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.]

Answer 1

Sol:- C(y) = 100 + 10y + y2

Average cost = C(y)/y = 100/y + 10 + y

This would be minimum when By diffferentiating with respect to x,(Average cost)/dy = 0

-100/y2 + 1 = 0

y = 10

Therefore, minimum ATC = 10 + 10 + 10 = 30

Demand curves:

i) :- Q = 130 - 4P
when Q = 10 ,
10 = 130 -4P
4P = 130-10
4P = 120
P = 120/4

Then P = 30

P = minimum ATC Therefore, this is not a natural monopoly

ii):- Q = 120-2P
when Q = 10 ,
10 = 120 -2P
2P = 120-10
2P = 110
P = 110/2

Then P = 55

P > minimum ATC Therefore, this is a natural monopoly

iii):- Q = 120-4P
when Q = 10 ,
10 = 120 -4P
4P = 120-10
4P = 110
P = 110/4

Then , P = 27.5

P < minimum ATC Therefore, this is not a natural monopoly.