Source: Strickland, Chapter 7, Texas A&M University
Tags: average fixed cost, AFC, average variable cost, AVC, average total cost, ATC, marginal cost, MC, cost functions, minimising ATC, U-shaped cost curves, ECON 323, microeconomic theory, cost of production
Average costs break total cost into a per-unit figure, and there are three flavours: AFC, AVC, and ATC. Marginal cost is the added cost of one more unit of output. The critical exam relationship is that MC intersects both AVC and ATC at their minimum points, and the shape of the ATC curve can be derived graphically from the slope of rays drawn from the origin to the TC curve.
Average fixed cost (AFC)
Fixed cost divided by quantity of output: AFC = FC / Q. AFC always declines as output rises, because a constant numerator is spread over more and more units.
Average variable cost (AVC)
Variable cost divided by quantity of output: AVC = VC / Q. Typically U-shaped: it falls at first (as the firm benefits from specialisation) then rises (as diminishing returns set in).
Average total cost (ATC)
Total cost divided by quantity of output: ATC = TC / Q. Equivalently, ATC = AFC + AVC. Also typically U-shaped.
Marginal cost (MC)
The change in total cost from producing one additional unit of output: MC = ΔTC / ΔQ. In calculus terms, MC = dTC/dQ.
Given a table like this:
Q | FC | VC | TC | AFC | AVC | ATC |
|---|---|---|---|---|---|---|
0 | 70 | 0 | 70 | – | – | – |
1 | 70 | 40 | 110 | 70.00 | 40.00 | 110.00 |
2 | 70 | 70 | 140 | 35.00 | 35.00 | 70.00 |
3 | 70 | 120 | 190 | 23.33 | 40.00 | 63.33 |
4 | 70 | 210 | 280 | 17.50 | 52.50 | 70.00 |
5 | 70 | 330 | 400 | 14.00 | 66.00 | 80.00 |
At Q = 4:
AFC = 70 / 4 = 17.50
AVC = 210 / 4 = 52.50
ATC = 280 / 4 = 70.00
This matches answer B on the exam.
If a graph shows ATC and AVC curves, AFC at any output level is the vertical gap between ATC and AVC, because ATC = AVC + AFC.
At 70 units, if ATC ≈ $25 and AVC ≈ $18, then AFC = $25 − $18 = $7.
The firm's total fixed cost is then AFC × Q = $7 × 70 = $490.
This matches answer D on the exam.
If a firm produces 50 units at a total cost of $1,000 and per-unit variable cost is $8:
Total variable cost = $8 × 50 = $400
Total fixed cost = TC − TVC = $1,000 − $400 = $600
Average fixed cost = FC / Q = $600 / 50 = $12
This is one of the most tested relationships in the chapter:
When MC < AVC, average variable cost is falling
When MC > AVC, average variable cost is rising
MC intersects AVC at AVC's minimum point
The same logic applies to ATC:
When MC < ATC, average total cost is falling
When MC > ATC, average total cost is rising
MC intersects ATC at ATC's minimum point
Think of it like a batting average: if your next at-bat (the marginal) is better than your current average, your average improves. If it is worse, your average worsens.
"If marginal cost is rising, the average total cost must be rising." FALSE. MC can be rising while still below ATC, in which case ATC is still falling. MC rising only means ATC is rising once MC has crossed above ATC.
"The marginal cost curve intersects both the average total and average variable cost curves at their minimum points." TRUE.
"If marginal cost is less than average variable cost, the average variable cost curve is negatively sloped." TRUE. When the marginal pulls the average down, the average is falling (negatively sloped).
Statements II and III are true, so the answer is D.
The average total cost at any output level equals the slope of a ray (straight line) drawn from the origin to the TC curve at that quantity.
Where the ray is steep, ATC is high
Where the ray is flattest, ATC is at its minimum
As you move along the TC curve, if the ray's slope first decreases then increases, the ATC curve is U-shaped
On Figure 7.5, point A is where a ray from the origin is tangent to the TC curve, which means the slope of the ray equals the slope of the curve there. That is, ATC = MC at that point. This is the minimum of the ATC curve.
At points before A (lower output), the ray is steeper, so ATC is higher and falling.
At points after A (higher output), the ray gets steeper again, so ATC is rising.
This produces a U-shaped ATC curve (panel d in the exam, which shows ATC declining then rising).
At point A: the ray from the origin is steeper than the tangent to TC, meaning ATC > MC ✓
At point B: the ray from the origin is tangent to TC, meaning ATC = MC ✓
At point C: the ray from the origin is less steep than the tangent, meaning ATC < MC ✓
All three statements (I, II, III) are true. Answer: B.
Given TC = 100 + 4Q + 2Q²:
FC = 100 (the constant term, the cost when Q = 0)
VC = 4Q + 2Q²
ATC = TC / Q = 100/Q + 4 + 2Q
AFC = FC / Q = 100/Q
AVC = VC / Q = 4 + 2Q
Check the exam statements:
I. ATC = 4Q + 2Q² — FALSE (this is VC, not ATC; ATC = 100/Q + 4 + 2Q)
II. AFC = 100/Q — TRUE
III. ATC = 2Q + 4 + 100/Q — TRUE
IV. FC = 100 + 4Q — FALSE (FC = 100 only)
Statements II and III are true. Answer: B.
ATC is minimised where MC = ATC. Set them equal and solve.
Example 1: TC = 18 + Q + 2Q², so MC = 1 + 4Q.
ATC = 18/Q + 1 + 2Q
Set MC = ATC: 1 + 4Q = 18/Q + 1 + 2Q
Simplify: 2Q = 18/Q
2Q² = 18
Q² = 9
Q = 3
Answer: B.
Example 2: TC = 192 + 10Q + 3Q², so MC = 10 + 6Q.
ATC = 192/Q + 10 + 3Q
Set MC = ATC: 10 + 6Q = 192/Q + 10 + 3Q
Simplify: 3Q = 192/Q
3Q² = 192
Q² = 64
Q = 8
Answer: A.
Given TC = 150 + 0.50Q + 1.5Q²:
VC = 0.50Q + 1.5Q² (everything except the constant)
AVC = VC / Q = 0.50 + 1.5Q
At Q = 10: AVC = 0.50 + 1.5(10) = 0.50 + 15.00 = $15.50
Answer: B.
Average cost formulas:
AFC = FC / Q
AVC = VC / Q
ATC = TC / Q = AFC + AVC
Marginal cost:
MC = ΔTC / ΔQ (discrete)
MC = dTC / dQ (continuous)
Minimising ATC (algebraic method):
Set MC = ATC, solve for Q.
From a TC function TC = a + bQ + cQ²:
FC = a
VC = bQ + cQ²
ATC = a/Q + b + cQ
AVC = b + cQ
MC = b + 2cQ
⚠️ AFC, AVC, and ATC are all per-unit measures. A very common error is forgetting to divide by Q.
⚠️ MC crosses AVC and ATC at their respective minimums. This is tested constantly. Remember: MC can be rising while ATC is still falling (as long as MC is still below ATC).
⚠️ When deriving ATC from a TC graph, the slope of the ray from the origin gives you ATC. The slope of the tangent line gives you MC. Where the ray is tangent to the curve, ATC = MC, which is the minimum ATC.
⚠️ For algebraic cost functions, the fixed cost is the constant term (the part with no Q in it). Everything else is variable cost.
⚠️ To find the ATC-minimising quantity, always set MC = ATC and solve. Do not set MC = 0 (that finds the minimum of the MC curve, which is a different thing).
Q: Given FC = 70 and VC = 210 at Q = 4, what are AFC, AVC, and ATC?
A: AFC = 70/4 = 17.50, AVC = 210/4 = 52.50, ATC = 280/4 = 70.00.
Q: If ATC = $25 and AVC = $18 at Q = 70, what is the firm's total fixed cost?
A: AFC = ATC − AVC = $7. Total FC = $7 × 70 = $490.
Q: If TC = 18 + Q + 2Q² and MC = 1 + 4Q, at what output is ATC minimised?
A: Set MC = ATC: 1 + 4Q = 18/Q + 1 + 2Q. Simplify to 2Q = 18/Q, so Q² = 9, Q = 3.
Q: MC is rising. Does that mean ATC is rising?
A: Not necessarily. ATC is rising only if MC is above ATC. MC can be rising while still below ATC, in which case ATC is still falling.
Q: Given TC = 150 + 0.50Q + 1.5Q², what is AVC at Q = 10?
A: AVC = 0.50 + 1.5Q = 0.50 + 15 = $15.50.
Q: The MC curve intersects the AVC curve at what point on the AVC curve?
A: At the minimum of the AVC curve.
average fixed cost, AFC, average variable cost, AVC, average total cost, ATC, marginal cost, MC, U-shaped cost curve, cost minimisation, cost functions, TC function, per-unit cost, spreading fixed costs, diminishing returns, ECON 323, Chapter 7, Strickland, microeconomic theory, Texas A&M