MR = MC

A firm’s profit is defined as its total revenue minus its total cost. In symbols, p
(*Q*) = *R*(*Q*) - *C*(*Q*). A firm that wishes to maximize its profits may find the corresponding output by differentiating p
(*Q*) with respect to output and finding the output that equates the derivative to zero:

dp
(*Q*)/d*Q* = d*R*(*Q*)/d*Q* - d*C*(*Q*)/d*Q* = *MR* - *MC *= 0. That is, profit maximization requires that, if the firm chooses to produce anything at all, it should equate marginal revenue and marginal cost. In the specific case of competitive firms, this takes the form *P* = *MC*. The second-order condition is:

- < 0, or that marginal revenue cuts marginal cost from below.

However, we know that the maximum found by the procedure above is only a local maximum. We also need to check that the output at the beginning of the range of possible output does not provide greater profits. Specifically, we need to check profits where output is zero. If output is zero, then revenue is zero, variable cost is zero, and profit equals p
(0) = 0 - (*FC* + 0); the firm loses an amount equal to its fixed costs. Naturally, then, the firm would only select this option if its losses at the *MR* = *MC* output were greater than this. Let *Q** be the output that satisfies the *MR* = *MC* rule. Total profits at this point will be p
(*Q**) = *PQ** - (*FC* + *VC*(*Q**)). Profits at *Q* = 0 will be higher than at *Q* = *Q* *if and only if p
(0) = -*FC* > *PQ** - (*FC* + *VC*(*Q**)) = p
(*Q**). Dividing both sides of this inequality by *Q** and rearranging, the firm maximizes profit at *Q* = 0 if and only if *P* < , that is, if and only if the firm’s price is less than its average variable cost at the *MR* = *MC* output. Since average variable cost is equal to marginal cost at the former’s minimum, we can state the complete short-run profit-maximizing rule as follows: produce at the output for which MR = MC, provided price is greater than minimum average variable cost; otherwise shut down.