Question

A monopolist sells two products x and y for which the inverse demand functions are: P_X=50-2Q_X and P_y=30-Q_y. The combined cost function is: C=〖Q_X〗^2+2Q_X Q_y+〖Q_y〗^2+20. Find the profit maximising levels of output for each product. What is the maximum profit?

Answer 1

The profit maximising levels of output for each product can be found at MR = MC.

If the inverse demand functions are

Px = 50 - 2Qx and Py = 30 - Qy, then:

$MRx = TR'(x) = 50 - 4Qx,$

$MRy = TR'(y) = 30 - 2Qy,$

$MCx = TC'(x) = 2Qx + 2Qy,$

$MCy = TC'(y) = 2Qx + 2Qy,$ so:

50 - 4Qx = 2Qx + 2Qy,

Qy = 25 - 3Qx,

30 - 2Qy = 2Qx + 2Qy,

Qy = 7.5 - 0.5Qx,

25 - 3Qx = 7.5 - 0.5Qx,

2.5Qx = 17.5,

Qx = 7 units,

Qy = 25 - 3×7 = 4 units.

Px = 50 - 2×7 = 36,

Py = 30 - 4 = 26.

Total profit is:

$TP = TR - C = 36×7 + 26×4 - (21 + 2×7×4 + 16 + 20) = 243.$