Question

If the demand functions for two products are Px = 40 – 5x and Py = 30 – 3y where x and y are the quantities of the two products, and total cost is TC = x2 + 2xy + 3y2, determine the quantities and prices that maximize profit and the maximum profit.

Answer 1

MR = MC = P,
MCx = TC' = 2x + 2y,
MCy = TC' = 2x + 6y,
MCy = Py, MCx = Px,

2x + 2y = 40 - 5x
2x + 6y = 30 - 3y
7x + 2y = 40
2x + 9y = 30

y = 20 - 3.5x

2x + 180 - 31.5x = 30
29.5x = 150,
x = 5.08,
y = 20 - 17.78 = 2.22
Px = 40 - 25.4 = $14.6
Py = 30 - 6.66 = $23.34

TP = TR - TC = Px*x + Py*y - TC = 14.6*5.08 + 23.34*2.22 - (25.8 + 22.56 + 14.79) = 74.17 + 51.81 - 63.15 = $62.83