Question

Round Answers to 2 decimal places.

Suppose that there is natural monopoly that faces a demand curve Q=100-P with total cost function C(Q)=400+15Q.

The profit maximizing quantity for the natural monopolist, in the absence of any regulation is ______ units.

The profit maximizing price that will set by the monopolist is $______

The average total cost per unit at the profit maximizing level of output is $______

The profit for the natural monopolist, given they set the profit maximizing price and level of output, is $________

Answer 1

Solution:

Given:

Total cost (TC) = 400+15Q

Demand Curve is given in terms of  quantity we will find it in terms of price

Q=100-P

or we can write this demand function like

P=100-Q

The profit is maximised where MR = MC

First we will calculate MC by finding the derivative of TC

TC = 400+15Q

MC= 15

Total Revenue = P*Q

TR = ( 100-Q) * Q

= 100Q-Q^2 (Q power 2)

MR is the derivative of TR

MR = 100 - 2Q

Profit Maximising Quantity is there where MR=MC

100-2Q =15

2Q = 85

Q= 42.5

Profit Maximising Quantity is 42.5

Now we will find Profit maximising Pcice by putting the value of profit maximising quantity in demand function:

P = 100 - Q

= 100 -42.5

= 57.5

So the Profit Maximising Price is $ 57.5

Average Total Cost per unit at profit maximised level = Total cost divided by profit maximised quantity

TC = 400 +15 Q

= 400 +15 (42.5)

= 1037.5

TC is $1037.5 and Profit maximised quantity is 42.5

Average total cost per unit at profit maximised level = 1037.5 / 42.5

= $24.41

Profit of monopolist at maximised level of quantity and price = TR - TC

TC = $1037.5

TR = 100Q - Q^2

= 100(42.5) - (42.5)^2

= $2443.75

TR - TC

2443.75 - 1037.5 = 1406.25

Profit of monopolist at maximised level of quantity and price is $ 1406.25

Summary of answers:

TR = $2443.75   

TC = $1037.5

Profit Maximising Quantity is 42.5

So the Profit Maximising Price is $ 57.5

Average total cost per unit at profit maximised level  = $24.41

Profit of monopolist at maximised level of quantity and price is $ 1406.25