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Lecture # 12
Cost Curves
Lecturer: Martin Paredes
1. Long Run Cost Functions
 Shifts
 Average and Marginal Cost Functions
 Economies of Scale
 Deadweight Loss
2. Long Run Cost Functions
 Relationship between Long Run and Short
Run Cost Functions
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Definition: The long run total cost function relates
the minimized total cost to output (Q) and the
factor prices (w and r).
TC(Q,w,r) = wL*(Q,w,r) + r K*(Q,w,r)
where L* and K* are the long run input
demand functions
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Example: Long Run Total Cost Function
 Suppose
Q = 50L0.5K0.5
 We found:
( )
( )
L*(Q,w,r) = Q . r
50 w
K*(Q,w,r) = Q . w
50
r
0.5
0.5
 Then
TC(Q,w,r) = wL*(Q,w,r) + rK*(Q,w,r)
= Q . (wr)0.5
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Definition: The long run total cost curve shows the
minimized total cost as output (Q) varies,
holding input prices (w and r) constant.
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Example: Long Run Cost Curve
 Recall TC(Q,w,r) = Q . (wr)0.5
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 What if r = 100 and w = 25?
TC(Q,w,r) = Q . (25100)0.5
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= 2Q
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TC (€ per year)
Example: A Total Cost Curve
TC(Q) = 2Q
Q (units per year)7
TC (€ per year)
Example: A Total Cost Curve
TC(Q) = 2Q
€2M.
1 M.
Q (units per year)8
TC (€ per year)
Example: A Total Cost Curve
TC(Q) = 2Q
€4M.
€2M.
1 M.
2 M.
Q (units per year)9
 We will observe a movement along the long
run cost curve when output (Q) varies.
 We will observe a shift in the long run cost
curve when any variable other than output (Q)
varies.
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K
Example: Movement Along LRTC
Q0
K0
0
•
L0
TC = TC0
L (labour services per year)
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K
Example: Movement Along LRTC
Q0
K0
0
TC = TC0
•
L (labour services per year)
L0
TC (€/yr)
LR Total Cost Curve
TC0=wL0+rK0
•
0
Q0
Q (units per year)
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K
Q1
Q0
K1
K0
0
•
•
L0 L1
Example: Movement Along LRTC
TC = TC0
TC = TC1
L (labour services per year)
TC (€/yr)
LR Total Cost Curve
TC0=wL0+rK0
•
0
Q0
Q (units per year)
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K
Q1
Q0
K1
K0
0
•
•
Example: Movement Along LRTC
TC = TC0
TC = TC1
L (labour services per year)
L0 L1
TC (€/yr)
TC1=wL1+rK1
TC0=wL0+rK0
•
0
Q0
•
Q1
LR Total Cost Curve
Q (units per year)
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Example: Shift of the long run cost curve

Suppose there is an increase in wages but the price of
capital remains fixed.
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K
Example: A Change in the Price of an Input
Q0
0
L
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K
Example: A Change in the Price of an Input
TC0/r
•
A
Q0
-w0/r
0
L
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K
Example: A Change in the Price of an Input
TC0/r
•
A
-w1/r
0
Q0
-w0/r
L
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K
Example: A Change in the Price of an Input
TC1/r
TC1 > TC0
B
TC0/r
•
•
A
-w1/r
0
Q0
-w0/r
L
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TC (€/yr)
Example: A Shift in the Total Cost Curve
TC(Q) ante
Q (units/yr)
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TC (€/yr)
Example: A Shift in the Total Cost Curve
TC(Q) ante
TC0
•
Q0
Q (units/yr)
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TC (€/yr)
Example: A Shift in the Total Cost Curve
TC(Q) post
TC1
TC0
•
•
Q0
TC(Q) ante
Q (units/yr)
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Definition: The long run average cost curve
indicates the firm’s cost per unit of output.
 It is simply the long run total cost function
divided by output.
AC(Q,w,r) = TC(Q,w,r)
Q
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Definition: The long run marginal cost curve
measures the rate of change of total cost as
output varies, holding all input prices constant.
MC(Q,w,r) = TC(Q,w,r)
Q
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Example: Average and Marginal Cost
 Recall
TC(Q,w,r) = Q . (wr)0.5
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 Then:
AC(Q,w,r) = (wr)0.5
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MC(Q,w,r) = (wr)0.5
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Example: Average and Marginal Cost
 If r = 100 and w = 25, then
TC(Q) = 2Q
AC(Q) = 2
MC(Q) = 2
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AC, MC (€ per unit)
Example: Average and Marginal Cost Curves
$2
0
AC(Q) =
MC(Q) = 2
Q (units/yr)
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AC, MC (€ per unit)
Example: Average and Marginal Cost Curves
AC(Q) =
MC(Q) = 2
$2
0
1M
Q (units/yr)
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AC, MC (€ per unit)
Example: Average and Marginal Cost Curves
AC(Q) =
MC(Q) = 2
$2
0
1M
2M
Q (units/yr)
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 When marginal cost equals average cost,
average cost does not change with output.
 I.e., if MC(Q) = AC(Q), then AC(Q) is flat
with respect to Q.
 However, oftentimes AC(Q) and MC(Q) are not
“flat” lines.
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 When marginal cost is less than average cost,
average cost is decreasing in quantity.
 I.e., if MC(Q) < AC(Q), AC(Q) decreases in Q.
 When marginal cost is greater than average cost,
average cost is increasing in quantity.
 I.e., if MC(Q) > AC(Q), AC(Q) increases in Q.
 We are implicitly assuming that all input prices
remain constant.
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AC, MC (€/yr)
Example: Average and Marginal Cost Curves
“Typical” shape of AC
AC
0
Q (units/yr)
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AC, MC (€/yr)
Example: Average and Marginal Cost Curves
“Typical” shape of MC
MC
AC
•
0
Q (units/yr)
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AC, MC (€/yr)
Example: Average and Marginal Cost Curves
MC
AC
•
AC at minimum when AC(Q)=MC(Q)
0
Q (units/yr)
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Definitions:
1. If the average cost decreases as output rises, all
else equal, the cost function exhibits economies
of scale.
2. If the average cost increases as output rises, all
else equal, the cost function exhibits
diseconomies of scale.
3. The smallest quantity at which the long run
average cost curve attains its minimum point is
called the minimum efficient scale.
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AC (€/yr)
Example: Minimum Efficient Scale
AC(Q)
0
Q (units/yr)
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AC (€/yr)
Example: Minimum Efficient Scale
AC(Q)
0
Q* = MES
Q (units/yr)
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AC (€/yr)
Example: Minimum Efficient Scale
AC(Q)
Diseconomies
of scale
0
Q* = MES
Q (units/yr)
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AC (€/yr)
Example: Minimum Efficient Scale
AC(Q)
Diseconomies
of scale
0
Economies
of scale
Q* = MES
Q (units/yr)
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Example:
Minimum Efficient Scale for Selected
US Food and Beverage Industries
Industry
Beet Sugar (processed)
Cane Sugar (processed)
Flour
Breakfast Cereal
Baby food
MES (% market output)
1.87
12.01
0.68
9.47
2.59
Source: Sutton, John, Sunk Costs and Market Structure. MIT Press,
Cambridge, MA, 1991.
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 There is a close relationship between the
concepts of returns to scale and economies of
scale.
1. When the production function exhibits
constant returns to scale, the long run average
cost function is flat: it neither increases nor
decreases with output.
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2. When the production function exhibits
increasing returns to scale, the long run
average cost function exhibits economies of
scale: AC(Q) increases with Q.
3. When . the production function exhibits
decreasing returns to scale, the long run
average cost function exhibits diseconomies of
scale: AC(Q) decreases with Q.
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Example:
Returns to Scale and Economies of Scale
Returns to Scale
Decreasing
Constant
Increasing
Production Function
Q = L0.5
Q=L
Q = L2
Labour Demand
L* = Q2
L* = Q
L* = Q0.5
Total Cost Function
TC = wQ2
TC = wQ
TC = wQ0.5
Average Cost Function
AC = wQ
AC = w
AC = wQ-0.5
Diseconomies
None
Economies
Economies of Scale
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Definition: The output elasticity of total cost is the
percentage change in total cost per one percent
change in output.
TC,Q = (% TC) = TC . Q = MC
(% Q)
Q
TC
AC
 It is a measure of the extent of economies of
scale
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 If TC,Q > 1, then MC > AC
 AC must be increasing in Q.
 The cost function exhibits economies of scale.
 If TC,Q < 1, then MC > AC
 AC must be increasing in Q
 The cost function exhibits diseconomies of
scale.
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Example:
Output Elasticities for Selected
Manufacturing Industries in India
Industry
TC,Q
Iron and Steel
Cotton Textiles
Cement
Electricity and Gas
0.553
1.211
1.162
0.3823
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Definition: The short run total cost function tells
us the minimized total cost of producing Q
units of output, when (at least) one input is
fixed at a particular level.
 It has two components: variable costs and fixed
costs:
STC(Q,K0) = TVC(Q,K0) + TFC(Q,K0)

(where K0 is the amount of the fixed input)
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Definitions:
1. The total variable cost function is the
minimised sum spent on variable inputs at the
input combinations that minimise short run
costs.
2. The total fixed cost function is the total
amount spent on the fixed input(s).
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TC ($/yr)
Example: Short Run Total Cost,
Total Variable Cost
Total Fixed Cost
TFC
Q (units/yr)
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TC ($/yr)
Example: Short Run Total Cost,
Total Variable Cost
Total Fixed Cost
TVC(Q, K0)
TFC
Q (units/yr)
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TC ($/yr)
Example: Short Run Total Cost,
Total Variable Cost
Total Fixed Cost
STC(Q, K0)
TVC(Q, K0)
TFC
Q (units/yr)
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TC ($/yr)
Example: Short Run Total Cost,
Total Variable Cost
Total Fixed Cost
STC(Q, K0)
rK0
TVC(Q, K0)
TFC
rK0
Q (units/yr)
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Example: Short Run Total Cost
 Suppose:
Q = K0.5L0.25M0.25
w = €16
m = €1
r = €2
 Recall the input demand functions:
LS* (Q,K0) = Q2
4K0
MS*(Q,K0) = 4Q2
K0
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Example (cont.):
 Short run total cost:
STC(Q,K0) = wLS* + mMS* + rK0
= 8Q2 + 2K0
K0
 Total fixed cost:
TFC(K0) = 2K0
 Total variable cost:
TVC(Q,K0) = 8Q2
K0
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 Compared to the short-run, in the long-run the
firm is “less constrained”.
 As a result, at any output level, long-run total
costs should be less than or equal to short-run
total costs:
TC(Q)  STC(Q,K0)
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 In other words, any short run total cost curve
should lie above the long run total cost curve.
 The short run total cost curve and the long run
total cost curve are equal only for some output
Q*, where the amount of the fixed input is also
the optimal amount of that input used in the
long-run.
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K
Example: Short Run and Long Run Total Costs
Q0
0
L
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K
TC0/r
Example: Short Run and Long Run Total Costs
Q0
A
•
0
TC0/w
L
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K
TC0/r
K0
0
Example: Short Run and Long Run Total Costs
Q0
A
•
TC0/w
L
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Example: Short Run and Long Run Total Costs
K
Q1
TC0/r
K0
0
Q0
A
•
TC0/w
L
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Example: Short Run and Long Run Total Costs
K
Q1
TC0/r
K0
0
Q0
A
•B
•
TC0/w
L
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Example: Short Run and Long Run Total Costs
K
TC2/r
Q1
TC0/r
K0
0
Q0
A
•B
•
TC0/w
TC2/w
L
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Example: Short Run and Long Run Total Costs
K
TC2/r
TC1/r
TC0/r
K0
0
Q1
C
•
A
•
Q0
•B
TC0/w
TC1/w
TC2/w
L
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Example: Short Run and Long Run Total Costs
K
TC2/r
TC1/r
TC0/r
K0
0
Expansion path
Q1
C
•
A
•
Q0
•B
TC0/w
TC1/w
TC2/w
L
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Total Cost (€/yr)
Example: Short Run and Long Run Total Costs
TC(Q)
0
Q (units/yr)
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Total Cost (€/yr)
Example: Short Run and Long Run Total Costs
TC(Q)
TC0
0
A
•
Q0
Q (units/yr)
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Total Cost (€/yr)
Example: Short Run and Long Run Total Costs
TC(Q)
•C
TC1
TC0
0
A
•
Q0
Q1
Q (units/yr)
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Total Cost (€/yr)
Example: Short Run and Long Run Total Costs
STC(Q,K0)
TC(Q)
•C
TC1
TC0
0
A
•
Q0
Q1
Q (units/yr)
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Total Cost (€/yr)
Example: Short Run and Long Run Total Costs
STC(Q,K0)
•B
•C
TC2
TC1
TC0
0
TC(Q)
A
•
Q0
Q1
Q (units/yr)
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Total Cost (€/yr)
Example: Short Run and Long Run Total Costs
STC(Q,K0)
•B
•C
TC2
TC1
TC0
TC(Q)
A
•
K0 is the LR cost-minimising
quantity of K for Q0
0
Q0
Q1
Q (units/yr)
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Definition: The short run average cost function
indicates the short run firm’s cost per unit of
output.
 It is simply the short run total cost function
divided by output, holding the input prices (w
and r) constant.
SAC(Q,K0) = STC(Q,K0)
Q
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Definition: The short run marginal cost curve
measures the rate of change of short run total
cost as output varies, holding all input prices
and fixed inputs constant.
SMC(Q,K0) = STC(Q,K0)
Q
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Notes:
 The short run average cost can be decomposed
into average variable cost and average fixed
cost.
SAC = AVC + AFC
where:
AVC = TVC/Q
AFC = TFC/Q
 When STC = TC, then also SMC = MC
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€ Per Unit
Example: Short Run Average Cost,
Average Variable Cost
Average Fixed Cost
AFC
0
Q (units per year)
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€ Per Unit
Example: Short Run Average Cost,
Average Variable Cost
Average Fixed Cost
AVC
AFC
0
Q (units per year)
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€ Per Unit
Example: Short Run Average Cost,
Average Variable Cost
Average Fixed Cost
SAC AVC
AFC
0
Q (units per year)
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€ Per Unit
Example: Short Run Average Cost,
Average Variable Cost
Average Fixed Cost
SMC SAC AVC
AFC
0
Q (units per year)
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 Just as with total costs curves, any short run
average cost curve should lie above the long run
average cost curve.
 In fact, the long run average cost curve forms a
boundary or envelope around the set of shortrun average cost curves.
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€ per unit
The Long Run Average Cost Curve
as an Envelope Curve
SAC(Q,K1)
0
Q (units per year)
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€ per unit
The Long Run Average Cost Curve
as an Envelope Curve
SAC(Q,K1)
SAC(Q,K2)
0
Q (units per year)
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€ per unit
The Long Run Average Cost Curve
as an Envelope Curve
SAC(Q,K3)
SAC(Q,K1)
SAC(Q,K2)
0
Q (units per year)
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€ per unit
The Long Run Average Cost Curve
as an Envelope Curve
SAC(Q,K3)
SAC(Q,K4)
SAC(Q,K1)
SAC(Q,K2)
0
Q (units per year)
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€ per unit
The Long Run Average Cost Curve
as an Envelope Curve
SAC(Q,K3)
SAC(Q,K4)
SAC(Q,K1)
AC(Q)
SAC(Q,K2)
•
0
Q1
•
Q2
•
Q3
•
Q4
Q (units per year)
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1. Long run total cost curves plot the minimized
total cost of the firm as output varies.
2. Movements along the long run total cost curve
occur as output changes.
3. Shifts in the long run total cost curve occur as
input prices change.
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4. Average costs tell us the firm’s cost per unit of
output.
5. Marginal costs tell us the rate of change in total
cost as output varies.
6. Relatively high marginal costs pull up average
costs, relatively low marginal costs pull average
costs down.
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