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
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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
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:
Q (units per year) 4
L (labor services per year) K
TC ($/yr)
0 0
LR Total Cost Curve
Q0 Q1
TC0 =wL0+rK0
• •
L0 L1 K0
K1
Q0
Q1
TC = TC1 TC = TC0
TC1=wL1+rK1
•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)
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L K
Q0
•
•
0
B TC0/r A
TC1/r
Slope=w
1/r
Slope=w
2/r
C2 C1 C3
C
1: Original isocost curve (TC = $200)
C
2: Isocost curve after
Price change (TC = $200) C
3: Isocost curve after
Price change (TC = $300)
Q (units/yr) TC ($/yr)
TC(Q) old TC(Q) new
Change in Input Prices ->
A Shift in the Total Cost Curve
Q 300
200
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Let Q=2(LK)
1/2MRTS=K/L, W=5, R=20, Q=40
What occurs to costs when rent falls to 5?
Initially:
MRTS=W/R K/L=5/20
4K=L
Q=2(LK)
1/240=2(4KK)
1/240=4K
10=K
40=L
Let Q=2(LK)
1/2MRTS=K/L, W=5, R=20, Q=40
What occurs to costs when rent falls to 5?
After Price Change:
MRTS=W/R K/L=5/5
L=K
Q=2(LK)
1/240=2(LL)
1/240=2L
20=L
20=K
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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 $200.
Q (units/yr) TC ($/yr)
TC(Q) final TC(Q) initial
Change in Rent
40 400
200
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To calculate total cost, simply substitute labour and capital demand into your cost expression:
Q= 50L
1/2K
1/2(From Chapter 7:) L*= (Q
0/50)(r/w)
1/2K* = (Q
0/50)(w/r)
1/2TC = wL +rK
TC= w [(Q
0/50)(r/w)
1/2] +r[(Q
0/50)(w/r)
1/2] TC= [(Q
0/50)(wr)
1/2] +[(Q
0/50)(wr)
1/2]
TC = 2Q
0(wr)
1/2/50
Let Q= L
1/2K
1/2,MP
L/MP
K=K/L, w=10, r=40.
Calculate total cost.
MRTS=w/r K/L=10/40 4K=L
Q=L
1/2K
1/2=(4K)
1/2K
1/2Q=2K K=Q/2
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Let Q= L
1/2K
1/2,MRTS=K/L, w=10, r=40.
Calculate total cost.
L=4K
L=4(Q/2) L=2Q
TC = wL +rK
TC = 10(2Q) +40(Q/2)
TC = 40Q
•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
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L (labor services/yr) K (capital services/yr)
0
•
AQ0
C
1=Isocost curve before ($200)
and after ($220) a 10% increase in input prices
C1
Q (units/yr) TC ($/yr)
TC(Q) old TC(Q) new
Example: A Shift in the Total Cost Curve When Input Prices Rise 10%
Q 220
200
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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…
Q Q Q TC
AC ( )
)
(
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
Q Q Q TC
MC
( ) )
(
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Q (units/yr) TC ($/yr)
TC(Q) post
Average vrs. Marginal Costs
Q0 TC0
Slope=LRMC
Slope=LRAC
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When marginal cost is less than average cost, average cost is decreasing in quantity. That is, if MC(Q) < AC(Q), AC(Q) decreases in Q.
When marginal cost is greater than average cost, average cost is increasing in quantity. That is, if MC(Q) > AC(Q), AC(Q) increases in Q.
When marginal cost equals average cost, average cost is at its minimum. That is, if MC(Q) = AC(Q),
AC(Q) is at its minimum.
Q (units/yr)22
AC, MC ($/yr)
0
MC AC
AC at minimum when AC(Q)=MC(Q)
“typical” shape of AC, MC
•
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
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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)
Why Diseconomies of scale?
-Diminishing Returns from Inputs -Managerial Diseconomies
-Growing in size requires a large expenditure on managers
-ie: The owner is very passionate, but can
only manage 1 or two branches
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0 Q (units/yr)
AC ($/yr)
Q*
AC(Q)
Typical Economies of Scale
Economies of scale Diseconomies of scale
Minimum Efficient Scale – smallest
Quantity where LRAC curve reaches
Its min.
Production Function Cost Function Increasing returns to
scale Economies of Scale
Decreasing returns to
scale Diseconomies of Scale
Constant Returns to
Scale Neither economies nor
diseconomies of scale
Production functions and cost functions are related:
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Example: Returns to Scale and Economies of Scale CRS IRS DRS
Production Function Q = L Q = L
2Q = L
1/2Labor Demand L*=Q L*=Q
1/2L*=Q
2Total Cost Function TC=wQ wQ
1/2wQ
2Average Cost Function AC=w w/Q
1/2wQ
Economies of Scale none EOS DOS
•Economies of Scale are also related to marginal cost and average cost:
If MC < AC, AC must be decreasing in Q.
Therefore, we have economies of scale.
If MC > AC, AC must be increasing in Q.
Therefore, we have diseconomies of scale.
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Let Cost=50+20Q
2MC=40Q
IF Q=1 or Q=2, determine economies of scale
(Let Q be thousands of units)
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TC=50+20Q
2MC=40Q
AC=TC/Q=50/Q+20Q Initially: MC=40(1)=40
AC=50/1+20(1)=70
MC<AC – Economies of Scale Finally: MC=40(2)=80
AC=50/2+20(2)=65
MC>AC – Diseconomies of Scale
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8.3 Short-Run Cost Curves
•In the short run, at least 1 input is fixed (ie: (K=K*)
•Total fixed costs (TFC) are the costs associated with this fixed input (ie: rk)
•Total variable costs (TVC) are the costs associated with variable inputs (ie:wL)
•Short-run total costs are fixed costs plus variable costs:
STC=TFC+TVC
Q (units/yr) TC ($/yr)
TVC(Q, K*)
TFC rK*
STC(Q, K*) rK*
Short Run Total Cost, Total Variable
Cost and Total Fixed Cost
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Short Run Costs
Example:
Minimize the cost to build 80 units if Q=2(KL)
1/2and K=25. If r=10 and
w=20, classify costs.
Q=2(KL)
1/280=2(25L)
1/280=10(L)
1/28=(L)
1/264=L
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
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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.
L K
TC0/w TC1/w TC2/w TC2/r
TC1/r TC0/r
Q0
•
•
•
Long Run Expansion path
0
A C
B Q1
Q0 K*
Only at point A is short run minimized as well as long run
Short Run
Expansion path
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Q (units/yr) TC ($/yr)
LRTC(Q) A
STC(Q)
•
rK*
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…
Q
Q Q STC
SAC ( )
)
(
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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
Q
Q Q STC
SMC
( )
)
(
In the short run, 2 additional average costs exist:
average variable costs (AVC) and average fixed costs (AFC)
Q
Q Q TVC
AVC
Q
Q Q TFC
AFC
) ) (
(
) ) (
(
AVC
42AFC SAC
Therefore
Q TVC Q
TFC Q
STC
TVC TFC
STC Note
:
:
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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
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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=P
E(E/Q)
$3.50=0.5 (E/Q) 7=E/Q
There were 7 eggs in each omelet.
Q (units per
$ Per Unit
0
AFC
Average fixed cost is
constantly decreasing, as
fixed costs don’t rise with
output.
Q (units per46
year)
$ Per Unit
0
AVC
AFC
Average variable cost generally
decreases then increases due to economies of
scale.
Q (units per
$ Per Unit
0
SAC AVC
AFC
SAC is the vertical sum of AVC and AFC
Equal
Q (units per48
year)
$ Per Unit
0
SMC
SAC AVC
AFC
SMC intersects SAC and
AVC at their minimum
points
•
•
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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
products (ie: no shipping between plants).
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If a firm can produce 2 products at a lower total cost than 2 firms each producing their own
product:
TC(Q
1,Q
2)<TC(Q
1,0)+TC(0,Q
2)
That firm experiences economies of scope.
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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(Q
1,Q
2)=$25 million
TC(Q
1,0)+TC(0,Q
2)=$15 million + $12 million TC(Q
1,0)+TC(0,Q
2)=$27 million
TC(Q
1,Q
2)<TC(Q
1,0)+TC(0,Q
2)
Economies of scope exist.
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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).
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.
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AVC
Eventually the curve Flattens out
The Experience Curve
Cumulative Output
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
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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
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
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