C tC
Cost
Curves
© 2010 W. W. Norton & Company, Inc.
Types of Cost Curves
A total cost curve is the graph of a
firm’s
firm
s total cost function.
function
 A variable cost curve is the graph of
a firm’s
fi ’ variable
i bl cost function.
f
i
 An average total cost curve is the
graph of a firm’s average total cost
function.
function

© 2010 W. W. Norton & Company, Inc.
2
Types of Cost Curves
An average variable cost curve is the
graph of a firm’s
firm s average variable
cost function.
 An
A average fi
fixed
d cost curve is
i the
h
graph of a firm’s average
g
g fixed cost
function.
 A marginal cost curve is the graph of
a firm’s marginal cost function.

© 2010 W. W. Norton & Company, Inc.
3
Types of Cost Curves
How are these cost curves related to
each other?
 How are a firm’s long-run and shortrun cost curves related?
l d?

© 2010 W. W. Norton & Company, Inc.
4
Fixed Variable & Total Cost Functions
Fixed,
F is the total cost to a firm of its shortrun fixed inputs. F, the firm’s fixed
cost, does not vary with the firm’s
firm s
output level.
 cv(y)
( ) iis the
th total
t t l costt to
t a firm
fi
off its
it
variable inputs when producing y
output units. cv(y) is the firm’s variable
cost function.
 cv(y) depends upon the levels of the
fixed inputs.
inputs

© 2010 W. W. Norton & Company, Inc.
5
Fixed Variable & Total Cost Functions
Fixed,

c(y) is the total cost of all inputs,
fixed and variable,
variable when producing y
output units. c(y) is the firm’s total
cost function;
c(y)  F cv(y).
© 2010 W. W. Norton & Company, Inc.
6
$
F
y
© 2010 W. W. Norton & Company, Inc.
7
$
cv(y)
y
© 2010 W. W. Norton & Company, Inc.
8
$
cv(y)
F
y
© 2010 W. W. Norton & Company, Inc.
9
$
c(y)
cv(y)
c(y)  F cv(y)
F
F
y
© 2010 W. W. Norton & Company, Inc.
10
Av. Fixed, Av. Variable & Av. Total
Cost Curves

The firm’s total cost function is
c(y)  F cv(y).
For y > 0, the firm’s average total
cost function is
F cv(y)
AC(y)  
y
y
 AFC(y)  AVC(y).
)
© 2010 W. W. Norton & Company, Inc.
11
Av. Fixed, Av. Variable & Av. Total
Cost Curves

What does an average fixed cost
curve look like?
F
AFC(y) 

y
AFC(y)
(y) is a rectangular
g
hyperbola
y
so
its graph looks like ...
© 2010 W. W. Norton & Company, Inc.
12
$/output
p unit
AFC(y)
(y)  0 as y 
AFC(y)
0
© 2010 W. W. Norton & Company, Inc.
y
13
Av. Fixed, Av. Variable & Av. Total
Cost Curves

In a short-run with a fixed amount of
at least one input,
input the Law of
Diminishing (Marginal) Returns must
apply causing the firm’s average
apply,
variable cost of production to
increase eventually.
© 2010 W. W. Norton & Company, Inc.
14
$/output
p unit
AVC(y)
0
© 2010 W. W. Norton & Company, Inc.
y
15
$/output
p unit
AVC(y)
AFC(y)
0
© 2010 W. W. Norton & Company, Inc.
y
16
Av. Fixed, Av. Variable & Av. Total
Cost Curves

And
ATC(y) = AFC(y) + AVC(y)
© 2010 W. W. Norton & Company, Inc.
17
$/output
p unit
ATC(y) = AFC(y) + AVC(y)
ATC(y)
AVC(y)
AFC(y)
0
© 2010 W. W. Norton & Company, Inc.
y
18
$/output
p unit
AFC(y) = ATC(y) - AVC(y)
ATC(y)
AFC
AVC(y)
AFC(y)
0
© 2010 W. W. Norton & Company, Inc.
y
19
$/output
p unit
Since AFC(y)  0 as y ,
ATC(y)  AVC(y) as y 
ATC(y)
AFC
AVC(y)
AFC(y)
0
© 2010 W. W. Norton & Company, Inc.
y
20
$/output
p unit
Since AFC(y)  0 as y ,
ATC(y)  AVC(y) as y 
And since short
short-run
run AVC(y) must
eventually increase, ATC(y) must
eventually increase in a short-run.
ATC(y)
AVC(y)
AFC(y)
0
© 2010 W. W. Norton & Company, Inc.
y
21
Marginal Cost Function

Marginal cost is the rate-of-change of
variable production cost as the
output level changes. That is,
 cv(y)
MC(y) 
.
y
© 2010 W. W. Norton & Company, Inc.
22
Marginal Cost Function

The firm’s total cost function is
c(y)  F cv(y)
and the fixed cost F does not change
with the output
level
y,
y so
(
c
y
)
 c(y)
v
MC(y) 

.
y

y
MC is the slope of both the variable
cost and the total cost functions.
© 2010 W. W. Norton & Company, Inc.
23
Marginal and Variable Cost Functions

Since MC(y) is the derivative of cv(y),
cv(y) must be the integral of MC(y).
MC(y)
(

c
y
)
v
That is, MC(y) 

y
y
cv(y)   MC(z)dz.
0
© 2010 W. W. Norton & Company, Inc.
24
Marginal and Variable Cost Functions
$/output unit
y
cv(y)   MC(z)dz
0
MC(y)
Area is the variable
cost of making y’ units
0
y
© 2010 W. W. Norton & Company, Inc.
y
25
Marginal & Average Cost Functions

How is marginal cost related to
average variable cost?
© 2010 W. W. Norton & Company, Inc.
26
Marginal & Average Cost Functions
cv(y)
AVC(y) 
,
Since
y
 AVC(y) y MC(y) 1 cv(y)

.
2
y
y
© 2010 W. W. Norton & Company, Inc.
27
Marginal & Average Cost Functions
cv(y)
AVC(y) 
,
Since
y
 AVC(y) y MC(y) 1 cv(y)

.
2
y
y
Therefore,

 AVC(y) 
0 as y MC(y)cv(y).
y


© 2010 W. W. Norton & Company, Inc.
28
Marginal & Average Cost Functions
cv(y)
AVC(y) 
,
Since
y
 AVC(y) y MC(y) 1 cv(y)

.
2
y
y
Therefore,

 AVC(y) 
0 as y MC(y)cv(y).
y


c (y)
 AVC(y) 
0 as MC(y) v
 AVC(y)).
y 
 y
© 2010 W. W. Norton & Company, Inc.
29
Marginal & Average Cost Functions

 AVC(y) 
0
0 as MC(y)AVC(y).
y 

© 2010 W. W. Norton & Company, Inc.
30
$/output
p unit
MC(y)
AVC(y)
© 2010 W. W. Norton & Company, Inc.
y
31
$/output
p unit
 AVC(y)
MC(y)  AVC(y) 
0
y
MC(y)
AVC(y)
© 2010 W. W. Norton & Company, Inc.
y
32
$/output
p unit
 AVC(y)
MC(y)  AVC(y) 
0
y
MC(y)
AVC(y)
© 2010 W. W. Norton & Company, Inc.
y
33
$/output
p unit
 AVC(y)
MC(y)  AVC(y) 
0
y
MC(y)
AVC(y)
© 2010 W. W. Norton & Company, Inc.
y
34
$/output
p unit
 AVC(y)
MC(y)  AVC(y) 
0
y
The short-run MC curve intersects
th short-run
the
h t
AVC curve ffrom
MC(y)
below at the AVC curve’s
minimum.
AVC(y)
© 2010 W. W. Norton & Company, Inc.
y
35
Marginal & Average Cost Functions
Similarly,
y, since
 ATC(y)

y
© 2010 W. W. Norton & Company, Inc.
c(y)
ATC(y) 
,
y
y MC(y) 1 c(y)
2
.
y
36
Marginal & Average Cost Functions
Similarly,
y, since
 ATC(y)

y
Therefore,
 ATC(y) 
0
y 
© 2010 W. W. Norton & Company, Inc.
c(y)
ATC(y) 
,
y
y MC(y) 1 c(y)
2
.
y
as

y MC(y)c(y).

37
Marginal & Average Cost Functions
Similarly,
y, since
 ATC(y)

y
c(y)
ATC(y) 
,
y
y MC(y) 1 c(y)
2
.
y
Therefore,
 ATC(y) 
0
y 

as y MC(y)c(y).

c(y)
 ATC(y) 
0 as MC(y)
 ATC(y)).
y 
 y
© 2010 W. W. Norton & Company, Inc.
38
$/output
p unit

 ATC(y) 
0
0 as MC(y)ATC(y)
y 

MC(y)
ATC(y)
© 2010 W. W. Norton & Company, Inc.
y
39
Marginal & Average Cost Functions
The short-run MC curve intersects
the short
short-run
run AVC curve from below
at the AVC curve’s minimum.
 And,
A d similarly,
i il l the
h short-run
h
MC
curve intersects the short-run ATC
curve from below at the ATC curve’s
minimum.

© 2010 W. W. Norton & Company, Inc.
40
$/output
p unit
MC(y)
ATC(y)
AVC(y)
© 2010 W. W. Norton & Company, Inc.
y
41
Short-Run & Long-Run Total Cost
Curves
A firm has a different short-run total
cost curve for each possible short
shortrun circumstance.
 Suppose
S
the
h firm
fi
can be
b in
i one off
jjust three short-runs;
x2 = x2
or
x2 = x2
x2 < x2 < x2..
or
x2 = x2.

© 2010 W. W. Norton & Company, Inc.
42
$
F = w2x2
F
cs(y;x2)
F
0
© 2010 W. W. Norton & Company, Inc.
y
43
$
F = w2x2
F
F = w2x2
cs(y;x2)
cs(y;x2)
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
44
$
F = w2x2
F
F = w2x2
A larger amount of the fixed
input increases the firm’s
firm s
fixed cost.
cs(y;x2)
cs(y;x2)
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
45
$
F =
F
F
x2w
 2x2
w2=
A larger amount of the fixed
input increases the firm’s
firm s
fixed cost.
cs(y;x2)
cs(y;x2)
Why does
a larger amount of
the fixed input reduce the
slope of the firm’s total cost curve?
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
46
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.
p 1
Therefore,, the extra amount of input
needed for 1 extra output unit is
© 2010 W. W. Norton & Company, Inc.
47
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.
p 1
Therefore,, the extra amount of input
needed for 1 extra output unit is
1/ MP1 units of input 1.
1
© 2010 W. W. Norton & Company, Inc.
48
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.
p 1
Therefore,, the extra amount of input
needed for 1 extra output unit is
1/ MP1 units of input 1.
1
Each unit of input 1 costs w1, so the firm’s
extra cost from producing one extra unit
of output is
© 2010 W. W. Norton & Company, Inc.
49
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.
p 1
Therefore,, the extra amount of input
needed for 1 extra output unit is
1/ MP1 units of input 1.
1
Each unit of input 1 costs w1, so the firm’s
extra cost from producing one extra unit
of output is MC w1 .
MP1
© 2010 W. W. Norton & Company, Inc.
50
Short-Run & Long-Run Total Cost
Curves
w1
MC
is the slope of the firm’s total
MP1
cost curve
curve.
© 2010 W. W. Norton & Company, Inc.
51
Short-Run & Long-Run Total Cost
Curves
w1
MC
is the slope of the firm’s total
MP1
cost curve
curve.
If input 2 is a complement to input 1 then
MP1 is higher for higher x2.
Hence MC is lower for higher x2.
Hence,
That is,, a short-run total cost curve starts
higher and has a lower slope if x2 is larger.
© 2010 W. W. Norton & Company, Inc.
52
$
F =
F
F
x2w
 2x2
w2=
F = w2x2
cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
53
Short-Run & Long-Run Total Cost
Curves
The firm has three short-run total
cost curves
curves.
 In the long-run the firm is free to
choose
h
amongst these
h
three
h
since
i
it
i
is free to select x2 equal to any
y of x2,
x2, or x2.
 How does the firm make this choice?

© 2010 W. W. Norton & Company, Inc.
54
$
For 0  y  y, choose x2 = ?
cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
y
y
55
$
For 0  y  y, choose x2 = x2. cs(y;x2)
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
y
y
56
$
For 0  y  y, choose x2 = x2. cs(y;x2)
For y  y  y, choose x2 = ?
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
y
y
57
$
For 0  y  y, choose x2 = x2. cs(y;x2)
For y  y  y, choose x2 = x2.
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
y
y
58
$
For 0  y  y, choose x2 = x2. cs(y;x2)
For y  y  y, choose x2 = x2.
For y
y  y,
y choose x2 = ?
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
y
y
59
$
For 0  y  y, choose x2 = x2. cs(y;x2)
For y  y  y, choose x2 = x2.
For y
y  y,
y choose x2 = x2.
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
y
y
60
$
For 0  y  y, choose x2 = x2. cs(y;x2)
For y  y  y, choose x2 = x2.
For y
y  y,
y choose x2 = x2.
cs(y;x2)
cs(y;x2)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
c(y), the
c(y)
firm’s longrun total
t t l
cost curve.
y
y
61
Short-Run & Long-Run Total Cost
Curves

The 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
long-run
envelope of the short-run total cost
curves.
© 2010 W. W. Norton & Company, Inc.
62
Short-Run & Long-Run Total Cost
Curves

If 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
long-run
lower envelope of all of the short-run
total cost curves.
© 2010 W. W. Norton & Company, Inc.
63
$
cs(y;x2)
cs(y;x2)
cs(y;x2)
c(y)
F
F
F
0
© 2010 W. W. Norton & Company, Inc.
y
64
Short-Run & Long-Run Average Total
Cost Curves
For an
any o
output
tp t le
level
el y, the long-run
long r n
total cost curve always gives the lowest
possible total production cost.
 Therefore,
Therefore the long
long-run
run av
av. total cost
curve must always give the lowest
possible av
av. total production cost.
cost
 The long-run av. total cost curve must
be the lower envelope of all of the
firm’s
firm
s short-run
short run av. total cost curves.

© 2010 W. W. Norton & Company, Inc.
65
Short-Run & Long-Run Average Total
Cost Curves

E.g.
E
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
firm s three short-run
short run
average total cost curves are ...
© 2010 W. W. Norton & Company, Inc.
66
$/output unit
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
© 2010 W. W. Norton & Company, Inc.
y
67
Short-Run & Long-Run Average Total
Cost Curves

The firm’s long-run average total
cost curve is the lower envelope of
the short-run average total cost
curves ...
© 2010 W. W. Norton & Company, Inc.
68
$/output unit
ACs(y;
(y;x2))
ACs(y;x2)
ACs(y;x2)
The long-run av. total cost
AC(y)
curve is the lower envelope
of the short-run av. total cost curves.
© 2010 W. W. Norton & Company, Inc.
y
69
Short-Run & Long-Run Marginal Cost
Curves

Q: Is the long-run marginal cost
curve the lower envelope of the
firm’s short-run marginal cost
curves?
© 2010 W. W. Norton & Company, Inc.
70
Short-Run & Long-Run Marginal Cost
Curves
Q: Is the long-run marginal cost
curve the lower envelope of the
firm’s short-run marginal cost
curves?
 A: No.

© 2010 W. W. Norton & Company, Inc.
71
Short-Run & Long-Run Marginal Cost
Curves

The firm’s three short-run average
total cost curves are ...
© 2010 W. W. Norton & Company, Inc.
72
$/output unit
ACs(y;x2)
ACs(y;x2)
ACs(y;x2))
© 2010 W. W. Norton & Company, Inc.
y
73
MCs(y;
(y;x2))
$/output unit
MCs(y;
(y;x2))
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
MCs(y;x
( 2)
© 2010 W. W. Norton & Company, Inc.
y
74
MCs(y;
(y;x2))
$/output unit
MCs(y;
(y;x2))
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
MCs(y;x
( 2)
© 2010 W. W. Norton & Company, Inc.
AC(y)
y
75
MCs(y;
(y;x2))
$/output unit
MCs(y;
(y;x2))
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
MCs(y;x
( 2)
© 2010 W. W. Norton & Company, Inc.
AC(y)
y
76
MCs(y;
(y;x2))
$/output unit
MCs(y;
(y;x2))
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
MCs(y;x
( 2)
MC(y), the long-run
MC(y)
long run marginal
cost curve.
© 2010 W. W. Norton & Company, Inc.
y
77
Short-Run & Long-Run Marginal Cost
Curves

For any output level y > 0, the longrun marginal cost of production is
the marginal cost of production for
the short-run
short run chosen by the firm
firm.
© 2010 W. W. Norton & Company, Inc.
78
MCs(y;
(y;x2))
$/output unit
MCs(y;
(y;x2))
ACs(y;x2)
ACs(y;x2)
ACs(y;x2)
MCs(y;x
( 2)
MC(y), the long-run
MC(y)
long run marginal
cost curve.
© 2010 W. W. Norton & Company, Inc.
y
79
Short-Run & Long-Run Marginal Cost
Curves
For any output level y > 0, the longrun marginal cost is the marginal
cost for the short-run chosen by the
firm.
firm
 This is always
y true, no matter how
many and which short-run
circumstances exist for the firm.

© 2010 W. W. Norton & Company, Inc.
80
Short-Run & Long-Run Marginal Cost
Curves
For any output level y > 0, the longrun marginal cost is the marginal
cost for the short-run chosen by the
firm.
firm
 So 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

© 2010 W. W. Norton & Company, Inc.
81
Short-Run & Long-Run Marginal Cost
Curves
$/output unit
SRACs
AC(y)
y
© 2010 W. W. Norton & Company, Inc.
82
Short-Run & Long-Run Marginal Cost
Curves
$/output unit
SRMCs
AC(y)
y
© 2010 W. W. Norton & Company, Inc.
83
Short-Run & Long-Run Marginal Cost
Curves
$/output unit
SRMCs
MC(y)
AC(y)
y
For each y > 0,
0 the long-run
long run MC equals the
MC for the short-run chosen by the firm. 84
© 2010 W. W. Norton & Company, Inc.