D oes publicinvestmentreduceprivateinvestment
risk?A realoptionapproach
¤
B runoCR U Z
yA udeP O M M ER ET
zA bstract
In this paper, the publicinvestmentprovision takes place in a stochasticenvi-ronnement. T heroleofthegovernmentis toremoveapartoftheuncertaintyfaced by the …rm. Ifthe governmentsimply maximizes the value ofthe …rm, then the optimaltaxis smallerunderimperfectcompetitionthenitisunderperfectcompeti-tionsincemorepubliccapitalreduces thesellingprice. B utifthegovernmentseeks tomaximize theconsumersurplus, tax and publiccapitalprovision areamean to correctthemarketand theoptimaltaxis then higher.
JEL: E22, E62,H 41
Keywords: irreversibleinvestment, publiccapital, uncertainty.
1. Introduction
IntheU S, largepublicexpenditures arecurrentlyundertakenwhichbearalargepartof themacroeconomicriskwhileintheEuropeanU nionthe”stabilitypact” whichconstrains publicexpenditures becomes strongly criticized. B uthowdoes publicinvestmenta¤ ect theeconomicperformance?Suchaquestionhas yetreceivedalargeattentionespecially sinceithas providedawaytoexplaintheproductivityslowdown: thegrowthrateofthe netstockofcapitalfellfrom 4.1% duringtheperiod19 50-19 7 0 to1.6% during19 7 1-19 85 andA schauer[19 89 ]showsthatanincreaseof1% inthepublic-privatecapitalratiowould increaseby0.39 % thetotalproductivityintheeconomy. T hishasgivenrisetoanumber ofempiricalworks as wellas todebates aboutthe productivity ofpublicspending(see G ramlich[19 9 4]and Shioji [2001]).
¤T heauthors thankStéphanieJametforherusefulcomments. A llremainingerrors aretheauthor’s. yU niversitécatholique de L ouvain and IR ES, P lace M ontesquieu, 3, B -1348 L ouvain-la-N euve (B
el-gium). E-mail: cruz@ ires.ucl.ac.be. T he authorwould like toacknowledge the …nnacialsupport from CA P ES andfrom theB elgianFrenchCommunity’s program ’A ctiondeR echerches Concertée’9 9 /04-235.
B arro [19 9 0]proposes a theoreticalapproach (following A rrowand Kurz [19 7 0]) in which publiccapitalappears as aproductivefactor. H econsiders an endogenous growth modelwith public expenditures thatenterthe production function as ‡ows ofservices and abalanced governmentbudgetforeach period. G rowth is then maximum when the tax rate equals the elasticity ofpubliccapitalwith respecttooutput. M any extensions have been proposed: forinstance, G lomm and R avikumar[19 9 4]introduce congestion, Cashin[19 9 5]considers theproductivityofthestockofpubliccapitalandnotofthe‡ow ofits services. M oreover, B urguetand Fernàndez-R uiz [19 9 8]relax the assumption on the governmentbudget, allowingforborrowingand analyzingthe possibility ofpoverty traps.
A llthis literature aboutpublicinvestmenta¤ ectingprivate production remains in a deterministic framework. B utwhatis the optimaltax rate ifthe productivity ofthe publicgoodisstochasticasitwouldbethecaseinanuncertainenvironment?T urnovsky (19 9 9 ) includes stochastic features into an endogenous growth model with productive publicspending. H owever, theauthorneglectsoneimportantcharacteristicofinvestment decision: the irreversible nature ofcapitalexpenditure. Itis nowlargely admitted (see D ixitandP indyck[19 9 4]) thattheassumptionofaperfectly‡exibleprivatecapitalis no longerrealisticin astochasticworld. Investmentdecisions are then forsure a¤ected by thejoined facts thatinvestmentis irreversibleand generates returns thatareuncertain. Infact, a…rm thatmayfacebadnews andwhichcannotselleasilyits capitalmayprefer topostponesomeinvestmentprojects, thatis, toaccumulatelesscapitalforagivenstate ofnature. T his signi…cantly alters the productivity ofthe private sectorand one may wonderhowthe public sectoris in turn a¤ ected: whathappens forpublic investment, thatis, fortheoptimaltaxrateandthepubliccapitalprovision?
Inthispaper, weextendusualmodelsonthisliteratureintroducingthestockofpublic capitalas an inputforthe private sector, which seems more realistic than considering the ‡ow of public spending or the services provided by the government. T he public investmenttakes place in a stochastic environment. P ublic capitalthen increases the productivityofprivatecapitalwhich is assumedtobefullyirreversible. Implications for the…rm canbegeneralizedfortheeconomyassumingarepresentative…rm. T his implies periodswithnoinvestmentattheaggregatelevelwhichisatleastrelevantfordeveloping countries. A more realistic modellingwould considerheterogeneity among…rms which wouldsigni…cantlycomplicatethemodelwithoutprobablya¤ectthenatureoftheresults weareinterested in. T hegovernmenthas an intertemporalbudgetconstraint, i.e. taxes arecollectedeachperiodtofundthepublicdebt. W eprovideapartialequilibrium analysis (asitisstandardinmodelsofirreversibleinvestmentunderuncertainty);we…rstconsider thecaseofperfectcompetitionandthenturntoissuesaboutimperfectcompetitionwhich allowfordi¤ erentobjectives forthe government: itmay seek tomaximize the value of the…rm orrathertheconsumersurplus. Evenunderuncertainty, theoptimaltaxrateis then constantand does notdepend on thesizeofuncertainty;itis exactlythesameas theonethatwould prevailin adeterministicworld. N otethatP arkand P hilippopoulos
(2002) showsthataconstanttaxrateavoidsanyintertemporaldistortionN everthelessthe optimalprovisionofpubliccapitalisnegativelya¤ectedbyuncertainty. W eshowthatthe governmenthasaninsurancerolesinceitremovespartoftheuncertaintyfacedbythe…rm. Such aroleforthegovernmenthas already been suggested byR odrik(19 9 8): observing thatthe positive correlation between openness ofeconomies and the governmentsize of theseeconomiesisstrongerwhenterms-of-traderiskishigher, hededucesthatgovernment spending may play a risk-reducing role. O urpapercan therefore be considered as an attempttoshowhowthepublicsectoruses publicinvestmentas arisk-reducingstrategy. Comparingcases underperfectcompetitionandunderimperfectcompetition, results depend on the objective ofthe government. Ifthe governmentsimply maximizes the valueofthe…rm, theoptimaltaxisthensmallerunderimperfectcompetitionthanunder perfectcompetitionbecausehavingmorepubliccapitalisnotsogoodforthe…rm because itreducesthesellingprice. B utifthegovernmentseekstomaximizetheconsumersurplus, tax and publiccapitalprovision aremeans tocorrectthemarketand theoptimaltaxis thenhigherthanunderperfectcompetition. Sectiontwopresentsthemodelunderperfect competition andsectionthreeextends ittoimperfectcompetition.
2. P rovidingpubliccapitalinastochasticworld
2.1. T heprogram ofthe …rmW e considerthe publiccapitalas akind ofpublicgood thatis provided by the govern-menttothe …rms. Followingthe literature, an increase in the amountofpubliccapital raises privateproductivity. Itis apurepublicgood sothereis nocongestion issue. T he productionfunctionhas aCobb-D ouglas form1:
Y(t) = A(t)K(t)®K g¯ (2.1)
with® + ¯ < 1.K g is theamountofpublicgood;productionis continuouslyperturbed byshockssinceparameterA(t)isstochasticandmovesaccordingtoageometricB rownian motion:
d A(t)
A(t) = ¹d t+ ¾ d z (t) (2.2) withd z = "pd twhere"~N (0;1);E("i"j) = 08i;jw ith i6= j:
T heprivatesectormustdealwith irreversibilityin theinstalled privatecapital;once theinvestmentis implementedthereis asunkcost, whichdoes notallowthem toreduce
1L aborcan beintroduced in theproduction function, onecould interprettheproduction function in
the totalstock. T he problem ofthe …rm2 is tomaximize its value (the discounted sum ofits cash ‡ows) by choosing the optimalstock ofprivate capital, given thatthere is uncertaintyand irreversibility: 8 > > > > > < > > > > > : maxI(t)V(0 ) = E0 R+1 0 h (1¡¿)A(t)K(t)® (K g)¯ ¡kI(t)ie¡rtd t sc. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ d A(t) = ¹ A(t)d t+ ¾ A(t)d z (t) I(t) = d K(t)¸0 K(t) = 0 8t< 0 K g and¿ given (2.3)
W e assume thatthe governmentand the private sectorpay the same constantprice kforthecapital.r is thediscountrateofbothgovernmentandprivate…rms3. T hegov-ernmenttaxes thecash-‡ows atarate¿ to…nancethepublicexpenditures. T heprivate sectortakes taxes andpublicexpenditureas giventosolvetheproblem.
2.1.1. D erivingthe desiredstockofcapital
T heB ellmanEquation ofthis problem is de…nedas follows (seeP indyck[19 88]): rv(t) = (1¡¿)® A(t)K(t)®¡1(K g)¯ + E(d v)
d t (2.4)
withv(t)beingthemarginalvalueofthe…rm. T hesolutionis then:
v(t) = (1¡¿)A(t)® K(t) ®¡1(K g)¯ r¡¹ | {z } D iscounted value of future cash-‡ows + (1¡¿)Z K(K(t))A(t)¸ | {z } O ption to invest (2.5)
whereZ k< 0 is the derivative of Z (K) with respecttoK(t) and¸ = 0:5¡¹ =¾2 +
p
(¹=¾2 ¡0:5)2 + 2r=¾2 > 1 thus, @¸ =@¾2 < 0. T he value ofthe lastunitofprivate
capitalis given byequation (2.5). Itencompasses thediscounted presentvalueoffuture cash-‡ows givenA(t), less the option to invest this unit later since having one more marginalunitofcapitalimplies to give up this option (and thus prevents to waitfor betterrealizations ofthe stochasticvariable). W e willhave tosolve forZ kand we will
showitis actuallynegative. N otethatthehighertheuncertainty, thelargerthevalueof theoptionthe…rm has togiveup toinvestinthemarginalunitandthus, thesmallerthe
2W estudythecaseofone…rm, buttheinclusionofnsymetrical…rms wouldnotchangesigni…cantly
ourresults sinceweconsiderapurepublicgood.
3In adi¤ erentsetting, A rrowand L ind (19 7 0) showthatuncertainty should notpreventfrom using
value ofthe marginalunit. T his is the main conclusion ofmodels ofirreversibility and uncertainty, andforplausiblevalues ofthemodel’s parameters, this optionvaluemaybe non-negligible, and may thus signi…cantly a¤ectthe optimalstockofcapital(again, see P indyck[19 88]).
T hedesiredcapitalstockisobtainedthrough”valuematching” and”smoothpasting” conditions thatare standard in the irreversible investmentunderuncertainty literature (seeD ixitandP indyck[19 9 4]) :
(1¡¿)Z K(K)A(t)¸ + (1¡¿)A(t)® K(t)®¡1(K g)¯ r¡¹ = k (2.6) (1¡¿)¸Z K(K)A(t)¸¡1+ (1¡¿)® K(t)®¡1(K g)¯ r¡¹ = 0 (2.7 )
T hese optimality conditions allowtoderive the value ofthe desired stock ofcapital K¤as wellasZ ¤
k, foragivenvalueofA(t). Firmsobservethe valueoftheparameterA(t)
andthencanchoosethedesired stockofcapitalas follows : Kd(t) = · ¸¡1 ¸k (1¡¿)A(t)® K g¯ (r¡¹) ¸1 1¡® (2.8) Zk(K(t)) = ¡ µ k (1¡¿)(¸¡1) ¶1¡¸ µ ® K g¯ (r¡¹)¸ ¶¸ Kd(t)¸(®¡1) (2.9 ) @ Kd(t) @¿ < 0 @Kd(t) @K g > 0 (2.10) @Z k @ ¿ < 0 @ Z k @K g > 0
Itis worth notingthattaxes which negatively a¤ectthe marginalcash ‡owreduce the desired capitalwhile theamountofpubliccapitalwhich increases themarginalproduc-tivityofcapitalleads toahigherdesiredstockofcapital. N evertheless, moretaxes allow governmenttoprovidemorepublicgood, butthe…rm doesnotinternalizetheexternality generatedbytaxes. T hetaxalsoreducesthevalueofZ k, thereforeitreducesthemarginal
optionvalue;thiscontrastswiththee¤ ectofthepublicgoodwhichhasapositiveimpact. A notherway tostudy the impactofthe governmentintervention is tofocus on the levelofthestochasticvariablerequiredtoinvestgivenaninstalledstockofprivatecapital, K(t). Suchathresholdmayalsobederivedfrom thevaluematchingandsmoothpasting conditions: A¤(t) = k (1¡¿) (r¡¹) ® K(t)®¡1K g¯ µ ¸ ¸¡1 ¶ (2.11)
E¤ ects ofthetaxrateand ofthestockofpubliccapitalareconsistentwith thoseon thedesired stockofcapitalthathavealreadybeenexamined: @A¤(t) @ ¿ > 0 @A¤(t) @K g < 0 2.1.2. D erivingthe initialvalueofthe…rm
G iventhedesiredcapitalstockderivedbefore, wecanexpressVd(t), thevalueofthe…rm
when the installed stockofcapitalis the desired one. Sincewe knowthe expression for ºd(t), the marginalvalueofthe…rm when theinstalled capitalis thedesired one, Vd(t)
maybecomputedas follows: V d(t) = Z+1 0 vd(K)d K = ZKd 0 (1¡¿)® A(t)K(t)®¡1(K g)¯ r¡¹ d K(t) + Z+1 Kd (1¡¿)Z K(K(t))A(t)¸d K(t) , Vd(t) = (1¡¿)A(t)Kd(t)®K g¯ r¡¹ (2.12) + µ 1 ¸(1¡®) + 1 ¶ µ k (¸¡1) ¶1¡¸µ ® A(t)(1¡¿)K g¯ (r¡¹)¸ ¶¸ Kd(t)¸(®¡1)+ 1 assuming¸(1¡®)> 1toensuretheconvergenceoftheintegral.
W e assume thatthe …rm has initially nocapital: itonly starts toinvestatthe time t= 0 when thegovernmentinstalls thecapitalK g. D uetothespeci…cation ofthecash ‡ows, thatarepositivewhatevertherealizationofthestochasticvariable, attimet= 0, the…rm willthen jump forsuretoits desired capitalstockKd(0 )> 0. T hus, theinitial
value ofthe …rm is such thatgiven the amountofpubliccapitaland the realization of thestochasticvariableattimet= 0, theinstalled capitalstockK(0 )corresponds tothe desiredstockKd(0 ). R eplacingKd(0 )byitsexpressiongivenbyequation(2.8), theinitial
valueofthe…rm is thus : V(0 ) = Vd(0 ) = µ A(0 ) r¡¹ ¶ 1 1¡® µ(¸ ¡1)® ¸k ¶ ® 1¡® (1¡¿)1¡1®K g ¯ 1¡® · 1 + ® ¸[(1¡®)¸+ 1] ¸ (2.13) N otsurprisingly, this valueis anincreasingfunctionofthestockofpubliccapitaland a decreasing function ofthe tax rate. T he e¤ ectofuncertainty on this initialvalue is givenby: @ V (0 ) @¾2 = @V(0 ) @ ¸ |{z } >0 @¸ @¾2 |{z} <0 < 0
T here exists two opposite e¤ects ofuncertainty onV(0 ). O n the one hand, more uncertainty induces the …rms toinstallless capital(Kd(0 )is smaller) thus reducingthe
currentcash-‡ow;ontheotherhand, alargeruncertaintyincreases theoptionvaluepart ofV(0 )whichrelates tofuturecash‡ows. Clearlyhere, the…rste¤ectprevails andmore uncertaintyreduces theinitialvalueofthe…rm.
2.2. T heprogram ofthe government
U ntil now, public expendituresKg and taxes ¿ are taken as given by the …rm when
deciding howmuch to invest. W e mustthen considerthe problem ofthe government: howshould theleveloftaxes and thestockofpubliccapitalbedetermined ? N otethat the modeldealthere is a partialequilibrium analysis. T herefore the objective ofthe governmentwillbetomaximizethevalueofthe…rm subjecttoitsinter-temporalbudget constraint.
2.2.1. T hegovernmentbudgetconstraint
A tthebeginningoftheprogram, thegovernmentde…nesK g, theoptimallevelofpublic capitaltobe provided once forall. Expenses are thenkK g withkbeingthe unitprice ofcapital. T his publicdebtis completely funded in future taxes on the currentpro…ts ofthe…rms. T herefore, wearesupposingthatthereis a ”R icardian equivalence” in the sense thatthe publicdebtis completely funded in future revenues4. Since the modelis stochastic, future tax revenues are subjectto uncertainty and the budgetconstraintis such thatthe expected presentvalue ofthe revenues mustbe equaltothe expenses in terms ofpubliccapital. M oreover, theexpectedpresentvalueoftherevenuederives from thetaxrateappliedtotheexpectedvalueofthefuturecash‡ows ofthe…rm. T his latter stream is given byV(0 ), which is the value ofthe …rm atperiod0. T he government budgetis thus:
¿V (0 ) = kK g (2.14)
N ote thatthis implies a precise time schedule in the realization ofprivate and public investments : attimet= 0 the governmentobserves the realization ofthe stochastic variable. Itcanthendeducetheamountofcapitalthe…rm wantstoinstallandtheinitial valueofthe…rm, dependingonthelevelsoftaxandpubliccapital. U singthisinformation thegovernmentdecides howmuch totaxand howmuch publicgoods toprovide, which implies (givenk) theinitialamountofdebt.
4H owever, thenameofR icardian Equivalenceis morerelated with resultthatthepublicdebtis not
2.2.2. D erivingthe optimaltaxrate
T heprogram ofthegovernmentis tochoosethelevels oftaxand ofpubliccapital, that maximizethevalueofthe…rm: 8 < : M axf¿;K ggV(0 ) = ³ A(0 ) r¡¹ ´ 1 1¡® ³(¸¡1)® ¸k ´ ® 1¡® (1¡¿)1¡1®K g ¯ 1¡® h 1 + ® ¸[(1¡®)¸+ 1] i sc.¿V (0 ) = kK g
SubstitutingforK g usingthebudgetconstraint, theproblem ofthepublicsectorcan berestated as: M ax¿V(0 ) = ª (1¡¿) 1 1¡®¡¯¿ ¯ 1¡®¡¯ (2.15) withª = ³(r¡¹A)(0 )k(®+¯) ´ 1 1¡®¡¯ ³(¸¡1)® ¸ ´ ® 1¡®¡¯ h 1 + ® ¸[(1¡®)¸+ 1] i 1¡® 1¡®¡¯
T he…rstordercondition ofthe problem is then (note thatª does notdepend upon ¿): @ V @¿ = ¡ ª 1¡® ¡¯(1¡¿) 1 1¡®¡¯¡1¿ ¯ 1¡®¡¯ + ¯ª 1¡® ¡¯(1¡¿) 1 1¡®¡¯¿ ¯ 1¡®¡¯¡1 = 0 (2.16)
M oreover, itcan beeasilybechecked that @2V @¿2 < 0 T heoptimaltaxrate¿¤ p is thus : ¿¤p = ¯ 1 + ¯ (2.17 ) T hemodelgivesrisetoaninverselyU -shapedrelationshipbetweentheleveloftaxandthe valueofthe…rm, as in B arro[19 9 0]and in thesubsequentextensions likeCashin [19 9 5] andB ajo-R ubio[2000]. T his arisesbecausetaxationplays anambiguousrolesinceitalso allows foraprovisionofpubliccapitalandthereforerepresents akindofexternality. O n theonehand, thetaxhas adirectnegativeimpactthrough(1¡¿)1¡®1¡¯, sinceitreduces
the after-tax value. O n theotherhand, ahighertax allows formore publiccapitaland this has apositiveimpacton thevalueofthe…rm through theparameter¿1¡®¯¡¯. Since
themodelexhibitsdecreasingreturnstothestockofpubliccapital, combiningbothe¤ects ofthetax, willgenerateaconcaverelationship : thepositivee¤ ectinitiallyprevails while fortaxeshigherthan¿¤
p, thereverseapplies. Since@¿¤p=@ ¯ > 0, thestrongertheimpact¯
ofthepubliccapitalon theproductionofthe…rm, thehighertheoptimaltax. Intuition isveryclear: thegreatertheimpactofthepublicgoodontheproductionlevel, thehigher
theprovisionofpublicgoodshould be, andthehigherthetaxneededto…nanceit. M oreover, equation (2.17 ) can be interpreted, like in B arro[19 9 0], as a naturale¢-ciency condition forthe public sector, where the governmentis equalizing the positive bene…ts ofits actions with the negative distortions itgenerates. H ere, the publicsector is equalizingthemarginalbene…ts ofthepublicexpenditures withthemarginalcostdue totaxation.
2.2.3. T heoptimalprovisionofpubliccapital
U sing equations (2.14) , (2.17 ) and (2.15), the optimal levelofpublic capital can be expressed as: K gp¤= (1 + ¯)1¡®®¡¡2¯¯1¡1¡®¡®¯ µ A(0 ) (r¡¹)k ¶ 1 1¡®¡¯ µ(¸ ¡1)® ¸ ¶ ® 1¡®¡¯ · 1 + ® ¸[(1¡®)¸+ 1] ¸1¡® 1¡®¡¯ (2.18) 2.3. E¤ectofuncertainty 2.3.1. E¤ ectuncertaintyontheoptimaltaxrate
T hee¤ectofuncertaintyontheoptimaltaxrateis given by: @¿¤
p
@¾2 = 0 (2.19 )
H enceanincreaseinuncertaintyhas noe¤ ectonthetaxrate, whichis quiteremarkable. T hishappensbecausetheimpactofthetax(eitherpositive, throughtheamountofpublic good itgenerates, ornegativesincereducingthecurrentpro…t) on thecurrentcash ‡ow is thesameas thatontheoptionvaluesincethetaxis leviedoncurrentcash‡owas well as futureones whateverthesizeinuncertainty. Sothetaxratewillalways representthe same proportion ofthe value atany time. Its optimallevelhas therefore nothingtodo withthesizeofuncertainty. Comparingwiththedeterministiccounterpartofthis model, itis clearthatthe introduction ofuncertainty does notchange the optimalleveloftax sincethetaxrateonlydepends onhowproductiveis thepubliccapital.
Incontrast, withtheresults foundbyJones, M anuelli & R ossi [19 9 7 ],theoptimaltax here is time independentand also has no correlation with the uncertainty in economy. T heresultalsodi¤ ers from thatofH asset& M etcalf[19 9 4],as arguedbyP ennings[2000] theyfoundacounter-intuitiveresultthatahigherpolicyuncertaintywouldleadtohigher levelofinvestment.
2.3.2. E¤ ectofuncertaintyontheoptimalprovisionofpubliccapital
Contrary tothe optimaltax rate, the correspondingoptimalprovision ofpubliccapital is a¤ ected by uncertainty. Itarises from the factthatthe amountofpublic good the governmentcan provide, given anytaxrate, directlydepends onthefuturetaxrevenues which aredetermined bythefuturecash ‡ows ofthe…rm (seeequation(2.14)).
@ K g¤ p @¾2 = ¿ k @V(0 ) @¾2 |{z } <0 + V(0 ) k @ ¿ @¾2 |{z} 0 < 0 (2.20)
T hisresultsfrom thetwoe¤ ectsofuncertaintyontheinitialvalue: aswehaveseenbelow, thenegativee¤ ectofuncertainty(which applies throughthecurrentcash ‡ow) prevails.
2.3.3. U ncertaintyandtherole ofthe government
T hegovernmenthas aninterestingrolein this modelsinceitbears theriskofstochastic tax revenues while o¤ ering as a counterparta deterministic initialamountK g to the privatesector.
First, thegovernmento¤ers less publicgoodinthestochasticworldwhenuncertainty is larger(see equation (2.20)). T his is due to the fact thatuncertainty generates an option value which reduces the marginalproductivity ofprivate capitalwhich in turn negatively a¤ ects the productivity ofthe publicgood. second, any smallergovernment intervention would reduce the desired stock ofprivate capitalforany realization ofthe stochasticvariable, orsymmetrically, foragiveninstalledstockofcapital, thelevelofthe stochasticvariable required toinvestwould behigher. T herefore, itcan be argued that the modelgenerates some positive relationship between the public investmentand the privateinvestment. A s the governmentdecides theprovision ofpublicgood onceforall on thebasis oftheexpected valueofits revenue, theprivatesectorbene…ts from akind ofinsurance scheme: duringbad realizations (A(t)< A(0 )e¹t) ofthe stochasticprocess,
the governmentis providingmore publicgood than the expected revenues, butin good times (A(t)> A(0 )e¹t), thegovernmentcollects moretaxes than theprovision ofpublic
good, this mechanism keepingthe intertemporalbudgetbalanced. T he governmenthas thereforeaninsurancerole.
N ote that in B arro [19 9 0]public capital positively a¤ ects the economy since it is …nanced byataxpaid byn …rms, soeach …rm onlybears asmallpartofthecostwhile thepubliccapitalentirelybene…ts toanyofthem. H ere, thepubliccapitalis …nancedby onlyone…rm. Sotheroleofthegovernmentis nottoprovideagoodwhich exhibits the specialcharacteristics ofapublicgood butrathertoremove apartofuncertainty from the…rm towards thegovernment.
3. M arketpowerandpubliccapitalprovisioninastochasticworld
Imperfect competition allows considering two di¤ erent objectives for the government. Itcould be assumed thatthe governmentsimply maximizes the value ofthe …rm, but underimperfectcompetition, thegovernmentmayratherseektomaximizetheconsumer surplus. B oth cases arestudiedinthis section.
A ssuminga monopolisticcompetition framework (see D ixitand Stiglitz [19 7 7 ]), the …rm is nolongerpricetakerbuttakes intoaccountthefollowingdemandfunction:
P(t) = bY(t)¡µ (3.1)
withµ < 1(thepriceelasticityis greaterthanunity). M oreover, thecash-‡owbecomes: P(t)Y(t) = bA(t)1¡µK(t)®(1¡µ)K g¯(1¡µ) (3.2)
A s before, the cash-‡owis a¤ected by uncertainty through the productivity parameter, butnowitis aconcavefunctionoftheuncertainvariable(notethatapowerfunction of aG eometricB rownian M otionis againaG eometricB rownianM otion).
3.1. T heprogram ofthe …rm
T heproblem of…rm maynowbewritten: m axItW(0 ) = E0 R+1 0 h (1¡¿)A0(t)K(t)®0(K g)¯0 ¡kI(t)ie¡rtd t sc. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ d A(t) = ¹ A(t)d t+ ¾ A(t)d z (t) I(t) = d K(t)¸0 K(0 ) = 0 K g and¿ given (3.3) wheretheA0(t) = bA(t)1¡µ;®0= ®(1¡µ)and¯0= ¯(1¡µ)
T heB ellmanequationis de…nedas:
rw(t) = (1¡¿)®0A0(t)K(t)®0¡1(K g)¯0+ E(d w)
d t (3.4)
withw(t)beingthe marginalvalue ofthe …rm. U singthe value matchingand smooth pastingconditions leads to:
Km cd¤= · (¸0¡1) ¸0k± (1¡¿)A 0(t)®0(K g)¯0 ¸ 1 1¡®0 (3.5) with±= r¡¹0, ¸0= 0:5¡¹0=¾02 + p(¹0=¾02 ¡0:5)2 + 2r=¾02 where¹0= (1¡µ)¹+ 1 2(1¡ µ)µ¾2 and¾02 = (1¡µ)2¾2
A ftervariables changes, the resultlooks like the previous one, exceptthatnow, un-certaintyappears attwolevels: as in thecompetitivecase, thereexists an irreversibility e¤ect(through¸0) which is nowcombined with a Jensen e¤ ect(through±). Since the cash-‡owis a concave function ofthe stochastic variableA(t), both e¤ects play in the same direction, and uncertainty unambiguously leads toasmallerdesired capitalstock. M oreover: W(0 ) = Wd(0 ) = µ A0(0 ) ± ¶ 1 1¡®0µ(¸0¡1)®0 ¸0k ¶ ®0 1¡®0 (1¡¿)1¡1®0K g ¯0 1¡®0 : · 1 + ® 0 ¸0[(1¡®0)¸0+ 1] ¸
3.2. T heprogram ofthe government
W econsidertwodi¤erentobjectivesforthegovernment: maximizingthevalueofthe…rm andmaximizingtheconsumer’s surplus.
3.2.1. M aximizingthe valueofthe…rm
Itcanbeseen thattheproblem is veryclosetotheprevious caseofperfectcompetition M ax¿W(0 ) = ª0(1¡¿) 1 1¡®0¡¯0¿ ¯0 1¡®0¡¯0 withª0= ³ A0(0 ) ±k(®0+¯0) ´ 1 1¡®0¡¯0³(¸0¡1)®0 ¸0 ´ ®0 1¡®0¡¯0h 1 + ®0 ¸0[(1¡®0)¸0+ 1] i 1¡®0 1¡®0¡¯0 (3.6) T hegovernmentmustthenfollowthefollowingoptimalrule: ¿¤m cf = ¯ 0 ¯0+ 1 = ¯(1¡µ) ¯(1¡µ) + 1 (3.7 ) @¿¤ m cf @µ < 0 (3.8)
T hehigherthepowerofthe…rm (whichis anincreasingfunction ofµ), theless e¤ ective thepubliccapital(¯0willdrop), andthusthesmallertheoptimaltaxrate. T heeconomic interpretationisasfollows. A highermarketpowermeansthatthe…rm wants toproduce lessatahigherprice, whilepublicexpendituresinducelowerpricesandhigherquantities. T herefore, thevalueofthe…rm thathasahighermarketpowerismaximizedbyasmaller governmentintervention (smallertax rate and less publiccapital). So…nally, imperfect competition leads toasmalleroptimaltax. M oreover,
lim
µ!0¿
¤
W hen the powerofthe …rms becomes negligible, the optimaltax rate underimperfect competition (¿¤ m cf) converges totheoptimaltaxrateunderperfectcompetition(¿¤). Itis alsopossibletoderivethestockofpubliccapital: K gm cf = µ (¯0)1¡®0 (1 + ¯0)2¡®0 ¶ 1 1¡®0¡¯0µA0(0 ) ±k ¶ 1 1¡®0¡¯0µ(¸0¡1)®0 ¸0 ¶ ®0 1¡®0¡¯0 (3.9 ) : · 1 + ® 0 ¸0[(1¡®0)¸0+ 1] ¸ 1¡®0 1¡®0¡¯0 (3.10) 3.2.2. M aximizingthe consumerSurplus
In an imperfectcompetition setup, thegovernmentmayseektomaximizetheexpected lifetime consumersurplus. Itis a bettermeasure forwelfare than the sole value ofthe …rms. B ut since we are in a stochastic world with irreversible investment, the exact expectedlifetimeconsumersurplus cannotbecomputed. N evertheless, itcanbeapprox-imated usingtheexpectedlongterm consumersurplus, giventheobservedrealization of theuncertain variable. T heproblem ofthegovernmentbecomes then:
M axf¿;K ggSLT(0 )
sc.¿W (0 ) = kK g
Itcanbeshown (seetheappendix) thattheproblem canberewrittenas:
M ax ¿ SLT (0 ) = Á(1¡¿) ¯0+®0 (1¡®0¡¯0)¿ ¯0 (1¡®0¡¯0) (3.11) where: Á = ³®0(±k¸¸0¡01) ´®0 1¡®0³ ½(1¡®0) ½(®0¡1)+®0 ´¡µ 1¡µ ¢ A0(0 )1¡1®0¡ª0 k ¢¯0 1¡®0 T he…rstorderconditiongives: ¿¤m cs= ¯ ® + 2¯ (3.12) @¿¤m cs @µ = 0 @¿¤m cs @¾2 = 0 (3.13) O bservethatthetaxratedependsneitherontheuncertaintynoronthedegreeofmarket imperfection. A s in the perfectcompetition case, the optimaltax rate is related tothe elasticityofpubliccapital;nevertheless, itnowdependsontheelasticityofprivatecapital as well: the more e¢cientthe private capital, the smallerthe optimaltax rate. T his comes from thefactthatthegovernmentconsiderstheconsumersurplusas theobjective;
itthus implements atax ratethatleads tothemaximum production each period, given its budget constraint(indeed we have: @Y(t)=@¿j¿=¿¤m cs = 0). T he governmentmust
therefore take intoaccountthe negative e¤ectofthe taxes on the currentproduction ; this is whytheoptimalvalue¿¤
m csdepends ontheelasticityofprivatecapital. M oreover,
the optimaltax rate is higherthan in the perfectcompetition modelwhateverofthe degreeofimperfection. T heoptimalamountofpubliccapitalis de…nedas: K gm cs= " µ A0(0 ) ±k ¶ µ (¸0¡1)®0 ¸0 ¶®0· 1 + ®0 ¸0[(1¡®0)¸0+ 1] ¸1¡®0à (® + ¯)¯1¡®0 (® + 2¯)2¡®0 ! # 1 1¡®0¡¯0 (3.14) 3.3. Comparingthethreecases
L ookingattheoptimaltaxrates underperfectcompetition, underimperfectcompetition withthe…rm’s valuemaximization, and underimperfectcompetitionwiththeconsumer surplus maximization, weget:
¿¤m cs> ¿¤p > ¿¤m cf (3.15) G overnment intervention generates a correction of in the level ofimperfection in the market: highertaxrevenuesimplyhigherpubliccapitalwhichinduceshigherproduction. Equation (3.15) thereforeexpresses thefactthatifthegovernmentmaximizes thevalue ofthe…rm underimperfectcompetition, theoptimaltaxrateis lowerthanunderperfect competition, whereasifthegovernmentmaximizes theconsumersurplus, theoptimaltax rateishigherthanunderperfectcompetition, inducingacorrectionofimperfectioninthe market. M oreover:
K gm cs> K gm cf (3.16)
T heprovisionofpubliccapitalis higherwhenthegovernmentmaximizes theconsumer’s surplus. In this lattercase moreover, as faras the optimallevelofprivate capitalis concerned, thepositivee¤ectduetoahigherpubliccapitalprevails onthenegativee¤ ect ofahighertaxation:
Kd¤
m cs> Km cd¤f (3.17 )
Ifthe governmentmaximizes the consumer’s surplus, outputis higheras well. U nder imperfectcompetition, thegovernmentcanthereforeinduceahigherproduction, through ahighertaxationandahigherprovision ofpubliccapital.
4. Conclusion
In this paper, wehavestudied theissueoftheoptimalprovision ofpubliccapitalunder uncertainty. W hateverthe competition in the market, the tax rate willnotdepend on thedegreeofuncertaintybutonlyon technologicalparameters and marketpower. N ev-ertheless, the optimalstock ofpubliccapitalwillbe negatively a¤ ected by uncertainty. T he governmenthas an insurancerolesince itcollects taxes from futurecash-‡ows that arestochasticand provides aninitialamountofpubliccapital.
U nderimperfectcompetition, twocases have been studied, the …rstkeeps the value ofthe …rm as the objective function ofthe government, while in the second case the governmentmaximizes the consumersurplus. Forthe …rstcase, the optimaltax rate depends on the technologicalparameters and on the marketpowerofthe …rms. T he higherthepowerofthe…rm, theless thetaxrate, sincemorepublicgood induces more productionandasmallersellingprice. B utinthesecondcasetheoptimaltaxrateishigher thanunderperfectcompetitionsinceitis ameantocorrectthemarketimperfection.
A morerealisticmodellingwithheterogenous …rms shouldnowbeconsideredinorder toavoid periods with noinvestmentin thecountry;nevertheless, thereis noreason for theinsuranceroleofthegovernmenttodisappearinsuchaframework. M oreinteresting wouldbetoallowforsuccessivepublicinvestments.
5. R eferences
A rrow, K.andKurz, M .(19 7 0),P ublicInvestment, theR ateofR eturnandO ptimalFiscal P olicy, JohnH opkins P ress, 19 7 0.
A rrowK.andL indR ., U ncertaintyandtheEvaluationofP ublicInvestmentD ecisions,
A merican EconomicR eview, p.364-37 8.
A schauer, D . (19 89 ), Is P ublicExpenditure P roductive?,JournalofM onetary Eco-nomics23, p. 17 7 -200.
B arro, R . (19 9 0), G overnmentSpendingin aSimpleM odelofEndogenous G rowth, . JournalofP oliticalEconomy, 9 8, p. s103-s125.
B ajo-R ubio, O . (2000). A FurtherG eneralization ofthe SolowG rowth M odel: T he R oleoftheP ublicSector.Economics L etters, 68 (1), p. 7 9 -84.
B ertolaG . (19 9 8), IrreversibleInvestment,R esearchin Economics, 52, 3-37 .
B urguet, R .and Ferná ndez-R uiz, J. (19 9 8), G rowth T hrough T axes orB orrowing?A M odelofD evelopmentT rapswithP ublicCapital,EuropeanJournalofP oliticalEconomy, 14(2), p.327 -344.
Cashin, P. (19 9 5), G overnmentSpending, T axes and Economic G rowth, IM F Sta¤ P apers, 42 (2), p. 237 -268.
D ixit, A .andP indyck, R . (19 9 4) InvestmentunderU ncertainty, P rincetonU niversity P ress.
D ixit, A . Stiglitz J. (19 7 7 ), M onopolisticCompetition and O ptimum P roductD iver-sity, A merican EconomicR eview, 67, 29 7 -308.
D rèze, J. (19 9 7 ) P ublicEconomics, P ublicP rojects and theirfunding. A sian D evel-opmentR eview, 41, 17 35-17 62.
G lomm, G .andB .R avikumar. (19 9 4) P ublicInvestmentinInfrastructureinaSimple G rowth Endogenous M odel, JournalofEconomic D ynamics and Control18, p. 117 3-1118.
G ramlichE., (19 9 4), InfrastructureInvestment: A R eviewEssay,JournalofEconomic L iterature, 32(3), 117 6-119 6.
Jones, L .E., M anuelli, R . and R ossi, P., (19 9 7 ), O n theO ptimalT axation ofCapital Income, JournalofEconomicT heory17 3(1), 9 3-117 .
M etcalf, G . and H assett, K., (19 9 4). InvestmentwithU ncertain T axP olicy,W orking paperno. 47 80, N B ER, Cambridge, M A .
P ark, H .& A .P hilippopoulos(2002) D ynamics ofT axes, publicServices andEndoge-nous G rowth.M acroeconomicD ynamics, 6,187 -201.
P ennings, E. (2000), T axes and Stimuli ofInvestmentU nderU ncertainty, European EconomicR eview, 44 (2) p. 383-39 1.
P indyckR .(19 88), IrreversibleInvestment;CapacityChoiceandtheV alueoftheFirm,
A merican EconomicR eview, p. 9 69-9 85.
Shoiji, E. P ublicCapitalandEconomicG rowth:A convergenceA pproach.Journalof EconomicG rowth, 6,p.205-227 .
T urnovsky, S.(19 9 9 )P roductiveG overnmentExpenditureinaStochatiscallyG rowing Economy.M acroeconomicD ynamics, 3, 544-57 0.
6.A ppendix: derivingthe expected conditionalconsumer’s
sur-plus (followingB ertola[19 9 8])
T heconsumersurplus is de…ned as: S(t) = ZQ(t) 0 bq(t)¡µ:d q¡P(Q(t))Q(t) = · b 1¡µq(t) 1¡µ ¸Q(t) 0 ¡bQ(t)1¡µ = b 1¡µQ(t) 1¡µ¡bQ(t)1¡µ = bµ 1¡µQ(t) 1¡µ = bµ 1¡µ £ A(t)K(t)®K g¯¤1¡µ(6.1) = µ 1¡µA 0(t)K(t)®0 K g¯0 (6.2) withA0(t) = bA(t)1¡µ;®0= ®(1¡µ)and¯0= ¯(1¡µ).
>From theoptimalityconditions forthe…rm weknowthat: ®0A0(t)K(t)®0¡1K g¯0= ±k¸
0
with: ± = r¡¹0+ 12µ(1¡µ)¾2 and¸0= 0:5¡¹0=¾02 + p(¹0=¾02 ¡0:5)2 + 2r=¾02 where ¹0= (1¡µ)¹+ 1 2(1¡µ)µ¾ 2 and¾02 = (1¡µ)2¾2. whichgivestheexpressionofthedesired capitalstock. L etus de…nethemarginalcash‡owX (t): X (t) = ®0A0(t)K(t)®0¡1K g¯0
T he marginalcash-‡ow thus follows a regulated B rownian motion. T hatis, when the …rm is notinvesting, X (t)follows a geometricB rownian motion with meanM = (1¡ µ)¹ ¡µ(1¡µ)¾22 and variance§ = (1 ¡µ)¾. W hen the …rm is investingthe marginal cash-‡owequals then theright-hand sideofequation (6.3) which is constant: X (t) = c withc= ±k¸0=(¸0¡1) D e…ning"(t) = lnX (t), then d "(t) = d X (t) X (t) ¡ 1 2 d2X (t) X (t)2 ) E[d "(t)]= M ¡ 1 2 § 2 > 0 , ¹ > ¾2 2 (6.4) which ensuretheexistenceofadensityfunctionwhoseexpressionis: w("(t)) = ½e½("(t)¡lnc) (6.5) with½= 2§M2
L etus expressS(t)as afunctionofX (t) : S(t) = µ 1¡µA 0(t)K(t)®0 K g¯0= z X (t)®®0¡01 (6.6) ) S(t) = z :e"(t)®®0¡01 (6.7 ) ) "(t) = ln 2 4 µ S(t) z ¶®0¡1 ®0 3 5 = g(S(t)) (6.8) where: z = µ 1¡µ £ (®0)®0A0(t)K g¯0¤1¡1®0
A variable change allows then todeduce the density function ofS(t)when knowing thatofX (t)(seeB ertola[19 9 8]orB entolilaand B ertola[19 9 0]):
f(S(t)) = ¡w[g(S(t))]@g(S(t)) @S(t) (6.9 ) = µ ½(1¡®0) ®0 ¶ z ½(1®¡0®0)c¡½S(t) ½(®0¡1) ®0 ¡1 (6.10)
W ethencan …ndtheexpectedlong-term valueoftheconsumersurplus: EL TS(0 ) = Z1 S(0 ) f(S(t))S(t)d S(t) = Z1 S(0 ) µ ½(1¡®0)) ®0 ¶ z ½(1®¡0®0)c¡½S(t) ½(®0¡1) ®0 d S(t)
which converges if®0< ½(1¡®0);then:
ELTS(0 ) = ·µ ½(1¡®0) ½(®0¡1) + ®0 ¶ c¡½z ½(1®¡0®0)S(t) ½(®0¡1) ®0 + 1 ¸1 S(0 ) (6.11) = µ ½(1¡®0) ½(®0¡1) + ®0 ¶ c¡½ µ S(0 ) z ¶½(®0¡1) ®0 S(0 ) (6.12) = µ ½(1¡®0) ½(®0¡1) + ®0 ¶ c¡½X ½(0 )S(0 ) (6.13) = µ ½(1¡®0) ½(®0¡1) + ®0 ¶ S(0 ) (6.14) =  A0(0 )1¡1®0K g ¯ 0 1¡®0 (6.15) with = ³®0(1¡±k¸¿)(0¸0¡1) ´ ®0 1¡®0³ ½(1¡®0) ½(®0¡1)+®0 ´¡µ 1¡µ ¢ T heintertemporalconstrainttothegovernmentis: ¿W(0 ) = kK g (6.16) B utweknowthatthevalueofthe…rm is givenby: W(0 ) = ª0(1¡¿)1¡®10¡¯0¿ ¯0 1¡®0¡¯0 (6.17 ) combiningboth equations: K g= ª 0(1¡¿)1¡®10¡¯0¿ 1¡®0 1¡®0¡¯0 k (6.18)
SubstitutingK gforitsexpressionintheexpectedlong-term consumersurplusgivesequa-tion(3.11) inthetext.
