Transcript Econ 281 Chapter08

Chapter 8: Cost Curves
•A firm aims to MAXIMIZE PROFITS
•In order to do this, one must understand
how to MINIMIZE COSTS
•Therefore understanding of cost curves is
essential to maximizing profits
1
Chapter 8: Costs Curves
In this chapter we will cover:
8.1 Long Run Cost Curves
8.1.1 Total Cost
8.1.2 Marginal Cost and Average Cost
8.2 Economies of Scale
8.3 Short Run Cost Curves
8.3.1 Total Cost, Variable Cost, Fixed Cost
8.3.2 Marginal Cost and Average Cost
8.4 Economies of Scope
8.5 Economies of Experience
2
8.1 Long Run Cost Curves
•In the long run, a firm’s costs equal zero
when zero production is undertaken
•As production (Q) increases, the firm must
use more inputs, thus increasing its cost
•By minimizing costs, a firm’s typical long
run cost curve is as follows:
3
K
Q1
Q0
K1
K0
0
TC ($/yr)
•
•
L0 L1
TC = TC0
TC = TC1
L (labor services per year)
LR Total Cost Curve
TC1=wL1+rK1
TC0 =wL0+rK0
0
Q0
Q1 Q (units per year)
4
•An increase in the price of only 1 input will
cause a firm to change its optimal choice of
inputs
•However, the increase in input costs will always
cause a firm’s costs to increase:
-(Unless inputs are perfect substitutes)
5
K
TC1/r
TC0/r
C1: Original isocost curve
(TC
=
$200)
Slope=w1/r
C2: Isocost curve after
Price change (TC = $200)
C3: Isocost curve after
A
•
Price change (TC = $300)
•
B
C2
0
Q0
C1
Slope=w2/r
C3
L
6
TC ($/yr)
Change in Input Prices ->
A Shift in the Total Cost Curve
TC(Q) new
TC(Q) old
300
200
Q0
Q (units/yr)
7
Let Q=2(LK)1/2 MRTS=K/L,
W=5, R=20, Q=40
What occurs to costs when rent falls to 5?
Initially:
1/2
Q=2(LK)
MRTS=W/R
1/2
40=2(4KK)
K/L=5/20
40=4K
4K=L
10=K
40=L
8
Let Q=2(LK)1/2 MRTS=K/L,
W=5, R=20, Q=40
What occurs to costs when rent falls to 5?
After Price Change:
1/2
Q=2(LK)
MRTS=W/R
1/2
40=2(LL)
K/L=5/5
40=2L
L=K
20=L
20=K
9
What occurs when rent falls to 5?
Initial: L=40, K=10 Final: W=5, R=20
Initial:
TC=wL+rK
TC=5(40)+20(10)
TC=400
Final:
TC=5(20)+5(20)
TC=200
Due to the fall in rent, total cost falls by $200.
10
TC ($/yr)
Change in Rent
TC(Q) initial
TC(Q) final
400
200
40
Q (units/yr)
11
To calculate total cost, simply substitute labour
and capital demand into your cost expression:
Q= 50L1/2K1/2 (From Chapter 7:)
L*(Q,w,r) = (Q0/50)(r/w)1/2
K*(Q,w,r) = (Q0/50)(w/r)1/2
TC = wL +rK
TC= w [(Q0/50)(r/w)1/2 ] +r[(Q0/50)(w/r)1/2 ]
TC= [(Q0/50)(wr)1/2 ] +[(Q0/50)(wr)1/2 ]
TC = 2Q0(wr)1/2 /50
12
Let Q= L1/2K1/2, MPL/MPK=K/L, w=10, r=40.
Calculate total cost.
MRTS=w/r
K/L=10/40
K=4L
Q=L1/2K1/2 =L1/2(4L)1/2
Q=2L
L=Q/2
13
Let Q= L1/2K1/2, MRTS=K/L, w=10, r=40.
Calculate total cost.
K=4L
L=K/4
L=Q/2
Q=L1/2K1/2
Q=(K/4)1/2K1/2
Q=1/2K
K=2Q
TC = wL +rK
TC = 10(Q/2) +40(2Q)
TC=85Q
14
•When the prices of all inputs change by the
same (percentage) amount, the optimal input
combination does not change
•The same combination of inputs are purchased
at higher prices
15
K (capital services/yr)
C1=Isocost curve
before ($200)
and after ($220)
a 10% increase
in input prices
•
A
Q0
C1
0
L (labor
16
services/yr)
TC ($/yr)
Example: A Shift in the Total Cost Curve
When Input Prices Rise 10%
TC(Q) new
TC(Q) old
220
200
Q0
Q (units/yr)
17
Definition: The long run average cost function is
the long run total cost function divided by
output, Q.
That is, the LRAC function tells us the firm’s cost
per unit of output…
TC (Q)
AC (Q) 
Q
18
Definition: The long run marginal cost function
is rate at which long run total cost changes with
a change in output
The (LR)MC curve is equal to the slope of the
(LR)TC curve
TC (Q)
MC (Q) 
Q
19
TC ($/yr)
Average vrs. Marginal Costs
Slope=LRMC
TC(Q) post
TC0
Slope=LRAC
Q0
Q (units/yr)
20
When marginal cost is less than average cost,
average cost is decreasing in quantity. That is, if
MC(Q) < AC(Q), AC(Q) decreases in Q.
When marginal cost is greater than average cost,
average cost is increasing in quantity. That is, if
MC(Q) > AC(Q), AC(Q) increases in Q.
When marginal cost equals average cost, average cost
is at its minimum. That is, if MC(Q) = AC(Q), AC(Q)
21
is at its minimum.
AC, MC ($/yr)
“typical” shape of AC, MC
MC
AC
•
AC at minimum when AC(Q)=MC(Q)
0
Q (units/yr)
22
If average cost decreases as output rises, all
else equal, the cost function exhibits
economies of scale.
-large scale operations have an advantage
If average cost increases as output rises, all
else equal, the cost function exhibits
diseconomies of scale.
-small scale operations have an advantage
23
Why Economies of scale?
-Increasing Returns to Scale for Inputs
-Specialization of Labour
-Indivisible Inputs (ie: one factory can produce
up to 1000 units, so increasing output up to
1000 decreases average costs for the factory)
24
Why Diseconomies of scale?
-Diminishing Returns from Inputs
-Managerial Diseconomies
-Growing in size requires a large
expenditure on managers
-ie: One genius cannot run more than 1
branch
25
AC ($/yr)
Typical Economies of Scale
Minimum Efficient Scale – smallest
AC(Q)
Quantity where LRAC curve reaches
Its min.
Economies of scale
0
Diseconomies of scale
Q*
Q (units/yr)
26
Production functions and cost functions are related:
Production Function
Cost Function
Increasing returns to
scale
Decreasing returns to
scale
Constant Returns to
Scale
Economies of Scale
Diseconomies of Scale
Neither economies nor
diseconomies of scale
27
Example: Returns to Scale and Economies of Scale
IRS
Q = L2
DRS
Q = L1/2
L*=Q1/2
L*=Q2
wQ1/2
wQ2
Average Cost Function AC=w
w/Q1/2
wQ
Economies of Scale
EOS
DOS
Production Function
Labor Demand
Total Cost Function
CRS
Q=L
L*=Q
TC=wQ
none
28
•Economies of Scale can be measured using
output elasticity of total cost; how cost changes
when output changes
%TC
TC ,Q 
%Output
TC Q
TC ,Q 
Q TC
29
•Economies of Scale are also related to
marginal cost and average cost:
TC TC
TC ,Q 
/
Q Q
TC ,Q  MC / AC
30
If TC,Q < 1, MC < AC, so AC must be
decreasing in Q. Therefore, we have
economies of scale.
If TC,Q > 1, MC > AC, so AC must be
increasing in Q. Therefore, we have
diseconomies of scale.
If TC,Q = 1, MC = AC, so AC is just flat
with respect to Q.
31
Let Cost=50+20Q2
MC=40Q
IF Q=1 or Q=2, determine economies of scale
(Let Q be thousands of units)
32
TC=50+20Q2
MC=40Q
AC=TC/Q=50/Q+20Q
Initially: MC=40(1)=40
AC=50/1+20(1)=70
Elasticity=MC/AC=40/70 – Economies of Scale
Finally: MC=40(2)=80
AC=50/2+20(2)=65
E=MC/AC=80/65 – Diseconomies of Scale
33
8.3 Short-Run Cost Curves
•In the short run, at least 1 input is fixed
(ie: (K=K*)
•Total fixed costs (TFC) are the costs
associated with this fixed input (ie: rk)
•Total variable costs (TVC) are the costs
associated with variable inputs (ie:wL)
•Short-run total costs are fixed costs plus
variable costs: STC=TFC+TVC
34
TC ($/yr)
Short Run Total Cost, Total Variable
Cost and Total Fixed Cost
STC(Q, K*)
rK*
TVC(Q, K*)
TFC
rK*
Q (units/yr)
35
Short Run Costs
Example:
Minimize the cost to build 80 units if Q=2(KL)1/2 and
K=25. If r=10 and w=20, classify costs.
Q=2(KL)1/2
80=2(25L)1/2
80=10(L)1/2
8=(L)1/2
64=L
36
Short Run Costs
Example:
K*=25, L=16. If r=10 and w=20, classify costs.
TFC=rK*=10(25)=250
TVC=wL=20(64)=1280
STC=TFC+TVC=1530
37
The firm can minimize costs better in
the long run because it is less
constrained.
Hence, the short run total cost curve
lies above the long run total cost
curve almost everywhere.
38
K
Only at point A is short run
minimized as well as long run
TC2/r
TC1/r
Q1
Long Run Expansion path
•
TC0/r
C
Q0
K*
0
•
A
Q0
B
•
Short Run
Expansion path
TC0/w TC1/w TC2/w
L
39
TC ($/yr)
STC(Q)
LRTC(Q)
A
•
rK*
Q (units/yr)
40
Definition: The short run average cost function
is the short run total cost function divided by
output, Q.
That is, the SAC function tells us the firm’s cost
per unit of output…
STC (Q)
SAC (Q) 
Q
41
Definition: The short run marginal cost function
is rate at which short run total cost changes with
a change in input
The SMC curve is equal to the slope of the STC
curve
STC (Q)
SMC (Q) 
Q
42
In the short run, 2 additional average costs exist:
average variable costs (AVC) and average fixed
costs (AFC)
TFC (Q)
AFC (Q) 
Q
TVC (Q)
AVC (Q) 
Q
43
Note :
STC  TFC  TVC
STC TFC TVC


Q
Q
Q
Therefore :
SAC  AFC  AVC
44
To make an omelet, one must crack a fixed number of
eggs (E) and add a variable number of other ingredients
(O). Total costs for 10 omelets were $50. Each omelet’s
average variable costs were $1.50. If eggs cost 50 cents,
how many eggs in each omelet?
AC=AVC+AFC
TC/Q=AVC+AFC
50/10=$1.50+AFC
$3.50=AFC
45
To make an omelet, one must crack a fixed number of
eggs (E) and add a variable number of other ingredients
(O). Total costs for 10 omelets were $50. Each omelet’s
average variable costs were $1.50. If eggs cost 50 cents,
how many eggs in each omelet?
$3.50=AFC
$3.50=PE (E/Q)
$3.50=0.5 (E/Q)
7=E/Q
There were 7 eggs in each omelet.
46
$ Per Unit
Average fixed cost is
constantly decreasing, as
fixed costs don’t rise with
output.
AFC
0
Q (units per
year) 47
$ Per Unit
Average variable
cost generally
decreases then
increases due to
economies of
scale.
AVC
AFC
0
Q (units per
year) 48
$ Per Unit
SAC is the
vertical sum
of AVC and
AFC
SAC
AVC
Equal
AFC
0
Q (units per
year) 49
$ Per Unit
SAC
SMC
AVC
•
•
SMC
intersects
SAC and
AVC at their
minimum
points
AFC
0
Q (units per
year) 50
In the long run, a firm can adjust its capital to a
level that is then fixed in the short run.
The long run average cost curve (LRAC) therefore
forms an “envelope” or boundary around the
various short run average cost curves (SAC)
corresponding to different capital levels.
51
$ per unit
SAC(Q,K3)
SAC(Q,K1)
SAC(Q,K2)
AC(Q)
•
0
•
•
Q1
Q2
Q3
Q (units per
year) 52
When a firm minimizes cost in the short run,
given capital chosen in the long run,
-AC=SAC (Point A, next slide)
-MC=SMC (Point B, next slide)
-SAC is not at its min (in general) (Point C,
next slide)
When a firm minimizes cost in the long run,
-AC=SAC=MC=SMC and SAC is at a
minimum (two slides hence)
53
$ per unit
MC(Q)
SAC(Q,K1)
AC(Q)
SMC(Q,K1)
A C
••
B
•
0
Q1
Q2
Q3
Q (units per
year) 54
$ per unit
Example: Putting It All
Together
MC(Q)
SAC(Q,K2)
AC(Q)
SMC(Q,K2)
•D
0
Q1
Q2
Q3
Q (units per
year) 55
Often a firm produces more than one product,
and often these products are related:
-Pepsi Cola makes Pepsi and Diet Pepsi
-HP makes Computers and Cameras
-Denny’s Serves Breakfast and Dinner
Often a firm benefits from economies of scope by
producing goods that are related; they share
common inputs (or good A is an input for good
B). Efficiencies often exist in producing related
56
products (ie: no shipping between plants).
If a firm can produce 2 products at a lower total
cost than 2 firms each producing their own
product:
TC(Q1,Q2)<TC(Q1,0)+TC(0,Q2)
That firm experiences economies of scope.
57
If the cities maintains local roads, it costs are $15
million a year. If a private firm covers park
maintenance, it costs are $12 million a year. If
the city does both, it costs $25 million a year.
TC(Q1,Q2)=$25 million
TC(Q1,0)+TC(0,Q2)=$15 million + $12 million
TC(Q1,0)+TC(0,Q2)=$27 million
TC(Q1,Q2)<TC(Q1,0)+TC(0,Q2)
Economies of scope exist.
58
Often with practice a firm “gets better” at
producing a given output; it cuts costs by being
able to produce the good faster and with fewer
defects.
Ie: The first time you worked on elasticities, each
question took you 10 minutes and 10% were
wrong. By the end of the course you’ll be able to
calculate elasticities in 4 minutes with only 5%
error (for example).
59
Economies of experience are efficiencies (cost
advantages) resulting from accumulated
experience (learning-by-doing).
The experience curve shows the relationship
between average variable cost and cumulative
production volume.
-As more is produced (more experience is
gained), average cost decreases.
60
AVC
The Experience Curve
Eventually the curve
Flattens out
Cumulative Output
61
Economies of experience occur once, while
economies of scale are ongoing.
A large producer benefiting from economies of
scale will increase average costs by decreasing
production.
A large producer benefiting from economies of
experience may safely decrease production
62
Chapter 8 Key Concepts
Long-Run Costs:
TC=wL+rK (if labor and capital are the
only inputs
AC=TC/Q
MC=∆TC/ ∆ Q
Economies of scale summarize how
average cost changes as Q increases
Economies of scale = AC decreases as
Q increases
Diseconomies of scale = AC increases
as Q increases
63
Chapter 8 Key Concepts
Short-Run Costs
TFC=All costs of the FIXED input
TVC=All total costs of the VARIABLE
input
STC=TFC+TVC
SAC=STC/Q
SMC=∆STC/ ∆Q
AFC=TFC/Q
AVC=TVC/Q
SAC=AFC+AVC
64
Chapter 8 Key Concepts
If one firm has lower costs producing two
goods than two firms producing the goods
individually, that firm enjoys ECONOMIES
OF SCOPE
If AC decreases as cumulative output
increases, a firm enjoys ECONOMIES OF
EXPERIENCE
This effect decreases over time
Calculators are important in Econ 281
65