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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(
),(
Cost Minimization:LongRun
Tosolvetheproblemgraphically,weneedtouseanewconcept:theisocostline.
Theisocostlinerepresentstheinputcombinationsthatcostthesame.
Cost Minimization:LongRun
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
Cost Minimization:LongRun
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
Interiorsolution:We solve the system formed by
Andwe obtain the conditional demands ofinputs:
LKKLFb =),()(
QLKQKLF
rwLKrwKLMRTS
=Þ=
=Þ=
),(
///),(
rwQK
wrQL == *;*
Cost Minimization inthe LongRun:Examples
Interiorsolution: Wesolvethesystemformedby
Theconditionaldemandsofinputsare:
3),()( LKKLFc =
QLKQKLF
rwLKrwKLMRTS
=Þ=
=Þ=3),(
///),(
rwQK
wrQL 2/32/3 *;* ==
Cost Minimization inthe LongRun:Examples
Interiorsolution:We solve the system formed by
Andwe obtain the conditional demands ofinputs:
{ }KLKLFd ,2min),()( =
{ } QKLQKLFKL
=Þ==
,2min),(2
QKQL == *;2
*
Cost Minimization inthe LongRun:Examples
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
Cost Minimization inthe LongRun:Examples
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)
Costs andEconomies ofScale:Examples
Intheexamplesabove,forw=1 andr=4 wehave:
QMC(Q)QATC(Q)
Q ) ,K*(Q, ) , L*(Q,(Q) T
LKF(L,K)
/2;/4
441441C
(a)
==
=+=
=
Costs andEconomies ofScale:Examples
4;4441441C
(b)
===+=
=
MC(Q)ATC(Q)Q ) ,K*(Q, ) , L*(Q,(Q) T
LKF(L,K)
Costs andEconomies ofScale:Examples
2/12/1
2/3
3
6;4441441C
(c)
QMC(Q)QATC(Q)Q ) ,K*(Q, ) , L*(Q,(Q) T
LKF(L,K)
==
=+=
=
Costs andEconomies ofScale:Examples
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)
Costs andEconomies ofScale:Examples
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
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
Cost Curvesinthe ShortRun
AVCminimization:dAVC(Q)/dQ=0d(VC/Q)/dQ=(1/Q)(dVC/dQ)– VC/Q2=0Therefore,attheminimumpointoftheAVC,itholdsthat:AVC=MC
MC
AVC
AVC, MC
Q
Cost Curvesinthe ShortRun
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.
Cost Curvesinthe Shortandthe LongRun
ATCL
ATC(Q)
Q
Inthelongrun,thecapitalisvariable.Theaveragetotalcostinthelongrunisthe“envelope”oftheaveragetotalcostcurvesintheshortrun.
Cost Curvesinthe Shortandthe LongRun
ATCL
ATC(Q),MC(Q)
Q
Intheshortrun,capitallevel cannot bemodified.The greencurvesdescribethe marginalcost inthe shortrun for K1 <K2 <K3
MCS1 MCS2 MCS3
Cost Curvesinthe Shortandthe LongRun
ATCL
ATC(Q),MC(Q)
Q
The marginalcost inthe long run is the envelope ofthe marginalcost functions inthe shortrun.
Cost Curvesinthe Shortandthe LongRun
ATCL
Q
ATC(Q) Inthis example,the ATCinthe long run is constant:there areneither economies nor diseconomies ofscale.
Cost Curvesinthe Shortandthe LongRun
MCS1 MCS2 MCS3
MC(Q)
Q
MCL=ATCL
If there areneither economies nor diseconomies ofscale,MCLis the same asATCL
Cost Curvesinthe Shortandthe LongRun
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.
Cost Curvesinthe Shortandthe LongRun
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