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Chapter 22
Cost Curves

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.

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.

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?

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.

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;

y
$
F

y
$
cv(y)

y
$
F
cv(y)

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

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

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 ...

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

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.

$/output unit
AVC(y)
y
0

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

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


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

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

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

$/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.

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

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.

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,


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

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


Marginal & Average Cost Functions
Since


Marginal & Average Cost Functions
Since
Therefore,
as

Marginal & Average Cost Functions
Since
Therefore,
as
as

Marginal & Average Cost Functions
as


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

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

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

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

$/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.

Marginal & Average Cost Functions
Similarly, since


Marginal & Average Cost Functions
Similarly, since
Therefore,
as

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

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

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.

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

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¢¢¢.

y
0
F¢ = w2x2¢
F¢
cs(y;x2¢)
$

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

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¢¢

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¢¢

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

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.

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

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

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


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.

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

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?

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

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

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¢¢

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¢¢

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¢¢

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¢¢

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¢¢

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.

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.

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

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.

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 ...

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

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 ...

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.

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?

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.

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


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

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

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

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

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.

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.

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.

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.

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 ...

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

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

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.