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Lecture # 12 Cost Curves Lecturer: Martin Paredes. Outline. Long Run Cost Functions Shifts Average and Marginal Cost Functions Economies of Scale Deadweight Loss Long Run Cost Functions Relationship between Long Run and Short Run Cost Functions. Long Run Cost Function.

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Lecture # 12 Cost Curves Lecturer: Martin Paredes

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Lecture # 12

Cost Curves

Lecturer: Martin Paredes


Outline

  • Long Run Cost Functions

    • Shifts

    • Average and Marginal Cost Functions

    • Economies of Scale

    • Deadweight Loss

  • Long Run Cost Functions

    • Relationship between Long Run and Short Run Cost Functions


Long Run Cost Function

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


Long Run Cost Function

Example: Long Run Total Cost Function

  • SupposeQ = 50L0.5K0.5

  • We found:L*(Q,w,r) = Q . r 0.5

    50 w

    K*(Q,w,r)= Q . w 0.5

    50 r

  • Then

    TC(Q,w,r) = wL*(Q,w,r) + rK*(Q,w,r)

    = Q . (wr)0.5

    25

( )

( )


Long Run Cost Curve

Definition: The long run total cost curve shows the minimized total cost as output (Q) varies, holding input prices (w and r) constant.


Example: Long Run Cost Curve

  • Recall TC(Q,w,r) = Q . (wr)0.5

    25

  • What if r = 100 and w = 25?

    TC(Q,w,r) = Q . (25100)0.5

    25

    = 2Q


TC (€ per year)

Example: A Total Cost Curve

TC(Q) = 2Q

Q (units per year)


TC (€ per year)

Example: A Total Cost Curve

TC(Q) = 2Q

€2M.

1 M.

Q (units per year)


TC (€ per year)

Example: A Total Cost Curve

TC(Q) = 2Q

€4M.

€2M.

1 M.

2 M.

Q (units per year)


Long Run Cost Curve

  • 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.


K

Example: Movement Along LRTC

Q0

TC = TC0

K0

0

L (labour services per year)

L0


K

Example: Movement Along LRTC

Q0

TC = TC0

K0

0

L (labour services per year)

L0

TC (€/yr)

LR Total Cost Curve

TC0=wL0+rK0

Q (units per year)

0

Q0


K

Example: Movement Along LRTC

Q1

Q0

TC = TC0

K1

K0

TC = TC1

0

L (labour services per year)

L0

L1

TC (€/yr)

LR Total Cost Curve

TC0=wL0+rK0

Q (units per year)

0

Q0


K

Example: Movement Along LRTC

Q1

Q0

TC = TC0

K1

K0

TC = TC1

0

L (labour services per year)

L0

L1

TC (€/yr)

LR Total Cost Curve

TC1=wL1+rK1

TC0=wL0+rK0

Q (units per year)

Q1

0

Q0


Long Run Cost Curve

Example: Shift of the long run cost curve

  • Suppose there is an increase in wages but the price of capital remains fixed.


K

Example: A Change in the Price of an Input

Q0

0

L


K

Example: A Change in the Price of an Input

TC0/r

A

Q0

-w0/r

0

L


K

Example: A Change in the Price of an Input

TC0/r

A

Q0

-w1/r

-w0/r

0

L


K

Example: A Change in the Price of an Input

TC1/r

B

TC1 > TC0

TC0/r

A

Q0

-w1/r

-w0/r

0

L


TC (€/yr)

Example: A Shift in the Total Cost Curve

TC(Q) ante

Q (units/yr)


TC (€/yr)

Example: A Shift in the Total Cost Curve

TC(Q) ante

TC0

Q0

Q (units/yr)


TC (€/yr)

Example: A Shift in the Total Cost Curve

TC(Q) post

TC(Q) ante

TC1

TC0

Q0

Q (units/yr)


Long Run Average Cost Function

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


Long Run Marginal Cost Function

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


Example: Average and Marginal Cost

  • RecallTC(Q,w,r) = Q . (wr)0.5

    25

  • Then: AC(Q,w,r) = (wr)0.5

    25

    MC(Q,w,r) = (wr)0.5

    25


Example: Average and Marginal Cost

  • If r = 100 and w = 25, then

    TC(Q) = 2Q

    AC(Q) = 2

    MC(Q) = 2


AC, MC (€ per unit)

Example: Average and Marginal Cost Curves

AC(Q) =

MC(Q) = 2

$2

0

Q (units/yr)


AC, MC (€ per unit)

Example: Average and Marginal Cost Curves

AC(Q) =

MC(Q) = 2

$2

0

1M

Q (units/yr)


AC, MC (€ per unit)

Example: Average and Marginal Cost Curves

AC(Q) =

MC(Q) = 2

$2

0

1M 2M

Q (units/yr)


Average and Marginal Cost

  • 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.


Average and Marginal Cost

  • 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.


AC, MC (€/yr)

Example: Average and Marginal Cost Curves

“Typical” shape of AC

AC

0

Q (units/yr)


AC, MC (€/yr)

Example: Average and Marginal Cost Curves

“Typical” shape of MC

MC

AC

0

Q (units/yr)


AC, MC (€/yr)

Example: Average and Marginal Cost Curves

MC

AC

AC at minimum when AC(Q)=MC(Q)

0

Q (units/yr)


Economies and Diseconomies of Scale

Definitions:

If the average cost decreases as output rises, all else equal, the cost function exhibits economies of scale.

If the average cost increases as output rises, all else equal, the cost function exhibits diseconomies of scale.

The smallest quantity at which the long run average cost curve attains its minimum point is called the minimum efficient scale.


AC (€/yr)

Example: Minimum Efficient Scale

AC(Q)

0

Q (units/yr)


AC (€/yr)

Example: Minimum Efficient Scale

AC(Q)

0

Q (units/yr)

Q* = MES


AC (€/yr)

Example: Minimum Efficient Scale

AC(Q)

Diseconomies of scale

0

Q (units/yr)

Q* = MES


AC (€/yr)

Example: Minimum Efficient Scale

AC(Q)

Diseconomies of scale

Economies of scale

0

Q (units/yr)

Q* = MES


Example: Minimum Efficient Scale for Selected

US Food and Beverage Industries

IndustryMES (% market output)

Beet Sugar (processed)1.87

Cane Sugar (processed)12.01

Flour0.68

Breakfast Cereal9.47

Baby food2.59

Source: Sutton, John, Sunk Costs and Market Structure. MIT Press, Cambridge, MA, 1991.


Returns to Scale and Economies of Scale

  • There is a close relationship between the concepts of returns to scale and economies of scale.

  • When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.


Returns to Scale and Economies of Scale

When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale: AC(Q) increases with Q.

When . the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale: AC(Q) decreases with Q.


Example: Returns to Scale and Economies of Scale


Output Elasticity of Total Cost

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


Output Elasticity of Total Cost

  • 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.


Example: Output Elasticities for Selected

Manufacturing Industries in India

IndustryTC,Q

Iron and Steel0.553

Cotton Textiles1.211

Cement1.162

Electricity and Gas0.3823


Short Run Cost Functions

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)


Short Run Cost Functions

Definitions:

The total variable cost function is the minimised sum spent on variable inputs at the input combinations that minimise short run costs.

The total fixed cost function is the total amount spent on the fixed input(s).


TC ($/yr)

Example: Short Run Total Cost,

Total Variable Cost

Total Fixed Cost

TFC

Q (units/yr)


TC ($/yr)

Example: Short Run Total Cost,

Total Variable Cost

Total Fixed Cost

TVC(Q, K0)

TFC

Q (units/yr)


TC ($/yr)

Example: Short Run Total Cost,

Total Variable Cost

Total Fixed Cost

STC(Q, K0)

TVC(Q, K0)

TFC

Q (units/yr)


TC ($/yr)

Example: Short Run Total Cost,

Total Variable Cost

Total Fixed Cost

STC(Q, K0)

TVC(Q, K0)

rK0

TFC

rK0

Q (units/yr)


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


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


Relationship Between Long Run and

Short Run Total Cost Functions

  • 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)


Relationship Between Long Run and

Short Run Total Cost Functions

  • 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.


Example: Short Run and Long Run Total Costs

K

Q0

0

L


Example: Short Run and Long Run Total Costs

K

TC0/r

Q0

A

0

L

TC0/w


Example: Short Run and Long Run Total Costs

K

TC0/r

Q0

A

K0

0

L

TC0/w


Example: Short Run and Long Run Total Costs

K

Q1

TC0/r

Q0

A

K0

0

L

TC0/w


Example: Short Run and Long Run Total Costs

K

Q1

TC0/r

Q0

A

B

K0

0

L

TC0/w


Example: Short Run and Long Run Total Costs

K

TC2/r

Q1

TC0/r

Q0

A

B

K0

0

L

TC0/w TC2/w


Example: Short Run and Long Run Total Costs

K

TC2/r

Q1

TC1/r

C

TC0/r

Q0

A

B

K0

0

L

TC0/w TC1/w TC2/w


Example: Short Run and Long Run Total Costs

K

TC2/r

Expansion path

Q1

TC1/r

C

TC0/r

Q0

A

B

K0

0

L

TC0/w TC1/w TC2/w


Total Cost (€/yr)

Example: Short Run and Long Run Total Costs

TC(Q)

0

Q (units/yr)


Total Cost (€/yr)

Example: Short Run and Long Run Total Costs

TC(Q)

A

TC0

0

Q0

Q (units/yr)


Total Cost (€/yr)

Example: Short Run and Long Run Total Costs

TC(Q)

TC1

C

A

TC0

0

Q0

Q1

Q (units/yr)


Total Cost (€/yr)

Example: Short Run and Long Run Total Costs

STC(Q,K0)

TC(Q)

TC1

C

A

TC0

0

Q0

Q1

Q (units/yr)


Total Cost (€/yr)

Example: Short Run and Long Run Total Costs

STC(Q,K0)

TC(Q)

B

TC2

TC1

C

A

TC0

0

Q0

Q1

Q (units/yr)


Total Cost (€/yr)

Example: Short Run and Long Run Total Costs

STC(Q,K0)

TC(Q)

B

TC2

TC1

C

A

TC0

K0 is the LR cost-minimising

quantity of K for Q0

0

Q0

Q1

Q (units/yr)


Short Run Average Cost Function

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


Short Run Marginal Cost Function

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


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


€ Per Unit

Example: Short Run Average Cost,

Average Variable Cost

Average Fixed Cost

AFC

0

Q (units per year)


€ Per Unit

Example: Short Run Average Cost,

Average Variable Cost

Average Fixed Cost

AVC

AFC

0

Q (units per year)


€ Per Unit

Example: Short Run Average Cost,

Average Variable Cost

Average Fixed Cost

SAC

AVC

AFC

0

Q (units per year)


€ Per Unit

Example: Short Run Average Cost,

Average Variable Cost

Average Fixed Cost

SMC

SAC

AVC

AFC

0

Q (units per year)


Relationship Between Long Run and

Short Run Average Cost Functions

  • 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 short-run average cost curves.


€ per unit

The Long Run Average Cost Curve as an Envelope Curve

SAC(Q,K1)

0

Q (units per year)


€ per unit

The Long Run Average Cost Curve as an Envelope Curve

SAC(Q,K1)

SAC(Q,K2)

0

Q (units per year)


€ 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)


€ 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)


€ 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)


Summary

Long run total cost curves plot the minimized total cost of the firm as output varies.

Movements along the long run total cost curve occur as output changes.

Shifts in the long run total cost curve occur as input prices change.


Summary

Average costs tell us the firm’s cost per unit of output.

Marginal costs tell us the rate of change in total cost as output varies.

Relatively high marginal costs pull up average costs, relatively low marginal costs pull average costs down.


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