Source: Lecture 13, Intermediate Microeconomic Theory, Arizona State University
Difficulty: Intermediate. Prerequisites: basic cost function c(r, k, y), fixed vs variable factors of production, profit maximisation concepts from earlier lectures.
Cost curves sit at the heart of producer theory. They translate the production function and input prices you studied earlier into a set of curves that describe how a firm's costs behave as output changes. Understanding these curves is what allows you to derive a firm's supply curve and to determine whether a firm should produce at all or shut down. If you are comfortable with cost functions, fixed vs variable factors, and basic differentiation, you are ready for this material.
A firm's total cost splits into fixed costs and variable costs. Dividing each by output gives you average cost curves (AC, AVC, AFC), while differentiating total cost gives you marginal cost (MC). The MC curve passes through the minimum of both AVC and AC, and the upward-sloping portion of MC above AVC is the firm's supply curve.
Fixed costs (F)
Costs associated with fixed factors of production that do not change with output. A liquor licence is a classic example.
In simple terms, these are the bills you pay regardless of whether you produce one unit or a thousand.
Quasi-fixed costs
Costs that are independent of output level but do not have to be paid if the firm produces nothing at all.
Think of it as a cost you can avoid entirely by shutting down, but that stays the same whether you produce 1 unit or 100.
Variable costs, c_v(y)
The portion of total cost that changes with the level of output.
In simple terms, these are the costs that rise as you make more stuff: raw materials, extra labour hours, electricity on the factory floor.
Average cost, AC(y)
Total cost divided by quantity of output: AC(y) = c(y) / y.
Think of it as the per-unit cost of production across all units produced so far.
Average variable cost, AVC(y)
Variable cost divided by quantity of output: AVC(y) = c_v(y) / y.
In simple terms, the per-unit cost once you strip out everything that is fixed.
Average fixed cost, AFC(y)
Fixed cost divided by quantity of output: AFC(y) = F / y.
This one just falls as you spread the same lump of fixed cost over more and more units.
Marginal cost, MC(y)
The derivative of total cost with respect to output: MC(y) = dc(y)/dy.
In simple terms, the cost of producing one more unit, right at the margin.
When input prices are fixed, the cost function simplifies to c(y), a function of output alone. Total cost decomposes as:
c(y) = c_v(y) + F
Divide through by y to get the three average cost measures:
AC(y) = c(y) / y = the full per-unit cost
AVC(y) = c_v(y) / y = the variable per-unit cost
AFC(y) = F / y = the fixed per-unit cost
So AC(y) = AVC(y) + AFC(y). This identity is worth memorising because it tells you exactly how the three curves relate.
AFC starts at infinity when output is near zero and falls continuously, approaching zero as output grows. It is a simple rectangular hyperbola (F/y).
AVC may initially slope downward (reflecting early efficiencies), but eventually rises as production pushes against capacity. The upturn comes from diminishing returns: at some point, expanding output is marginally very expensive.
AC is the vertical sum of AVC and AFC. Because AFC is large at low output and AVC is large at high output, AC is U-shaped: it falls at first (fixed costs being spread), reaches a minimum, then rises as variable costs dominate.
Given c(y) = (1/5)(y - 5)^3 + 50, which expands to (1/5)y^3 - 3y^2 + 15y + 25:
The fixed cost F = 25 (the portion that remains when y = 0, noting the quasi-fixed component brings total to 50 at zero output)
Variable cost c_v(y) = (1/5)y^3 - 3y^2 + 15y
AC(y) = [(1/5)y^3 - 3y^2 + 15y] / y + 25/y = (1/5)y^2 - 3y + 15 + 25/y
Marginal cost is the derivative of total cost with respect to output:
MC(y) = dc(y)/dy = dc_v(y)/dy
Because fixed costs vanish under differentiation, marginal cost equals marginal variable cost. Fixed costs have no effect on MC whatsoever.
Using c(y) = (1/5)(y - 5)^3 + 25:
MC(y) = (3/5)(y - 5)^2
This is a parabola with its minimum at y = 5 (where MC = 0). It is U-shaped for this particular function, though MC curves can take other shapes depending on the cost function.
These relationships are examined constantly and are essential for the course:
When MC < AC, average cost is falling. Each additional unit costs less than the current average, so it pulls the average down.
When MC > AC, average cost is rising. Each additional unit costs more than the current average, pulling it up.
MC passes through the minimum of AC. This is the crossing point where MC = AC.
MC also passes through the minimum of AVC, by the same logic.
At the very first unit of output, MC and AVC are equal.
The intuition is the same as a batting average: if your next at-bat (marginal) is better than your season average, it pulls the average up. If worse, it pulls it down.
Expression | Formula | Notes |
|---|---|---|
Total cost | c(y) = c_v(y) + F | Decomposes into variable + fixed |
Average cost | AC(y) = c(y) / y | U-shaped curve |
Average variable cost | AVC(y) = c_v(y) / y | Eventually rises |
Average fixed cost | AFC(y) = F / y | Rectangular hyperbola, always falling |
Marginal cost | MC(y) = dc(y)/dy | Equals marginal variable cost |
AC identity | AC(y) = AVC(y) + AFC(y) | Vertical sum of the two components |
For the lecture's running example, c(y) = (1/5)(y - 5)^3 + 50:
Expanded: (1/5)y^3 - 3y^2 + 15y + 25 (plus the quasi-fixed 25)
MC(y) = (3/5)(y - 5)^2
AVC(y) = (1/5)y^2 - 3y + 15
AFC(y) = 25/y
The U-shaped average cost curve explains why factories have an ideal operating scale. Run a car plant below capacity and per-unit costs are high because you are still paying for the building, the tooling and the management layer. Push it well beyond design capacity and overtime, breakdowns and cramped floor space drive per-unit costs back up.
The MC = supply result is the foundation of competitive market analysis. When you see a supply curve in any applied context (energy markets, agriculture, manufacturing), it traces the marginal cost schedules of the firms in that market.
Students often think fixed costs affect marginal cost. They do not. Differentiating a constant gives zero, so MC depends only on variable costs.
Students sometimes assume AVC must always slope downward before sloping upward. The lecture notes are clear: AVC may initially fall but does not have to. It is assumed to eventually rise.
A common error is believing MC intersects AC at the point where AC starts to fall. MC intersects AC at AC's minimum, the point where AC stops falling and begins to rise.
Students confuse the shutdown condition (p < AVC) with the break-even condition (p < AC). A firm making losses can still be better off producing than shutting down, provided price covers variable costs.
⚠️ You will almost certainly be asked to derive AC, AVC, AFC and MC from a given cost function. Practise the algebra until it is automatic.
⚠️ The relationship "MC passes through the minimum of AC and AVC" is a perennial exam favourite. Be ready to explain it both mathematically (set dAC/dy = 0 and show MC = AC at that point) and intuitively (the batting average analogy).
⚠️ Know the AC = AVC + AFC decomposition cold. Questions may give you two of the three and ask for the third.
⚠️ Graphical questions: be able to sketch AC, AVC, AFC and MC on the same axes, showing where MC crosses AVC and AC at their respective minima.
True or false: Average fixed cost eventually reaches zero as output increases. (Answer: False. AFC approaches zero but never reaches it.)
Fill in the blank: MC passes through the ______ of both the AVC and AC curves. (Answer: minimum)
True or false: If MC is below AC, then AC is rising. (Answer: False. If MC < AC, AC is falling.)
Fill in the blank: Marginal cost equals marginal ______ cost because fixed costs differentiate to zero. (Answer: variable)
True or false: The AVC curve must always slope downward before it slopes upward. (Answer: False. It may, but it does not have to.)
Q: Given c(y) = (1/5)(y - 5)^3 + 50, what are the fixed costs?
A: Plug in y = 0: c(0) = (1/5)(0 - 5)^3 + 50 = (1/5)(-125) + 50 = -25 + 50 = 25. The fixed cost is 25, plus the quasi-fixed component of 25, for a total of 50 when the firm operates. (Note: the lecture uses F = 25 for the truly fixed portion.)
Q: Derive the marginal cost function from c(y) = (1/5)(y - 5)^3 + 25.
A: MC(y) = dc/dy = (3/5)(y - 5)^2. This is a U-shaped parabola with minimum at y = 5.
Q: At what output level does MC equal AVC, and why does this matter?
A: MC equals AVC at the first unit of output. This matters because it establishes the starting relationship between the two curves. More importantly, MC crosses AVC at AVC's minimum, which determines the shutdown price.
Q: Explain intuitively why MC passes through the minimum of AC.
A: When the cost of the next unit (MC) is less than the average so far (AC), producing that unit pulls the average down, so AC is falling. When MC exceeds AC, the next unit pulls the average up. The crossover, where MC = AC, is the turning point: the minimum of AC.
Q: If AC(y) = (1/5)y^2 - 3y + 15 + 25/y, how would you find the output that minimises AC?
A: Set dAC/dy = 0: (2/5)y - 3 - 25/y^2 = 0. Solve for y. At that y, MC = AC.
This material connects directly to supply theory (covered in the second half of this lecture and in subsequent lectures), because the supply curve is derived from the MC curve. It also ties back to production functions and isoquants: the shape of the cost curves reflects the underlying technology and returns to scale. If the production function exhibits diminishing marginal returns, that is exactly what drives MC upward and gives the cost curves their characteristic shapes.
Cost curves, average cost, average variable cost, average fixed cost, marginal cost, U-shaped cost curve, AC AVC AFC MC, cost function, fixed costs, variable costs, quasi-fixed costs, per-unit cost, cost minimisation, producer theory, intermediate microeconomics, ECN 312, Arizona State University