A Note on Profit Maximization

In 101, you learn about the concept of allocative efficiency and the necessary condition of producing at the point where P=MC. Today, I'll show how this condition is derived. Recall that in a perfectly competitive market, producers and consumers alike are price takers: there are so many producers and consumers that a single one of either cannot alter the market price by a significant amount. In this case, market price (P) can be treated as a constant. As such, your profit function is as follows:

π = PQ - C(Q)

where π is profit, P is the market price, Q is the quantity produced by an INDIVIDUAL firm, and C(Q) is the total cost of the firm as a function of Q. Taking the first order condition (Maximize π, i.e. derivate with respect to Q and set the function equal to zero), we get:

P - MC = 0

where MC is the marginal cost (as an exercise, work out and see for yourself why the derivative of C(Q) is equal to MC).

From this step, it becomes obvious where the condition P = MC comes from. As seen, setting P = MC simply the profit maximizing condition in a scenario where P is a constant.

With this in mind, we can generalize the equation for cases where P is not a constant. When is this scenario possible? When P is NOT a constant, a firm has pricing power: that is, a producer does not need to take a price as given, but influences price as a function of the quantity that he produces. We denote this as P(Q).

With this in mind, we can write the firm's profit function as:

 π = P(Q)*Q - C(Q)

Noting that P(Q)*Q is equal to total revenue, I will now re-write the function as follows:

 π = R(Q) - C(Q)

where R(Q) is a total revenue as a function of quantity produced (Q). Taking the first order condition (maximize profit with respect to Q), we get:

MR - MC = 0

From this, we get the condition MR = MC, the condition a producer with pricing power uses to maximize profit. Note that this is not a contradiction to the P = MC condition; it is merely a generalization of it because in a perfectly competitive market, P =MR.

If the calculus is above you, fear not: you'll get there. I'm merely trying to show that all of these profit maximizing conditions are not merely theoretical, but mathematically derived.