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Production FunctionProduction Function

Q t= (inputs t)Q t=output rateinput t=input rate

where is technology?

Firms try to be on the surface of the PF.Inside the function implies there iswaste, or technological inefficiency.

Production FunctionQ= (K t,L t)

Q t

Kt

Lt

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D ifference between LR and SRD ifference between LR and SR

LR is time period where all inputs can be varied.L abor, land, capital, entrepreneurial effort, etc.

SR is time period when at least some inputs are fixed.Usually think of capital (i.e., plant size) as the fixed input, and labor

as the variable input.

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L ong R un: Q = f (K, L )L ong R un: Q = f (K, L )Suppose there are two different sizedSuppose there are two different sizedplants, K plants, K 11 and K and K 22..One Short R un:One Short R un:

Q = f ( K Q = f ( K 11,L ),L ) i.e., K fixed at K i.e., K fixed at K 11A second Short R un:A second Short R un:

Q = f ( K Q = f ( K 22,L ),L ) i.e., K fixed at K i.e., K fixed at K 22Show this graphicallyShow this graphically

LR production function as many SR productionLR production function as many SR production

functions.functions.

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Two Separa te SR Produc tion Func tionsTwo Separa te SR Produc tion Func tions

QQ

LL

Q = f( KQ = f( K 22, L ), L )Q = f( KQ = f( K 11, L ), L )

KK22 > K> K 11

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Wh at Happens w h en Tec h nology C h anges?Wh at Happens w h en Tec h nology C h anges?

T his shifts the entire production function, both in the S R and inthe LR .

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Tec h nology C h angesTec h nology C h anges

QQ

LL

TP before computer TP before computer

TP after computer TP after computer

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SR Production Function in More D etailSR Production Function in More D etail

E xpress this in two dimensions, L and Q, since K is fixed.Define Marginal Product of L abor.Slope is MP L =dQ/d L

Identify three rangesI: MP L >0 and risingII: MP L >0 and fallingIII: MP L

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Wh ere D iminis h ing Returns Sets InWh ere D iminis h ing Returns Sets In

A s you add more and more variableinpu ts to fixed inpu ts, even tuallymarginal produc tivity begins tofall .

As you move into zone II,diminishing returns sets in!

Why does this occur?

L

Q Q

I II I II

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Wh y D iminis h ing Returns Sets InWh y D iminis h ing Returns Sets In

Since plant size (i.e., capital) is fixed,labor has to start competing for thefixed capital.E ven though Q still increases with L for a while, the change in Q is smaller.

L

Q Q

I II I II

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Average Product = Q / L output per unit of labor.frequently reported in press.

Marginal Product = dQ/d L output attributable to last unit of labor used.

what economists think of.

D efine APD efine AP LL and MPand MP LL

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Average Productivity Grap h icallyAverage Productivity Grap h ically

T ake ray from origin to the S R production function.Derive slope of that ray

( Q=Q 1 ( L=L 1

T hus,( Q/ ( L =Q1 /L1

Q Q 11

(( Q Q

Q Q

LL

Q=f(K Q=f(K FIXEDFIXED ,L),L)

LL11(( LL

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AP L rises until L2B eyond L2 , the AP L begins tofall.T hat is, the average productivityrises, reaches a peak, and thendeclines

Q Q

Q Q 2 2

LL

Q=f(K Q=f(K FIXEDFIXED ,L),L)

LL22

LL22

Q/LQ/L

AP AP LL

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Av erage & Marg inal Produc tiv ity Av erage & Marg inal Produc tiv ity

T here is a relationship between the productivity of the average worker, andthe productivity of the marginal worker.T hink of a batting average.

T hink of your marginal productivity in the most recent game.T hink of average productivity from beginning of year.

Wh en MP > A P, th en A P is RISINGWh

en MP < A

P, th

enA

P is FA

LL INGWh en MP = A P, th en A P is a t its MAX

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MP L rises until L1B eyond L1 , the MP L begins tofall.L ook at AP

i. U ntil L2, MPL >A PL and th us A PLrises.

ii. A t L2, MPL=A PL and th us A PL peaks.iii. Beyond L2, MPL

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Anytime you add a marginal unit to an average unit, it either pulls theaverage up, keeps it the same, or pulls it down.

Wh en MP > A P, th en A P is rising since i t pulls i t th e average up.

Wh en MP < A P, th en A P is falling since i t pulls th e average down.Wh en MP = A P, th en A P stays th e same.

T hink of softball batting average example.

Intuitive explanationIntuitive explanation

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LR Produc tion Func tionLR Produc tion Func tion

Q t

Lt

Kt

IsoquantsIsoquants(i.e.,constant (i.e.,constant quantity)quantity)

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Def ine Isoquan t

Def ine Isoquan t

Different combinations of K t and L t which generate thesame level of output, Q t.

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Isoquan ts & LR Produc tion Func tionsIsoquan ts & LR Produc tion Func tions

QQ tt = Q(K = Q(K tt, L, L tt))Output rate increases as you move to higherOutput rate increases as you move to higherisoquants.isoquants.

Slope represents ability to tradeoff inputs whileSlope represents ability to tradeoff inputs whileholding output constant.holding output constant.

Marginal Rate of Technical Substitution Marginal Rate of Technical Substitution ..C loseness represents steepness of productionC loseness represents steepness of productionhill.hill.

ISOQU A NT M A PISOQU A NT M A P

Q1

Q2

Q3

K

L

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Slope of IsoquantSlope of Isoquant

Slope is typically not constant.Tradeoff between K and L dependson level of each.

C an derive slope by totallydifferentiating the LR production

function.Marginal rate of technicalsubstitution is MP L /MP K

KKtt

LLtt

QQ

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Ex treme CasesEx treme Cases

N o Substitutability Perfect Substitutability

LL

KK

QQ 11

QQ

22

Inputs used in fixed Inputs used in fixed

proportions! proportions!

KK

QQ 11

QQ 22

LL

Tradeoff is constant Tradeoff is constant

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Subs titu tab ilitySubs titu tab ility

L ow Substitutability High Substitutability

LL

KK

QQ 11

KK

QQ 11

LLSl ope of Isoquant Sl ope of Isoquant

changes very l itt l echanges very l itt l e

Sl ope of Isoquant Sl ope of Isoquant

changes a l ot changes a l ot

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Isoquants and Returns to ScaleIsoquants and Returns to Scale

R eturns to scale are cost savings associated with a firmgetting larger.

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Increasing Returns to ScaleIncreasing Returns to Scale

Production hill is rising quickly.L ines on the contour map getcloser with equal increments inQ.KK

LLQ =10Q =10

Q =20Q =20Q =30Q =30

Q =40Q =40

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D ecreasing Returns to ScaleD ecreasing Returns to Scale

Production hill is rising slowly.L ines on the contour map getfurther apart with equal

increments in Q.

KK

LLQ =10Q =10

Q =20Q =20

Q =30Q =30

Q =40Q =40

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How Can You Tell if a PF h as IRS, D RS, or CRS?How Can You Tell if a PF h as IRS, D RS, or CRS?

It is possible that it has all three, along various ranges of production.However, you can also look at a special kind of function, called ahomogeneous function.

Degree of homogeneity is an indicator returns to scale.

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Homogeneous Functions of D egreeHomogeneous Functions of D egree HH

A function is homogeneous of degree k if multiplying all inputs by P, increases the dependent variable by PH

Q = f ( K, L )So, P H Q = f ( P K, P L) is homogenous of degree k.

C obb-Douglas Production Functions are homogeneous of degreeE

+ F

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CobbCobb--D ouglas Produc tion Func tionsD ouglas Produc tion Func tions

Q = A K E L F is a C obb-Douglas Production FunctionDegree of Homogeneity is derived by increasing all the inputs by P

PH

Q = A ( P K)E

( PL

) F

PHQ = A P E K E P F L FPHQ = P E F A K E L F

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CobbCobb- -D ouglas Production FunctionsD ouglas Production Functions

Th is is a Cons tant Elas ticity Func tionElas ticity of subs titution W = 1

Coefficien ts are elas ticitiesE is the capital elasticity of output, E K F is the labor elasticity of output, E L

If E k or L < 1 then that input is subject to Diminishing R eturns.C-D PF can be IRS , DRS or CRS

if E + F ! 1, th en CRSif E + F < 1, th en DRS

if E + F > 1, th en IRS

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Tec h nical C h ange in LRTec h nical C h ange in LR

T echnical change causes isoquants to shift inwardL ess inputs for given output

May cause slope to change along ray from originL abor savingC apital saving

May not change slopeN eutral implies parallel shift

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Tec h nical c h angeTec h nical c h ange

L abor Saving C apital Saving

K

L

K

L

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Lets now turn to t h e Cost SideLets now turn to t h e Cost Side

What is Goal of Firm?

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TC constant along Isocost line.TC constant along Isocost line.

KK

LL

TCTC 11 /r /r

TCTC 11 /w /w

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in TC parallel s h ifts Isocostin TC parallel s h ifts Isocost

KK

LL

TCTC 11 /r /r

TCTC 11 /w /w

TCTC 22 /r /r

TCTC 22 /w /w

TCTC 22 > TC> TC 11

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Ch ange in input price rotates IsocostCh ange in input price rotates Isocost

KK

LL

TC/r TC/r

TC/wTC/w 11 TC/wTC/w 22

ww 22 < w< w 11

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O ptimal Input Levels in LRO ptimal Input Levels in LR

Suppose Optimal Output level isdetermined (Q 1).Suppose w and r fixed.What is least costly way toproduce Q 1?

KK

LL

QQ11

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Cos t Min im izing Cond itionCos t Min im izing Cond ition

Slopes of Isoquant and Isocost are equalSlope of Isoquant=M R T S=- MP L / MP K Slope of Isocost=input price ratio=-w/r

At tangency, - MP L / MP K = -w/rR earranging gives: MP L /w= MP K /rIn words:

Additional output from last $ spent on L = additional output from last $ spent onK.

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Th e LR Expansion Pat hTh e LR Expansion Pat h

C osts increase when outputincreases in LR !L ook at increase from Q 1 to Q 2.B oth L abor and C apital adjust.C onnecting these points gives theexpansion path.

K

L

Q 1

Q 2

L1 L2

K 1K 2

expansion path

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W e can s h ow t h at LR adjustment along t h eW e can s h ow t h at LR adjustment along t h eexpansion pat h is less costly t h an SR adjustmentexpansion pat h is less costly t h an SR adjustment

h olding K fixed!h olding K fixed!

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Start at an original LR equilibrium (i.e., at QStart at an original LR equilibrium (i.e., at Q 11).).

K

L

Q 1

L1

K 1

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LR AdjustmentLR Adjustment

LR adjustment:K increases (K 1 to K 2)L increases ( L 1 to L 2)TC increases from black to blue isocost.

K

L

Q 1

Q 2

L1 L2

K 1K 2

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SR AdjustmentSR Adjustment

SR adjustment.K constant at K 1.L increases ( L 1 to L 3 )TC increases from black to whiteisocost.

K

L

Q 1

Q 2

L1

K 1

L3

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LR Adjustment less CostlyLR Adjustment less Costly

White Isocost (i.e., S R ) is further fromthe origin than the B lue Isocost ( LR ).T hus, the more flexible LR is less costlythan the less flexible S R .

K

L

Q 1

Q 2

L1 L2

K 1K 2

L3

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Impact of Input Price C h angeImpact of Input Price C h ange

Start at equilibrium.R ecall slope of isocost= ( K/ ( L = -w/r

Suppose w and optimal Q stayssame (i.e., Q 1)R otate budget line out, and thenshift back inward!

K

L

Q 1

L1

K 1

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D erivation of Labor D emand from SubstitutionD erivation of Labor D emand from SubstitutionEffectEffect

Wage falls w

K

L

Q 1

L1

K 1

K 2

L2

a b

LL1 L2

w 1

w 2

DL1

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Th ere is also a scale effectTh ere is also a scale effect

Scale effect is increase in output thatresults from lower costsScale effect: b-c K

L

Q 1

L1

K 1 a

b

c

Q 2

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Scale Effect S h ifts D emandScale Effect S h ifts D emand

Wage falls w

K

L

Q 1

L1

K 1

K 2

L2

a b

LL1 L2

w 1

w 2 c

L3L3

DL1DL2

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Recall t h e Isocost LineRecall t h e Isocost LineTC=w*L + r*KTC=w*L + r*K

Thus, TC=TVC+TFC Lets relate the cost relationships to the

production relationships.Recall the Law of Diminishing Returns.

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Law of D iminis h ing Marginal ReturnsLaw of D iminis h ing Marginal Returns

As you add more and more variable inputs ( L ) to your fixed inputs(K), marginal additions to output eventually fall (i.e., MP L =( Q/ ( L falls)What does this say about the shape of cost curves?

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Marginal Productivity (MPMarginal Productivity (MP LL) and Marginal Cost (MC)) and Marginal Cost (MC)

L ook at how TC changes when output changes.Assume w and r are fixed.Since TC =w* L + r*K

then ( TC = w* ( L + r* ( K How does K change in S R ?

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Ch anges in TC in SR must come from c h anges inCh anges in TC in SR must come from c h anges inLabor.Labor.

( TC = w* ( L

Divide through by change in Q (ie. ( Q)( TC /( Q = w* ( ( L /( Q)

( TC /( Q = Marginal C ost = M CWhat is MP L ?

MP L =( ( Q/ ( L )T hus: ( TC /( Q = w* 1/( ( Q/ ( L )T his gives: M C =w/MP L

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MC=w/MPMC=w/MP LL

MP L

LL1

MC

Q

Look at where Diminishing Returns sets in.

MP L

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MC=w/MPMC=w/MP LL

MP L MC

S ubstitute L 1 into S R Production FunctionQ 1=f(K FIXED ,L1 )

LL1 Q

MC

Q 1

MP L

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Alternatively: TC and TPAlternatively: TC and TP

Q TC

S ubstitute L 1 into S R Production FunctionQ 1=f(K FIXED ,L1 )

LL1 Q

TC

Q 1

MP L

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AVC and APAVC and AP LL

AVC =w* L /QR earranging: A VC =w/(Q/ L )Since Q/ L =AP LAVC =w/AP L

Diagram is similar.

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AVC=w/APAVC=w/AP LL

AP L

S ubstitute L 2 into S R Production FunctionQ 2 =f(K FIXED ,L2 )

LL2

AVC

Q

AVC

Q 2

AP L

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Put SR Cost Curves Toget h er Put SR Cost Curves Toget h er

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Average Cost CurvesAverage Cost Curves

$

Q

ATC

AVC

AFC

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S h ort Run Average Costs and Marginal CostS h ort Run Average Costs and Marginal Cost

$

Q

ATC

AVC MC

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Cost Curve S h iftersCost Curve S h ifters(Variable Cost Increases)(Variable Cost Increases)

A change in the wage shifts theAVC and M C curves.T hus, the A TC curve also shiftsupward. $

Q

ATC

MC

ATC MC

AVC

AVC

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Cost Curve S h iftersCost Curve S h ifters(Fixed Cost Increases)(Fixed Cost Increases)

An increase in price of capitalincreases fixed costs, but notvariable costs.T hus, A VC and M C are fixed,but A TC increases.

$

Q

AVC

MC ATC ATC

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Costs in t h e LRCosts in t h e LR

Why did S R cost curves have the shape they did?Why do LR cost curves have the shape they do?

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LR Total Costs Grap h icallyLR Total Costs Grap h ically

TC TC

IR S IR S DR S DR S

QQ

Cost Cost CR S CR S

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Wh y are t h ere Economies of Scale?Wh y are t h ere Economies of Scale?

Specializa tion in use of inpu ts.Less th an propor tiona te ma terials use as plan t sizeincrease.Some tech nologies are no t feasible a t small scales.

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Wh y do D iseconomies of Scale Set In?Wh y do D iseconomies of Scale Set In?

Even tually , large scale opera tions become more cos tly toopera te (i.e. , th ey use more resources ) due to problems of coordina tion and con trol.e.g. , red tape in th e bureaucracy .Graphical R epresentation

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Economies and D iseconomies of ScaleEconomies and D iseconomies of Scale

A ssume Q increases 10 units for eac h isoquan t

IR S

K

L

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IR S

K

L

DR S

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LR AC curve

IR S

K

L

DR S

$

Q

IR S DR S

CR S

Q M E S

CR S

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LRMC and LRAC CurvesLRMC and LRAC Curves

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LRAC and LRMCLRAC and LRMC

$

Q

LR ACLR M C

LR M C is ( TC /( Q (i.e., change inTC from a change in Q) whe n all inputs ar e variabl e inputs.When LR M C is above LR AC , itpulls the average up, and vice-

versa.

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Relating SR to LR curvesRelating SR to LR curves

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Relations h ip between SR ATC and LRAC curvesRelations h ip between SR ATC and LRAC curves ..

At Q 1, the S R plant size which givesATC 1 is least costly.

$

Q

LR ACATC 1

Q 1

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Relations h ip between SR ATC and LRAC curves.Relations h ip between SR ATC and LRAC curves.

At Q 1, the S R plant size which givesATC 1 is least costly.SR ATC is tangent to LR AC at onepoint.

$

Q

LR ACATC 1

Q 1

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Adjustments in SR are still more costly t h an LRAdjustments in SR are still more costly t h an LR

At Q 2, the S R plant size which givesATC 1 is no longer least costly.

$

Q

LR ACATC 1

Q 2

atc 1

lrac 1

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Adjustments in SR are still more costly t h an LRAdjustments in SR are still more costly t h an LR

At Q 2, the S R plant size which givesATC 1 is no longer least costly.Optimal move would be to largerplant size!

$

Q

LR ACATC 1

Q 2

atc 1

lrac 1

LRAC is lower envelope of family of SRATCLRAC is lower envelope of family of SRATC

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8 0

LRAC is lower envelope of family of SRATCLRAC is lower envelope of family of SRATCcurvescurves

$

Q

LR ACATC 1 A

TC 3

ATC 2

Q 1 Q 2=Q M E S Q 3

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SRMC and LRMCSRMC and LRMC

q1 q2 q3

SR ATC1

SR ATC2

SR A

TC3

SRMC 1

SRMC 2

SRMC 3 LR A C

LRMC$

q

costs 2

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