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\u00d77 + 26\u00d74 - (21 + 2\u00d77\u00d74 + 16 + 20) = 243."
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