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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 CurvesIn this chapter we will cover:

8.1 Long Run Cost Curves 8.1.1 Total Cost

8.1.2 Marginal Cost and Average Cost8.2 Economies of Scale8.3 Short Run Cost Curves

8.3.1 Total Cost, Variable Cost, Fixed Cost8.3.2 Marginal Cost and Average Cost

8.4 Economies of Scope8.5 Economies of Experience

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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 long run cost curve is as follows:

4Q (units per year)

L (labor services per year)

K

TC ($/yr)

0

0

LR Total Cost Curve

Q0Q1

TC0 =wL0+rK0

••

L0 L1

K0

K1

Q0

Q1

TC = TC1

TC = TC0

TC1=wL1+rK1

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•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:

-(This is only not true in the case of perfect substitutes when the productivity per dollar of each substitute is originally equal)

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L

K

Q0•

•

0

B

ATC0/r

TC1/rSlope=w1/r

Slope=w2/r

C2 C1C3

C1: Original isocost curve (TC = $200)C2: Isocost curve after Price change (TC = $200)C3: Isocost curve after Price change (TC = $300)

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Q (units/yr)

TC ($/yr)

TC(Q) old

TC(Q) new

Change in Input Prices ->A Shift in the Total Cost Curve

Q0

300

200

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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:MRTS=W/RK/L=5/20 4K=L

Q=2(LK)1/2

40=2(4KK)1/2

40=4K10=K40=L

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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:MRTS=W/RK/L=5/5 L=K

Q=2(LK)1/2

40=2(LL)1/2

40=2L20=L20=K

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What occurs when rent falls to 5?Initial: L=40, K=10 Final: L=K=20W=5, R=20, Q=40

Initial: TC=wL+rKTC=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.

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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= 50L1/2K1/2 (From Chapter 7, slide 38:)L*(Q,w,r) = (Q0/50)(r/w)1/2 K*(Q,w,r) = (Q0/50)(w/r)1/2

TC = wL +rKTC= 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

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Let Q= L1/2K1/2, MPL/MPK=K/L, w=10, r=40.

Calculate total cost.

MRTS=w/rK/L=10/40K=4L

Q=L1/2K1/2 =L1/2(4L)1/2

Q=2LL=Q/2

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Let Q= L1/2K1/2, MRTS=K/L, w=10, r=40.

Calculate total cost.

K=4LL=K/4L=Q/2

Q=L1/2K1/2 Q=(K/4)1/2K1/2

Q=1/2KK=2Q

TC = wL +rKTC = 10(Q/2) +40(2Q)TC=85Q

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•When the price 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

•A

Q0

C1=Isocost curve before ($200) and after ($220) a 10% increase in input prices

C1

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Q (units/yr)

TC ($/yr)

TC(Q) old

TC(Q) new

Example: A Shift in the Total Cost Curve When Input Prices Rise 10%

Q0

220200

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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

QTCQAC

)()(

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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

QTCQMC

)(

)(

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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 does not change with quantity. That is, if MC(Q) = AC(Q), AC(Q) is flat with respect to Q.

22Q (units/yr)

AC, MC ($/yr)

0

MC AC

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

“typical” shape of AC, MC

•

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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)

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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

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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 – smallestQuantity where LRAC curve reachesIts min.

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When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale so that AC(Q) decreases with Q, all else equal.

When the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale so that AC(Q) increases with Q, all else equal.

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

Production Function => Returns to ScaleCosts => Economies of Scale

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Example: Returns to Scale and Economies of Scale

CRS IRS DRSProduction Function Q = L Q = L2 Q = L1/2

Labor Demand L*=Q L*=Q1/2 L*=Q2

Total Cost Function TC=wQ wQ1/2 wQ2

Average Cost Function AC=w w/Q1/2 wQ

Economies of Scale none EOS DOS

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•Economies of Scale can be measured using output elasticity of total cost; how cost changes when output changes

TC

Q

Q

TC

Output

TC

QTC

QTC

,

, %

%

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•Economies of Scale are also related to marginal cost and average cost

ACMC

Q

TC

Q

TC

QTC

QTC

/

/

,

,

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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.

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Let Cost=50+20Q2

MC=40Q

IF Q=1 or Q=2, determine economies of scale

(Let Q be thousands of units)

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TC=50+20Q2

MC=40QAC=TC/Q=50/Q+20Q

Initially: MC=40(1)=40AC=50/1+20(1)=70

Elasticity=MC/AC=40/70 – Economies of Scale

Finally: MC=40(2)=80AC=50/2+20(2)=65

E=MC/AC=80/65 – 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

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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 CostsExample: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

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Short Run CostsExample:K*=25, L=16. If r=10 and w=20, classify costs.

TFC=rK*=10(25)=250TVC=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.

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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*

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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

QSTCQSAC

)()(

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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

QSTCQSMC

)(

)(

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In the short run, 2 additional average costs exist: average variable costs (AVC) and average fixed costs (AFC)

Q

QTVCQAVC

Q

QTFCQAFC

)()(

)()(

45AVCAFCSAC

Therefore

Q

TVC

Q

TFC

Q

STC

TVCTFCSTC

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+AFCTC/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=PE (E/Q)$3.50=0.5 (E/Q)

7=E/Q

There were 7 eggs in each omelet.

48Q (units peryear)

$ Per Unit

0

AFC

Average fixed cost is constantly decreasing, as fixed costs don’t rise with output.

49Q (units peryear)

$ Per Unit

0

AVC

AFC

Average variable cost generally decreases then increases due to economies of scale.

50Q (units peryear)

$ Per Unit

0

SAC

AVC

AFC

SAC is the vertical sum of AVC and AFC

Equal

51Q (units peryear)

$ Per Unit

0

SMCSAC

AVC

AFC

SMC intersects SAC and AVC at their minimum points

••

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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.

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Q (units per year)

$ per unit

0

• ••

AC(Q)

SAC(Q,K1)

SAC(Q,K2)

SAC(Q,K3)

Q1 Q2 Q3

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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)

At the MES:-AC=SAC=MC=SMC and SAC is at a

minimum (two slides hence)

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Q (units per year)

$ per unit

0

••

AC(Q)

SAC(Q,K1)

Q1 Q2 Q3

MC(Q)

SMC(Q,K1)

A

B

C•

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Q (units per year)

$ per unit

0

•

AC(Q)SAC(Q,K2)

Q1 Q2 Q3

MC(Q)Example: Putting It All Together

SMC(Q,K2)

D

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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(Q1,Q2)<TC(Q1,0)+TC(0,Q2)

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(Q1,Q2)=$25 millionTC(Q1,0)+TC(0,Q2)=$15 million + $12

millionTC(Q1,0)+TC(0,Q2)=$27 million

TC(Q1,Q2)<TC(Q1,0)+TC(0,Q2)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).

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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 curveFlattens out

The Experience Curve

Cumulative Output

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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 ConceptsLong-Run Costs:

TC=wL+rK (if labor and capital are the only inputsAC=TC/QMC=∆TC/ ∆ Q

Economies of scale summarize how average cost changes as Q increases

Economies of scale = AC decreases as Q increasesDiseconomies of scale = AC increases as Q increases

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Chapter 8 Key ConceptsShort-Run Costs

TFC=All costs of the FIXED inputTVC=All total costs of the VARIABLE inputSTC=TFC+TVCSAC=STC/QSMC=∆STC/ ∆QAFC=TFC/QAVC=TVC/QSAC=AFC+AVC

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Chapter 8 Key ConceptsIf one firm has lower costs producing two goods than two firms producing the goods individually, that firm enjoys ECONOMIES OF SCOPEIf AC decreases as cumulative output increases, a firm enjoys ECONOMIES OF EXPERIENCE

This effect decreases over timeCalculators are important in Econ 281