Question

Suppose that a monopoly has a profit function π(q) = pq - q2 where p denotes price and q denotes quantity. The demand function is faces is q(p) = 10 - p/2
Find:

the inverse demand function
the profit when the firm produces 5 units of output
the level of output that maximizes the firm’s profit

Answer 1

a. R\p = (10 - q)*2

The inverse demand function is just the inverse function of the demand function. In other words, we just have to isolate p in the demand function:

p = (10 - q)*2

b. R\25

The price for 5 units of output is given by the inverse demand function:

p = (10 - 5)*2 = 10

We replace p in the profit function:

π(q) = 10 * 5 - 5² = 25

c. 3

For this one, we replace the inverse demand function in the profit function and derivate for q, then equate to 0 and solve:

π(q) = ((10 - q)*2)*q - q² = 20q - 2q² - q² = 20q - 3q²

dπ/dq = 20 - 6*q

20 - 6q = 0

q = 20/6 = 3.33333

Now, a decimal level of output makes no sense. So, now we try the nearest integers 3 and 4, and find the respectives profits. The output that has that maximum profit will be the one that maximizes the profit. Keep in mind, that this will only be true in this particular case because the profit function has the form of a quadratic equation:

π(3) = 20 * 3 - 3*(3)² = 33

π(4) = 20 * 4 - 3*(4)² = 32

The answer is 3.