zero to get X* because it is required to calculate numbers of hours of skilled labors for minimum cost.

F.O.C = dC ÷ dX = – 3+4X+Y = 0

X* = 1/4 (3 – Y)

Checking whether X* gives minimum cost: Find second order derivative of C w.r.t X

d2C ÷ dX2 = 4>0. Second order derivative gives value more than 0 hence case of minimization exists at X*.

Calculation of numbers of hours of unskilled labors:

First calculate the first order Condition (F.O.C). Derive the Cost function with respect to Y and holding X constant and then equate it with zero to get Y* because it is required to calculate the numbers of hours of Unskilled labors for minimum cost.

dC ÷ dY = – 4 + 6Y + X = 0

Y* = 1/6(4 – X)

Checking whether Y* gives minimum cost: Find second order derivative of C w.r.t Y

d2C ÷ dY2 = 6>0. Second order derivative gives value more than 0 hence case of minimization exists at Y*.

Hence Number of hours (in thousands) of skilled labors to minimize cost = 1/4(3-no. of hours of unskilled labors)

Number of hours (in thousands) of Unskilled labors to minimize cost = 1/6(4-no. of hours of skilled labors).

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.