PUBLIC DEBT AND ECONOMIC GROWTH IN UZAWA S TWO-SECTOR MODEL WITH PUBLIC GOODS

Transcription

1 Inernaional Journal of Economic Sciences DOI: /ES PUBLIC DEBT AND ECONOMIC GROWTH IN UZAWA S TWO-SECTOR MODEL WITH PUBLIC GOODS WEI-BIN ZHANG Absrac: This paper sudies public deb dynamics in a neoclassical growh model. The economy consiss of one capial, one service and one public secor. The model is a synhesis of Solow s growh, Uzawa s wo-secor, and Diamond s deb models. The public secor supplies services which direcly affec produciviy of he wo secors and welfare of he populaion. The governmen finances public expendiure by axing he oupus, consumpion, wealh income, and wage households. The sudy focuses on he effecs of changes in governmen s expendiure, he public secor s produciviy, and differen axes on he dynamics of public deb and economic growh. Comparaive dynamic analysis examines he effecs of changes in preference and polices. Keywords: public deb, ax raes, public good, growh JEL Classificaion: O41, H41, H63 Auhors: WEI-BIN ZHANG, Risumeikan Asia Pacific Universiy, Japan, wbz1@apu.ac.jp Ciaion: WEI-BIN ZHANG (216). Public Deb and Economic Growh in Uzawa s Two-Secor Model wih Public Goods. Inernaional Journal of Economic Sciences, Vol. V(4), pp , /ES Acknowledgemens: The auhor is graeful o he consrucive commens of he wo anonymous referees. The auhor also hank Japan Sociey for he Promoion of Science for he financial suppor from he Grans-in-Aid for Scienific Research (C), Projec No Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 51

2 Inernaional Journal of Economic Sciences 1 Inroducion Many models have been developed o sudy relaions beween naional debs and growh. The emergence of huge public debs in he las hree decades in differen economies has simulaed heoreical as well as empirical analyses of naional debs in he lieraure of economics. The purpose of his sudy is o address issues relaed o public deb in he neoclassical growh framework. The main feaures of economic growh are based on he neoclassical growh heory. The neoclassical heory is based on he pioneering works of Solow (1956) and Swan (1956). In he sand neoclassical growh model, capial and labor are subsiues for one anoher wih he resul ha he long-run growh pah of he economy is one of full employmen. The hree facors, capial, labor and echnology, are he key deerminans of economic growh. Alhough he earlier models of he heory rea labor and echnology as exogenous variables, many effors have been made o relax he radiional assumpions. The Solow model shows ha he razor-edge growh pah of he Domar model is primarily a resul of he paricular producion funcion assumpion adoped herein and ha he need for delicae balancing may no arise when he producion funcion is aken on a differen ype. The Solow model has been exended and generalized in numerous direcions (Burmeiser and Dobell, 197; Zhang, 25). This paper follows he radiional wo-secor economy iniially proposed by Uzawa (1961). The Uzawa model exends he Solow model by a breakdown of he producive sysem ino wo secors using capial and labor, one of which produces capial goods, he oher consumpion goods (Solow, 1962). This paper inroduces an alernaive approach o consumer decision o examine srucural change wihin Uzawa s wo-secor economy. This sudy is concerned wih issues relaed o he role of fiscal policies on economic growh. We are concerned wih dynamic inerdependence beween economic growh and public invesmen. Some recen endogenous growh models have emphasized he role of producive fiscal policy as a deerminan of persisen economic growh (Barro, 199; Turnovsky, 2, 24; Glomm and Ravikumar, 1997; Gómez, 28; and Park, 29). As in Hochman (1981) and Wijkander (1984), his paper is concerned wih provision of public goods. We assume ha only he governmen is responsible for he provision of public goods. The governmen chooses he values of a se of conrol measures according o some predeermined rules. The governmen provides public goods, minimizing he cos of public goods provision under hese rules. The se of conrol measures a he governmen s disposal includes he oal expendiures and ax raes on he indusrial secor s oupu, he service secor s oupu, he wage income, he consumpion and he ineres income. The modeling of his sudy is also influenced by Lin (2). This sudy is differen from Lin s model in ha he Lin s model is developed in he discree Uzawa-Lucas wo-secor model, while his model is a coninuous Uzawa s wo-secor growh model. In Lin s model, he governmen expendiure is spen on educaion, while in his sudy he public expendiure is spen on providing public services which are used by he wo producion secors and he household. In Lin s model, here is only one lump sum ax on he household, while in his sudy he governmen may ax on he household s wage Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 52

3 Inernaional Journal of Economic Sciences income, wealh income, consumpion, as well as on he wo producion secors oupus. Lin s examinaion is focused on he effecs of changes in some parameers on he seady sae; while his sudy is focused on effecs of changes in some parameers on he dynamic pahs of he economic sysem as well as on he equilibrium poin. Almos of all he recen heoreical lieraure of dynamic ineracions beween economic growh and public debs use eiher he Ramsey framework in coninuous ime (Cohen and Sachs, 1986; Blanchard and Fischer, 1989; Barro e al. 1995; Semmler and Sieveking, 2; Guo and Harrison, 24; and Giannisarou, 27) or he OLG modeling framework in discree ime (Diamond, 1965; Farmer, 1986; Turnovsky and Sen, 1991; Azariadis, 1993; de la Croix and Michel, 22; and Chalk, 2). Differen from he radiional approaches o household decision, his sudy uses Zhang s approach o household decision o re-examine he deb issues addressed by Diamond (1965). Zhang proposed he alernaive approach in he early 199s (Zhang, 1993). The implicaions of his approach are similar o hose in he Keynesian consumpion funcion and models based on he permanen income hypohesis, which are empirically much more valid han he approaches in he Solow model or he in Ramsey model. The approach is discussed a lengh by Zhang (25, 28). Zhang (25) has also examined he relaions beween his approach and he Solow growh heory, he Ramsey growh heory, he permanen income hypohesis, and he Keynesian consumpion funcion in deails. I can be shown ha he behavior generaed by he radiional approaches can also be observed in Zhang s approach by specifying cerain paerns of preference changes. As i assumes a fixed proporion of disposable income (which is he curren income in Zhang s approach) is saved, he Solow model does no ake accoun of how changes in wealh affec saving behavior. Alhough i can be generalized o ake accoun of he possible impac of wealh on saving behavior, he radiional Keynesian consumpion funcion lacks a raional micro foundaion. The approach becomes less effecive when one has o deal wih muliple goods and saving in a consisen manner. The Ramsey approach is based on he assumpion ha he uiliy is addiional for he same person over he person s life ime. There exensive sudies on he limiaions of his formaion. A comprehensive survey of he lieraure is referred o Frederick e al. (22). Zhang proposes an alernaive approach o he household behavior by aking accoun of he valid poins and overcoming he shorcomings in he radiional approaches. The res paper is organized as follows. Secion 2 defines he basic model. Secion 3 shows how we solve he dynamics and simulaes he model. Secion 4 examines effecs of changes in some parameers on he economic sysem over ime. Secion 5 concludes he sudy. The appendix proves he main resuls in Secion 3. 2 The growh model wih governmen deb The model is based on he basic feaures of hree well-known models, he growh model of Solow (1956), he wo-secor growh model of Uzawa (1961), and he growh model wih public deb of Diamond (1965). The household decision is based on Zhang s approach (Zhang, 1993). Following he radiional wo-secor growh model, we consider an economy Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 53

4 Inernaional Journal of Economic Sciences which produces capial good and consumpion good. We exend he wo-secor growh model by inroducing public secor. The public secor uses capial and labor as inpus and supplies public services which are freely available o consumers and producers. The public secor is financially by he governmen which axes he household and he wo producion secors. The price of he indusrial good is uniy. Capial depreciaes a a consan exponenial rae, k, which is independen of he manner of use. Technologies of he producion secors are characerized of consan reurns o scale. All markes are perfecly compeiive and capial and labor are compleely mobile among he secors. All privae asses are held by he households eiher as capial or in he form of governmen deb. We assume ha labor is homogeneous and is fixed. We use subscrip index, i, s, and p, o denoe respecively he indusrial and service secors. Le i, s, w, and k, sand for, respecively, he fixed ax raes on he indusrial oupu, he service oupu, he wage income, and he ineres income. We inroduce x 1 x, where x i, s, w, k. Le K j and N j sand for he capial socks and labor force employed by secor j, j i, s, p, a ime. We use F j o represen he oupu level of secor j. Indusrial secor The producion funcion of he indusrial secor is given by Fi Ai Fp i Ki i Ni i, i, i, i i 1, (1) where i, and i are parameers. Markes are compeiive; hus labor and capial earn heir marginal producs, and firms earn zero profis. We inerpre he variables as public goods such as physical and insiuional infrasrucures (Zhang, 29, 211). The aggregae public goods Fp is supplied by he governmen and is aken as given by he firms. Despie increasing social reurns o scale, he funcion allows o mainain he assumpion of perfec compeiion in he goods marke since he echnology exhibis consan reurns o scale for any given level of public goods, which firms canno conrol. We use w and r o sand for he wage rae and rae of ineres. The profi is i 1 i Fi r k r w N i. The marginal condiions for maximizing he profi are r i i Ai Fp i ki i, w i i Ai Fp i ki i, Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp (2) 54

5 Inernaional Journal of Economic Sciences where ki Ki / Ni, and r r k. Service secor The producion funcion of he service secor is Fs As Fp s K s s N s s, s, s, s s 1, 1 s (3) where s and s are parameers. We use p o sand for he price of service. The profi is s 1 s p Fs r k r w N s. The marginal condiions for he service secor are r s s As Fp s p ks s 1, w s s As Fp s p ks s, (4) where ks Ks / Ns. The public secor We now describe he public secor. In his model, we assume ha he public secor is financially suppored by he governmen. The capial socks and workers employed by he public secor are paid a he same raes ha he privae secors pay he services of hese facors. We assume ha he governmen uses he resources effecively in he sense ha he budge is used o maximize public services. The producion of public services is o combine capial K p and labor force N p as follows Fp Ap K p p N p p, p, p, Ap. Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp (5) 55

6 Inernaional Journal of Economic Sciences For simpliciy, we assume ha he oal expendiure on supplying he public good Yp is consan, i.e. Yp Y, (6) where Y is a posiive consan. The public secor is faced wih he following budge consrain w N p r K p Yp. (7) Maximizaion of public services under he budge consrain yields r K p p Yp, w N p p Yp, (8) in which p p p, p. p p p p The governmen budge Following Diamond (1965) and Barro (1974), we assume ha he governmen issues an amoun deb D, which may be considered as real-valued bonds. I is supposed ha asse holders regard equiy and governmen bonds as perfec subsiues. The deb pays he amoun of real ineres r D. The governmen finances curren spending by collecing axes and issuing ineres-bearing deb. Le Tp sand for he governmen s ax income. The dynamics of public deb is D r D Yp Tp. Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp (9) 56

7 Inernaional Journal of Economic Sciences I should be noed ha if D is negaive, hen he negaive privae ownership of capial is less han he economy s capial. This may occur, for insance, he governmen purchases shares in privae firms and issues zero governmen deb. Behavior of domesic households This sudy uses an alernaive approach for modeling consumers behavior. The implicaions of his approach are similar o hose in he Keynesian consumpion funcion and models based on he permanen income hypohesis, which are empirically much more valid han he approaches in he Solow model or he in Ramsey model. The approach is discussed a lengh by Zhang (25). I should be noed ha Zhang (25) has also examined he relaions beween his approach and he Solow growh heory, he Ramsey growh heory, he permanen income hypohesis, and he Keynesian consumpion funcion in deails. I can be shown ha he behavior generaed by he radiional approaches can also be observed in Zhang s approach by specifying cerain paerns of preference changes. Firs, we use k and d o represen respecively he real wealh and governmen deb owned by he represenaive household. The curren income is y k r k w w r d, (1) where r *k is he ineres paymen, w he wage paymen and d D / N. We call y he curren income in he sense ha i comes from consumers wages and curren earnings from ownership of wealh and deb. The sum of income ha consumers are using for consuming and saving are no necessarily equal o he curren income because consumers can sell wealh o pay, for insance, he curren consumpion if he curren income is no sufficien for consuming. The oal value of he wealh ha a consumer can sell o purchase goods and o save is equal o k d. The disposable income a any poin of ime is yˆ y k d. (11) The disposable income is used for saving and consumpion. A ime he consumer has he oal amoun of income equaling y o disribue beween consuming and saving. A each poin in ime, a consumer disribues he oal available budge beween he consumpion of services cs, indusrial goods ci, and saving s. The budge consrain is Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 57

8 Inernaional Journal of Economic Sciences 1 ~s p cs 1 ~c ci s yˆ, (12) where ~s and ~c are respecively he ax raes on he consumpion of services and indusrial good. Equaion (12) means ha he consumpion and saving exhaus he consumers disposable personal income. We assume ha uiliy level U of he household is dependen on cs, ci and s as follows U Fpd cs ci s, d,,,, in which,, and are a ypical person s elasiciy of uiliy wih regard o services, indusrial goods, and saving. We call,, and o consume services, o consume indusrial goods, and o hold wealh, respecively. I should be remarked ha Blanchard (1983) inroduces he disuiliy of deb in he radiional uiliy funcion. Maximizing U subjec o (12) yields cs yˆ p, ci yˆ, s yˆ, (13) where 1,,,. ~ ~ 1 s 1 c The household s oal wealh is given by a k d. According o he definiion of s, he wealh accumulaion for he household is a s a. (14) This equaion saes ha he change in wealh equals he saving minus he dissaving. The governmen s ax income The governmen s ax income comes from axing he wo secors, he ownership of wealh, and consumpion. We have Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 58

9 Inernaional Journal of Economic Sciences Tp i Fi s p Fs k r k N ~s p cs N ~i ci N, (15) where i Fi is he governmen s income from axing he capial secor, s p Fs is from axing he service secor, k r k N from axing he households ineres income, ~s p cs N from axing he service consumpion, and ~i ci N from axing he good consumpion. Demand of and supply for services The equilibrium condiion for services is cs N Fs. (16) Full employmen of capial and labor The oal capial socks employed by he counry, K, is employed by he hree secors. The full employmen of labor and capial is represened by Ki K s K p K, Ni N s N p N. (17) We have hus buil he dynamic growh model wih naional deb. 3 The Dynamics of he Economy The appendix shows ha he moion of he economic sysem is deermined by wo differenial equaions wih z and k as he variables, where z w / r k. The following lemma shows how we can deermine he moion of all he variables in he dynamic sysem. Lemma The moion of k and z is deermine by deermined by he following wo differenial equaions ~ k z, k, Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 59

10 Inernaional Journal of Economic Sciences ˆ z, k, z (18) ~ in which and are funcions of k and z defined in he appendix. We deermine all he oher variables as funcions of k and z as follows: D by (A12) r by (A11) r r k K s by (A9) K i by (A9) K p by (A8) k j z / j N j K j / k j K i by (A1) K s by (A6) N j K j / k j by (A15) w z r Fj by he specified forms p by (A9) yˆ p Fs / N cs, ci, and s by (8) Tp by (14) K k N Yp by (1) a k d. The lemma implies ha he moion of economic sysem a any poin in ime can be uniquely described as funcions of he wo variables, k and z. If we deermine he moion of (18), i is sraighforward o deermine he moion of he whole sysem. As he expressions are oo edious, i is difficul o ge explici conclusions. For inerpreaion, we simulae he model. We specify parameer values as follows N 1, Y 3, Ai 1.1, As, Ap, i, s 5, p, p.5,.7,.15,.6, pi.7, ps.6, ~s.3, ~i.8, k.15, w.5, i.6, s.6, k.5. (19) The populaion is 1. The governmen spends 3 on supplying public goods. The propensiy o save is.7. The propensiy o consume goods is.15, which is much higher han he propensiy o consume services. The ax raes on consumpion goods, service and ineres income are respecively 8 percen, 3 percen, and 13 percen. The ax raes on he wo secors are 6 percen. I should be remarked ha alhough he specified values are no based on empirical observaions, he choice does no seem o be unrealisic. For insance, some empirical sudies on he US economy demonsrae ha he value of he parameer,, in he Cobb-Douglas producion is approximaely equal o. The depreciaion rae of physical capial is fixed a 3 percen. Alhough he choice of he parameer values are no based on a special economy, his will no affec our main purpose of providing ino inerdependence beween various variables by examining differen changes in exogenous condiions such as policies and preferences on he economic dynamics. To simulae he model, we choose he iniial condiions k.5, z 1. Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 6

11 Inernaional Journal of Economic Sciences We plo he moion of he dynamic sysem for a shor period of ime in Figure 1. The deb rises and he ax income falls over ime. The labor employed by he public secor is increased bu he capial is reduced. The price of service and wage rae rise over ime. The rae of ineres falls. The simulaion demonsraes ha he dynamic sysem does no approach o he equilibrium poin. We will demonsrae ha he equilibrium poin is a saddle poin. I is known ha some neoclassical growh models wih debs have saddle poins (e.g., Turnovsky and Sen, 1991, Lin, 2). For his kind of models, i is imporan o follow shifs of pahs wih changes in public policies and oher parameers since he focus on unsable seady saes does no provide enough informaion for behavior of he sysem. I should be noed ha differen policies for sabilizing governmen debs are suggesed in he lieraure. For insance, according o Michel e al. (21: 925), To operaionalize he noion of unsable governmen deb dynamics, we consider seady saes which are locally unsable under he assumpion of a permanenly balanced primary budge. However, he economy can be sabilized a hese seady saes if one allows for appropriae budgeary adjusmens. For racabiliy, we consider deb sabilizing rules ha specify hese adjusmens as a linear funcion of he wo sae variables of he model (physical capial and real governmen deb). Moreover, we assume ha such adjusmens can be brough abou by wo differen insrumens (governmen consumpion or a lump-sum ax on young agens) Fi D k.42 cs ci p w.142 r Tp.15 Kp Fp Ks Np 1 6 Ki 34 Fs K 98 Ns Ni.263 Figure 1. The Moion of he Naional Economy We calculae he equilibrium values of he variables as follows D 26.29, Tp 5.91, r.11, p 1, w.15, N i 63.5, N s 24.19, N p 12.76, K i 24.72, K s 11.92, K p 7., Fi 59.2, Fs 18.88, Fp 5.76, ki 9, Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 61

12 Inernaional Journal of Economic Sciences ks.49, k p.55, k.44, ci.14, cs.19. (2) We see ha he deb in Figure 1 is far from is equilibrium poin and moves away from is equilibrium poin. The eigenvalue a he equilibrium poin are respecively.23 and.191. The equilibrium poin is a saddle poin. I is sraighforward o show ha if we neglec axaion and deb in he model, he sysem has a sable unique equilibrium (Zhang, 25). The inroducion of dynamics of he public deb makes he neoclassical growh model unsable. I is well known ha in radiional growh models wih deb equilibria can be locally unique or indeerminae, depending on wheher he governmen uses income axes, consumpion axes or oher policies (e.g., Judd, 1987; Turnovsky,199; SchmiGrohe and Uribe, 1997; and Mankiw and Weinzierl, 26). 4 Comparaive Dynamic Analysis The previous secion plos he moion of he variables. As we have shown how o simulae he moion of he sysem, i is sraighforward o make comparaive dynamic analysis. As he equilibrium poin is a saddle poin, he sysem will no approach he equilibrium poin excep he iniial condiion is locaed on some special pah. As Michel e al. (21: 923) emphasize, Unsable governmen deb dynamics can ypically be sabilized around a cerain arge level of deb by appropriae budgeary adjusmens. To achieve he needed budgeary correcions a governmen can normally adjus a broad range of fiscal insrumens, like governmen spending, axes or ransfers. Ye, depending on he iming of acions and he paricular insrumen (or subse of insrumens) ha ges adjused, successful sabilizaions can be associaed wih a broad range of possible sequences of he budgeary balance during he adjusmen period. The difficuly of describing possible dynamic adjusmens is ha one has o be follow dynamics under differen exogenous changes. As our model can simulae he moion of he dynamic sysem, we can examine he dynamics of he whole sysem under any combinaion of fiscal policies and oher parameers. A rise in he governmen s expendiure Firs, we examine he impac of he following change in he governmen s expendiure: Y : We inroduce a variable, x, o sand for he change rae of he variable, x, in percenage due o changes in he parameer value. Firs, we examine he effecs of he increase in he governmen expendiure on he equilibrium. The effecs are lised in Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 62

13 Inernaional Journal of Economic Sciences D 2.19, Tp.15, r 6, p.4, w.54, N i 5, N s.55, N p 2.78, K i.85, K s 6, K p 4.2, Fi.19, Fs.3, Fp 2.59, ki ks k p 1.21, k 1.31, ci.1, cs.3. (21) We see ha as he expendiure is increased, he public secor enlarges is producion scale, employing more labor and capial. As he public secor supplies more public goods, he produciviies of he wo producion secors are increased. Accordingly, he wo secors oupu levels are reduced even hough he labor inpus of he wo secors are reduced. The physical wealh is increased, which resuls in he increases of he capial socks employed by all he secors. The increased wealh is associaed wih he falling rae of ineres and increasing wage rae. In associaion wih he falling price of service, he consumpion level of service is slighly increased. I should be noed ha as he equilibrium poin is a saddle poin, in general one canno guaranee ha he sysem will approach is equilibrium. Hence, i is imporan o follow how he sysem moves when some exogenous change is added. Figure 2 plos how he sysem is affeced over ime. The public deb and ax income are reduced. Some of he labor force is shifed from he service secor o he goods secor and public secor. The oal capial and he capial socks employed by he good and public secors are increased, while he capial sock used by he service secor is reduced. The oupu levels of he public and goods secors are increased, while ha of he service secor is reduced. The wage rae and capial inensiies rise. The rae of ineres and he price of service rise iniially bu fall lae on. The consumpion levels of good and service are reduced. I should be remarked ha he change direcions of some variables in he equilibrium are no he same as hose of he corresponding variables in he ransiional processes. I should be noed ha here are many sudies concerning condiions for deb neuraliy (Yaari, 1965; Blanchard, 1984, 1985; and Buier, 1988). The model by Barro (1974) demonsraes he Ricardian equivalence hypohesis which implies ha any mix of public deb and a lump-sum ax o finance governmen lump sum ransfers has no real effec. I has been demonsraed ha he deb neuraliy is obained from specified frameworks. In our approach as he governmen expendiure affecs economic produciviies a change in he governmen policy will affec he equilibrium of he economic sysem. Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 63

14 Inernaional Journal of Economic Sciences D Tp r 1.1 p cs w ci k Fs Ns Ks Fp.5 Ni.5 Np Fi Ki Kp 2.6 K Figure 2. A Rise in he Governmen Expendiure A rise in he public secor s oal produciviy We now allow he oal produciviy of he public secor o be improved as follows: Ap : 1. The effecs on he equilibrium are given in (22) D 1.8, Tp 1.4, r 1, p.7, w 2, N i.12, N s.18, N p 1, K i.82, K s.88, K p.21, Fi 1.3, Fs 1.3, Fp 1.54, ki ks k p.7, k 7, ci 1.9, cs 1.3. (22) As he produciviy is improved, more public service is supplied by he public secor and less labor force and capial socks are employed by he secor wih he unchanged expendiure. As he produciviies of he good and service secors are increased, he oupu levels of he wo secors are increased. The rae of ineres, he price of service, and wage rae are all increased. The wealh, he capial inensiies, he consumpion levels of good and service are all increased. Figure 3 plos how he sysem is affeced over ime. Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 64

15 Inernaional Journal of Economic Sciences D 1.49 K.5 1 Fi Np Kp Tp.1.13 p k.115 w r cs ci Ns Ks Fs Fp 1 Ni Ki 1.43 Figure 3. A Rise in he Toal Produciviy of he Public Secor A rise in he ax rae on he goods secor We now increase he ax rae on he goods secor as follows: i :.6.8. The effecs on he equilibrium are given in (23) D 17.7, Tp 14.28, r 9.59, p 1.52, w 5.56, N i 3.39, N s 5.74, N p 5.89, K i 14.42, K s 6.32, K p 6.2, Fi 6.78, Fs 1.41, Fp 5, ki ks k p 11.41, k 1.89, ci.14, cs (23) As he ax rae on he goods secor is increased, he oal ax income is increased. As i has o pay more ax ou of uni of is oupu, he goods secor reduces is oupu. As he good oupu is reduced, he rae of ineres is increased. Some of he labor force is shifed from he goods secor o he public and service secors. The wealh, he oal capial and capial socks employed by he hree secors are all reduced. As a consequence of he increase in he rae of ineres and he fall in he goods secor, he deb is increased even hough he expendiure is no changed and he ax income is increased. Figure 4 plos how he sysem is affeced over ime. Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 65

16 Inernaional Journal of Economic Sciences D Kp 3 Fp K 11.2 Np 2.1 p 1 k Fs Ns w cs ci r Ni Tp Ki 3 1 Ks 1 Fi 3.6 Figure 4. A Rise in he Tax Rae on he Goods secor 5 Conclusions This paper developed a neoclassical wo-secor economic growh model wih public secor and deb in a compeiive economy. The model describes nonlinear dynamic ineracions among economic srucural change, capial accumulaion and public deb under differen combinaions of axes on he goods secor, he service secor, he wage income, he ineres reurn of wealh, consumpion of good, and consumpion of service. The model has a unique saddle equilibrium poin. The sudy focuses on effecs of changes in some policies on he equilibrium and ransiional processes of he economic dynamics. I should be remarked ha he economic srucures and ineracions in our model are delicaely inerrelaed. Our comparaive dynamic analysis is limied o a few cases. We migh ge more insighs from furher simulaion. I is possible o exend he model in some direcions. The Solow model and he Uzawa wo-secor growh model are mos well-known models in he lieraure of growh heory. Many limiaions of our model and possible exensions and generalizaions become apparen in he ligh of he sophisicaion of he lieraure. Appendix: Proving Lemma 1 From (2), (4) and (8), we ge Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 66

17 Inernaional Journal of Economic Sciences z w i ki s k s p k p, r (A1) where k p K p / N p, j j / j, j i, s, p. Inser k j z / j and N j K j / k j in (17) i Ki s K s p K p z N. (A2) From (A2) and Ki K K s K p, we have K p x N z x i K s i x K s, Ki N x z x p K s p x K s, (A3) where x 1 / p i. From (13) and (16), we have yˆ N p Fs. Subsiuing r s s p Fs / K s ino he above equaion yields yˆ N r K s s s (A4). From (1) and (11) yˆ 1 k r k w w 1 r d. (A5) Inser he above equaion and w z r in (A4) k r K w N z r r D r K s s s Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp, (A6) 67

18 Inernaional Journal of Economic Sciences where we use r r k, 1 k k and 1 k. From (A6) we solve r K N D s k k r k w z r. s s N r (A7) From (6) and (8), we have Kp p Y. r (A8) From (A3) and (A8), we solve p Y 1 K s x N z x i N k, r s i x Ki N x z x p K s p x K s, (A9) From (2), we have p r i i Ai Ap pi K p pi p N p pi ki i, (A1) where we also use (5). Inser ki z / i and N p p K p / z in (A1) pi pi p r i i Ai Ap K p p i p p Kp z i i. z (A11) Inser (A8) in (A1) Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 68

19 Inernaional Journal of Economic Sciences p i p i i pi p i p p p r z i i Ai Ap p Y, z z (A11) where 1 / 1 pi p p. From (A12)-(A14) and (A16), we know ha r is a funcion of z. Inser (A16) and (A12) in (A7) r K N D z, k s s k k r k w z r. s s N z r (A12) By he following procedure we can deermine all he variables as funcions of z and k : D by (A12) r by (A11) r r k K s by (A9) K i by (A9) K p by (A8) k j z / j, j i, s, p N j K j / k j K i by (A1) K s by (A6) N j K j / k j, j i, s, p by (A15) w z r Fj by he specified forms p by (A9) yˆ p Fs / N cs, ci and s by (13) Tp by (15) K k N Yp Y a k D / N. By (9), (14) and his procedure, we have D z, k r D Yp Tp, (A18) D ~ k z, k s k. N N (A19) Taking derivaives of (A15) wih respec o ime yields D z k. z k (A2) From (A18) and (A2), we solve Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 69

20 Inernaional Journal of Economic Sciences 1 ~ ˆ z, k, z k z (A21) where we also use (A19). We hus proved Lemma 1. References Azariadis, C. (1993) Ineremporal Macroeconomics. MA., Cambridge: Blackwell. Barro, R.J. (1974) Are Governmen Bonds Ne Wealh? Journal of Poliical Economy 82, Barro, R.J. (199) Governmen Spending in a Simple Model of Endogenous Growh. Journal of Poliical Economy 98, S Blanchard, O.J. (1983). Deb and Curren Accoun Defici in Brazil, in: Armella, P.A., Dornbusch, R., Obsfeld, M. (Eds.) Financial Policies and he World Capial Marke: The Problem of Lain American Counries. Chicago: Universiy of Chicago Press. Blanchard, O. J. (I984) Curren and Anicipaed Deficis, Ineres Raes and Economic Aciviy. European Economic Review 25, Blanchard, O. J. (I985) Deb, Deficis and Finie Horizons. Journal of Poliical Economy 93, Buier, W.H. (1988) Deah, Birh, Produciviy Growh and Deb Neuraliy. The Economic Journal 98, Burmeiser, E. and Dobell, A.R. (197) Mahemaical Theories of Economic Growh. London: Collier Macmillan Publishers. Diamond, P. (1965) Naional Deb in a Neoclassical Growh Model. American Economic Review 55, Barro, R., Mankiw, N.G., Sala-i-Marin, X. (1995) Capial Mobiliy in Neoclassical Models of Growh. American Economic Review 85, Blanchard, O.J., Fischer, S., Lecures in Macroeconomics. MA., Cambridge: MIT Press. Buier, W.H. (1988) Deah, Birh, Produciviy Growh and Deb Neuraliy. The Economic Journal 98, Chalk, N. (2) The Susainabiliy of Bond-Financed Deficis: An Overlapping Generaions Approach. Journal of Moneary Economics 45, Cohen, D., Sachs, J. (1986) Growh and Exernal Deb under Risk of Deb Repudiaion. European Economic Review 3, De la Croix, D. and Michel, P. (22) A Theory of Economic Growh, Dynamics and Policy in Overlapping Generaions. Cambridge: Cambridge Universiy Press. Diamond, P. (1965) Naional Deb in a Neoclassical Growh Model. American Economic Review 55, Farmer, R. (1986) Deficis and Cycles. Journal of Economic Theory 4, Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 7

21 Inernaional Journal of Economic Sciences Frederick, S., Loewensein, G., and O Donoghue, T. (22) Time Disconinuing and Time Preference: A Criical Review. Journal of Economic Lieraure 11, Giannisarou, C. (27) Balanced Budge Rules and Aggregae Insabiliy: The Role of Consumpion Taxes. Economic Journal 117, Gómez, M.A. (28) Fiscal Policy, Congesion, and Endogenous Growh. Journal of Public Economic Theory 1, Guo, J., and Harrison, S. (24) Balanced-Budge Rules and Macroeconomic (In)Sabiliy. Journal of Economic Theory 119, Hochman, O. (1981) Land Rens, Opimal Taxaion and Local Fiscal Independence in an Economy wih Local Public Goods. Journal of Public Economics 15, Judd, K. (1987) A Dynamic Theory of Facor Taxaion. American Economic Review 77, Lin, S.L. (2) Governmen Deb and Economic Growh in an Overlapping Generaions Model. Souhern Economic Journal 66, Mankiw, G. and Weinzierl, M. (26) Dynamic Scoring: A Back-of-he-Envelope Guide. Journal of Public Economics 9, Michel, P., Thadden, L., and Vidal, J.P. (21) Deb Sabilizing Fiscal Rules. Journal of Public Economic Theory 12, Park, H. (29) Ramsey Fiscal Policy and Endogenous Growh. Economic Theory 39, Schmi-Grohe, S., and Uribe, M. (1997) Balanced-Budge Rules, Disorionary Taxes and Aggregae Insabiliy. Journal of Poliical Economy 15, Semmler, W. and Sieveking, M. (2) Criical Deb and Deb Dynamics. Journal of Economic Dynamics & Conrol 24, Solow, R. (1956) A Conribuion o he Theory of Growh. Quarerly Journal of Economics 7, Solow, R. M. (1962) Noe on Uzawa s Two-Secor Model of Economic Growh. Review of Economic Sudies 29, Swan, T.W. (1956) Economic Growh and Capial Accumulaion. Economic Record 32, Turnovsky, S. (199) The Effecs of Taxes and Dividend Policy on Capial Accumulaion and Macroeconomic Behavior. Journal of Economic Dynamics and Conrol 14, Turnovsky, S.J. (2) Fiscal Policy, Elasic Labor Supply, and Endogenous Growh. Journal of Moneary Economics 45, Turnovsky, S.J. (24) The Transiional Dynamics of Fiscal Policy: Long-Run Capial Accumulaion and Growh. Journal of Money, Credi, and Banking 36, Turnovsky, B.J. and Sen, P. (1991) Fiscal Policy, Capial Accumulaion, and Deb in an Open Economy. Oxford Economic Papers 43, Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 71

22 Inernaional Journal of Economic Sciences Uzawa, H. (1961) On a Two-Secor Model of Economic Growh. Review of Economic Sudies 29, Wijkander, H. (1984) Provision of Public Goods in Congesed Ciies. Journal of Public Economics 25, Yaari, M. E. (I985) The Uncerain Lifeime, Life Insurance and he Theory of he Consumer. Review of Economic Sudies 32, I37-5. Zhang, W.B. (25) Economic Growh Theory. London: Ashgae. Zhang, W.B. (28) Inernaional Trade Theory: Capial, Knowledge, Economic Srucure, Money and Prices over Time and Space. Berlin: Springer. Zhang, W.B. (29) Agglomeraion and Reurns o Scale wih Capial and Public Goods in a Muli-Regional Economy. Inernaional Economic Journal 23, Zhang, W.B. (211) Economic Growh wih Space and Fiscal Policies wih Housing and Public Goods. Journal of Economic Sudies 38, Copyrigh 216, WEI-BIN ZHANG, wbz1@apu.ac.jp 72