Firm cost function - microeconomics

Question

A firm's cost function is C(q) = q^2 + 3q + 25 and it sells units at a price of $55 per unit. q denotes the number of units. What is the profit-maximising number of units? type your answer to the closest whole number.

Answer 1

Revenue = (Price/unit)(number of units) = 55q

Profit = Revenue - Cost

So P(q) = (55q) - (q² + 3q + 25) or:

P(q) = 55q - q² - 3q - 25

P(q) = -q² + 52q - 25

To find the maximum, take the derivative and set that equal to zero:

P'(q) = -2q + 52

-2q + 52 = 0

2q = 52

q = 26

To be thorough, you would need to analyze this local extreme to determine if it is a maximum, minimum, or point of inflection. In this case, the function is an inverted parabola (-q² + 52q - 25) so it would have a maximum. That analysis is all that is necessary in this case.

Note that there are Algebra methods to find this maximum. The maximum or minimum of a parabola occurs at the vertex for which the independent variable (typically "x") can be found using -b/(2a) which in this case means q_max = (-52)/(2(-1)) = -52/-2 = 26

Profit is maximized when the number of units sold is 26