Question

) Suppose that the firm’s total cost function is of the form C(y) = 100 + 10y + y^2 . Where does the average cost function reach a minimum? Consider the following demand curves:

i. D(P) = 130 − 4P

ii. D(P) = 120 − 2P

iii. D(P) = 120 − 4P

For which of these examples is the firm a natural monopoly? [There may be multiple tests to answer this question. Clearly state which one you are using. That is give a specific mathematical criterion for this and the intuition for using that choice.

Can you please explain thoroughly, thank you.

Answer 1

AC = C(y)/y = (100/y) + 10 + 2y

AC is minimum when dAC/dy = 0.

dAC/dy = - (100/y2) + 2 = 0

100/y2 = 2

y2 = 50

y = 7.07

A natural monopoly is that firm, for which the profit-maximizing output at intersection of MR & MC curves is less than 7.07 (i.e. that raneg of AC where AC is decreasing with increasing in output).

MC = dc(y)/y = 10 + 2y

(i) y = D(p) = 130 - 4p

p = (130 - y)/4 = 32.5 - 0.25y

TR = p x y = 32.5y - 0.25y2

MR = dTR/dy = 32.5 - 0.5y

32.5 - 0.5y = 10 + 2y

2.5y = 22.5

y = 9

Since profit-maximizing output is higher than 7.07, this is not a natural monopoly.

(ii) y = D(p) = 120 - 2p

p = (120 - y)/2 = 60 - 0.5y

TR = p x y = 60y - 0.5y2

MR = dTR/dy = 60 - y

60 - y = 10 + 2y

3y = 50

y = 16.67

Since profit-maximizing output is higher than 7.07, this is not a natural monopoly.

(iii) y = D(p) = 120 - 4p

p = (120 - y)/4 = 30 - 0.25y

TR = p x y = 30y - 0.25y2

MR = dTR/dy = 30 - 0.5y

30 - 0.5y = 10 + 2y

2.5y = 20

y = 8

Since profit-maximizing output is higher than 7.07, this is not a natural monopoly.