Lesson 07: Production & Short-Run Costs

Warm-Up

A food truck has one grill (fixed for the summer — that's the short run). One cook makes 30 tacos an hour. A second cook — one preps while one grills — output jumps to 70. A third: 100. A fourth: 115, they're bumping elbows. A fifth: 120, mostly standing around. Each extra cook adds less than the last. This crowding — diminishing marginal returns — shapes every cost curve in this lesson, and those cost curves are the stage on which Units 3 and 4 (a quarter or more of the exam) are performed.

Core Concept

Short run vs. long run

• Short run: at least one input is fixed (the grill, the factory). Output changes only by varying labor/materials.
• Long run: all inputs are variable — firms can build factories, and firms can enter or exit the industry.

These are capability horizons, not calendar lengths.

Production: marginal and average product

• Marginal product (MP) = Δtotal product / Δlabor — the extra output from one more worker.
• Average product (AP) = total product / labor.

Law of diminishing marginal returns: as more of a variable input is added to a fixed input, MP eventually falls (here, starting with worker 3). It's a short-run law — it requires a fixed input. Early specialization can make MP rise first (worker 2), but crowding always wins.

The cost family

Fixed vs. variable: - Total fixed cost (TFC): doesn't change with output — rent, insurance, loan payments. Paid even at Q = 0. - Total variable cost (TVC): rises with output — labor, ingredients. - Total cost: TC = TFC + TVC.

Per-unit costs: - AFC = TFC/Q — always falling as Q rises ("spreading the overhead") - AVC = TVC/Q - ATC = TC/Q = AFC + AVC - Marginal cost: MC = ΔTC/ΔQ = ΔTVC/ΔQ (fixed cost never affects MC)

The geometry (must be automatic)

[GRAPH: Cost curves. X-axis "Quantity", Y-axis "Cost per unit ($)". MC is J-shaped: briefly falling, then rising steeply. AVC and ATC are U-shaped, ATC above AVC, the vertical gap between them (= AFC) shrinking as Q grows. MC passes through the minimum point of AVC, then the minimum point of ATC. AFC drawn separately, continuously declining.]

Three relationships the exam tests constantly:

1. MC intersects AVC and ATC at their minimum points. The average-marginal rule: when the marginal is below the average, it pulls the average down; when above, it pulls it up (like a test score vs. your GPA). So each average curve bottoms exactly where MC crosses it.
2. The gap between ATC and AVC = AFC, which shrinks as output rises but never reaches zero.
3. MC mirrors MP inverted: when MP rises, MC falls; when MP diminishes, MC rises; MC = wage / MP. Diminishing returns is rising marginal cost.

Building the table (the exam's favorite puzzle)

Given scraps of information, reconstruct everything. Anchors: - TC at Q = 0 equals TFC (TVC = 0 with no output). - MC entries stack: TVC at Q = sum of MCs up to Q. - Any average × Q gives its total: ATC × Q = TC.

Worked Examples

Example 1 (easy): Diminishing returns

Total product with 0–4 workers: 0, 12, 30, 42, 48. Where do diminishing marginal returns set in?

Solution: MP = 12, 18, 12, 6. MP rises to worker 2, falls starting with worker 3. Diminishing marginal returns begin at the 3rd worker — even though total product is still rising.

Interpretation: Diminishing returns ≠ falling output. Output keeps rising while MP > 0; it just rises more slowly.

Example 2 (medium): Complete the cost table

A firm's TFC = $60. TVC: Q=1 → $30; Q=2 → $50; Q=3 → $80; Q=4 → $130. Build MC, AVC, ATC.

Solution:

Interpretation: Check the average-marginal rule in the numbers: ATC falls while MC < ATC (Q = 2, 3) and rises once MC (50) exceeds ATC (Q = 4). AVC bottoms at Q = 2, exactly where MC crosses from below (20 < 25) to above (30 > 26.67).

Example 3 (AP-style): Working backward

At Q = 10, a firm's ATC = $12 and AVC = $9. (i) Find TFC. (ii) If MC of the 11th unit is $10 and rising, is ATC at Q = 11 higher or lower than $12?

Solution: - (i) AFC = ATC − AVC = $3 → TFC = 3 × 10 = $30. - (ii) MC ($10) < ATC ($12) → the 11th unit costs less than the current average → ATC falls below $12.

Interpretation: You don't need the whole table — the marginal-average relationship answers direction questions instantly.

Example 4 (AP-style): Fixed costs and decisions

A bakery signs a $2,000/month lease. Halfway through the month, it considers whether to bake an extra batch (ingredients + labor: $40; expected revenue: $55). A consultant says, "You must factor in the lease." Evaluate.

Solution: The lease is a sunk/fixed cost — unchanged whether or not the batch is baked. Marginal analysis: MR ($55) > MC ($40) → bake it. The consultant is wrong; fixed costs are irrelevant to short-run output decisions.

Interpretation: "Fixed costs don't affect marginal decisions" is one of the top-five ideas of the course; it returns in the shutdown rule (Lesson 9).

Common Mistakes

1. "Diminishing returns means output falls." It means MP falls. Output declines only if MP goes negative.
2. Including fixed cost in MC. MC = ΔTVC/ΔQ. If TC jumps from 90 to 110, MC = 20, whatever TFC is.
3. Forgetting TC = TFC at zero output. In the short run a closed-but-not-exited firm still pays fixed costs.
4. Drawing MC through the wrong points. MC must pass through both minimums: AVC's first, then ATC's. Sloppy graphs lose FRQ points.
5. Confusing AP and MP. AP is the team average; MP is the newest hire's contribution. MP crosses AP at AP's maximum (same average-marginal rule).

Practice Problems

11. (FRQ-style) Denny's Dumplings operates with fixed costs of $200 per day. Daily production:

(a) Calculate marginal cost for each batch. (b) Calculate AVC and ATC at Q = 4. (c) Between which batches do diminishing marginal returns appear to set in? Explain the link between marginal product and marginal cost. (d) Draw a correctly labeled graph of MC, ATC, and AVC, showing where MC crosses the average curves.