profit maximization and cost minimization · 2017-04-20 · cost minimization: short run in our...

Theoryofthefirm

ProfitMaximizationand

CostMinimization

TheProblemoftheFirmWeconsiderafirmproducingasinglegoodQ, usinglabour(L)andcapital(K),andatechnologydescribedbytheproductionfunction,F(L,K).

Thefirmisapricetaker competitiveinthelabourandcapitalmarkets,inwhichthepricesarewand r,respectively.(Thisassumptionisreasonableifthelabourandcapitalmarketsarelargerelativetothefirm’soutputmarket.)

Letp denotethemarketpriceofgoodQ.

ProblemoftheFirmThefirm’sprofitmaximizationproblemis:

maxpQ– wL– rKs.t.

F(L,K)≥QQ≥0,L≥0,K≥0.

HerepQ isthefirm’srevenue,andwL+rKisthecostoftheinputsusedbythefirm.

Whatarethefirm’sdecisionvariables?Q,L,K,p?

ProblemoftheFirm

Inacompetitivemarket,thesupplyofasinglefirmisverysmallcomparedtothemarketsupply.Inthiscase,asinglefirmhasanegligibleimpactonthemarketprice,p,andisthefeforereasonabletoassumethatthefirmactsasaprice-taker.

Butifthefirm’soutputislargerelativetothemarketsupply,thatis,ifthefirmhasmarketpower,thenassumingthatthefirmactsasapricetakerwouldbeamistake.

Fornow,letuspostponetheprofit-maximizationproblemandletustreatthe“internal”problemofthefirmtakingtheproductionlevelasgiven:Q0.

FixingQ0,thentheobjectiveofmaximizingprofitsimplies,asanintermediateobjective,minimizingthecostofproducingthelevelQ0.

CostMinimization

CostMinimization

Thereareseveraltypesofcostconcepts:

Accounting cost:purchasepricenetofdepreciation.

Opportunitycost:valueofthebestalternativeuse.

SunkCost:unrecoverablecostsasociatedwithpastdecisions.

CostMinimization

Fromaneconomicpointofview,relevantcostsareopportunitycost.Sunk costs areirrelevantinmakingoptimaldecisions.

Example: Afirmownsabuildingthatisnotbeingusedintheproductionprocess.Asthefirmdoesnotpayanyrent,thereisnoaccountingcost.Nevertheless,theopportunitycostisgreaterthanzero– thebuildingmaybeputupforrent.

CostMinimization

ShortRunandLongRunCost

Longrun:allinputsarevariable.

Shortrun:someinputsarefixed– capital,forexample.

FixedandVariableCosts

Thevariablecostisthecostoftheinputsthatmaybevariedintheshortrundependingonthedesiredlevelofoutput,whereasthefixedcostisthecostofthoseinputsthatarefixedintheshortindependentlyofthelevelofoutput.

CostMinimization

MaxL³0,K³0 pQ0 -wL - rK

MaxL³0,K³0 - (wL+ rK)

MinL³0,K³0 wL+ rK

SincepQ0 isaconstant,thisproblemisequivalentto:

Whichinturnsisequivalentto:

CostMinimization:ShortRun

Inourframeworkthereareonlytwoinputs,labourandcapital.IfweassumecapitalisfixedK0intheshortrun,thentheshortruncost-minimizationproblemis

minwL– rK0s.t.F(L,K0)≥Q,L≥0.

Here rK0 isthefixedcost(FC).

Cost Minimization:ShortRun

Thesolutiontothisprobleminvolvesusingamountoflabor(theonlyvariableinput)thatsolvestheequation

F(L,K0)=Q.

Thatis,thesolutiontothecostminimizationconsistofchoosingtheminimumamountoflaborthatallowsforproducingQ unitsofthegoodgiventhatwehaveK0 unitsofcapital.

Bysolvingthisequation,wefindtheshortrunconditional labordemand

L*=L(K0,Q).

CostMinimizationintheShortRun

Example. Afirm’sproductionfunctionis

F(L,K)=(LK)1/2

ItscapitalisfixedintheshortruntoK0=36.Henceitsshortrunproductionfunctionis

F(L,36)=(L36)1/2 =6L1/2.

Thereforeitsshortrunconditionaldemandoflaboris

L(Q)=Q2/36.

CostMinimization:LongRun

Inthelongrun,bothinputs,labourandcapital,arevariable.Thus,thecost-minimizationproblemcanbewrittenas

Solvingtheproblem,wefindtheconditional demandfunctionsofinputs:

L*=L(w,r,Q)and K*=K(w,r,Q)

minL³0,K³0

wL + rK

F(L,K) ³Q

Cost Minimization:LongRun

Asinthe consumer theory,the cost-minimization problemmay have interiorand/or corner solutions,depending onthe features ofthe production function.

(a) Interiorsolution

QKLFrwKLMRTS

=

=

),(

),(

Cost Minimization:LongRun(b)Cornersolution(b1)Onlycapitalisused(L*=0)

(b2)Onlylabourisused(K*=0)

QLFrwKLMRTS

=

³

)0,(

),(

QKFrwKLMRTS

=

£

),0(

),(


Tosolvetheproblemgraphically,weneedtouseanewconcept:theisocostline.

Theisocostlinerepresentstheinputcombinationsthatcostthesame.


wL+rK=C Thecostincreasesalongallthenortheastdirections:C1<C2

L

K

C / w

C / r

C1C2

L

K

C1 /r

C2 /r

C1 /w C2 /w

Slope = w/r


A

F(L,K)=Q

L

K

C / r

C / w

K*

L*


K*=0 L*=0

L

K

B

F

L

K

F

C


Inputsubstitutionwhenthepriceofoneoftheinputschanges

C2

K2

L2

B

C1

K1

L1

A

F

L

KIfthelaborpriceincreases,theisocostlinebecomessteeper.Slope =w/r

TheoptimalchoicenowisB. Thefirmreactstotheincreaseofthelaborpriceusingmorecapitalandlesslabor.

Interiorsolution:We solve the system formed by

Andwe obtain the conditional demands ofinputs:

LKKLFa =),()(

QLKQKLFrwLKrwKLMRTS

=Þ==Þ=

),(///),(

QrwKQ

wrL == *;*

Cost Minimization inthe LongRun:Examples



LKKLFb =),()(

QLKQKLF

rwLKrwKLMRTS

=Þ=

=Þ=

),(

///),(

rwQK

wrQL == *;*


Interiorsolution: Wesolvethesystemformedby

Theconditionaldemandsofinputsare:

3),()( LKKLFc =

QLKQKLF

rwLKrwKLMRTS

=Þ=

=Þ=3),(

///),(

rwQK

wrQL 2/32/3 *;* ==




{ }KLKLFd ,2min),()( =

{ } QKLQKLFKL

=Þ==

,2min),(2

QKQL == *;2

*


L

K

A

L*

K*


Cornersolution:Inthiscasetheconditionaldemandsofinputsare:

KLKLFe 2),()( +=

ïþ

ïý

ü

ïî

ïí

ì

=Î><

=2/1/],0[

2/1/02/1/

*rwifQ

rwifrwifQ

Lg ï

þ

ïý

ü

ïî

ïí

ì

=->

<=

2/1/2/)(2/1/2/

2/1/0*

rwifQrwifQ

rwifK

g


CostMinimizationintheLongRun:Examples

F(·)=L+2K,w=1, r=3 F(·)=L+2K,w=r=1

L

K

A L

K

B

CostfunctionsThetotalcostfunctionistheminimumcostofproductionforagivenlevelofoutputQ,andinputpricesw andr:

C(Q,w,r)=wL(Q,w,r)+rK(Q,w,r).

Forgiveninputprices,thelongruntotalcostislessthanorequaltothetotalcostintheshortrun– why?

Thetotalcostmaybedecomposedasthesumofthevariablecost(thecostofthevariableinputs),VC(Q,w,r),andthefixcost(thecostofthefixinputs),FC,whichisindependentofthelevelofoutput.

C(Q,w,r)=VC(Q,w,r)+FC=wL0(Q,w)+rK0

Inthelongrunthetotalandvariablecostcoincide.

CostfunctionsTheaverage(total)costmeasuresaveragecostofproduction,

AC(Q,w,r)=C(Q,w,r)/Q.Forgiveninputprices,thelongrunaveragecostislessthanorequaltotheshortrunaveragecost.

Likewise,theaveragevariablecostis

AVC(Q,w,r)=VC(Q,w,r)/Q.

Inthelongruntheaveragetotalandvariablecostscoincide.

Theaveragetotalcostcanbedecomposeasthesumoftheaveragevariablecostandtheaveragefixedcost

AC(Q,w,r)=AVC(Q,w,r)+FC/Q.

CostfunctionsThemarginalcostmeasuresthecostincreaseduetoamarginal(infinitesimal)increaseofoutput,

MC(Q,w,r)=dC(Q,w,r)/dQ.

Forgiveninputpricesthelongrunmarginalcostcostmaybelargerorsmallerthanthetheshortrunmarginalcost.

Economies ofScaleEconomiesofscale: costincreaseslessthanproportionallywiththelevelofoutput;thatis,forλ>1,

C(λQ)<λC(Q).

Equivalently,theaveragecostdecreaseswiththelevelofoutput;thatis,

dAC(Q,w,r)/dQ<0.

Ifthefirm’stechnologyexhibitsincreasingreturnstoscale,thenthefirmhaseconomiesofscale.

Economies ofScaleDiseconomiesofscale:costincreasesmorethanproportionallywiththelevelofoutput;thatis,forλ>1,

C(λQ)>λC(Q).

Equivalently,theaveragecostincreaseswiththelevelofoutput;thatis,

dAC(Q,w,r)/dQ>0.

Athefirm’stechnologyexhibitsdecreasingreturnstoscale,thefirmhasdiseconomiesofscale.

Economies ofScale

Economiesofscalemayresultfromtheexistenceoffixedcosts,forexample.

TC

ATCFixedCost

Q

TC,ATC

Economies ofScaleConstanteconomiesofscale:costincreasesproportionallywiththelevelofoutput;thatis,forλ>1,

C(λQ)=λC(Q).

Equivalently,theaveragecostdecreasesisconstant;i.e.,

dAC(Q,w,r)/dQ=0.

Inthefirm’stechnologyexhibitsconstantreturnstoscale,thenthefirmhasconstanteconomiesofscale.

Economies ofScale

EconomiesanddiseconomiesofscaleWITHOUTfixed costs

Economiesofscale(concavecost)

Diseconomiesofscale(convexcost)

Q

TC

Costs andEconomies ofScale:Examples

Intheexampleabove,forw=1 andr=1 wehave:

18;36

36;36

3636

2

0

QMC(Q)Q

QATC(Q)Q L*(Q) TC(Q)

LKF(L,K)

=+=+=+=

=

0 1 2 3 4 535.5

36.0

36.5

Q

CT(Q)

0 10 200

5

10

15

Q

CTMe(Q), CMa(Q)


Intheexamplesabove,forw=1 andr=4 wehave:

QMC(Q)QATC(Q)

Q ) ,K*(Q, ) , L*(Q,(Q) T

LKF(L,K)

/2;/4

441441C

(a)

==

=+=

=


4;4441441C

(b)

===+=

=

MC(Q)ATC(Q)Q ) ,K*(Q, ) , L*(Q,(Q) T

LKF(L,K)


2/12/1

2/3

3

6;4441441C

(c)

QMC(Q)QATC(Q)Q ) ,K*(Q, ) , L*(Q,(Q) T

LKF(L,K)

==

=+=

=


5.4;5.45.441441C

},2min{ (d)

===+=

=

MC(Q)ATC(Q)Q ) ,K*(Q, ) , L*(Q,(Q) T

KLF(L,K)

0 2 4 60

2

4

6

Q

CTMe(Q), CMa(Q)

0 1 2 30

5

10

Q

CT(Q)


1;141441C

2 (e)

===+=

+=

MC(Q)ATC(Q) Q) ,K*(Q, ) , L*(Q,(Q) T

KLF(L,K)

0 1 2 3 4 50

2

4

Q

CT(Q)

0 1 2 3 4 50

1

2

Q

CTMe(Q), CMa(Q)

Cost Curvesinthe ShortRun

FC

VC

TC

Q

TC, VC,FC

Total cost is the vertical sum of

FC and VC.

Variable cost increases withthe production

Fixed cost does not change with the production

TCTC

Q

ATC=slopeof0B.

MC=slopeofCB

B

C

0


ATCminimization:dATC(Q)/dQ=0d(TC/Q)/dQ=(1/Q)(dTC/dQ)– TC/Q2=0

Therefore,attheminimumpointoftheATC,itholdsthat:ATC=MC

ATC

MC

Q

ATC, MC


VCVC

Q

B

0

AVCistheslopeof0B.

MCisthetangentofVCatB


AVCminimization:dAVC(Q)/dQ=0d(VC/Q)/dQ=(1/Q)(dVC/dQ)– VC/Q2=0Therefore,attheminimumpointoftheAVC,itholdsthat:AVC=MC

MC

AVC

AVC, MC

Q


Q

ATC, AVC, AFC

ATC

AVC

AFC

ATC=AFC+AVC


Cost Curvesinthe ShortRunATC, AVC, AFC,MC

Q

AFC

AVC

ATCMC

TheminimumvalueofATCislocatedaboveandtotherightoftheminimumvalueofAVCbecauseATC>AVCandMCisincreasing.

DiseconomiesofScaleEconomiesofScale

ATC

ATC

Q

Inthelongrun,firmshaveeconomiesofscaleforrelativelylowproductionlevels,anddiseconomiesofscaleforhighproductionlevels.TheaveragetotalcostisU-shaped.Intheshortrun,ATCisU-shapedtoo,butthisiscausedbytheincreasinganddecreasinginputreturns.

Cost Curvesinthe LongRun

MC<ATC→ATCisdecreasing

MC>ATC→ ATCisincreasing

MC=ATC→ ATCisminimized

ATC

MC

Q

ATC,MC

Cost Curvesinthe LongRun

Cost Curvesinthe Shortandthe LongRun

ATCS1 ATCS2 ATCS3

Q

ATC(Q) Intheshortrun,thecapitallevelcannotbemodified.ThethreecurvesofthegraphdescribetheaveragetotalcostintheshortrunforK1<K2 <K3.


ATCL

ATC(Q)

Q

Inthelongrun,thecapitalisvariable.Theaveragetotalcostinthelongrunisthe“envelope”oftheaveragetotalcostcurvesintheshortrun.


ATCL

ATC(Q),MC(Q)

Q

Intheshortrun,capitallevel cannot bemodified.The greencurvesdescribethe marginalcost inthe shortrun for K1 <K2 <K3

MCS1 MCS2 MCS3


ATCL

ATC(Q),MC(Q)

Q

The marginalcost inthe long run is the envelope ofthe marginalcost functions inthe shortrun.


ATCL

Q

ATC(Q) Inthis example,the ATCinthe long run is constant:there areneither economies nor diseconomies ofscale.


MCS1 MCS2 MCS3

MC(Q)

Q

MCL=ATCL

If there areneither economies nor diseconomies ofscale,MCLis the same asATCL


ATCS ATCL

Q

ATC(Q) Inthis example,there areeconomies anddiseconomies inthe long run.

For each given level ofK,there is alevel ofQ(for which Kis the optimalamount ofKinthe long run)inwhich ATCSis tangent to ATCL.

The minimum values ofATCSarenot on the ATCLcurve.


MCSATCS

MCL

ATCL

ATCL(Q),MCL(Q),

ATCS(Q),MCS(Q)

Q

ATCSistangenttoATCLatQ*suchthatMCS=MCL

Q*

ShortRun andLongRun

• EveryfixedinputintheSRrepresentstheresultsoftheLRdecisionsmadepreviouslybyfirms.ThesepreviousLRdecisionsareafunctionoftheforecastabouttheamountofgoodthatwouldbeprofitabletoproduce.

• DecisionsmadeintheSRandintheLRareverydifferent.

• TheperiodtodifferentiatebetweentheSRandtheLRdependsonthesector.

Expansion path inthe long run

L

K

L2

Q2

K2

Q1

L1

K1A

BL

C2

Assume afirm wants to increaseits production level from Q1 toQ2.

Inthe long run,all the factorsarevariable.The firm increasesthe capitalfrom K1 to K2 andthelabour from L1 to L2.The costincreases from C1 to C2.

C1

ShortRun andLongRun:Expansion Path

ShortRun andLongRun:Expansion Path

Expansion pathin the short run

L

K

A

BL

BC

Q2

Q1

AssumethatintheshortrunthecapitalisfixedinK1.

ToincreasetheproductiontoQ2 thefirmhastoincreasethelabourlevelfromL1 toL3.ThecostincreasesfromC1 toC3.C3 ishigherthanC2 becauseinthelongrun,thefirmcansubstitutelabourbycapital,thatisrelativelycheaper.

C3

C2

C1

K2

K1

L1 L2 L3

profit maximization and cost minimization · 2017-04-20 · cost minimization: short run in our...

Documents