Public Capital Maintenance and Congestion: Long-Run Growth and Fiscal Policies

Transcription

1 Public Cpitl Mintennce nd Congestion: Long-Run Growth nd Fiscl Policies Evngelos Dioikitopoulos nd Srntis Klyvitis y Athens University of Economics nd Business Mrch 2008 z Forthcoming in Journl of Economic Dynmics nd Control Abstrct In this pper we study n endogenous growth model, in which public mintennce expenditures ect the deprecition rte of public cpitl nd the ltter is subject to congestion. We nd tht economies with low congestion in public infrstructure will require threshold level of public cpitl mintennce for ongoing growth. We lso exmine the scl implictions of public cpitl mintennce policies nd we nd tht the composition of public cpitl expenditures under congestion is crucil determinnt of optiml nd growth-mximizing scl policies. The government cn ect the return of public cpitl by re-llocting public expenditures between new public investment nd mintennce nd hence void excessive txtion tht is required under incresing congestion. JEL clssi ction: O41, H54, E62. Keywords: public cpitl mintennce; congestion; optiml scl policies. Deprtment of Interntionl nd Europen Economic Studies, Athens University of Economics nd Business, Ptision Str 76, Athens 10434, Greece. e-mil: evngelos.dioikitopoulos@ueb.gr y Corresponding uthor: Deprtment of Interntionl nd Europen Economic Studies, Athens University of Economics nd Business, Ptision Str 76, Athens 10434, Greece. Tel: Fx: e-mil: sklyvitis@ueb.gr z Acknowledgments: Finncil support under the PYTHAGORAS II project co- nnced by the Europen Socil Fund nd the Greek Ministry of Ntionl Eduction nd Religious A irs is grtefully cknowledged. We hve bene ted from comments nd suggestions by n nonymous referee, J.C. Cones, G. Economides, O. Licndro, I. Mourmours, K. Nenides, T. Plivos, E. Ppp, A. Philippopoulos, T. Xeppdes, nd seminr prticipnts t the Athens University of Economics nd Business, University of Mcedoni in Thesloniki, Universitt Autonom Brcelon, the Conference on Smll Open Economies in Globlized World, the 8th Conference of the Society for the Advncement of Economic Theory, nd the 6th Conference on Reserch on Economic Theory nd Econometrics. The usul disclimer pplies.

2 1. Introduction Public cpitl mintennce expenditures re importnt for infrstructure services, which in turn re key instrument for long-run growth. In prticulr, outlys on public cpitl mintennce cover ctivities required for infrstructure to function nd re necessry for its repir nd sfe opertion, compred to spending for the purchse, construction, rehbilittion, nd improvement of physicl infrstructure tht comprises new investment. The two types of cpitl spending re expected to operte vi di erent chnnels on the cpitl ccumultion process: new public investment embeds the stndrd concept of resources tht dd to the existing cpitl stock, wheres public cpitl mintennce opertes by ecting the deprecition rte, nd consequently the service life, of public cpitl goods. 1 Severl country reports nd cse studies from developing countries demostrte tht the lck of su cient public cpitl mintennce hs been criticl fctor for the observed growth stgntion, since in most cses infrstructure building is given higher priority thn mintennce. According to the World Bnk (1994), lck of mintennce in key infrstructure sectors (like rods, rilwys, power, nd wter) in developing countries cused losses equivlent to qurter of their nnul investment in infrstructure in the erly 1990s, wheres similr sitution ws fced in most South Estern Europen countries during the lst decde (World Bnk, 2000). A lrge frction of public cpitl mintennce involves spending on rods, for which mintennce expenditures typiclly 1 A mjor obstcle in ssessing the role of cpitl mintennce is tht there is no systemtic recording of mintennce expenditures, becuse they re treted s current expenditure nd re not ssigned seprte ctegory in the ntionl ccounts or ny other mcroeconomic dt source. However, some evidence suggests tht public cpitl mintennce comprises non-negligible shre of output. In the United Sttes report by the Congressionl Budget O ce (2007) provides detiled time series dt for public infrstructure on trnsporttion nd wter covering the period According to the evidence, the verge size of these expenditures hs been round 2.6% of GDP with the shre of O&M expenditures mounting to 49% of totl. Also, dt on cpitl spending in newly purchsed ssets nd mintennce from the Cndin survey on Cpitl nd Repir Expenditures show tht totl public cpitl mintennce nd repir expenditures in Cnd mounted on verge to 1.5% of GDP for the period nd comprised 21% of totl public cpitl spending; see McGrttn nd Schmitz (1999) nd Klitzidkis nd Klyvitis (2005) for more detiled presenttion of this dtset. Yepes (2004) estimtes tht infrstructure mintennce in Est Asin countries mounted to 2.2% of GDP over the period nd covered roughly 30% of totl cpitl expenditures. 1

3 ccount between 30 to 60% of totl expenditures nd s much s 0.5% of GNP (Gwillim nd Shlizi, 1999). Hrrl nd Fiz (1988) estimted tht the mintennce level required to prevent rod deteriortion mounted to 0.2% of GDP for Est Asi nd Pci c countries nd to 1% for West Africn countries for ; in turn, the bcklog of mintennce work vried from 1.6% of GDP in Est Asi nd the Pci c to 3.5% in South Asi. 2 The in uence tht public cpitl mintennce my hve on the growth process should therefore mke scl policies relted to cpitl expenditures nd their lloction n importnt fctor in models tht ttempt to understnd the behvior of the mcroeconomy. Despite the intuitive consensus on the crucil impct of mintennce expenditures in public cpitl formtion, there hve been only recently some systemtic ttempts to investigte their growth impct. 3 Rioj (2003) hs set up growth model where domestic tx revenues nnce mintennce expenditures for public cpitl, wheres public infrstructure is nnced solely by foreign donors. The uthor shows tht the optiml mintennce level (s shre of GDP) depends upon vrious prmeters nd presents clibrtion results from Ltin Americn countries tht con rm the importnce of mintennce for the pttern of growth in these countries. Klitzidkis nd Klyvitis (2004) hve extended Rioj s (2003) model by concentrting on the growth implictions of public cpitl mintennce expenditures. In their model, both types of expenditures re nnced by tx on output s in Brro (1990); by ltering their lloction the government cn use the shre of mintennce s policy instrument to rise the shdow vlue of privte cpitl nd the growth rte of the economy. An ppeling impliction is tht the growth-mximizing tx rte is higher thn 2 These estimtes concur with necdotl evidence from vrious studies. The World Development Report (World Bnk, 1994) stted tht n dditionl $12 billion spent on timely rod mintennce in Afric could hve sved $45 billion spent in reconstruction. Also, the lloction of recurrent expenditures in the 1993/4 budget for Ethiopi ws estimted to be less thn hlf of wht would be needed for regulr rod mintennce (Interntionl Monetry Fund, 1995); see lso Tble 1 in Gwillim nd Shlizi (1999) for dditionl necdotl evidence. The study by Heggie nd Vickers (1998) corrobortes these ssessments by stting tht the return for rod mintennce projects between 1961 nd 1988 ws 38.6 percent compred to 26 percent for ll trnsport projects nd 21 percent for ll World Bnk investment projects. 3 The ssessment of the impct of public cpitl mintennce is usully performed in the context of cost-bene t nlysis nd exmines primrily the issue of rod dmge nd optiml user chrges, which rely on required repirs nd their timing (see Newbery, 1988). 2

4 the elsticity of public cpitl in the production function (which reverses stndrd rgument put forwrd by, mong others, Brro (1990), Glomm nd Rvikumr (1994), Devrjn et l. (1998)), s the dditionl positive e ect of mintennce expenditure on the ccumultion of public cpitl rises the bene ts of txtion compred to the stndrd models. 4 A nturl extension of this frmework envisges the role of public cpitl s rivl nd nonexcludble good tht is subject to congestion; s rgued by Brro nd Sl-I-Mrtin (1992), virtully ll services provided by the public sector, like trnsport nd energy, re subject to congestion. 5 In turn, models of productive government services, such s those developed by Brro nd Sl-I- Mrtin (1992), Turnovsky (1997, 1997b), nd Fisher nd Turnovsky (1998), hve ssumed tht the input of public cpitl to privte production is subject to congestion nd hve estblished tht the degree of congestion is importnt in ssessing the linkges between growth, public nd privte cpitl, nd represents key determinnt of tx policies. The tsk of integrting public cpitl mintennce in growth model with congestion in public cpitl ppers therefore to be stimulting chllenge. 6 The purpose of the present pper is to provide uni ed frmework for nlyzing the mcroeconomic implictions of these issues by formulting growth model with endogenous public cpitl deprecition nd congestion in public cpitl services, in order to explore the stedy-stte properties of the economy nd the ssocited rst-best nd growth-mximizing scl policies. To this end, we follow the minstrem clss of infrstructure-led endogenous growth models developed by Brro (1990) nd Brro nd Sl-I- 4 In n empiricl context, Klitzidkis nd Klyvitis (2005) provide evidence tht the Cndin economy would bene t in terms of growth from relloction between new public investment nd mintennce expenditures. 5 Although the concept of congestion ppers theoreticlly sound nd rising congestion levels in public services re typiclly observed in most countries nd regions, it does not trnslte esily into opertionl guidelines regrding its growth e ects. Dt on congestion re not systemticlly collected on cross-country bsis nd the existing evidence is bsed on locl or ntionl studies involving minly the trnsporttion sector. For instnce, Schrnk nd Lomx (2005) report tht the cost of congestion in the US trnsporttion sector, which ccounts for more thn 30% of totl infrstructure, hs incresed from 12.5 billion Dollrs in 1982 to 63.1 in 2003 (in constnt 2003 prices). Some cross-country estimtes indicte tht the ensuing costs of congestion re found to exceed 3.5 per cent of GDP (United Ntions, 2002). 6 Interestingly, severl reports from developed nd developing countries often ssocite the rising costs of congestion with insu cient infrstructure mintennce; see United Ntions (2006). 3

5 Mrtin (1992), which hve stressed the role of government productive ctivities s key determinnts of long-run growth, by ssuming tht privte sector productivity is ected by government productive services in blnced budget frmework. In our pproch the production side of the economy follows Turnovsky (1997, 1997b) nd Fisher nd Turnovsky (1998) by dopting the view tht the public cpitl stock rther thn the ow of government expenditures is crucil for economic ctivity. In this setup, where the public cpitl stock is subject to congestion, the government cn decide on the public cpitl ccumultion rte by investing either in new public cpitl, or by retrding the rte of decy of instlled cpitl through public mintennce expenditures. Our mjor ndings cn be summrized s follows. We nd tht threshold level of expenditures for public cpitl mintennce, which depends qulittively on the level of congestion in the economy, is necessry for the existence of the blnced growth pth. The key mechnism is tht rise in new public investment nd, consequently, output lso rises the public cpitl deprecition rte nd reduces public cpitl ccumultion due to incresed public cpitl usge. The mgnitude of the rise in public cpitl usge depends upon congestion nd hence economies with low (high) congestion in public infrstructure, in which the rise in output is high (low), will require threshold level of public cpitl mintennce ( new public investment) for ongoing growth. We lso estblish tht, in the presence of congestion, the shre of mintennce expenditures in totl public cpitl expenditures is criticl determinnt of the optiml nd the growth-mximizing government size. Consequently, if the level of congestion in the economy chnges the government cn use the two scl policy instruments, nmely the tx rte nd the composition of public cpitl expenditures, for the design of scl policies to improve welfre nd growth. In prticulr, the optiml tx rte is positively relted to the degree of congestion becuse of the negtive externlity leding to n over-ccumultion of privte cpitl. The optiml tx rte will be even higher in the presence of public cpitl mintennce, becuse the implied rise in public cpitl mintennce long with ggregte public cpitl expenditures reduces the deprecition rte of public cpitl nd, s result, the dditionl tx revenues cn nnce new public investment t greter proportion. Hence, the government lso bene ts from the re-lloction between new public investment nd mintennce to improve welfre. These results hve some importnt policy implictions. First, the requirement of minimum 4

6 level for public cpitl mintennce (nd lso for new public investment) in order for the economy to ttin sustinble growth conforms with the common view tht stgnnt growth is often ssocited with the lck of criticl level of public cpitl mintennce required to boost growth. Second, our results underline the importnce of the e cient policy mix between public cpitl mintennce nd new public investment for the design of scl policies. In prticulr, our ndings imply tht when congestion rises (flls), the government cn bene t by steering public cpitl expenditures wy from (towrds) mintennce nd towrds (wy from) new public investment without necessrily ltering the size of public expenditures. In the sme vein, government tht fces crowded rods nd empty ports cn steer the mix of the existing public cpitl expenditures towrds building new rods nd mintining the existing ports, in order to improve public sector e ciency. The rest of the pper is structured s follows. Section 2 solves the decentrlized equilibrium problem nd section 3 studies the stedy stte nd the dynmic properties of the model. Section 4 presents the socil plnner problem nd derives the rst-best scl policies. Section 5 investigtes the growth-mximizing scl policies nd, nlly, section 6 discusses the results nd concludes the pper. 2. The model This section presents representtive gent model with congestion in public cpitl, in which mintennce expenditures by the government ect the deprecition of the public cpitl stock. The min fetures of the model re s follows: () the production function of the rm depends upon the privte nd public cpitl stocks, with the ltter providing positive production externlity to privte rms, (b) the production function exhibits congestion e ect in the cquisition of public cpitl services by the privte sector, (c) public mintennce expenditures ect negtively the decy of public cpitl, nd (d) public cpitl services in the form of expenditures on new public investment nd public cpitl mintennce re nnced by tx on output. Lower- nd upper-cse vribles denote individul nd ggregte quntities, respectively The representtive gent Consider n economy populted by N homogenous gents with no popultion growth. The 5

7 representtive consumer-producer in this economy consumes, c, of the production good, in order to mximize the following intertemporl logrithmic utility function: mx 1Z 0 log(c)e t dt (1) The representtive gent produces single trded good, y, by fcing the following Cobb-Dougls production function: K y = f(k; Kg) s = y = (k) s 1 g N 1 ; 0 < < 1 (2) where k denotes the individul privte cpitl stock nd K s g denotes the public cpitl services. We ssume tht public cpitl services re subject to congestion nd prmeter 2 (0; 1) mesures the degree of congestion. To eliminte ny scle e ects we ssume tht the productivity of individul privte cpitl depends on the verge level of public cpitl services, which is in turn determined by the number of individuls tht uses them nd the degree of congestion. For instnce, under no congestion ( = 0) ll individuls mke e cient use of the public services nd the productivity of privte cpitl depends on the verge public cpitl services given by Ks g N. We ssume tht the services derived by the gent from the public cpitl stock re given by: K s g = K g k K (3) where K denotes the ggregte privte cpitl stock. Speci ction (3) follows concept put forwrd by Edwrds (1990) nd formlized by, mong others, Turnovsky (1997, 1997b), Fisher nd Turnovsky (1998), nd Gomez (2004), nd implies tht public cpitl is congested by the use of privte cpitl. This speci ction embodies reltive congestion where the level of services derived by the gent from the provision of public good is in terms of the usge of the individul cpitl stock reltive to the ggregte privte cpitl stock (s, for instnce, in the trnsport sector). 7 Within 7 Reltive congestion is opposed to bsolute congestion where the public cpitl stock is congested by the ggregte privte cpitl stock (like, for instnce, in the cse of police protection); see Eicher nd Turnovsky (2000) for n extensive discussion on these two concepts of congestion. We opted for the speci ction with reltive congestion for 6

8 this context, = 0 corresponds to the no-congestion cse, wheres = 1 implies full congestion nd the privte cpitl hs to rise in direct proportion to the ggregte privte cpitl stock to mintin xed level of public services vilble to the rm. The representtive gent is endowed with n initil cpitl stock k(0) > 0 nd ccumultes privte cpitl by spending on investment, i. The privte cpitl ccumultion constrint is then given by the following lw of motion: : k = i k k (4) where k denotes the constnt privte cpitl deprecition rte Government Sector The government invests in new public cpitl, I g, nd mintins the public cpitl stock, M g, to ect its decy rte. We ssume tht the lw of motion for public cpitl ccumultion is given by: : K g = I g g ( M g Y )K g (5) where the deprecition function g () is continuous nd hs the following properties: 0 g() < 0, 00 g() > 0, lim g () = 1 nd lim Mg Y!0 Mg Y g () = g where g 2 (0; 1). This setup covers the generl!1 de nition of cpitl mintennce s the... deliberte utiliztion of ll resources tht preserve the opertive stte of cpitl goods (Bitros, 1976), where s fr s public cpitl is concerned it encompsses the... wide rnge of ctivities iming t keeping infrstructure t serviceble condition (Heller, 1991). We ssume tht the deprecition rte cnnot be diminished below threshold, g, determined by the physicl deprecition due to cpitl geing nd other technicl two resons. First, reltive congestion is ssocited with sectors of economic ctivity tht re closer to infrstructurl cpitl. Second, bsolute congestion in the production function yields scle e ect tht is empiriclly implusible. Notice tht the results derived lter on re not qulittively ected if the speci ction with bsolute congestion is dopted. 8 Assuming tht privte cpitl deprecition is endogenously determined would not ect qulittively the results derived lter on, provided tht privte mintennce enters s rtio of the privte cpitl stock, rther thn output, in the deprecition function. This would be plusible ssumption since it implies tht privte cpitl deprecition is ected by the usge of the privte cpitl stock, in contrst to public cpitl deprecition tht depends upon its ggregte usge given here by output; see the next subsection. 7

9 fctors. We lso ssume tht the public cpitl deprecition rte is negtive function of public mintennce expenditures s rtio of ggregte economic ctivity, Y. Hence, public cpitl deprecition is positive function of the usge of public cpitl by ll gents of the economy (including the public sector) given by totl economic ctivity. This ssumption implies, for instnce, tht public rods will deteriorte fster if the ggregte production ctivity by both the privte nd public sectors in the economy is high. In turn, the government cn choose the level of mintennce expenditures s shre of output to determine the deprecition rte of its cpitl stock. The government nnces expenditures by levying t tx rte on privte output, 2 (0; 1). Assuming blnced budget the government budget constrint is given by: Y = I g + M g (6) To ese exposition, we prmeterize public mintennce M g s shre on tx revenues by g 2 (0; 1) nd thus the corresponding shre for new public investment is given by (1 g ). The internl lloction of government expenditures cn then be written s: I g = (1 g )Y (7) M g = g Y (8) 2.3. The representtive gent problem The representtive gent in the economy receives income from fter-tx production (income) tht is llocted to investment nd consumption: (1 )y = i + c (9) The ggregte production of the economy is llocted to consumption, privte nd public investment, nd public cpitl mintennce. The economy-wide resource constrint is given by: Y = C + I k + I g + M g (10) 8

10 By summrizing the bove constrints nd using (4) in (9) we get tht the ow budget constrint is given by: : k = (1 )y c k k (11) given the initil cpitl endowment k(0) > 0: The intertemporl problem of the representtive gent is to mximize lifetime utility (1) subject to (11). The current-vlue Hmiltonin of the problem is given by: H = log(c) + ((1 )y c k k) The rst-order conditions re given by: 1 c = (12) (1 " # : 1 (1 ) ) ( + (1 ))k (1 )(1 ) (K g ) 1 N (1 )(1 ) = + k K (13) lim (t)k(t)e t = 0 (14) t!1 Eqution (12) is the stndrd condition stting tht t the optimum the mrginl utility of consumption equls the shdow price of welth,. Eqution (13) sttes tht t the optimum the fter-tx mrginl product of cpitl is equl to the opportunity cost of investing in cpitl, which in turn equls the deprecition of physicl cpitl, nd the rte of time preference minus the cpitl gin. The fter-tx mrginl product of privte cpitl is ugmented by the level of congestion becuse investment in privte cpitl lso rises the mrginl bene ts tht the individul derives from the public cpitl stock given the ggregte privte cpitl stock. Congestion opertes then s n externlity of the public cpitl stock on privte decisions; see lso Turnovsky (1997, 1997b). 3. Blnced growth nd equilibrium dynmics in the decentrlized economy The Blnced Growth Pth (BGP) is de ned stte where the vribles of the economy grow t constnt rte. Since ll gents in the economy re identicl, the equilibrium reltionship K = Nk links the ggregte nd individul cpitl stocks. De nition 1. The competitive equilibrium of the economy is de ned for the exogenous policy 9

11 instruments ; g, nd ggregte lloctions I k ; I g ; M g ; C; K; K g such tht the individuls solve their intertemporl utility mximiztion problem by choosing c nd i given nd g. Tking the time derivtive of (12) nd substituting in (13), we get fter ggregting tht the equilibrium growth rte of ggregte consumption, C, is given by: C C = (1 )(1 (1 )(1 )) (1 ) Kg k (15) K Also, fter ggregting the fesibility constrint of our economy, (11), we get tht the ggregte privte cpitl stock evolves s: K K = (1 ) (1 ) Kg C K K k (16) Using (7) nd (8) in (5) the equilibrium growth rte of public cpitl is given by: K g K g = (1 g ) Kg g K g (17) nd the trnsverslity condition under (12) is now modi ed to: k(t) lim c(t) e t = 1 (18) t!1 At the BGP the growth rtes of consumption, public nd privte cpitl hve to grow necessrily t the sme rte, i.e. C C = K K = K g K g = g. This result is esily obtined by investigting the equilibrium growth rtes of these vribles seprtely. In prticulr, for the consumption growth rte, given by (15), to be constnt in the stedy-stte we see tht both K nd K g hve to grow t the sme constnt rte, sy g k = g kg = g. Since the stedy-stte rtio of public to privte cpitl will be constnt, the equilibrium growth rte of consumption, g c, will be constnt too. Then, by inspection of (16), in order for the growth rte of privte cpitl to be constnt we need tht g k = g kg nd g c = g k (for K K g nd C K to be constnt respectively). Tken together with the previous condition on the consumption growth rte to be constnt these conditions imply tht 10

12 g c = g k = g kg = g. Since this condition lso stis es (17), s well s the trnsverslity condition, (18), the necessry condition for the existence of BGP in this economy is tht ll vribles grow t the sme rte, g. Given the bove necessry condition for BGP we cn now derive the equilibrium growth rte of the economy. We rst de ne the following uxiliry sttionry vribles, nmely! C K )! C K C K, z Kg K ) z z K K. By solving (15) for z we obtin tht t the BGP: K g K g! z = g + + k (1 )(1 (1 )(1 )) 1 (1 ) (19) Substituting then (19) in (17) we obtin the equilibrium growth rte of the economy implicitly s: g + + k 1 (g) g (1 g ) + g (1 )(1 (1 )(1 )) g = 0 (20) where the solution of this continuous function for g > 0, such tht (g) = 0, determines the existence nd the properties of the equilibrium growth rte. Proposition 1. is given by (20) i g( g ) (1 g ) (1 ) exogenous policy instruments. Proof. See Appendix 1. A unique positive equilibrium growth rte, g, exists in our economy nd 1 < 1 (1 )(1 ) + k 1 holds for given prmeter vlues nd Notice tht our economy will exhibit zero equilibrium growth rte if the necessry nd su cient prmetric condition holds with equlity. A crucil remrk is tht the two scl instruments, nd g, re criticl not only for the quntittive determintion of the equilibrium growth rte, but lso for the existence of non-negtive BGP. In fct, direct consequence of Proposition 1 is tht there exists subset in the domin of the policy instruments tht forms set of su cient vlues for sustinble growth. The following Corollry formlizes this point. Corollry to Proposition 1. Given the prmetric chrcteristics of the economy, there exists rnge ( g ; ^ g ) 2 (0; 1) of public mintennce expenditures s shre of public cpitl 11

13 expenditures tht hs to be implemented in order for Proposition 1 to hold. The boundries of the public mintennce expenditure shre re given by the following conditions. (I) Lower bound of mintennce shre: The shre of mintennce expenditures in totl public cpitl expenditures hs to exceed minimum vlue, g > 0, where g( ) g (1 g ) (1 ) 1 = 1 (1 )(1 ) + k 1 shre exists i 1 >, in order for the economy to ttin the BGP for ny vlue of 2 (0; 1). This (1 )(1 (1 )(1 )) 1. A su cient prmetric condition for non-zero + k lower bound for mintennce expenditures to exist is given by < + k 1. (II) Upper bound of mintennce expenditures: There exists n upper bound for the shre of mintennce expenditures in public cpitl expenditures, g < ^ g < 1, where g(^ ) g (1 ^ g ) (1 ) 1 = 1 (1 )(1 ) + k 1 Proof. See Appendix 1. for ny vlue of 2 (0; 1), in order for the economy to ttin the BGP. Proposition 1 in conjunction with cses (I) nd (II) of the Corollry show tht there exists rnge g < g < ^ g tht is necessry for the existence of positive long-run growth rte in the economy for ny vlue of 2 (0; 1). In prticulr, economies tht re described by (I) will fce non-positive stedy-stte growth regrdless of their government size, unless dequte resources on public cpitl mintennce re spent. For su ciently high levels of public investment n dditionl unit of new public cpitl boosts ggregte economic ctivity nd hence the usge of the existing public cpitl stock, which depends upon the degree of congestion, rises. This genertes the need for dditionl public cpitl mintennce becuse the deprecition rte of public cpitl is endogenously determined by the shre of mintennce expenditures in output, g. Due to the convexity of the deprecition function, t su ciently low levels of mintennce ctivity the positive e ect of public investment on public cpitl ccumultion is outweighed by the negtive e ect generted through the rise on the usge of public cpitl through its deprecition rte. As result, minimum level of mintennce is necessry to sustin positive growth rte of public cpitl in the long run. The inverse hppens for su ciently high levels of mintennce ctivities: the bene ts on the deprecition rte re diminishing nd, moreover, re bounded by the nturl deprecition rte. A threshold level of new public investment is therefore necessry s well for ttining long-run growth. 12

14 These e ects depend on the level of congestion in the economy through the degree of public cpitl usge determined by economic ctivity, Y. As indicted by Prt I of Corollry to Proposition 1, when congestion is su ciently low the usge of the public cpitl stock will be high. This rises the deprecition of the existing cpitl stock nd in order for the growth rte of public cpitl to follow blnced growth pth minimum level of mintennce is required to preserve the existing cpitl stock t the higher usge. 9 Hence, when the minimum level of public cpitl mintennce is not reched, new public investment will not be dequte to ttin long-run growth due to the high deprecition of public cpitl. This result extends the point by Rioj (2003) nd Klyvitis nd Klitzidkis (2004) on the importnce of these expenditures for growth mximiztion by showing tht public cpitl mintennce expenditures re necessry for the existence of sustinble growth. On the ipside, the condition on the upper bound for mintennce expenditures sttes tht country hs to set public cpitl mintennce expenditures t level tht will ensure minimum level of investment in new public cpitl. 10 Finlly, given the initil vlue of public to privte cpitl rtio, z 0, we cn exmine the locl stbility of the economy tht is determined by the two-dimensionl system:!! = (1 )(1 )(1 )z1 +! (21) z z = (1 g)z g g (1 )z 1 +! + k (22) 9 Notice tht the other structurl chrcteristics of the economy re importnt for the BGP s well. In prticulr, the elsticity of public cpitl in the production function, the deprecition rte of privte cpitl, the rte of time preference, nd congestion ect ceteris pribus the threshold level of public mintennce expenditures to ttin positive growth. For instnce, high rte of time preference (low propensity to sve) nd high deprecition rte of privte cpitl ect negtively the growth rtes of consumption growth nd privte cpitl. In these cses the privte gents cnnot bene t dequtely from public cpitl services nd the high usge of the public cpitl stock requires threshold level of mintennce expenditures s shre of output in order for cpitl ccumultion to generte positive growth. 10 Notice tht, s in Brro (1990), similr resoning pplies for the lower nd upper bound of the government size required to ensure positive growth. It is strightforwrd to show tht Proposition 1 imposes boundries for the tx rte since it does not hold for = 0 or = 1. 13

15 where dot bove vrible denotes the corresponding derivtives with respect to time. In mtrix nottion we cn write: 2 6 4! z = 2 6 4! ((1 )(1 )(1 )2 (z)! z (1 g )( )(z) 1 (1 )(1 )(z) z !! z z The determinnt of the bove system is given by: J =!z (1 g )(z) 1 (1 (1 )(1 ))(1 )(1 )(z) with J < 0, since (1 g )(z) 1 + (1 (1 )(1 ))(1 )(1 )(z) > 0 for ny positive vlue of z (which cn be esily shown to exist). Thus the system comprised by (21) nd (22) is loclly sddle-pth stble. 4. Optiml policies In this section we solve the socil plnner problem in order to determine the optiml scl policy. To this end, we will derive the optimlity conditions for the socil plnner economy nd we will then determine the optiml government size (tx rte) with the shres of its components, new public investment nd public cpitl mintennce, chosen optimlly in order to replicte the rst-best environment The Socil Plnner problem The objective of the socil plnner is to mximize socil welfre given by: mx 1Z 0 log(c)e t dt subject to the economy s ggregte resource constrint: Y = K Kg 1 = C + I k + I g + M g (23) 14

16 nd the lw of motion for the ggregte public nd privte cpitl stocks: _K = I k K (24) _K g = I g g ( M g Y )K g (25) given the initil public nd privte cpitl stocks, K(0) nd K g (0). Eqution (23) denotes the ggregte resource constrint in our economy. In contrst to the decentrlized equilibrium, the production technology in the socil plnner problem internlizes the congestion externlity, since the socil plnner solves for the ggregte quntities. The current vlue Hmiltonin for the bove problem is given by: H = log(c) + (K Kg 1 C k K I g M g ) + (I g g ( M g Y )K g) where the choice vribles re C, I g, M g, nd the stte vribles of the economy re K, K g. The rst-order conditions re given by: 1 C = (26) = (27) 0 g( M g Y )K g Y = (28) K 1 g K k + 0 g( M g Y )M g Kg Y K = : + (29) K g (1 ) K + (1 ) 0 g( M g Y )M g Y g ( M g Y ) = : + (30) Equtions (26) to (30) yield the optiml lloctions in the socil plnner economy given the initil conditions nd the ssocited trnsverslity conditions. Speci clly, eqution (26) shows tht t the optimum the mrginl utility of consumption equls the mrginl chnge of welth. Eqution (27) implies tht the bene t of incresing public investment mesured by the shdow price of public cpitl equls the cost in welth by llocting expenditures of consumption nd mintennce on investment. As opposed to the decentrlized equilibrium, the socil plnner decides for the optiml 15

17 lloction of public investment nd mintennce expenditures with the ltter given by (28). The government fces then trde-o in llocting resources to mintennce nd investment s both expenditures ect positively the ccumultion rte of public cpitl nd re relted through the government resource constrint. Substituting (26), (28), nd (27) in (29) we cn obtin the consumption growth rte in the socilly plnned economy, g sp c : g sp c C _ K 1 C = g K [1 M g Y ] k (31) Also, by the fesibility constrint nd the privte nd public cpitl ccumultion rtes, nd by (7) nd (8), we obtin tht: k K _ K 1 g C = (1 ) K K K k (32) g sp g sp g _K g K g = (1 g ) K g K g ( M g Y ) (33) where g sp k nd gsp g denote the equilibrium growth rtes of public nd privte cpitl in the socil plnner economy. In the presence of congestion the shre of public expenditures for cpitl mintennce ects the growth rtes of consumption nd public cpitl in the socil plnner economy. This occurs becuse of the required txtion imposed on privte gents, since the return to privte investment exceeds the socil one. To internlize the distortion, the individul producer who ccumultes k (nd thus contributes positively to ggregte usge given by Y ) hs to provide dditionl resources to mintin the public services vilble to other gents, where the required compenstion should equl M g =Y times the ddition to Y. In turn, by substituting (27) in (28) we cn obtin the optiml lloction of mintennce expenditures to output by the following condition: 0 g( M g Y ) = Y K g (z ) (34) Eqution (34) determines the optiml public to privte cpitl rtio nd sttes tht it is optiml for the government to equte the chnge in the deprecition rte of public cpitl generted by 16

18 public cpitl mintennce reltive to its usge with the verge product of public cpitl. Furthermore, (29) nd (30) represent the optiml conditions with respect to the privte nd public cpitl stocks respectively. Using (27) nd (28) in (29) nd (30), nd by virtue of (8), we cn obtin tht: (1 g )(z ) 1 k = (1 )(1 g )(z ) g g (35) Eqution (35) yields the rtio of public to privte cpitl in the socil plnner economy s function of the prmeters of the economy nd the policy instruments nd shows tht the net returns of privte nd public cpitl re equlized t equilibrium. The mrginl products of privte nd public cpitl re now reduced s the output shre of public cpitl mintennce increses. The mrginl bene t of spending one unit in public cpitl mintennce, Mg, is given by the fll in g g nd the rise in the net return on K g. At the sme time, there re less vilble resources (output units) for C + I k + I g nd s M g rises the mrginl products of the cpitl stocks re reduced by frction ( C+I k+i g Y ), since the lst unit of output spent on mintennce hs to hve the sme impct on welfre with the corresponding one spent on the other components of output. Notice tht if the exogenous deprecition rte of privte cpitl hppens to coincide with the endogenous deprecition rte of public cpitl evluted t the optimum, then the rtio of public to privte cpitl is given by the fmilir condition z = 1, which sttes tht it depends upon the rtio of the corresponding cpitl stock elsticities in the production function. This condition however is only specil cse here, s the public to privte cpitl rtio will depend on other vribles nd prmeters of the economy s well, such s the mintennce to output rtio, the endogenous deprecition rte of public cpitl nd the privte deprecition rte. It is strightforwrd to show tht well-de ned z > 0 exists for ny prmeter vlue of our economy in the ssumed domin nd tht it is function of the structurl chrcteristics of our economy Eqution (35) cn be written s (z ) = (1 g )(z ) 1 k (1 )(1 g )z + g g = 0 nd hs the following properties: (1) (z ) is continuous function for z > 0 since it is the ddition of continuous functions. = (1 )(1 g )(z ) (1 + (z ) 1 ) > 0 (3) lim z!1 (z ) = +1 17

19 4.2. Optiml scl policies Given the solution of the socil plnner nd the decentrlized equilibrium, the government ims t choosing tht tx rte nd the composition of public cpitl expenditures tht enble the competitive equilibrium to chieve rst-best lloction of resources. Using (8) the consumption growth rte in the socil plnner economy cn be written s: g sp c = 1 g (z ) 1 k (36) Since, by (36),(32), nd (33), t the BGP consumption, public cpitl, nd privte cpitl hve to grow t the sme rte, in order to enble the decentrlized equilibrium to replicte the rstbest outcome the government hs to set the tx rte t the level tht equtes the growth rte of consumption of the decentrlized economy with the one ttined by the socil plnner. 12 From (15) nd (36) we obtin the optiml tx rte with public cpitl mintennce under congestion: = (1 ) (1 ) + (1 g) (37) The optiml tx rte is clerly fesible for ny vlue of congestion nd mintennce in their domin. Under public cpitl mintennce nd congestion, the optiml tx rte di ers from tht derived by Brro (1990) nd is more generl cse of the policy rule derived by Klitzidkis nd Klyvitis (2004). To highlight eqution (37), we rst consider some limit cses. If there is no congestion in the economy ( = 0), the optiml tx rte is equl to zero since there is no distortion on individul decisions; this is mnifesttion of the well-know Chmley (1986) result tht cpitl should be untxed t the optimum. On the opposite, under full congestion the optiml tx rte is given by =1 = 1 1, which sttes tht the optiml tx rte depends positively on the public g (4) lim z!0 (z ) = 1 From 1-4 it follows tht there exists z > 0 tht solves (z ) such tht (z ) = 0 nd it is unique. 12 Notice tht the growth rtes of public nd privte cpitl re quntittively di erent thn the ones obtined in the decentrlized economy becuse the corresponding consumption growth rtes re di erent. By equting the consumption growth rte of the socil plnner with the competitive equilibrium through the optiml tx rte, the decentrlized economy will chieve the outcome of the socil plnner in terms of the growth rtes of public nd privte cpitl s well. 18

20 mintennce expenditures to output rtio. 13 As hs been pointed out in Klyvitis nd Klitzidkis (2004), the endogeneity of public cpitl deprecition renders the Brro (1990) optiml txtion rule suboptiml, s mintennce expenditures increse the mrginl cost of public funds nd the government size hs to exceed the elsticity of public cpitl in the production function to equl the corresponding mrginl bene t. In the cse of prtil congestion (0 < < 1), the tx rte is lower nd we cn estblish tht it is positive function of the level of congestion,, for given shre of public = (1 )(1 g) 1 (1 )(1 ) g 2 > 0 (38) Equtions (37) nd (38) extend the optiml txtion rule derived by Gomez (2004), which lso sttes tht the government hs to implement positive time-invrint tx rte in order to drive the decentrlized economy to the rst best outcome. The optiml tx rte is positively relted to the degree of congestion in the economy s result of the implied intensity of the negtive externlity: the higher return to privte investment nd the over-ccumultion of privte cpitl triggered by congestion hve to be diminished by imposing higher tx on output in order to ttin the socil optimum. Moreover, the optiml tx rte is higher in the presence of public cpitl mintennce becuse gents do not tke into ccount the positive e ect on ggregte ctivity generted by their over-investment in privte cpitl. Hence, the optiml tx rte will be positively relted to mintennce expenditures, which hve to be rised to sustin the high usge of public cpitl. However, the e ect of congestion on the optiml tx rte is now mitigted s the government cn bene t from re-lloction between new public investment nd public cpitl mintennce to chieve the rst-best outcome. In prticulr, we cn investigte the long-run impct of congestion by using equtions (34), (35) nd (37), which determine jointly the three endogenous vribles of the model, ; z ; g, in terms of the exogenous prmeters. 14 The totl e ects of congestion on 13 Note tht in the cse of full congestion the optiml txtion rule coincides with the growth-mximizing rule derived lter on, becuse the economy hs to grow t the mximum rte, in order for the decentrlized economy to eliminte the externlity nd replicte the rst-best environment. 14 Since this is perfect foresight model, our comprtive sttic exercises re equivlent to n unnticipted chnge in congestion. 19

21 the optiml tx rte nd the optiml lloction of government revenues to mintennce re given by: d d j totl = A B g + > 0 (39) d g d = A g B g + < 0 (40) > 0 is the prtil derivtive of the optiml tx rte w.r.t. given by (38) = (1 ) > 0 is the prtil derivtive of the optiml tx rte w.r.t. g (1 (1 )(1 ) g g )2 determined by (37). Tking the totl derivtive of the optiml tx rte given in (37), we obtin eqution (39) lterntively s: d d j totl = A + B d g d (41) Here, A represents the positive direct e ect of congestion on the optiml tx rte ( txtion e ect) nd (B d g d ) represents the negtive indirect e ect triggered by congestion through the chnge in the optiml shre of mintennce expenditures ( relloction e ect). The txtion e ect clerly domintes the relloction e ect, since the totl e ect of congestion on the optiml tx rte is positive, s shown by (39). These results re summrized in the following Proposition. Proposition 2. The txtion e ect of congestion on the optiml tx rte is positive. The relloction e ect of congestion on the optiml shre of public cpitl mintennce in totl public cpitl expenditures is negtive. The totl e ect of congestion on the optiml tx rte is positive. Proof. See Appendix 2. Proposition 2 generlizes the impct of congestion on optiml scl policy. The policy chnge in the optiml lloction of resources brought bout by congestion, which is cptured by the term (B d g d ), mitigtes the required increse in the optiml tx rte, becuse the government cn now bene t from the optiml re-lloction of public cpitl expenditures between new investment nd public cpitl mintennce. Intuitively, rise in congestion reduces the mount of public services extrcted by the privte gent nd increses their mrginl vlue, thus incresing the optiml tx rte ( txtion e ect). At the sme time, the rise in ggregte public cpitl expenditures reduces the deprecition rte of public cpitl. With constnt di erentil between the mrginl products 20

22 of privte nd public cpitl, eqution (35) implies tht the shre of public cpitl mintennce in output hs to fll to equlize the net returns of the cpitl stocks. As result, the dditionl tx revenues nnce new public investment t greter proportion ( relloction e ect). We cn lso investigte the impct of congestion on the lloction of public cpitl expenditures when the government ims t holding the tx rte constnt. In such cse, the government cn use the internl lloction of resources s n lterntive policy instrument, wheres the tx rte remins un ected by congestion. This e ect is given by: d g d j d =0 = 1 g < 0 (42) Eqution (42) gives the optiml response of government expenditures in mintennce s shre of totl public cpitl expenditures following mrginl chnge in the level of congestion for xed government size. In prticulr, following rise (fll) in congestion the government should reduce (increse) the level of public cpitl mintennce. Therefore, following rise in congestion the government cn improve e ciency by ltering the composition of public cpitl expenditures between mintennce nd new investment insted of imposing higher tx on production to rise public services, s Proposition 2 would imply. A direct consequence of (42) is tht countries or regions fcing scl limittions cn re-llocte the existing public resources by steering expenditures towrds (wy from) new public investment nd wy (towrds) public cpitl mintennce in response to higher (lower) congestion. Therefore, in the presence of incresing congestion in public cpitl (nd ssuming tht mintennce expenditures nd new investment exceed their threshold vlues dictted by the Corollry to Proposition 1), the government is equipped with n extr policy instrument for the conduct of optiml scl policy. 5. Growth-mximizing policies In this section we nlyze growth-mximizing scl policy rules. Modern growth theory hs shown prticulr interest in growth-enhncing policies, s the understnding of the forces of economic growth is crucil in order to identify the reltive merits nd synergies of government interventions in res like the formtion nd lloction of public cpitl. Moreover, the growth rte is usully the min mesurble objective of the government nd hence it is useful to ssess the 21

23 contribution of the components of public cpitl expenditures iming t long-run growth. To nlyze the scl policies tht im t growth-mximiztion in the context of the present model, we extend the pproch dopted by Klyvitis nd Klitzidkis (2004) by tking into ccount the impct of congestion on the e ects on the growth-mximizing government size nd the shre of mintennce expenditures s follows. De nition 2. Growth-mximizing policies in the competitive equilibrium of the economy re given under De nition 1 when the government chooses ~ nd ~ g in order to mximize the long-run growth rte of the economy by tking into ccount the ggregte mximizing behvior of individuls, nd the government budget constrints nd the fesibility nd technologicl conditions re met. The problem of long-run growth-mximizing policies cn then be formulted s: mx ~g(; g ; z) = (1 )(1 (1 )(1 ))~z 1 k subject to the decentrlized equilibrium response by the privte gent given by the system (21) nd (22). Tking the rst-order conditions, we cn obtin fter some lgebr the following growthmximizing conditions: ~ = 1 1 ~ g (43) 0 g ~ g ~ = ~z (44) (1 ~ g )~z ~ g ~ g ~ (1 ~)(1 (1 )(1 ))~z 1 + k + = 0 (45) Equtions (43), (44), nd (45) express the three endogenous vribles, ~, ~ g, ~z, in terms of the model prmeters nd determine the rules tht the government hs to stisfy to ttin growthmximiztion. Eqution (43) coincides with the growth-mximizing tx rte obtined by Klitzidkis nd Klyvitis (2004) nd sttes tht public mintennce expenditures ect positively the growth-mximizing size. The size of these expenditures s shre of totl public cpitl expenditures is then determined by the curvture of the public cpitl deprecition function, determined in (44) in terms of the verge product of public cpitl determined. As in Klitzidkis nd Klyvitis (2004), the Brro (1990) growth-mximizing tx rte is suboptiml nd depends upon public 22

24 mintennce expenditures. However, the growth-mximizing tx rte does not depend here solely on the growth-mximizing shre of expenditures in public cpitl mintennce. Tking the totl derivtives in the bove system we cn ssess the response of the policy vribles with respect to congestion, which re summrized in the following Proposition. Proposition 3. The e ects of congestion on the growth-mximizing tx rte nd the growthmximizing shre of public cpitl mintennce in totl public cpitl expenditures re mbiguous nd depend upon the mgnitude of the elsticity of the public cpitl deprecition rte with respect to the shre of public cpitl mintennce expenditures in totl public expenditures. Proof. See Appendix 3. Congestion impcts on the growth-mximizing government size through the chnge in the growth-mximizing lloction of government expenditures on mintennce. However, the outcome di ers from the rst-best solution where the plnner tkes the public to privte cpitl rtio, z, s given by (34). The tx rte depends now on the response of the public cpitl deprecition rte ( ected by the equilibrium level of mintennce to output rtio) nd the ssocited verge product of public cpitl. For instnce, rise in congestion increses initilly the verge product of public cpitl s privte gents over-ccumulte privte cpitl. This lowers the public to privte cpitl rtio nd rises output directly through the production function. As output rises the usge of public cpitl rises too, which in turn rises the public cpitl deprecition rte nd reduces the public cpitl stock. If the elsticity of the deprecition function is su ciently low the positive chnge in the mrginl bene t of public funds through congestion is outweighed by the fll in the mrginl bene t of public funds nd mintennce expenditures, nd the government size hs to fll. On the ip side, fll in congestion will rise the mrginl bene t of mintennce expenditures nd the verge product of public cpitl if the response of the deprecition rte w.r.t. public cpitl mintennce (mesured by the corresponding elsticity) is lrge enough. This will generte the need for dditionl expenditures on public cpitl mintennce, which in turn requires higher tx rte to nnce these outlys By substituting (43) nd (45) in (44), it is strightforwrd to show tht the totl e ect of congestion on growthmximizing scl policies under congestion depends on the initil level of congestion nd the structurl prmeters of 23

25 6. Discussion nd conclusions The im of the pper ws to explore the stedy stte nd scl policy implictions of public cpitl mintennce in n endogenous growth model under the presence of congestion in public cpitl services. We showed tht minimum mount of cpitl expenditures hs to be devoted in public cpitl mintennce nd new public investment in order for the economy to ttin stedystte growth. This result extends the set of mechnisms tht underline the importnce of threshold levels for scl vribles in order to ttin long-run growth. These thresholds stem now from the composition of public productive expenditures in two seprte ctegories with di erentil impcts on public cpitl ccumultion nd growth, nd s in the cse of the tx rte in the Brro (1990) model, the threshold levels rise in the form of sine qu non conditions (rther thn s priori ssumptions) for long-run growth. In prctice, however, the threshold level for mintennce expenditures cn be more di cult policy trget to chieve: public cpitl mintennce often hs low priority in government budgets, s this form of outlys re politiclly less ppeling nd visible thn new investment projects, nd hrdly ect the condition of the public cpitl stock for some time. This hs often led to mintennce deferrl, s the consequence of deferring mny types of public cpitl re not visible in the short or medium-run to myopic voters nd further hrdens the provision of dequte resources for public cpitl mintennce. 16 We lso found tht in the presence of congestion, mintennce expenditures re importnt for the rst-best txtion rule. The shre of mintennce expenditures ects positively the optiml government size due to the chnge on the mrginl cost of public funds. The government cn lso bene t from the re-lloction between new public investment nd mintennce to improve e ciency t the existing tx rte in the presence of incresing (decresing) congestion by shifting cpitl expenditures wy from (towrds) public cpitl mintennce. Thus, following the spirit of Devrjn et l. (1996) our ndings extend the set of mechnisms tht highlight the importnce of the composition of public productive expenditures for growth. 17 Given tht most countries the economy. For instnce, if the initil congestion level nd the intensity of public cpitl in the production function re low, the e ect of congestion on the growth-mximizing shre of mintennce is more likely to be positive. 16 Hulten nd Peterson (1984) discuss in more detil the issue of mintennce deferrl in the public sector. 17 The existing growth literture on the optiml composition of public spending typiclly distinguishes between 24