Short-run cost curves and break-even

Production and costs
What does each unit cost me on average, what does one more unit cost, and how many units must I sell to cover my fixed costs?

Learning goals

After this page, you can:

  • Build AFC, AVC, ATC and MC from a production table, a wage and a fixed cost, and use MC = wage ÷ MP.
  • Explain why MC pulls the averages: when MC is below an average, the average falls; when it is above, the average rises — so MC crosses AVC and ATC at their lowest points.
  • Compute the manager’s versions: contribution margin per unit (price − AVC), CM ratio, and break-even quantity F ÷ (price − AVC).

The idea

The pie-shop owner runs four bakers and sells 64 pies a day. Her average total cost is $17.50 a pie. A café asks for 12 extra pies a day at $15 each. That is below her average cost. Should she say no?

No. The average is the wrong number for this question. A fifth baker would make exactly those 12 pies for $160 a day. That is $13.33 a pie. The order brings in 12 × $15 = $180 and costs $160, so it adds $20 a day. The right number is the cost of the extra pies. That is the marginal cost. This page shows where each cost number comes from, and which one to use when.

From the production table to costs. Start with the one-oven pie shop. Each baker costs a wage (w) of $160 a day. The rent, oven lease and insurance add up to a fixed cost (F) of $480 a day.

  • Variable cost: VC = w × L, where L is the number of bakers. Four bakers: 4 × 160 = $640.
  • Total cost: TC = F + VC. Four bakers: 480 + 640 = $1,120.

Four “per pie” numbers. Divide by output (q) to get the averages, and look at the change to get the marginal:

  • Average fixed cost AFC = F ÷ q. It always falls as output grows: the same $480 is spread over more pies.
  • Average variable cost AVC = VC ÷ q.
  • Average total cost ATC = TC ÷ q = AFC + AVC. This is the full cost of the average pie.
  • Marginal cost (MC) is the cost of one more unit. Here we hire whole bakers, so we measure it per baker: the extra wage divided by the extra pies. MC = w ÷ MP.

Falling MP means rising MC. The last formula links this page to the production page. When a baker adds many pies (high MP), each extra pie is cheap. When a baker adds few pies (low MP), each extra pie is expensive. Baker 3 adds 20 pies: MC = 160 ÷ 20 = $8. Baker 6 adds only 8: MC = 160 ÷ 8 = $20. Diminishing returns in production show up as rising marginal cost.

The class-average rule: MC pulls the averages. Think about a class average. A new student who scores below the average pulls it down. A new student who scores above it pulls it up. Costs work the same way.

  • When MC is below ATC, each new pie is cheaper than the average, so ATC falls.
  • When MC is above ATC, each new pie is dearer than the average, so ATC rises.
  • So MC crosses ATC exactly at the bottom of ATC. The same holds for AVC.

That is why AVC and ATC are U-shaped, and why MC cuts through both at their lowest points. In the pie shop, baker 5’s pies cost $13.33 each, below the old average of $17.50, so ATC falls to $16.84. Baker 6’s pies cost $20 each, above $16.84, so ATC turns up to $17.14.

The manager’s version: contribution margin and break-even. Managers rarely draw U-curves. They use three numbers built from the same pieces:

  • Contribution margin per unit = price − AVC. It is what each unit leaves over to cover the fixed cost. At $20 a pie and AVC of $10.53, each pie leaves $9.47.
  • CM ratio = (price − AVC) ÷ price. The share of each sales dollar that is kept after variable cost. Here 9.47 ÷ 20 = 47.4%.
  • Break-even quantity = F ÷ (price − AVC). How many units must sell to cover the fixed cost. Here 480 ÷ 9.47 ≈ 51 pies a day.

These are the same ideas as on the cost-concepts page, written per unit instead of for the whole company. The cost-structure builder below works with company totals from a 10-K; add a price per unit to get break-even in units.

Which number for which question?

  • Should I take one more order, or hire one more worker? Use marginal cost.
  • Am I making money at today’s output? Compare price with ATC.
  • How many must I sell before I make any profit? Use the break-even quantity.

Using the average where you need the marginal is the most common cost mistake. It makes managers turn down orders that would add profit, as in the café story above.

Key formulas

VC = w × L · TC = F + w × L · AFC = F ÷ q · AVC = VC ÷ q · ATC = TC ÷ q = AFC + AVC

MC of worker n’s extra output = w ÷ MP(n)

CM ratio = (p − AVC) ÷ p · Break-even quantity QBE = F ÷ (p − AVC)

Try it

Part 1: the cost curves. The chart uses the one-oven pie table from the production page.

  1. Move the slider from 1 to 7 bakers. For which baker does the sentence switch from “MC below ATC → ATC falling” to “MC above ATC → ATC rising”?
  2. Find the lowest point of the AVC line. What is MC there?
  3. Raise the wage to $200. What happens to MC, ATC and the break-even number of pies?
  4. Lower the price to $12. Is the shop still covering AVC? Is it covering ATC?

Part 2: your company’s cost structure. This is the same builder as on the cost-concepts page, with the same saved entries. If you know a price per unit, type it to see break-even in units.

Worked example

Problem. The one-oven pie shop pays $160 per baker per day. Its fixed cost is $480 a day (rent $240, oven lease $120, insurance $120). A pie sells for $20. Output with 1 to 7 bakers is 12, 28, 48, 64, 76, 84 and 88 pies. Build the cost table, find where AVC and ATC are lowest, and work out the manager’s numbers with 5 bakers.

Step 1. Marginal cost = 160 ÷ MP. The MPs are 12, 16, 20, 16, 12, 8, 4. So MC is 13.33, 10.00, 8.00, 10.00, 13.33, 20.00 and 40.00 dollars a pie.

Step 2. AVC = 160 × L ÷ q. 160 ÷ 12 = 13.33; 320 ÷ 28 = 11.43; 480 ÷ 48 = 10.00; 640 ÷ 64 = 10.00; 800 ÷ 76 = 10.53; 960 ÷ 84 = 11.43; 1,120 ÷ 88 = 12.73. AVC is lowest at 48–64 pies ($10.00), where MC is $10 too.

Step 3. ATC = (480 + 160 × L) ÷ q. 640 ÷ 12 = 53.33; 800 ÷ 28 = 28.57; 960 ÷ 48 = 20.00; 1,120 ÷ 64 = 17.50; 1,280 ÷ 76 = 16.84; 1,440 ÷ 84 = 17.14; 1,600 ÷ 88 = 18.18. ATC is lowest at 76 pies ($16.84). Baker 5’s MC ($13.33) is below ATC, so the average keeps falling. Baker 6’s MC ($20) is above it, so ATC turns up.

Step 4. The manager’s numbers with 5 bakers. AVC = 800 ÷ 76 = $10.53. Contribution margin = 20 − 10.53 = $9.47 a pie. CM ratio = 9.47 ÷ 20 = 47.4%. Break-even = 480 ÷ 9.47 ≈ 51 pies a day. Profit = 76 × 20 − 1,280 = $240 a day.

Step 5. A hint of next week. With 6 bakers, profit is also 84 × 20 − 1,440 = $240. Baker 6’s pies cost exactly the $20 price, so hiring baker 6 neither adds nor loses profit. That is next week’s rule in disguise.

Check yourself

Check 1.

A cake shop has a fixed cost of $300 a day and pays 4 bakers $150 each. Together they make 100 cakes a day. What is the average total cost per cake?

ATC = (fixed cost + variable cost) ÷ quantity.

Check 2.

At a shop’s current output, marginal cost is $13, average total cost is $17 and average variable cost is $10. If it produces a few more units, what happens to the two averages?

MC below an average pulls it down; MC above an average pulls it up.

Check 3.

The 6th baker costs $160 a day and adds 8 pies a day. What is the marginal cost of those extra pies?

MC = wage ÷ marginal product; fewer extra units for the same wage means higher MC.

Check 4.

A cake sells for $12 and average variable cost is $9 a cake. What is the contribution-margin ratio?

CM ratio = (price − AVC) ÷ price.

Check 5.

A shop has fixed costs of $600 a day, sells at $10 a unit and has an average variable cost of $6 a unit. How many units a day must it sell to break even?

Break-even quantity = fixed cost ÷ (price − AVC).

Video

Video coming soon.