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21
Cost Curves

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Types of Cost Curves
uA total cost curve is the graph of a firm’s total cost function.
uA variable cost curve is the graph of a firm’s variable cost function.
uAn average total cost curve is the graph of a firm’s average total cost function.

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Types of Cost Curves
uAn average variable cost curve is the graph of a firm’s average variable cost function.
uAn average fixed cost curve is the graph of a firm’s average fixed cost function.
uA marginal cost curve is the graph of a firm’s marginal cost function.

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Types of Cost Curves
uHow are these cost curves related to each other?
uHow are a firm’s long-run and short-run cost curves related?

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Fixed, Variable & Total Cost Functions
uF is the total cost to a firm of its short-run fixed inputs.  F, the firm’s fixed cost, does not
vary with the firm’s output level.
ucv(y) is the total cost to a firm of its variable inputs when producing y output units.  cv(y) is
the firm’s variable cost function.
ucv(y) depends upon the levels of the fixed inputs.

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Fixed, Variable & Total Cost Functions
uc(y) is the total cost of all inputs, fixed and variable, when producing y output units.  c(y) is
the firm’s total cost function;

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y
$
F

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y
$
cv(y)

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y
$
F
cv(y)

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y
$
F
cv(y)
c(y)
F

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Av. Fixed, Av. Variable & Av. Total Cost Curves
uThe firm’s total cost function is
For y > 0, the firm’s average total cost function is

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Av. Fixed, Av. Variable & Av. Total Cost Curves
uWhat does an average fixed cost curve look like?
uAFC(y) is a rectangular hyperbola so its graph looks like ...

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$/output unit
AFC(y)
y
0
AFC(y) ® 0 as y ® ¥

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Av. Fixed, Av. Variable & Av. Total Cost Curves
uIn a short-run with a fixed amount of at least one input, the Law of Diminishing (Marginal)
Returns must apply, causing the firm’s average variable cost of production to increase eventually.

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$/output unit
AVC(y)
y
0

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$/output unit
AFC(y)
AVC(y)
y
0

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Av. Fixed, Av. Variable & Av. Total Cost Curves
uAnd    ATC(y) = AFC(y) + AVC(y)

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$/output unit
AFC(y)
AVC(y)
ATC(y)
y
0
ATC(y) = AFC(y) + AVC(y)

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$/output unit
AFC(y)
AVC(y)
ATC(y)
y
0
AFC(y) = ATC(y) - AVC(y)
AFC

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$/output unit
AFC(y)
AVC(y)
ATC(y)
y
0
Since AFC(y) ® 0 as y ® ¥,
ATC(y) ® AVC(y) as y ® ¥.
AFC

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$/output unit
AFC(y)
AVC(y)
ATC(y)
y
0
Since AFC(y) ® 0 as y ® ¥,
ATC(y) ® AVC(y) as y ® ¥.
And since short-run AVC(y) must
eventually increase, ATC(y) must
eventually increase in a short-run.

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Marginal Cost Function
uMarginal cost is the rate-of-change of variable production cost as the output level changes.  That
is,

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Marginal Cost Function
uThe firm’s total cost function is
and the fixed cost F does not change with the output level y, so
uMC is the slope of both the variable cost and the total cost functions.

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Marginal and Variable Cost Functions
uSince MC(y) is the derivative of cv(y), cv(y) must be the integral of MC(y).  That is,

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Marginal and Variable Cost Functions
MC(y)
y
0
Area is the variable
cost of making y’ units
$/output unit

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Marginal & Average Cost Functions
uHow is marginal cost related to  average variable cost?

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Marginal & Average Cost Functions
Since

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Marginal & Average Cost Functions
Since
Therefore,
as

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Marginal & Average Cost Functions
Since
Therefore,
as
as

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Marginal & Average Cost Functions
as

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$/output unit
y
AVC(y)
MC(y)

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$/output unit
y
AVC(y)
MC(y)

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$/output unit
y
AVC(y)
MC(y)

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$/output unit
y
AVC(y)
MC(y)

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$/output unit
y
AVC(y)
MC(y)
The short-run MC curve intersects
the short-run AVC curve from
below at the AVC curve’s
     minimum.

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Marginal & Average Cost Functions
Similarly, since

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Marginal & Average Cost Functions
Similarly, since
Therefore,
as

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Marginal & Average Cost Functions
Similarly, since
Therefore,
as
as

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$/output unit
y
MC(y)
ATC(y)
as

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Marginal & Average Cost Functions
uThe short-run MC curve intersects the short-run AVC curve from below at the AVC curve’s minimum.
uAnd, similarly, the short-run MC curve intersects the short-run ATC curve from below at the ATC
curve’s minimum.

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$/output unit
y
AVC(y)
MC(y)
ATC(y)

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Short-Run & Long-Run Total Cost Curves
uA firm has a different short-run total cost curve for each possible short-run circumstance.
uSuppose the firm can be in one of just three short-runs;
x2 = x2¢
or  x2 = x2¢¢            x2¢ < x2¢¢ < x2¢¢¢.
or  x2 = x2¢¢¢.

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y
0
F¢ = w2x2¢
F¢
cs(y;x2¢)
$

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y
F¢
0
F¢ = w2x2¢
F¢¢
F¢¢ = w2x2¢¢
cs(y;x2¢)
cs(y;x2¢¢)
$

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y
F¢
0
F¢ = w2x2¢
F¢¢ = w2x2¢¢
A larger amount of the fixed
input increases the firm’s
fixed cost.
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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y
F¢
0
F¢ = w2x2¢
F¢¢ = w2x2¢¢
A larger amount of the fixed
input increases the firm’s
fixed cost.
                                        Why does
                          a larger amount of
             the fixed input reduce the slope of the firm’s total cost curve?
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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MP1 is the marginal physical productivity
of the variable input 1, so one extra unit of
input 1 gives MP1 extra output units.
Therefore, the extra amount of input 1
needed for 1 extra output unit is
Short-Run & Long-Run Total Cost Curves

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MP1 is the marginal physical productivity
of the variable input 1, so one extra unit of
input 1 gives MP1 extra output units.
Therefore, the extra amount of input 1
needed for 1 extra output unit is
Short-Run & Long-Run Total Cost Curves
            units of input 1.

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MP1 is the marginal physical productivity
of the variable input 1, so one extra unit of
input 1 gives MP1 extra output units.
Therefore, the extra amount of input 1
needed for 1 extra output unit is
Short-Run & Long-Run Total Cost Curves
            units of input 1.
Each unit of input 1 costs w1, so the firm’s
extra cost from producing one extra unit
of output is

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MP1 is the marginal physical productivity
of the variable input 1, so one extra unit of
input 1 gives MP1 extra output units.
Therefore, the extra amount of input 1
needed for 1 extra output unit is
Short-Run & Long-Run Total Cost Curves
            units of input 1.
Each unit of input 1 costs w1, so the firm’s
extra cost from producing one extra unit
of output is

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Short-Run & Long-Run Total Cost Curves
is the slope of the firm’s total cost curve.

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Short-Run & Long-Run Total Cost Curves
is the slope of the firm’s total cost curve.
If input 2 is a complement to input 1 then
MP1 is higher for higher x2.
Hence, MC is lower for higher x2.
That is, a short-run total cost curve starts
higher and has a lower slope if x2 is larger.

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y
F¢
0
F¢ = w2x2¢
F¢¢ = w2x2¢¢
F¢¢¢
F¢¢¢ = w2x2¢¢¢
cs(y;x2¢¢¢)
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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Short-Run & Long-Run Total Cost Curves
uThe firm has three short-run total cost curves.
uIn the long-run the firm is free to choose amongst these three since it is free to select x2 equal
to any of x2¢, x2¢¢, or x2¢¢¢.
uHow does the firm make this choice?

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y
F¢
0
F¢¢¢
y¢
y¢¢
For 0 £ y £ y¢, choose x2 = ?
cs(y;x2¢¢¢)
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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y
F¢
0
F¢¢¢
y¢
y¢¢
For 0 £ y £ y¢, choose x2 = x2¢.
cs(y;x2¢¢¢)
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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y
F¢
0
F¢¢¢
y¢
y¢¢
For 0 £ y £ y¢, choose x2 = x2¢.
For y¢ £ y £ y¢¢, choose x2 = ?
cs(y;x2¢¢¢)
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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y
F¢
0
F¢¢¢
y¢
y¢¢
For 0 £ y £ y¢, choose x2 = x2¢.
For y¢ £ y £ y¢¢, choose x2 = x2¢¢.
cs(y;x2¢¢¢)
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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y
F¢
0
F¢¢¢
y¢
y¢¢
For 0 £ y £ y¢, choose x2 = x2¢.
For y¢ £ y £ y¢¢, choose x2 = x2¢¢.
For y¢¢ < y, choose x2 = ?
cs(y;x2¢¢¢)
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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y
F¢
0
F¢¢¢
cs(y;x2¢¢¢)
y¢
y¢¢
For 0 £ y £ y¢, choose x2 = x2¢.
For y¢ £ y £ y¢¢, choose x2 = x2¢¢.
For y¢¢ < y, choose x2 = x2¢¢¢.
cs(y;x2¢)
cs(y;x2¢¢)
$
F¢¢

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y
F¢
0
cs(y;x2¢)
cs(y;x2¢¢)
F¢¢¢
cs(y;x2¢¢¢)
y¢
y¢¢
For 0 £ y £ y¢, choose x2 = x2¢.
For y¢ £ y £ y¢¢, choose x2 = x2¢¢.
For y¢¢ < y, choose x2 = x2¢¢¢.
c(y), the
firm’s long-
run total
cost curve.
$
F¢¢

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Short-Run & Long-Run Total Cost Curves
uThe firm’s long-run total cost curve consists of the lowest parts of the  short-run total cost
curves.  The long-run total cost curve is the lower envelope of the short-run total cost curves.

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Short-Run & Long-Run Total Cost Curves
uIf input 2 is available in continuous amounts then there is an infinity of short-run total cost
curves but the long-run total cost curve is still the lower envelope of all of the short-run total
cost curves.

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$
y
F¢
0
F¢¢¢
cs(y;x2¢)
cs(y;x2¢¢)
cs(y;x2¢¢¢)
c(y)
F¢¢

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Short-Run & Long-Run Average Total Cost Curves
uFor any output level y, the long-run total cost curve always gives the lowest possible total
production cost.
uTherefore, the long-run av. total cost curve must always give the lowest possible av. total
production cost.
uThe long-run av. total cost curve must be the lower envelope of all of the firm’s short-run av.
total cost curves.

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Short-Run & Long-Run Average Total Cost Curves
u
uE.g. suppose again that the firm can be in one of just three short-runs;
x2 = x2¢
or  x2 = x2¢¢          (x2¢ < x2¢¢ < x2¢¢¢)
or  x2 = x2¢¢¢
then the firm’s three short-run average total cost curves are ...

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)

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Short-Run & Long-Run Average Total Cost Curves
uThe firm’s long-run average total cost curve is the lower envelope of the short-run average total
cost curves ...

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)
AC(y)
The long-run av. total cost
curve is the lower envelope
of the short-run av. total cost curves.

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Short-Run & Long-Run Marginal Cost Curves
uQ: Is the long-run marginal cost curve the lower envelope of the firm’s short-run marginal cost
curves?

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Short-Run & Long-Run Marginal Cost Curves
uQ: Is the long-run marginal cost curve the lower envelope of the firm’s short-run marginal cost
curves?
uA:  No.

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Short-Run & Long-Run Marginal Cost Curves
uThe firm’s three short-run average total cost curves are ...

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)
MCs(y;x2¢)
MCs(y;x2¢¢)
MCs(y;x2¢¢¢)

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)
MCs(y;x2¢)
MCs(y;x2¢¢)
MCs(y;x2¢¢¢)
AC(y)

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)
MCs(y;x2¢)
MCs(y;x2¢¢)
MCs(y;x2¢¢¢)
AC(y)

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)
MCs(y;x2¢)
MCs(y;x2¢¢)
MCs(y;x2¢¢¢)
MC(y), the long-run marginal
cost curve.

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Short-Run & Long-Run Marginal Cost Curves
uFor any output level y > 0, the long-run marginal cost of production is the marginal cost of
production for the short-run chosen by the firm.

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y
$/output unit
ACs(y;x2¢¢¢)
ACs(y;x2¢¢)
ACs(y;x2¢)
MCs(y;x2¢)
MCs(y;x2¢¢)
MCs(y;x2¢¢¢)
MC(y), the long-run marginal
cost curve.

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Short-Run & Long-Run Marginal Cost Curves
uFor any output level y > 0, the long-run marginal cost is the marginal cost for the short-run
chosen by the firm.
uThis is always true, no matter how many and which short-run circumstances exist for the firm.

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Short-Run & Long-Run Marginal Cost Curves
uFor any output level y > 0, the long-run marginal cost is the marginal cost for the short-run
chosen by the firm.
uSo for the continuous case, where x2 can be fixed at any value of zero or more, the relationship
between the long-run marginal cost and all of the short-run marginal costs is ...

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Short-Run & Long-Run Marginal Cost Curves
AC(y)
$/output unit
y
SRACs

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Short-Run & Long-Run Marginal Cost Curves
AC(y)
$/output unit
y
SRMCs

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Short-Run & Long-Run Marginal Cost Curves
AC(y)
MC(y)
$/output unit
y
SRMCs
uFor each y > 0, the long-run MC equals the
MC for the short-run chosen by the firm.