Cost curves

1
Lecture 6

• Cost curves

2
Types of Cost Curves

• A total cost curve is the graph of a firms total
  cost function.
• A variable cost curve is the graph of a firms
  variable cost function.
• An average total cost curve is the graph of a
  firms average total cost function.

3
Types of Cost Curves

• An average variable cost curve is the graph of a
  firms average variable cost function.
• An average fixed cost curve is the graph of a
  firms average fixed cost function.
• A marginal cost curve is the graph of a firms
  marginal cost function.

4
Types of Cost Curves

• How are these cost curves related to each other?
• How are a firms long-run and short-run cost
  curves related?

5
Fixed, Variable Total Cost Functions

• F is the total cost to a firm of its short-run
  fixed inputs. F, the firms fixed cost, does not
  vary with the firms output level.
• cv(q) is the total cost to a firm of its variable
  inputs when producing y output units. cv(q) is
  the firms variable cost function.

6
Fixed, Variable Total Cost Functions

• c(q) is the total cost of all inputs, fixed and
  variable, when producing y output units. c(q) is
  the firms total cost function

7

F
q
8
cv(q)

F
q
9
c(q)
cv(q)

F
F
q
10
Av. Fixed, Av. Variable Av. Total Cost Curves

• The firms total cost function is
• For q gt 0, the firms average total cost function
  is

11
Av. Fixed, Av. Variable Av. Total Cost Curves

• What does an average fixed cost curve look
  like?
• AFC(q) is a rectangular hyperbola so its graph
  looks like ...

12
/output unit
AFC(q) 0 as q
AFC(q)
q
0
13
Av. Fixed, Av. Variable Av. Total Cost Curves

• In a short-run with a fixed amount of at least
  one input, the Law of Diminishing (Marginal)
  Returns must apply, causing the firms average
  variable cost of production to increase
  eventually.

14
/output unit
AVC(q)
q
0
15
/output unit
AVC(q)
AFC(q)
q
0
16
Av. Fixed, Av. Variable Av. Total Cost Curves

• And ATC(q) AFC(q) AVC(q)

17
/output unit
ATC(q) AFC(q) AVC(q)
ATC(q)
AVC(q)
AFC(q)
q
0
18
/output unit
AFC(q) ATC(q) - AVC(q)
ATC(q)
AVC(q)
AFC
AFC(q)
q
0
19
Marginal Cost Function

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

20
Marginal Average Cost Functions

• How is marginal cost related to average variable
  cost?

21
/output unit
The short-run MC curve intersectsthe short-run
AVC curve frombelow at the AVC curves
minimum.
MC(q)
AVC(q)
q
22
Marginal Average Cost Functions

• The short-run MC curve intersects the short-run
  AVC curve from below at the AVC curves minimum.
• And, similarly, the short-run MC curve intersects
  the short-run ATC curve from below at the ATC
  curves minimum.

23
/output unit
MC(q)
ATC(q)
AVC(q)
q
24
For example
25
Short-Run Long-Run Total Cost Curves

• A firm has a different short-run total cost curve
  for each possible short-run circumstance.
• Suppose the firm can be in one of just three
  short-runs x2 x2 or x2 x2
  x2 lt x2 lt x2. or x2 x2.

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cs(qx2)
F w2x2
F
q
0
27

cs(qx2)
F w2x2
F w2x2
cs(qx2)
F
F
q
0
28

cs(qx2)
F w2x2
F w2x2
cs(qx2)
F
F
q
0
29
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
units of input 1.
30
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
units of input 1. Each unit of input
1 costs w1, so the firms extra cost from
producing one extra unit of output is
31
Short-Run Long-Run Total Cost Curves
is the slope of the firms total cost curve.
32
Short-Run Long-Run Total Cost Curves
is the slope of the firms total cost curve.
If input 2 is a complement to input 1 thenMP1 is
higher for higher x2. Hence, MC is lower for
higher x2.
That is, a short-run total cost curve
startshigher and has a lower slope if x2 is
larger.
33

cs(qx2)
F w2x2
F w2x2
F w2x2
cs(qx2)
cs(qx2)
F
F
F
q
0
34
Short-Run Long-Run Total Cost Curves

• The firm has three short-run total cost curves.
• In 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.
• How does the firm make this choice?

35

For 0 q q, choose x2 ?
cs(qx2)
cs(qx2)
cs(qx2)
F
F
F
q
q
q
0
36

For 0 q q, choose x2 x2.
cs(qx2)
cs(qx2)
cs(qx2)
F
F
F
q
q
q
0
37

For 0 q q, choose x2 x2.
cs(qx2)
For q q q, choose x2 ?
cs(qx2)
cs(qx2)
F
F
F
q
q
0
q
38

For 0 q q, choose x2 x2.
cs(qx2)
For q q q, choose x2 x2.
cs(qx2)
cs(qx2)
F
F
F
q
q
q
0
39

For 0 q q, choose x2 x2.
cs(qx2)
For q q q, choose x2 x2.
For q lt q, choose x2 ?
cs(qx2)
cs(qx2)
F
F
F
q
q
q
0
40

For 0 q q, choose x2 x2.
cs(qx2)
For q q q, choose x2 x2.
For q lt q, choose x2 x2.
cs(qx2)
cs(qx2)
F
F
F
q
q
q
0
41

For 0 q q, choose x2 x2.
cs(qx2)
For q q q, choose x2 x2.
For q lt q, choose x2 x2.
cs(qx2)
cs(qx2)
c(q), thefirms long-run totalcost curve.
F
F
F
q
q
q
0
42
Short-Run Long-Run Total Cost Curves

• The firms 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.

43
Short-Run Long-Run Average Total Cost Curves

• For any output level y, the long-run total cost
  curve always gives the lowest possible total
  production cost.
• Therefore, the long-run av. total cost curve must
  always give the lowest possible av. total
  production cost.
• The long-run av. total cost curve must be the
  lower envelope of all of the firms short-run av.
  total cost curves.

44
Short-Run Long-Run Average Total Cost Curves

• E.g. suppose again that the firm can be in one of
  just three short-runs x2 x2 or x2
  x2 (x2 lt x2 lt x2) or x2
  x2then the firms three short-run average
  total cost curves are ...

45
/output unit
ACs(qx2)
ACs(qx2)
ACs(qx2)
q
46
Short-Run Long-Run Average Total Cost Curves

• The firms long-run average total cost curve is
  the lower envelope of the short-run average total
  cost curves ...

47
/output unit
ACs(qx2)
ACs(qx2)
ACs(qx2)
The long-run av. total costcurve is the lower
envelopeof the short-run av. total cost curves.
AC(q)
q
48
Short-Run Long-Run Marginal Cost Curves

• Q Is the long-run marginal cost curve the lower
  envelope of the firms short-run marginal cost
  curves?

49
Short-Run Long-Run Marginal Cost Curves

• Q Is the long-run marginal cost curve the lower
  envelope of the firms short-run marginal cost
  curves?
• A No.

50
/output unit
MCs(qx2)
MCs(qx2)
ACs(qx2)
ACs(qx2)
ACs(qx2)
MCs(qx2)
q
51
/output unit
MCs(qx2)
MCs(qx2)
ACs(qx2)
ACs(qx2)
ACs(qx2)
AC(q)
MCs(qx2)
q
52
/output unit
MCs(qx2)
MCs(qx2)
ACs(qx2)
ACs(qx2)
ACs(qx2)
MCs(qx2)
MC(q), the long-run marginalcost curve.
q
53
Short-Run Long-Run Marginal Cost Curves

• For any output level q gt 0, the long-run marginal
  cost is the marginal cost for the short-run
  chosen by the 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 ...

54
Short-Run Long-Run Marginal Cost Curves
/output unit
SRACs
AC(q)
q
55
Short-Run Long-Run Marginal Cost Curves
/output unit
SRMCs
AC(q)
q
56
Short-Run Long-Run Marginal Cost Curves
/output unit
MC(q)
SRMCs
AC(q)
q

• For each q gt 0, the long-run MC equals theMC for
  the short-run chosen by the firm.