Question

Let the inverse demand function and the cost function be given by P = 50 − 2Q and C = 10 + 2q respectively, where Q is total industry output and q is the firm’s output.

a) First consider first the case of uniform-pricing monopoly, as a benchmark. Then in this case Q = q. Find out the firm’s profit, price and quantity. (10p)
Please with detail.

Answer 1

In the case of uniform-pricing monopoly, the firm maximizes its profit by setting its output level where marginal revenue (MR) equals marginal cost (MC).

To find the firm's MR, we first need to find the inverse demand function in terms of q, which is the firm's output. We can do this by solving the inverse demand function for Q, which is the total industry output, and then substituting Q with q:

P = 50 - 2Q

Q = (50 - P)/2

Q = (50 - (50 - 2q))/2

Q = q

Now, we can find the firm's MR by taking the derivative of the inverse demand function with respect to q:

MR = d(50 - 2q)/dq

MR = -2

The firm's MC is given as C = 10 + 2q. Setting MR equal to MC, we get:

-2 = d(10 + 2q)/dq

-2 = 2

q = 4

So the firm's profit-maximizing output level is q = 4. To find the price, we can substitute this output level into the inverse demand function:

P = 50 - 2Q

P = 50 - 2q

P = 50 - 2(4)

P = 42

Therefore, the firm's profit is:

Profit = Total revenue - Total cost

Profit = P*q - C

Profit = 42*4 - (10 + 2*4)

Profit = 152

The firm's price is 42 and its output level is 4. Its profit is 152.