Difficulty: Introductory-Intermediate | Prerequisites: Basic algebra, familiarity with what a function is. If you are comfortable plugging values into equations and taking simple derivatives, you are ready.
Cost functions are the bridge between a firm's production technology and its profit-maximising decisions. Once you know how costs behave as output changes, you can answer the questions that drive the rest of intermediate micro: what price to charge, how much to produce, and whether to stay in business.
This set of notes covers the mechanics of total cost (TC), average total cost (ATC), and marginal cost (MC), with particular attention to where MC crosses ATC and what that crossing point means. You should already be comfortable with the idea that firms face fixed costs (which do not change with output) and variable costs (which do). If those terms are fuzzy, revisit Ch. 6 before continuing.
Total cost = fixed cost + variable cost. Average total cost is U-shaped: it falls while fixed costs are being spread over more units, then rises once variable costs dominate. Marginal cost always crosses ATC at ATC's minimum, and that crossing tells you whether producing one more unit is pulling average cost down or pushing it up.
Fixed cost (F or FC)
The portion of total cost that does not change with the quantity produced. It must be paid even if the firm produces nothing.
Think of it as the rent you owe whether you open the shop or not.
Variable cost (VC)
The portion of total cost that changes as the firm produces more or fewer units. It equals zero when output is zero.
In simple terms, this is the cost that scales with how busy you are.
Total cost (TC or C)
The sum of fixed and variable costs: TC = F + VC. It captures everything the firm spends to produce a given quantity.
Average total cost (ATC)
Total cost divided by the number of units produced: ATC = TC / q. It tells you the cost per unit at a given output level.
Think of it as: "If I spread all my spending evenly across every unit, what does each one cost me?"
Marginal cost (MC)
The additional cost of producing one more unit. Formally, MC = dTC/dq (the derivative of total cost with respect to quantity).
In simple terms, this is the price tag on the next unit off the line.
Minimum efficient scale
The quantity at which ATC reaches its lowest point. At this output, the firm produces at the lowest possible cost per unit. MC = ATC at this point.
Start from the two components: fixed cost F and variable cost VC(q).
Total cost is simply their sum: TC(q) = F + VC(q).
To identify fixed vs. variable from a combined TC function, look for the constant term (that is fixed cost) and the term(s) that depend on q (that is variable cost).
Fixed cost: the term that remains when q = 0.
Variable cost: everything else.
Worked example: F = 81, VC = q². So TC = 81 + q².
At q = 0, TC = 81. That 81 is the fixed cost.
The q² term grows with output. That is the variable cost.
ATC = TC / q.
Using the example: ATC = (81 + q²) / q = 81/q + q.
The first piece (81/q) falls as q rises, because you are spreading fixed cost over more units.
The second piece (q) rises as q rises, reflecting growing variable cost per unit.
The tug of war between these two pieces gives ATC its characteristic U shape.
MC is the derivative of TC with respect to q. For TC = 81 + q², MC = 2q.
The key relationship: MC intersects ATC at ATC's minimum.
When MC < ATC, producing another unit pulls the average down. ATC is falling.
When MC > ATC, producing another unit pushes the average up. ATC is rising.
When MC = ATC, the average is neither rising nor falling. That is the minimum.
To find the crossing point, set MC = ATC: 2q = 81/q + q. Solving: q = 81/q, so q² = 81, so q = 9.
Minimum ATC = 81/9 + 9 = 9 + 9 = 18.
At q = 3: ATC = 81/3 + 3 = 27 + 3 = 30. MC = 2(3) = 6. Since MC (6) < ATC (30), producing a bit more pulls ATC down.
At q = 27: ATC = 81/27 + 27 = 3 + 27 = 30. MC = 2(27) = 54. Since MC (54) > ATC (30), producing a bit more pushes ATC up.
Notice both quantities give ATC = 30, but the direction of change is opposite. The MC comparison explains why.
ATC is U-shaped, reaching its minimum of 18 at q = 9.
MC is a straight line through the origin with slope 2.
MC cuts through ATC from below at the minimum point (q = 9, ATC = 18).
To the left of q = 9, MC lies below ATC (ATC falling). To the right, MC lies above ATC (ATC rising).
TC(q) = F + VC(q)ATC(q) = \frac{TC(q)}{q} = \frac{F}{q} + \frac{VC(q)}{q}MC(q) = \frac{dTC}{dq}For the worked example (F = 81, VC = q²):
TC = 81 + q^2ATC = \frac{81}{q} + qMC = 2qMC = ATC crossing: set 2q = 81/q + q, solve to get q = 9, ATC_min = 18.
The U-shaped ATC is why startups obsess over "scaling." A software company's fixed costs (servers, salaries, office lease) are enormous relative to revenue at low output. As the customer base grows, those fixed costs spread over more users, pulling ATC down, until eventually the variable costs of support, bandwidth, and infrastructure begin to push it back up.
The MC-ATC relationship also explains why airlines add flights on popular routes. As long as the marginal cost of one more flight is below the average cost, filling that flight lowers the airline's cost per passenger across the board.
Students often think MC = ATC means total cost is minimised. It does not. MC = ATC is where average cost per unit is minimised. Total cost keeps rising with every additional unit.
Students sometimes believe ATC falls forever as output grows ("just keep spreading the fixed cost"). In reality, variable costs eventually dominate, causing ATC to rise again.
Confusing MC with ATC is common. MC is the cost of the next unit only. ATC is the cost per unit averaged across all units produced so far. They measure different things.
Some students set MC = 0 to find the minimum of ATC. The correct condition is MC = ATC. Setting ATC's derivative equal to zero is equivalent, but MC = ATC is the more intuitive framing.
⚠️ "At what quantity is ATC minimised?" is a near-certain exam question. The answer is always where MC = ATC. Be able to solve this algebraically.
⚠️ Expect a question asking you to compare MC and ATC at a specific quantity, then explain whether ATC is rising or falling. The logic: MC < ATC means ATC falls; MC > ATC means ATC rises.
⚠️ You may be asked to sketch ATC and MC on the same graph. Make sure MC passes through the minimum of ATC from below, and that ATC is U-shaped.
⚠️ Know how to decompose a given TC function into its fixed and variable components, and be able to explain the economic meaning of each.
True or false: Fixed cost changes when the firm produces more output. (False.)
Fill in the blank: ATC reaches its minimum where MC ______ ATC. (equals / intersects)
True or false: If MC is below ATC, producing one more unit will increase ATC. (False. It will decrease ATC.)
If TC = 100 + 5q², what is the fixed cost? (100.)
True or false: MC is the slope of the total cost curve. (True.)
Q: A firm has F = 81 and VC = q². What is its total cost function, and how would you identify the fixed and variable components from the combined expression?
A: TC = 81 + q². The constant term (81) is fixed cost because it does not depend on q. The q² term is variable cost because it changes with output.
Q: Given TC = 81 + q², what is the average total cost function?
A: ATC = TC/q = 81/q + q.
Q: If MC = 2q and ATC = 81/q + q, at what quantity does MC intersect ATC? What is the minimum ATC?
A: Set MC = ATC: 2q = 81/q + q. This simplifies to q = 81/q, giving q² = 81, so q = 9. Minimum ATC = 81/9 + 9 = 18.
Q: At q = 3, ATC = 30 and MC = 6. Would producing slightly more raise or lower ATC? Explain.
A: MC (6) < ATC (30), so the next unit costs less than the current average. Producing more pulls ATC down.
Q: At q = 27, ATC = 30 and MC = 54. Would producing slightly more raise or lower ATC? Explain.
A: MC (54) > ATC (30), so the next unit costs more than the current average. Producing more pushes ATC up.
Q: Describe the shape of the ATC curve and explain why MC must cross it at the minimum.
A: ATC is U-shaped: it falls at first (fixed costs are spread over more units) and then rises (variable costs dominate). MC crosses ATC at the bottom of the U because when MC is below ATC the average is being pulled down, and when MC is above ATC the average is being pushed up. The crossover is exactly where the pulling stops and the pushing starts.
This material connects directly to the theory of the firm's supply curve (Ch. 8+), because a competitive firm produces where P = MC, but only if P is at or above minimum ATC. Without understanding the MC-ATC relationship here, the shutdown and entry/exit decisions in later chapters will not click.
It also connects to short-run vs. long-run cost structures (covered in the next set of notes). In the short run, some costs are fixed; in the long run, all costs are variable, which changes the shape of the cost curves.
Cost functions, total cost, fixed cost, variable cost, average total cost, ATC, marginal cost, MC, U-shaped ATC, minimum efficient scale, cost curves, MC crosses ATC, cost per unit, economies of scale, ECON 300, intermediate microeconomics, Chapter 7, University of Washington