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Prof George Alogoskoufis, Dynamic Macroeconomic Theory, 2015

DynamicModelsofInvestment

ConvexAdjustmentCostsandtheDeterminantsofInvestmentunderCertaintyandUncertainty


TheOp>malCapitalStock• Sofarwehavebeenassumingthatfirmschoosetheircapitalstocksothatthemarginalproductofcapitalequalstheusercostofcapital,asdeterminedbytherealinterestrateandtherateofdeprecia>on.

• Thistheoryinfactdeterminestheop>malcapitalstockandnottheamountofinvestment.

• Investmentspecifieshowquicklyafirmmovesfromitscurrenttotheop>malcapitalstock.Whenfirmscanadjusttheircapitalstockimmediatelyandwithoutcost,theflowofinvestmentisnotdefined,asthecapitalstockjumpsimmediatelytoitsop>mallevel.

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AdjustmentCostsandtheInvestmentFunc>on

• Infact,however,thechangeofthecapitalstockinvolvesadjustmentcosts.

• Afirmthatchoosestoraiseitsstockofproduc>vecapitalshouldrentorbuyaddi>onalspace,buyandinstallnewequipment,andtrainemployeestousetheextraequipment.Inaddi>onaltherearedeliverylagsandinstalla>oncosts,makingitmorecostlytoadjustthecapitalstockquickly.Allthesecostsarebeyondthecostofbuyingaddi>onalcapitalgoods.

• Inaddi>on,itistobeexpectedthattheseadjustmentcostswillbeconvex,i.e.theywilldependonthesizeofthenewinvestment.Thehigherthesizeofnewinvestment,thegreaterwillbetheaverageadjustmentcostofinstalling(orde-installing)anaddi>onalunitofcapital.

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TheJorgensonModeloftheFlexibleAccelerator

• Jorgenson(1963)assumedthat,preciselybecauseoftheexistenceofadjustmentcosts,firmsarenotimmediatelybutonlygraduallyadjus>ngtheirstockofcapitaltowardsits“op>mal”level,asdeterminedbytheusercostofcapitalandthemarginalproductofcapital.

• Hethuspostulatedaninvestmentfunc>onwhichdeterminedcurrentinvestmentasafrac>onofthedifferencebetweenthecurrentandthe“op>mal”(desired)capitalstock.

• However,Jorgensondidnotderivethespeedofadjustment,andthustheflowofinvestment,fromafullydynamicop>miza>onproblem.ThiswasaccomplishedlaterbyLucas(1967),Gould(1968)andTreadway(1969),who,insteadofpostula>ngtheinvestmentfunc>on,asJorgensenhaddone,solvedfortheop>malinvestmentfunc>onfromthedynamicproblemofafirmmaximizingthepresentvalueofitsprofits,subjecttoconvexcostsofadjus>ngitscapitalstock.

• Soona]erwards,LucasandPresco_(1971)extendedthisframeworktoexaminethedetermina>onofinvestmentunderuncertainty.

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TheqApproachofTobin:MarketValuetoReplacementValueofInstalled

Capital• Tobin(1969)comparedthera>oofthemarketvalueofinstalledcapitalofafirm,tothereplacementcostofcapital,namingthisra>oq.

• Tobinarguedthatifthealreadyinstalledcapitalstockofafirmhasahighervaluethanthecostofreplacingthecapitalgoodsthatcomposeit,i.e.ifqisgreaterthanone,thenitwillbeprofitableforthefirmtoinvest,i.e.purchaseandinstallnewcapitalgoods.

• Tobinarguedthattherateofinvestmentwillbeanincreasingfunc>onofq,i.ethera>oofthevalueofthealreadyinstalledcapitalstocktoitsreplacementcost(“Tobin’sq”).

• However,muchlikeJorgenson,Tobindidnotderivehisinvestmentfunc>onfromadynamicop>miza>onproblemeither.

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TheAbelandHayashiSynthesis:AdjustmentCostsandtheDetermina>onofq

• Abel(1982)andHayashi(1982)showedthatTobin’s“qtheory”andthetheoryof“adjustmentcosts”forinvestmentofJorgenson,asmodeledbyLucas,Presco_,GouldandTreadway,canbecombinedinaunifiedframework.Thissynthesisofthetwotheoriesisnowconsideredasthemainneoclassicaldynamicmodelofinvestment.

• Thefirmdoesnotchoosethelevelofitscapitalstock,byequa>ngatany>methemarginalproductofcapitaltothesumoftherealinterestrateandthedeprecia>onrate,butitchoosestheamountofinvestment,takingintoaccounttheadjustmentcostsofthecapitalstock.Sincemarginaladjustmentcostsincreasewiththeamountofinvestment,investmentresultsinagradualadjustmentofthecapitalstocktowardsitssteadystatevalue.

• Ontheadjustmentpath,thefirmtakesintoaccountboththecurrentandfutureeffectsofitsinvestmentdecisions.Thus,investmentdependsonbothcurrentandexpectedfuturedevelopmentsinthevalueofthemarginalproductofcapitalandtheusercostofcapital.

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ASimpleDynamicModelofInvestmentinthePresenceofConvexAdjustmentCosts

Weconsideracompe>>vefirmproducingagoodY.Theproduc>onfunc>onofthefirmisgivenby,

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Y (t) = AF K(t)( )

whereAistotalfactorproduc>vityandKthecapitalstock.Theproduc>onfunc>onischaracterizedbydiminishingreturns.Themarketpriceofoutputandthecapitalstockisequaltounity.


ASimpleDynamicModeloftheDetermina>onofInvestment

Inordertochangeitscapitalstock,thefirmmustundertakegrossinvestmentI.Thechangeinitscapitalstockisthusdeterminedby,

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whereδ>0isaconstantdeprecia>onrate.

Weassumethatthecostofgrossinvestmentforthefirmisequalto,

I(t) = K•(t)+δK(t)

I(t)+ψ I(t)( )whereψisaconvexfunc>on,forwhichψ(0)=0,ψ’>0καιψ’’≥0.


TheInstalla>onCostofInvestment

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TheOp>malChoiceofInvestment

Assumethata>me0thefirmchoosesaninvestmentpaththatmaximizesthepresentvalueofcurrentandfutureprofits.Assuminganinfinite>mehorizon,thepresentvalueoftheprofitsofthefirmisequalto,

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V (0) = e−rt Y (t)− I(t)−ψ (I(t))( )t=0

∞

∫ dt

ristherealinterestrate,assumedexogenousandconstant.Thepresentvalueismaximizedundertheconstraintoftheproduc>onfunc>onandtheinvestmentrela>onwhichlinksgrossinvestmenttotheaccumula>onofcapital.ThecurrentvalueHamiltonianisdefinedby,

AF K(t)( )− I(t)−ψ (I(t))( )+ q(t) I(t)−δK(t)( )whereq(t)isthemul>plierofthecapitalaccumula>onconstraint.q(t)istheshadowvalueofanaddi>onalunitofcapitalatt.


FirstOrderCondi>onsfortheOp>malChoiceofInvestment

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Interpreta:on:theshadowvalueofanaddi>onalunitofcapitalqisequaltothemarginalcostofinvestment.Thisisequaltothepurchasepriceofcapitalgoods(assumedequaltounity),plusthemarginalinstalla>oncostψ΄(I(t)).

Interpreta:on:Thefirmwillinvestun>ltheusercostofcapital(onthele]handside)isequaltothemarginalproductofcapital(ontherighthandside).Theusercostofcapitalistherealinterestrate,plusthedeprecia>onrate,minustheexpectedapprecia>onrateofthecapitalstock,mul>pliedbytheshadowvalueofcapital.

q(t) = 1+ ′ψ I(t)( ) = 1+ ′ψ K•(t)+δK(t)⎛

⎝⎞⎠

r +δ − q•(t)q(t)

⎛

⎝⎜⎜

⎞

⎠⎟⎟q(t) = A ∂F K(t)( )

∂K(t)= AFK K(t)( )


TheCaseofZeroAdjustmentCosts

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Thesearetheusualfirstordercondi>onswehaveu>lizedsofar.

ThevariablesqandKjumpimmediatelytotheirequilibriumvalues.

Theshadowvalueofcapitaliscon>nuouslyequaltounity,i.e.thepurchasepriceofcapitalgoods,andthecapitalstockadjustsimmediatelytothelevelwherethemarginalproductofcapitalisequaltotherealinterestrateplusthedeprecia>onrate,r+δ.

Thereisnoinvestmentflow,asthecapitalstockadjustsimmediately.Withoutadjustmentcosts,thismodeldoesnotdeterminegrossinvestment,butonlytheequilibriumcapitalstock.

q(t) = 1

AFK K(t)( ) = r +δ


TheInvestmentFunc>oninthePresenceofAdjustmentCosts

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Grossinvestmentdependsonlyonthedifferenceoftheshadowpriceofinstalledcapitalqfromunity,asassumedbyTobin.

Thisdependenceisposi>vebecausethemarginalcostofinvestmentisposi>ve.

Forthisreason,thetheorythatdependsonarisingadjustmentcostoninvestmentisreferredtoastheqtheoryofinvestment.

Inthegeneralcase,wherethereisastrictlyconvexadjustmentcostfunc>onforthecapitalstock,investmentisaposi>vefunc>onofq-1.Solvingthefirstordercondi>onforI,wegetthat,

I(t) = ′ψ( )−1 q(t)−1( )


TheDeterminantsofMarginalq

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Thisisafirstorderlineardifferen>alequa>onwithvariablecoefficients,whosesolu>onis,

qisequaltothemarginalcostofinvestment.Wehavealreadyusedthiscondi>ontoderivetheinvestmentfunc>on.Toanalyzethedeterminantsofq,wemustlookintothesecondfirstordercondi>onfortheusercostofcapital.Thiscanbere-wri_enas,

q•(t) = (r +δ )q(t)− AFK K(t)( )

q(t) = e−(r+δ )(s−t )As=t

∞

∫ FK K(s)( )dsqisthepresentvalueofallfuturemarginalproductsofcapital.Asaresult,qdependsnega>velyontherealinterestrateandthedeprecia>onrate,aswellasfactorsthatreducethemarginalproductofcapital,suchasthecapitalstock.qdependsposi>velyonfactorsthatincreasethemarginalproductofcapital,suchastotalfactorproduc>vity.


TheDynamicEvolu>onofqandtheCapitalStockΚ

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Foraconstantshadowpriceofcapitalq,thesecondofthecondi>onsimpliesthat

Thedetermina>onoftheshadowpriceofanaddi>onalunitofcapitalq,andthestockofcapitalK,canbeinferredfromthetwofirstordercondi>ons.Sincebothofthesearenonlineardifferen>alequa>onsarenonlinear,theirsolu>oncanbedescribedbyaphasediagram.

ForaconstantcapitalstockKthefirstofthecondi>onsimpliesthat,

Thesteadystateisdeterminedattheintersec>onofthetwocurves.Thissteadystateequilibriumisasaddlepoint,sinceqisanonpredeterminedvariableandKisapredeterminedvariable.Theadjustmentpathisunique.

q = 1+ ′ψ δK( )

q = 1r +δ

AFK (K )


TheDynamicAdjustmentofqandtheCapitalStockΚ

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EffectsofaPermanentIncreaseintheRealInterestRate

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EffectsofaPermanentIncreaseinTotalFactorProduc>vity

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InvestmentandAdjustmentCostsfortheCapitalStock

• Wehaveanalyzedthebasicneoclassicalmodelofinvestmentwithconvexadjustmentcostsofinvestment.

• Thismodelcanbegeneralizedsothattheadjustmentcostfunc>ondependsnotonlyongrossinvestment,butalsoonthestockofcapital.Itcanalsobegeneralizedtosimultaneouslyanalyzeinvestmentandlabordemand.

• Itcanalsobegeneralizedtoallowforproductmarketimperfec>ons.

• Finally,itcanbegeneralizedtothecaseofuncertainly(LucasandPresco_1971).

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TheAppropriateDiscountRateforChoosingOp>malInvestmentunderUncertainty

• Justasundercertainty,weshallassumethatthefirmchoosesitsinvestmentpathinordertomaximizeitsvaluetoitsowners.Thevalueisequaltothepresentvalueoftheprofitsthatthefirmgenerates.

• Whereasundercertaintytheproblemofthemaximiza>onofthepresentvalueoftheprofitsofthefirmiseasilydefined,underuncertainty,theques>onthatarisesiswhatshouldbethediscountrateatwhichfirmsshoulddiscountfutureprofits.

• Inprac>ce,itiso]enassumedthatfirmsmaximizethepresentdiscountedvalueofprofitsbyusingadeterminis>cdiscountrate.,butthiscanonlybejus>fiedunderveryspecificassump>ons.

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MaximizingtheValueofaFirmunderUncertainty

LetusassumethatVtisthevalueofthefirminperiodt,andΠtisitsperperiodrevenue,netofinvestmentexpenditures.Thentherateofreturn1+πtfromholdingthefirmforoneperiod,willbegivenby,

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1+π t =Vt+1 +Πt+1

VtForaconsumerthatinvestsinthefirmunderuncertainty,therateofreturnfromholdingthefirm1+πt,andhenceVtandΠtmustsa>sfy,

′u (Ct ) =1

1+ ρEt 1+π t( ) ′u (Ct+1)⎡⎣ ⎤⎦ =

11+ ρ

EtVt+1 +Πt

Vt

⎛⎝⎜

⎞⎠⎟

′u (Ct+1)⎡

⎣⎢

⎤

⎦⎥


TheAppropriateDiscountFactorandtheMarginalU>lityofConsump>on

• Thiscondi>onisthesameasthefirstordercondi>onforinves>ngina“risky”asset,intheconsumerproblemunderuncertainty.

• Thereturnsgeneratedbythefirmineachstateofnatureareweightedbythemarginalu>lityofconsump>oninthatstate.

• Thus,thediscountfactorthatmustbeappliedmusttakeintoaccountthecorrela>onofthefirm’sprofitswiththemarginalu>lityofconsump>onateachstate.

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TheValueoftheFirmunderUncertainty

Solvingthefirstordercondi>onrecursivelyforward,assumingawaybubbles,wegetthe“fundamentalsolu>on”forthevalueofthefirmVtas,

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Vt = Et1

1+ ρ⎛⎝⎜

⎞⎠⎟s=1

∞∑s

′u (Ct+s )′u (Ct )

Πt+s

⎛

⎝⎜

⎞

⎠⎟

Thevalueofthefirmisequaltothepresentdiscountedvalueofexpectedfutureprofits.Thediscountrateforeachperiodandforeachstateofnatureisthemarginalrateofsubs:tu:onbetweenconsump>onat>metandconsump>onatthatperiodandthatstateofnature.Thehigherthecorrela>onbetweenafirm’sprofitsandconsump>on,thehigherwillbethediscountfactorapplied,andthelowerthevalueofthefirm.


MaximizingtheValueofaFirmUsingaDeterminis>cDiscountRateInprac>ce,itiso]enassumedthatfirmsmaximizethepresentdiscountedvalueofprofitsbyusingadeterminis>cdiscountrate.Inthiscase,

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Vt = Et1

1+ rt+zz=1

s∏⎛⎝⎜⎞⎠⎟s=1

∞∑ Πt+s

⎛

⎝⎜⎞

⎠⎟

wherert+zisadeterminis>cinterestrateinperiodt+z.Althoughwidelyused,aspecifica>onsuchasthisisgenerallyinappropriate,becauseitsuggeststhatateachdate,thesamediscountfactorisusedtoevaluatereturnsindifferentstatesofnature.Thiscanonlybejus>fiedunderveryspecificassump>ons.


TheCaseofRiskNeutralityOnesetofassump>onsthatcanbeusedtojus>fyitistheassump>onofriskneutralityonthepartofconsumers.Ifconsumersareriskneutral,sotheiru>lityislinearinconsump>onandtheirmarginalu>lityofconsump>onisconstant,thenthediscountrateisnotonlydeterminis>c,butalsoconstant,andequaltothepurerateof>mepreferenceρ.Thus,inthecaseofriskneutralitythevalueofthefirmsimplifiesfurtherto,

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Vt = Et1

1+ ρ⎛⎝⎜

⎞⎠⎟s=1

∞∑s

Πt+s

⎛

⎝⎜

⎞

⎠⎟


TheCaseofIndependencebetweenInvestmentDecisionsandtheRela>veDistribu>onof

ReturnsAcrossStatesofNature

Anothersetofassump>onsthatcanbeusedtojus>fyadeterminis>cdiscountrateistoassumethatinvestmentdecisionsdonotaffecttherela>vedistribu>onofreturnsacrossstatesofnature,butonlythescaleofthefirm.Inthiscasethefirmcanuseaconstantdiscountrate,equaltotheriskfreerate,plusariskpremiumthatreflectsthespecificriskassociatedwiththefirm’sac>vi>es.

Bothsetsofassump>onsareunlikelytoholdingeneral,buttheyareo]enusedasconvenientapproxima>ons.Itisworthno>nghoweverthattheyaregoodapproxima>onsonlywhenconsidera>onsofriskaversionarenotcentraltotheproblemanalyzed.

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TheLucasandPresco_ModelofInvestmentunderUncertainty

Wenextturntoanexamina>onoftheinvestmentdecisionsofacompe>>vefirmunderuncertainty,assumingthattheobjec>veofthefirmistomaximizevaluewithadeterminis>cdiscountrate.

Themodelweanalyzeisalinearquadra>cvariantoftheclassofmodelsintroducedbyLucasandPresco_(1971),andissimilarinmanyrespectstotheqmodelweanalyzedundercertainty.

Weshallusethismodeltoderiveaggregateinvestmentinara>onalexpecta>onsequilibrium.

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TheInvestmentDecisionofaCompe>>veFirm

Weassumeacompe>>vefirmithattakesmarketpricesasgiven.Itsprofitinperiodtisdefinedby,

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Πit = ptYit − Iit −ψ2Iit( )2⎛

⎝⎜⎞⎠⎟

Outputisproducedusingcapital,throughalinearproduc>onfunc>onoftheform,

Yit = AKit

Theevolu>onofthecapitalstockisdeterminedby,

Kit+1 = Iit + (1−δ )Kit


TheIndustryandtheRepresenta>veFirm

iiscon>nuousintheinterval[0,1].Industryoutputisthusgivenby,

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Yt = Yiti=0

1

∫ di

Sinceallfirmsfacethesametechnologyandthesamemarketprices,theoutputofallfirmswillbethesame.Thus,fromnowonwetreatYastheoutputoftherepresenta>vefirm.Thesamegoesforallothervariables,suchasinvestmentandthecapitalstock.


TheProblemoftheRepresenta>veFirm

Therepresenta>vefirmmaximizesitspresentvalue,

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Vt = Et11+ r

⎛⎝⎜

⎞⎠⎟s=1

∞∑s

Πt+s

⎛

⎝⎜⎞

⎠⎟= Et

11+ r

⎛⎝⎜

⎞⎠⎟s=1

∞∑s

pt+sAKt+s − It+s −ψ2It+s( )2⎛

⎝⎜⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟

subjecttothesequenceofaccumula>onequa>onsoftheform,

Kt+1 = It + (1−δ )Kt

Thestochas>cprocessesdrivingthemarketprice,therela>vepriceofcapitalgoodsandtotalfactorproduc>vityaretakenasgiven.


TheLargangianandtheFirstOrderCondi>ons

TheLagrangianfortheproblemoftherepresenta>vefirm,

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Et11+ r

⎛⎝⎜

⎞⎠⎟s=1

∞∑s

pt+sAKt+s − It+s −ψ2Iit+s( )2⎛

⎝⎜⎞⎠⎟ + qt+s It+s + (1−δ )Kt+s − Kt+s+1( )⎛

⎝⎜⎞⎠⎟

⎛

⎝⎜⎞

⎠⎟

Fromthefirstordercondi>onsforamaximumwegetthetwofamiliarfirstordercondi>ons,withthefamiliarinterpreta>ons.

qt = 1+ψ It

(1+ r)qt − (1−δ )Etqt+1 = Et pt+1A


TheDetermina>onofqandindustryinvestmentI

Thetwofirstordercondi>onsimplythat,

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It =1ψ

qt −1( ) qt =11+ r

Et1−δ1+ r

⎛⎝⎜

⎞⎠⎟s=0

∞∑s

pt+s+1A

Investmentdependsposi>velyonthedifferenceofqfromunity,whichisthepurchasepriceofinvestmentgoods.

qturnsouttobethediscountedvalueofallexpectedfuturevaluesofthemarginalproductofcapital.Itdependsposi>velyontheexpectedfutureevolu>onoftherela>vepricefortheproductofthefirmandthemarginalproduc>vityofcapitalA.Italsodependsnega>velyonthediscountraterandthedeprecia>onrateδ.


TheDeterminantsofInvestmentandtheCapitalStock

Subs>tu>ngforq,investmentthusdependsposi>velyonthediscountedvalueofallexpectedfuturechangesinthevalueofthemarginalproductofcapital,andnega>velyontherealinterestrate,thedeprecia>onrateandtheadjustmentcostparameterψ.

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Fromthedefini>onofinvestment,thecapitalstockthusevolvesaccordingto,

It =1ψ

1(1+ r)

Et1−δ1+ r

⎛⎝⎜

⎞⎠⎟s=0

∞∑s

pt+s+1A⎛

⎝⎜⎞

⎠⎟−1

⎛

⎝⎜

⎞

⎠⎟

Kt+1 = (1−δ )Kt +1ψ

1(1+ r)

Et1−δ1+ r

⎛⎝⎜

⎞⎠⎟s

s=0

∞∑ pt+s+1A⎛

⎝⎜⎞

⎠⎟−1

⎛

⎝⎜

⎞

⎠⎟


FromIndustryDemandtoEquilibriumPrices

Althoughforeachcompe>>vefirmthepriceofoutputistakenasgiven,fortheindustry,themarketpricewillbedeterminedendogenously,fromtheequa>onoftotaldemandforitsproductandindustrysupply.Industrysupplywilldependoninvestmentandtheevolu>onofthecapitalstock.Thisallowsustosolvefortheequilibriumpriceendogenously,asafunc>onofthecapitalstock,andcharacterizingtheevolu>onofthecapitalstockandtheequilibriumpriceinara>onalexpecta>onsequilibrium.

Assumethatindustrydemandislinearinthepriceandgivenby,

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Yt = D − bpt + vt

pt =1bD −Yt + vt( ) = 1

bD − AKt + vt( )

Then,theoutputpriceisdeterminedby,


TheDeterminantsofInvestmentinRa>onalExpecta>onsEquilibrium

Wecanusethepriceequa>ontosubs>tutefortheexpectedequilibriumpriceinthecapitalaccumula>onequa>on,andsolvefortheevolu>onofthecapitalstockasafunc>onofonlyexogenousshocks.Thisresultsin,

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where,λ<1,and,

λ + µ =ψ (1+ r) 1+ (1−δ )2( )+ A2

ψ (1+ r)(1−δ )> 2 λµ = 1+ r

Kt+1 = λKt +λ

ψ (1−δ )(1+ r − λ)ADb

− (r +δ )⎛⎝⎜

⎞⎠⎟ +

Ab

λψ (1−δ )(1+ r)

Etλ1+ r

⎛⎝⎜

⎞⎠⎟s

vt+s+1s=0

∞∑


TheDeterminantsofEquilibriumInvestmentandtheCapitalStock

Theevolu>onoftheindustrycapitalstockinra>onalexpecta>onsequilibriumdependsoncurrentexpecta>onsaboutthewholefuturepathofdisturbancestoindustrydemandv,andparameterssuchasthediscountrater,theproduc>vityofcapitalΑ,theadjustmentcostparameterψ,thedeprecia>onrateδ,thesizeofthemarketDandthepriceresponsivenessofindustrydemandb.

1-λ,thespeedofadjustment,dependsonlyonthediscountrateandtechnologicalparameters.

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TheStructureoftheLucasandPresco_Model

• Therepresenta>vefirmchoosesinvestment,andimplicitlythecapitalstockandoutput,tomaximizethepresentvalueofitsprofits,takingasgiventhemarketpriceofitsoutput,theexogenousrela>vepriceofcapitalgoodsandexogenousproduc>vityA.

• Inordertocomputethefullequilibrium,onceinvestmentandoutputoftherepresenta>vefirmaredetermined,industryoutputisreplacedintheindustrydemandfunc>ontosolvefortheequilibriumpriceintermsoftheexogenousstochas>cprocessdrivingindustrydemand.

• Thefullequilibriumisthusdescribedbyapairofinterrelatedcapitalaccumula>onandpriceequa>ons,whichareconsistentwithcon>nuousmarketclearing.

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