Optimization in Economics

Where the Derivative Says Stop

Economics turns optimization into a two-step ritual. First-order condition (FOC): at an interior optimum, the derivative vanishes:

$$f'(x^*) = 0$$

At the top of a hill, the ground is flat. Second-order condition (SOC) determines the shape:

$$f''(x^*) < 0: \text{maximum} \quad f''(x^*) > 0: \text{minimum}$$

Economic translation (marginal analysis): profit $\pi(q) = R(q) - C(q)$ peaks where marginal revenue equals marginal cost, $MR = MC$. Expand until the next step stops paying.

Common pitfall: Stopping at the FOC. $MR = MC$ also holds at profit minima. Without the second-order check and boundary scan, your "optimum" may be the worst point.

The Margin Is a Crossing

The optimum lives where marginal revenue and marginal cost cross:

$$q^*: \quad MR(q^*) = MC(q^*)$$

• Left of $q^*$: $MR > MC$. The gap is profit left on the table.
• Right of $q^*$: $MR < MC$. Extra units destroy value.
• At $q^*$: Total profit is the accumulated area between the curves:

$$\pi(q^*) = \int_0^{q^*}\big(MR - MC\big)\,dq$$

Classic error: 'we are profitable, so produce more.' Total profit being positive says nothing about the margin.

Common pitfall: Maximizing revenue instead of profit. Revenue peaks where $MR = 0$, well past the profit optimum at $MR = MC$. The units between those two quantities all sell, and all lose money.