The MBA Math Monday series helps prospective MBA students to self assess their proficiency with the quantitative building blocks of the MBA first year curriculum.
The first two economics exercises examined marginal analysis. which discovers a firm’s optimal production quantity and profit. Marginal analysis problems can be posed equivalently in terms of tables, formulas, or charts. The first exercise used data in tables. The second exercise used formulas but without invoking calculus. This exercise uses formula and calculus.
The calculus bit is the only aspect of this exercise that is different from the second one. The key idea, which is explained in the full MBA Math course, is that the derivative of the total cost function is the marginal cost function. The derivative examines the incremental change in a function (e.g., y = f(x)) given a small change in the underlying value (e.g., x). That’s just what we want when we are considering the marginal cost, which shows how much total cost (e.g., TC = f(q)) has changed for a small change in quantity (e.g., q).
Similarly, the derivative of the total profit function is the marginal profit function and the derivative of the total revenue function is the marginal revenue. At least it’s consistent.
Scary as calculus may be to many incoming b-school students, the main idea is quite straightforward if you can keep your brain from locking up in fear. And the mechanics of taking the derivative of a polynomial, which is a fancy name for the functions determining cost, are refreshingly simple, taking no more than 15 minutes to recover from the memory vault of lost ideas from high school.
Suppose a competitive firm has as its total cost function:
TC = 24 + 2q2
Suppose the firm’s output can be sold at $66 per unit.
Using calculus and formulas (but no tables) to find a solution, how many units should the firm produce to maximize profit?
Solution (with audio commentary): click here