Question

2. A firm produces two different kinds, A and B, of a commodity. The daily cost of producing x units of A and y units of Bis C(x,y)=2x² - 4xy + 4y2 - 40x - 20y + 514 Suppose that the firm sells all its output at a price per unit of $24 for A and $12 for B. (a) Find the daily production levels x and y that maximize profit. (b) The firm is required to produce exactly 54 units per day of the two kinds combined. What is the optimal production plan now?

Answer 1

(a)

Total revenue (R) = x.Px + y.Py = 24x + 12y

Profit (Z) = R - C

Z = 24x + 12y - 2x² + 4xy - 4y² + 40x + 20y - 514 = 64x + 32y - 2x² + 4xy - 4y² - 514

Profit is maximized when $\dpi{80} \partial$ Z/$\dpi{80} \partial$x = 0 and $\dpi{80} \partial$ Z/$\dpi{80} \partial$y = 0.

$\dpi{80} \partial$Z/$\dpi{80} \partial$​​​​​​​x = 64 - 4x + 4y = 0

4x - 4y = 64

x - y = 16..........(1)

$\dpi{80} \partial$Z/$\dpi{80} \partial$​​​​​​​y = 32 + 4x - 8y = 0

4x - 8y = - 32

x - 2y = - 16..........(2)

(1) - (2) yields:

y = 32

x = 16 + y [from (1)] = 16 + 32 = 48

(b)

x + y = 54

y = 54 - x

Substituting in profit function,

Z = 64x + 32.(54 - x) - 2x² + 4x.(54 -x) - 4.(54 - x)² - 514

Z = 64x + 1728 - 32x - 2x² + 216 - 4x - 4.(2916 - 108x + x²) - 514

Z = 28x + 1430 - 2x² - 11664 + 432x - 4x²

Z = 460x - 10234 - 6x²

Profit is maximized when dZ/dx = 0

dZ/d​​​​​​​x = 460 - 12x = 0

12x = 460

x = 38.33

y = 54 - 38.33 = 15.67