# Optimization in Economics

Business I · Mathematics I · https://tryals.app/learn/business-i/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.

| Step | Condition | What it does |
| --- | --- | --- |
| FOC | $f'(x^*) = 0$ | Finds flat candidates |
| SOC | Sign of $f''(x^*)$ | Peak, valley, or shelf |
| Boundaries | Check endpoints | Corners beat interior flats |
| Compare | Evaluate $f$ | Crowns the winner |

> **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.

## Practice questions

6 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort each managerial rule by whether it is marginal reasoning done right, or a totals fallacy.

**Answer:**

- Correct marginal reasoning: Hire while the next worker adds more value than her wage, Advertise until the marginal euro returns exactly one euro, Stop discounting when the marginal sale loses money
- Totals fallacy: We are profitable overall, so expand output, This product line made money last year, so double it

**Why:** Decisions live at the margin. Totals report the past; only the *next* unit’s revenue and cost say whether to take one more step.

Page: https://tryals.app/practice/business-i/optimization-in-economics/sort-each-managerial-rule-by-whether-it-is-marginal-reasoning-done

### 2. Match each condition to what it establishes.

**Answer:**

- $f'(x^*) = 0$ → Candidate point: no marginal gain remains
- $f''(x^*) < 0$ → The candidate is a local maximum
- Corner of the feasible range → Optimum may hide where no derivative vanishes
- $MR > MC$ → Producing more still adds profit

**Why:** FOC nominates, SOC classifies, boundaries lurk, and the margin’s sign gives the direction to walk.

Page: https://tryals.app/practice/business-i/optimization-in-economics/match-each-condition-to-what-it-establishes

### 3. A manager finds the output where marginal profit is zero and stops there. What has been skipped, and what could go wrong?

A. The second-order condition, since zero marks three possibilities
B. The comparison with fixed costs, which determine whether to produce at all
C. The constraint check, since the output may exceed capacity
D. Nothing; zero marginal profit is the definition of the optimum

**Answer:** A. The second-order condition, since zero marks three possibilities

**Why:** Flatness is shared by peaks and troughs, so the sign of the second derivative is what tells them apart, and a manager who skips it can confidently produce at the profit minimum. The FOC finds candidates and the SOC selects among them, which is why the ritual has two steps.

Page: https://tryals.app/practice/business-i/optimization-in-economics/a-manager-finds-the-output-where-marginal-profit-is-zero-and-stops

### 4. A bakery’s profit satisfies $MR = MC$ at two outputs: $q = 3$ and $q = 14$. At $q = 3$, $\pi^{\prime\prime} > 0$; at $q = 14$, $\pi^{\prime\prime} < 0$. Where should it produce?

A. At $q = 3$, the first crossing always wins
B. At both, alternating days
C. At $q = 14$, confirmed by the SOC
D. Wherever total revenue is highest

**Answer:** C. At $q = 14$, confirmed by the SOC

**Why:** $q = 3$ is a profit *minimum* dressed in the same FOC; $q = 14$ passes the second-order test. This is precisely why the ritual has two steps, the first equation alone cannot tell a peak from a pit.

Page: https://tryals.app/practice/business-i/optimization-in-economics/a-bakerys-profit-satisfies-mr-mc-at-two-outputs-q-3-and-q-14

### 5. Profit is currently at an output where $MR < MC$. Which statements are true?

A. Cutting output raises profit
B. The firm is necessarily making losses overall
C. The firm has overshot the optimal quantity
D. The last units produced are destroying value

**Answer:** A. Cutting output raises profit; C. The firm has overshot the optimal quantity; D. The last units produced are destroying value

**Why:** $MR < MC$ means overshoot: trim back toward the crossing. But totals can remain healthy well past the peak, being past the optimum and being unprofitable are different diseases.

Page: https://tryals.app/practice/business-i/optimization-in-economics/profit-is-currently-at-an-output-where-mr-mc-which-statements-are

### 6. A firm at $MR = MC$ wonders about one more unit: producing it would bring 18€ of revenue against 24€ of cost. Set the slider to the profit change (€) from producing it anyway.

**Answer:** -6 (within ±1)

**Why:** $18 - 24 = -6$€: past the crossing, each unit is a small act of self-harm. The optimum is not where production is impossible, it is where the next step, precisely measured, subtracts.

Page: https://tryals.app/practice/business-i/optimization-in-economics/a-firm-at-mr-mc-wonders-about-one-more-unit-producing-it-would