Question

A product is produced using two inputs x1 and x2 costing P1=$10 and P2 = $5 per unit respectively. The production function is y = 2(x1)1.5 (x2)0.2 where y is the quantity of output, and x1, x2 are the quantities of the two inputs. A)What input quantities (x1, x2) minimize the cost of producing 10,000 units of output? (3 points) B)What is the optimal mix of x1 and x2 if the company has a total budget of $1000 and what is the optimal output? What if P1=$5 and P2=5 (3 points) Hints: 1) if y = 2 (x1)a (x2)b , MPx1 = 2a (x1)a-1 (x2)b

Answer 1

Given

A)

Total Output Y=10000

10000=2*(X1)^1.5 *(X2)^0.2

X2=(5000)^5/(X1)^7.5 Eq 1

Total Cost TC=P1*X1+P2*X2

TC=10X1+5X2 Eq 2

From equation 1 and 2

TC=10X1+5*{(5000)^5/(X1)^7.5}

For Cost Minimization

dTC/dX1=0

10-37.5*{(5000)^5/(X1)^8.5}=0

X1=175.15

X2=(5000)^5/(X1)^7.5 =5000^5/(175.15)^7.5=46.71

B)

Total Budget B=$1000

B=P1*X1+P2*X2

1000=5X1+5X2

X1+X2=200

Y=2*(X1)^1.5*(X2)^0.2

Y=2*(200-X2)^1.5*(X2)^0.2

For optimal solution

dY/dX2=0

2*(-1.5)*(200-X2)^0.5*(X2)^0.2+2*0.2*(200-X2)^1.5*(X2)^-0.8=0

1.5*X2=0.2(200-X2)

1.5X2=40-0.2X2

1.7X2=40

X2=23.53

X1=200-X2=200-23.53=176.47

Y=2*(X1)^1.5*(X2)^0.2=2*(176.47)^1.5*(23.53)^0.2=$8817.81