C = 100 + 30q - 8q2 + q3
(a)
AC = C/q = (100/q) + 30 - 8q + q2
TVC = 30q - 8q2 + q3, so
AVC = TVC/q = 30 - 8q + q2
MC = dC/dq = 30 - 16q + 3q2
(b)
(I) AC is minimum when dAC/dq = 0
- (100/q2) - 8 + 2q = 0
- 100 - 8q2 + 2q3 = 0
q3 - 4q2 - 50 = 0
Solving this cubic equation using online polynomial solver we find that the only real root is
q = 5.60
(II) AVC is minimum when dAVC/dq = 0
- 8 + 2q = 0
2q = 8
q = 4
(c)
For each firm, Price = MC.
30 - 16q + 3q2 = 40
3q2 - 16q - 10 = 0
Solving this quadratic equation using online quadratic solver we find that the only positive root is
q = 5.9
(d)
TR = p x q = 40 x 5.9 = 236
TC = 100 + 30 x 5.9 - 8 x (5.9)2 + (5.9)3 = 100 + 177 - 278.48 + 205.38 = 203.9
Profit = TR - TC = 236 - 203.9 = 32.1
In following graph, profit is area of rectanle P0ABC.
2. An industry consists of many identical firms, each with the cost function C(q) = 100...
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