Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Econ 401 Price Theory
Chapter 21: Cost Curves
Instructor: Hiroki Watanabe
Summer 2009
1 / 53
Intro
Duality
1
2
3
4
5
6
7
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Introduction
Question
Duality
Duality between Profit Maximization Problem &
Cost Minimization Problem
Example of Duality
Total Cost in Terms of y
Cost Curves
Variable Cost & Fixed Cost
Marginal Cost
Average Cost & Marginal Cost
Average Variable Cost & Marginal Cost
STC & LTC
Short-Run Total Cost under Different Fixed Inputs
Long-Run Total Cost Is Smaller Than Short-Run
Total Cost
SATC & LATC
Short-Run Marginal Cost & Long-Run Marginal Cost
Summary
2 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Question
It might be easier to discuss Kayak’s production
cost in terms of y (one variable) rather than in
terms of (xC , xK ) (two variables).
Q: Is cost function in terms of (xC , xK ) different from
a cost function in terms of y?
Luckily, the answer is no.
3 / 53
Intro
Duality
1
2
3
4
5
6
7
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Introduction
Question
Duality
Duality between Profit Maximization Problem &
Cost Minimization Problem
Example of Duality
Total Cost in Terms of y
Cost Curves
Variable Cost & Fixed Cost
Marginal Cost
Average Cost & Marginal Cost
Average Variable Cost & Marginal Cost
STC & LTC
Short-Run Total Cost under Different Fixed Inputs
Long-Run Total Cost Is Smaller Than Short-Run
Total Cost
SATC & LATC
Short-Run Marginal Cost & Long-Run Marginal Cost
Summary
4 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Duality between Profit Maximization Problem & Cost Minimization Problem
Q: How does profit maximization problem differ
from cost minimization problem?
Fact: Duality
Let f (x) be decreasing returns–to-scale technology.
Suppose optimal production plan to
PMP: max pf (xC , xK ) − wC xC − wK xK
xC ,xK
is (xC∗ , xK∗ , y∗ ). Then the optimal bundle to
CMP: min wC xC + wK xK
xC ,xK
s.t. f (xC , xK ) = y∗
is (xC∗ , xK∗ ).
5 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Duality between Profit Maximization Problem & Cost Minimization Problem
If you ask Kayak’s to maximize their profit, they
choose y∗ with the input bundle (xC∗ , xK∗ ).
If instead you ask Kayak’s produce y∗ at the
smallest cost possible, they choose (xC∗ , xK∗ ) as well.
∗
ƒ ()
PMP
y∗

ƒ ()
CMP
∗ (y ∗ )

6 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Example of Duality
Duality
1
Profit Maximization Problem:
∗ , x∗ , y∗ ) for
Find the optimal production plan (xC
K
Kayak’s with production function
p
p
f (xC , xK ) = xC + xK , w = (1, 1) and p = 4. Marginal
products are MPC (xC , xK ) = 2 p1x and
1
MPK (xC , xK ) =
Compute the cost to produce y∗ .
2
2
C
1
p .
2 xK
Cost Minimization Problem:
0 , x0 ) to
Find the cost minimizing input bundle (xC
K
produce y∗ . Technical rate ofp substitution is
−MP (x ,x )
− x
TRS(xC , xK ) = MP C(x C,x K) = px K .
1
K
C
K
C
Compute the minimized cost to produce y∗ .
2
7 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Example of Duality
Isoquant
10
√ 6 √
y = xC + xK
xC + xK = 8
9
5
8
6
5
5
3
Size of Kitchen (xK)
4
7
4
4
5
3
2
1
4
3
2
4
3
0
0
1
1
2
2
3
4
5
6
Chefs (xC)
3
7
8
9
10
8 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Example of Duality
1
Profit Maximization Problem:
Write the profit maximization problem:
p
p
max π(x) = 4( xC + xK ) − xC − xK .
1
xC ,xK
Tangency conditions:
2
1
1
= ,
p
2 xC
4
1
1
=
.
p
2 xK
4
∗ , x∗ ) = (4, 4).
Optimal input bundle (xC
K
∗
∗ , x∗ ) = 4.
Optimal output level y = f (xC
K
∗
∗
Cost associated with x is TC(xC = 4, xK∗ = 4) = 8.
To produce y = 4, it costs at least $8. So,
TC(y = 4) = 8.
3
4
5
6
9 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Example of Duality
Iso−Cost
10
12
Size of Kitchen (xK)
16T
√C
= x√
C + xK
xC + xK = 4
10
9
8
14
18
8
12
7
6
14
16
10
6
8
5
12
14
10
4
4
6
3
8
2
10
2
1
4
6
0
0
1
2
3
4
5
6
Chefs (xC)
8
7
8
9
10
10 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Example of Duality
2
Cost Minimization Problem:
Write the profit maximization problem:
p
p
min xC + xK s.t.
xC + xK = 4.
1
xC ,xK
Tangency conditions:
p
− xK
−1
=
⇒ xC = xK .
p
xC
1
2
Constraint:
3
p
xC +
p
∗
xC = 8 ⇒ xC
= 4.
0 (y = 4), x0 (y = 4)) = (4, 4).
Optimal input bundle (xC
K
x0 coincides with x∗ .
Cost associated with x0 is
4
5
6
0
TC(xC
(y = 4) = 4, xK0 (y = 4) = 4) = 8.
To produce y = 4, it costs at least $8. Instead of (1),
you can write
TC(y = 4) = 8.
7
Intro
(1)
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
11 / 53
Σ
Total Cost in Terms of y
Minimized cost from CMP is
TC(y) = wC xC∗ (y) + wK xK∗ (y).
For each y, TC(y) automatically computes the
minimum total cost of production for Kayak’s.
So, Kayak’s (or we) does not have to worry about
the combination of (xC , xK ) once they have TC(y)
handy.
Thanks to duality, the following yields the same
answer:
Profit Maximization Problem (Ch18)
max π(xC , xK ) = pf (xC , xK ) − wC xC − wK xC .
xC ,xK
Profit Maximization Problem in Terms of y
max π(y) = py − TC(y).
y
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Intro
Duality
1
2
3
4
5
6
7
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Introduction
Question
Duality
Duality between Profit Maximization Problem &
Cost Minimization Problem
Example of Duality
Total Cost in Terms of y
Cost Curves
Variable Cost & Fixed Cost
Marginal Cost
Average Cost & Marginal Cost
Average Variable Cost & Marginal Cost
STC & LTC
Short-Run Total Cost under Different Fixed Inputs
Long-Run Total Cost Is Smaller Than Short-Run
Total Cost
SATC & LATC
Short-Run Marginal Cost & Long-Run Marginal Cost
Summary
13 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Variable Cost & Fixed Cost
Total cost has two components.
Short-Run Cost Minimization Problem
min wC xC + wK x̄K
xC
s.t. f (xC , x̄K ) = y.
Cost-minimizing input bundle (xC∗ , x̄K ) = (xC∗ (y), x̄K )
ƒ ()
y
CMP

̄K
∗
(y)
C
14 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Variable Cost & Fixed Cost
Rent for kitchen space cannot be changed on a
daily basis:
TC(y) = wC xC∗ (y) + wK x̄K .
Kayak’s has to pay wK x̄K even when y = 0.
Write
TC(y) = VC(y) + FC.
Fixed cost can be found by plugging y = 0:
TC(y = 0) = 0 + FC.
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Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Marginal Cost
Marginal cost denotes the additional cost required to
produce one more slice of cheesecake:
MC(y) =
∆TC(y)
∆y
.
16 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Total Cost & Margianl Cost
8
7
Total Cost TC(y)
Margianl Cost MC(y)
6
$
5
4
3
2
1
0
0
Intro
Duality
1
2
3
4
5
Cheesecakes (y)
Cost Curves
STC & LTC
6
7
8
17 / 53
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Recall marginal product is an additional cheesecake
produced by hiring one more chef.
Then MP (x1 ,x̄ ) is additional number of chefs to
C C K
produce one more slice of cheesecake.
Since it cost wC to hire one chef,
wC
1
MPC (xC , x̄K )
= MC(y).
Confirm:
MPC (xC , x̄K ) =
∆y
∆xC
∆TC(y) ∆wC xC
∆xC
MC(y) =
=
=w
∆y
∆y
∆y
18 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Diminishing marginal product: additional chef
contributes less as Kayak’s increases the number of
chefs.
Diminishing marginal product implies increasing
marginal cost:
MC(y) = wC
1
MPC (xC∗ (y), x̄K )
.
(2)
(You don’t have to remember (2). Just know the
relationship, i.e., they’re inversely related.)
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Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Production Function and Marginal Product
√
Production Function f(xC ) = xC
Marginal Product MPC (xC ) = 2√1xC
Cheesecakes (y)
3
2
1
0
0
1
2
3
4
5
6
Chefs (xC)
7
8
9
10
20 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Margianl Cost
100
Margianl Cost MC(y)=w y
90
2
C
80
70
$
60
50
40
30
20
10
0
0
Intro
Duality
1
2
3
Cost Curves
4
5
6
7
Cheesecakes (y)
STC & LTC
8
9
10
21 / 53
SATC & LATC
SMC & LMC
Σ
Marginal Cost
On the contrary, increasing marginal product
implies diminishing marginal cost:
MC(y) = wC
1
MPC (xC∗ (y), x̄K )
.
22 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Production Function and Marginal Product
100
Production Function f(x ) = x2
90
C
Marginal Product MP (x )=2x
C
80
Cheesecakes (y)
C
C
C
70
60
50
40
30
20
10
0
0
Intro
1
Duality
2
3
Cost Curves
4
5
6
Chefs (xC)
7
8
9
10
23 / 53
STC & LTC
SATC & LATC
SMC & LMC
Σ
Marginal Cost
Margianl Cost
10
9
8
7
2
$
6
Margianl Cost MC(y)=w /(2y )
C
5
4
3
2
1
0
0
1
2
3
4
5
6
7
Cheesecakes (y)
8
9
10
24 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Average Cost & Marginal Cost
Average Cost & Marginal Cost
Average cost takes its smallest value when
AC(yBE ) = MC(yBE ).
Superscript
BE
stands for break even (c.f. Ch23).
25 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Average Cost & Marginal Cost
1
If AC(y) > MC(y):
increasing y costs Kayak’s only MC(y), which is less
than the ongoing unit production cost. AC(y) goes
down if Kayak’s expands its production.
If your current GPA is B and receive A, your GPA
goes up.
2
If AC(y) < MC(y):
increasing y costs Kayak’s MC(y), which is more
than the current unit production cost.
An additional cheesecake raises average cost than
before.
If your current GPA is B and receive C, your GPA
goes down.
3
If AC(y) = MC(y):
Unit production cost becomes smallest when it is
the same as the cost of producing an additional
cheesecake.
If your current GPA is B and receive B, your GPA
doesn’t change.
26 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Average Cost & Marginal Cost
Cost Functions
8
Total Cost TC(y)
Margianl Cost MC(y)
Average Total Cost ATC(y)
7
6
$
5
4
3
2
1
0
0
Intro
Duality
1
2
3
4
5
6
7
Cheesecakes (y)
Cost Curves
STC & LTC
8
9
10
27 / 53
SATC & LATC
SMC & LMC
Σ
Average Variable Cost & Marginal Cost
Average Variable Cost
Average variable cost is unit production cost associated
with variable component of total cost:
AVC(y) =
VC(y)
y
=
TC(y) − FC
y
.
Average Variable Cost & Marginal Cost
Average cost takes its smallest value when
AVC(ySD ) = MC(ySD ).
Superscript
SD
stands for shut down (c.f. Ch22).
28 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Average Variable Cost & Marginal Cost
1
If AVC(y) > MC(y):
increasing y costs Kayak’s only MC(y), which is less
than the ongoing unit production cost without FC.
AVC(y) goes down if Kayak’s expands its production.
If your current GPA is B and receive A, your GPA
goes up.
2
If AVC(y) < MC(y):
increasing y costs Kayak’s MC(y), which is more
than the current unit production cost without FC.
An additional cheesecake raises average cost than
before.
If your current GPA is B and receive C, your GPA
goes down.
3
If AVC(y) = MC(y):
Unit production cost (less FC) becomes smallest
when it is the same as the cost of producing an
additional cheesecake.
If your current GPA is B and receive B, your GPA
doesn’t change.
29 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Average Variable Cost & Marginal Cost
Cost Functions
8
Total Cost TC(y)
Margianl Cost MC(y)
Average Variable Cost ATC(y)
7
6
$
5
4
3
2
1
0
0
1
2
3
4
5
6
7
Cheesecakes (y)
8
9
10
30 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Average Variable Cost & Marginal Cost
AC(yBE ) = MC(yBE ) at the bottom of AC(y).
AVC(ySD ) = MC(ySD ) at the bottom of AVC(y).
1
2
Cost Functions
8
Total Cost TC(y)
Margianl Cost MC(y)
Average Cost AC(y)
Average Variable Cost ATC(y)
7
6
$
5
4
3
2
1
0
0
1
2
3
4
5
6
7
Cheesecakes (y)
8
9
10
31 / 53
Intro
Duality
1
2
3
4
5
6
7
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Introduction
Question
Duality
Duality between Profit Maximization Problem &
Cost Minimization Problem
Example of Duality
Total Cost in Terms of y
Cost Curves
Variable Cost & Fixed Cost
Marginal Cost
Average Cost & Marginal Cost
Average Variable Cost & Marginal Cost
STC & LTC
Short-Run Total Cost under Different Fixed Inputs
Long-Run Total Cost Is Smaller Than Short-Run
Total Cost
SATC & LATC
Short-Run Marginal Cost & Long-Run Marginal Cost
Summary
32 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Short-Run Total Cost under Different Fixed Inputs
A firm has a different shot-run total cost curve for
each possible short-run circumstances.
Recall
STC(y) = wC xC∗ (y) + wK x̄K .
Suppose the firm can be in one of just three
short-run environments:
x̄K0 < x̄K00 < x̄K00 .
33 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Short-Run Total Cost under Different Fixed Inputs
34 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Short-Run Total Cost under Different Fixed Inputs
If Kayak’s has a small kitchen, their initial
investment is small (small xK and small FC), but
then their kitchen will get crowded early on.
⇒ small FC, rapid short-run total cost increase, high
MC(y).
If Kayak’s has a large kitchen, their initial
investment is large (large xK and large FC), but
their kitchen will not get crowded early on.
⇒ large FC, slow short-run total cost increase, low
MC(y).
35 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Long-Run Total Cost Is Smaller Than Short-Run Total Cost
Kayak’s has three short-run total cost curves.
1
2
3
Start with a small kitchen x̄K0 .
Start with a medium size kitchen x̄K00 .
Start with a large kitchen x̄K000 .
In the long-run, Kayak’s is free to choose among
three short-run total cost curves.
I.e., they can choose any of x̄K0 < x̄K00 < x̄K00 .
What does their long-run total cost look like?
36 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Long-Run Total Cost Is Smaller Than Short-Run Total Cost
37 / 53
Intro
Duality
Cost Curves
STC & LTC
Figure:
SATC & LATC
SMC & LMC
Σ
Long-Run Total Cost Is Smaller Than Short-Run Total Cost
Kayak’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 cost curves.
If the kitchen size comes not just in three sizes but
in any size...
38 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Long-Run Total Cost Is Smaller Than Short-Run Total Cost
Short−Run Total Cost under Different xK
3
25
1. .5
1
Size of Kitchen (x )
Short−Run Total Cost STC(y)
K
1.7
5
2
1.75
1.75
1.5
1
1.5
1.25
1
1.25
0.75
0.75
5
0.
1.2.55
01.251
5
0.7
1
0.5
0.5
0.25
0.25
0
0
Intro
Duality
1
2
3
4
5
6
7
1
Cost Curves
2
Cheesecakes (y)
STC & LTC
3
4
39 / 53
SATC & LATC
SMC & LMC
Σ
Introduction
Question
Duality
Duality between Profit Maximization Problem &
Cost Minimization Problem
Example of Duality
Total Cost in Terms of y
Cost Curves
Variable Cost & Fixed Cost
Marginal Cost
Average Cost & Marginal Cost
Average Variable Cost & Marginal Cost
STC & LTC
Short-Run Total Cost under Different Fixed Inputs
Long-Run Total Cost Is Smaller Than Short-Run
Total Cost
SATC & LATC
Short-Run Marginal Cost & Long-Run Marginal Cost
Summary
40 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
For any output y, the long-run total cost curve
always gives the lowest possible total production
cost.
Therefore, the long-run average total cost curve
must always give the lowest possible average total
production cost.
The long-run average cost curve must be the lower
envelope of all of the firm’s short-run average total
cost curves.
41 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
42 / 53
Intro
Duality
1
2
3
4
5
6
7
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Introduction
Question
Duality
Duality between Profit Maximization Problem &
Cost Minimization Problem
Example of Duality
Total Cost in Terms of y
Cost Curves
Variable Cost & Fixed Cost
Marginal Cost
Average Cost & Marginal Cost
Average Variable Cost & Marginal Cost
STC & LTC
Short-Run Total Cost under Different Fixed Inputs
Long-Run Total Cost Is Smaller Than Short-Run
Total Cost
SATC & LATC
Short-Run Marginal Cost & Long-Run Marginal Cost
Summary
43 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Q: Is the long-run marginal cost curve the lower
envelope of the firm’s short-run marginal cost
curves?
A: Sadly, no.
44 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
45 / 53
Intro
Duality
Cost Curves
STC & LTC
Figure:
SATC & LATC
SMC & LMC
Σ
The long-run marginal cost is the marginal cost for
the short-run environment chosen by Kayak’s.
This is always true, no matter how many and which
chort-run circumstances exist for Kayak’s.
46 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
47 / 53
Intro
Duality
Cost Curves
STC & LTC
Figure:
SATC & LATC
SMC & LMC
Σ
If the kitchen size x̄K can take any number,
48 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
49 / 53
Intro
Duality
Cost Curves
STC & LTC
Figure:
SATC & LATC
SMC & LMC
Σ
50 / 53
Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Figure:
Intro
Duality
1
2
3
4
5
6
7
Cost Curves
STC & LTC
Σ
51 / 53
SATC & LATC
SMC & LMC
Σ
Introduction
Question
Duality
Duality between Profit Maximization Problem &
Cost Minimization Problem
Example of Duality
Total Cost in Terms of y
Cost Curves
Variable Cost & Fixed Cost
Marginal Cost
Average Cost & Marginal Cost
Average Variable Cost & Marginal Cost
STC & LTC
Short-Run Total Cost under Different Fixed Inputs
Long-Run Total Cost Is Smaller Than Short-Run
Total Cost
SATC & LATC
Short-Run Marginal Cost & Long-Run Marginal Cost
Summary
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Intro
Duality
Cost Curves
STC & LTC
SATC & LATC
SMC & LMC
Σ
Duality.
Average cost and its relation to marginal cost in
short-run and long-run environments.
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