On the dynamics of growth and fiscal policy with redistributive transfers

Transcription

1 Journal of Publc Economcs 87 (2003) locate/ econbase On the dynamcs of growth and fscal polcy wth redstrbutve transfers a b, * Hyun Park, Apostols Phlppopoulos a Kyung Hee Unversty, Seoul, South Korea b Athens Unversty of Economcs and Busness, 76Patsson street, Athens 10434, Greece Receved 2 June 1999; receved n revsed form 1 March 2000; accepted 9 September 2000 Abstract Ths paper contnues the study of optmal fscal polcy n a growng economy, by studyng the case n whch the government smultaneously provdes three man categores of expendtures; publc producton servces, publc consumpton servces and state-contngent redstrbutve transfers. We make two contrbutons to the lterature. Frstly, we set up a dynamc general equlbrum model of endogenous growth, n whch ratonal voters choose the captal tax rate and the allocaton of total tax revenues between productve government expendtures (publc producton servces) and non-productve ones (publc consumpton servces and redstrbutve transfers). We study the propertes of these polces and ther mpact on growth. Secondly, we delver analytc results for the long-run and transtonal dynamcs of optmal fscal polces and endogenous growth. In partcular, we establsh exstence and nvestgate the possblty of unqueness and dynamc determnacy, and how ths possblty s affected by the chosen allocatve and redstrbutve polces Elsever Scence B.V. All rghts reserved. Keywords: Redstrbuton; Publc goods; Economc growth; Dynamcs JEL classfcaton: H2; H4; O4; E62 1. Introducton The lnk between economc growth and economc polcy has long been a central *Correspondng author. Tel.: ; fax: E-mal address: aphl@aueb.gr (A. Phlppopoulos) /00/$ see front matter 2000 Elsever Scence B.V. All rghts reserved. do: /s (00)

2 516 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) area of research. Interest n ths area has surged recently, thanks to the theory of endogenous (long-term) growth. Studes have become more refned n ther predctons of the mpact of dfferent categores of government expendtures and taxes on growth. Ths n turn rases questons regardng the optmal sze of the publc sector, and the optmal structure of both expendture and taxaton. Barro (1990), Barro and Sala--Martn (1992), Glomm and Ravkumar (1994) and others have studed the optmal level of government expendtures and the assocated optmal tax polcy, when government expendtures take the form of publc producton servces that rase the productvty of prvate frms. Devereux and Wen (1998) and many others have studed the more conventonal case n whch government expendtures take the form of publc consumpton servces that provde drect utlty to households. Persson and Tabelln (1994) and Alesna and Rodrk (1994) have developed poltco-economc models to study how optmal redstrbutve tax polcy affects growth. Parallel to the theoretcal lterature, recent emprcal work (see e.g. Devarajan et al. (1996) and Kneller et al. (1999)) has provded evdence that dfferent types of government expendtures and taxes have dfferent mplcatons for growth. Most of the exstng theoretcal models have focused on one category of government expendture and gnored the others. The goal of the present paper s to contnue the study of optmal fscal polcy n a growng economy, by nvestgatng the case n whch the government smultaneously provdes three man categores of expendtures; publc producton servces, publc consumpton servces and state- 1 contngent redstrbutve transfers. In other words, we combne both allocatve and redstrbutve actvtes on the part of the government. Total government expendtures are fnanced by dstortng taxes. Gven the drect role such expendtures and 2 taxes play n captal formaton and growth, we focus on effcency ssues n a 3 general equlbrum set-up. 1 Mueller (1989, chapter 17) argues that the man explanatons for why governments exst are that they provde publc goods (e.g. publc producton servces and publc consumpton servces) and redstrbute ncome or wealth. He emphasses (see Mueller (1989, p. 333)) that an mportant explanaton for the growth of government les n government s redstrbutonal actvtes, so substantal has been the growth n the transfer component of government budgets. Smlarly, Sandmo (1995) states that to access the role of the publc sector n growth, we have to consder a somewhat wder role for fscal polcy. Specfcally, he says that the man focus of the lterature has been on cash transfers, whereas effects of the provson of socal goods have not receved the attenton that they probably deserve. 2 Of course, there are other ssues too, e.g. equty. However, effcency (growth) s mportant (see also Atknson (1999, p. 18)). 3 As Atknson (1999) argues, any fscal decson requres us to look at both sdes of the balance sheet. Frstly, what s the tax burden of government expendture? Secondly, how do the specfc features of each type of government expendtures affect economc performance? For nstance, transfers may adversely affect the ncentves of economc agents. To study these ssues smultaneously, we have to use a general equlbrum set-up. Also, Sandmo (1995) ponts out that tax fnancng of publc expendtures wll presumably ncrease the qualty of the labor force, so that an assessment of the effects of the welfare state polcy... n effcency terms ought to consder these effects jontly.

3 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) In partcular, we make two contrbutons to the lterature. Frstly, we set up a dynamc general equlbrum model of endogenous growth, n whch ratonal voters choose the captal tax rate and the allocaton of total tax revenues between productve government expendtures (.e. publc producton servces) and non- 4 productve ones (.e. publc consumpton servces and redstrbutve transfers). We then study the propertes of these polces and ther mpact on growth. Secondly, we delver analytc results for the long-run and transtonal dynamcs of optmal fscal polces and endogenous growth. We establsh exstence and nvestgate the possblty of unqueness and dynamc determnacy, and how ths possblty s affected by the chosen allocatve and redstrbutve polces. Our set-up s as follows. A decentralzed economy s populated by prvate agents (households and frms) and a government. Dfferent households can start wth dfferent captal endowments. The government taxes captal ncome to fnance transfer payments, publc consumpton servces and publc producton servces. Transfer payments are state-contngent and redstrbute from those who are endowed more than the average to those who are endowed less. Publc consumpton servces provde drect utlty to households. Publc producton servces provde producton externaltes to prvate frms; hence, they are the engne of long-run growth as n Barro (1990). Publc consumpton and producton servces are pure publc goods. When economc polcy s exogenous, a decentralzed equlbrum has the followng features: frstly, when households nternalse redstrbuton polcy, moral hazard behavor reduces growth. The larger the number of ndvduals, the stronger the ncentve to free rde on each other, and the stronger the moral hazard problem. Secondly, the relaton between the economy s growth rate and the captal tax rate s an nverse U-curve. At low (resp. hgh) tax rates, an ncrease n the tax rate ncreases (resp. decreases) growth. Ths s as n Barro (1990) and Alesna and Rodrk (1994). We then endogenze economc polcy. To do so, we suppose that ndvduals voters choose ther preferred polcy to maxmze ther own utlty functon by playng ash vs-a-vs each other. Then, fve features of optmal polcy are of partcular nterest. Frstly, the two polcy nstruments (e.g. the captal tax rate and the share of total tax revenues used to fnance productve government expendtures) are substtutes along the optmal path. Intutvely, when the government allocates a larger share of tax revenues to publc producton servces, t can afford a lower tax rate snce publc producton servces stmulate nvestment and so ncrease the tax base. Secondly, snce there are also non-productve government expendtures, the sze of both polcy nstruments should be all the tme hgher than the productvty of publc producton servces. Ths generalzes the standard 4 The classfcaton as productve and non-productve expendtures s consstent wth the emprcal lterature (see Kneller et al. (1999) for a functonal classfcaton of expendtures and taxes). Of course, our classfcaton s not a detaled one. Ths apples n partcular to welfare state actvtes where redstrbutve transfers are only one of many welfare state type actvtes (see Atknson (1999)).

4 518 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) Barro-type publc fnance model of growth, whch predcts that t s optmal to keep the tax rate constant over tme and equal to the productvty of publc servces. Thrdly, t s optmal to redstrbute ncome from the rch to the poor, only when the government also provdes publc producton servces. Recall that t s the latter that generates long-term growth. In other words only growng economes can afford wealth redstrbuton. Fourthly, from the vewpont of the poor (resp. rch) ndvdual, taxes should ncrease (resp. decrease) wth nequalty. Ths s because redstrbuton requres taxes. In turn, snce the growth rate s negatvely affected by the tax rate along the optmal path, t follows that ntal 5 nequalty hurts growth. Ths s smlar to Persson and Tabelln (1994) and Alesna and Rodrk (1994) but, n contrast to these papers, here we explctly 6 nclude redstrbutve transfers. Ffthly, provson of publc consumpton servces mples that optmal polcy changes over tme and hence there are transtonal dynamcs. Ths enrches Barro (1990), Barro and Sala--Martn (1992) and Alesna and Rodrk (1994) who do not get transtonal growth dynamcs. Concernng the long-run, we focus on a steady state n whch all ndvduals voters are alke ex post (see also Bewley (1982)). Then, we fnd a suffcent condton on the parameter values under whch there exst unque, well-defned long-run values for the polcy nstruments and the assocated long-term growth path (known as Balanced Growth Path) along whch consumpton and captal can grow at the same constant postve rate. When we compare the long-run equlbrum n the case n whch economc polcy s chosen by selfsh ndvduals voters to the case n whch economc polcy s chosen by a benevolent government (whch maxmses the sum of all ndvduals well-beng and nternalses all cross-ndvdual effects), the long-term growth rate s hgher n the latter benevolent case. Fnally, we study dynamc stablty around steady state. We can clam that the antcpaton of large redstrbutve transfers opens the door for ndetermnacy. That s, once we explctly nclude redstrbuton polcy, there can be many (nfnte) 7 possble equlbrum paths for fscal polcy and growth. Indvduals who start wth smlar endowments may consume and save at completely dfferent rates over tme, even f they eventually converge to the same long-term rate of consumpton and savng. Ths mples that the antcpaton of redstrbutve transfers, and the resultng moral hazard behavor, can themselves generate ncome nequalty. On 5 Ths happens when the medan voter s less captal-endowed than the average. 6 In most of the nequalty-growth lterature based on the medan voter model, redstrbuton s not modelled explctly. It just takes the form of taxes on captal. In partcular, as Alesna and Rodrk (1994, p. 466) say, taxes on captal must be nterpreted as a metaphor for any knd of redstrbutve polcy that transfers ncome to unsklled labor. The medan-voter approach s not the only one. Other explanatons of the nequalty-growth lnk rely on mperfect captal markets, rent-seekng etc. Ths lnk s stll an open ssue. For a survey, see Benabou (1996) and Aghon and Howtt (1998). 7 There s always determnacy f the government s concerned only wth allocaton (.e. t provdes consumpton and producton publc servces).

5 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) the other hand, when economc polcy s not chosen by selfsh ndvduals voters but by a benevolent government, there s determnacy,.e. a unque path to steady 8 state. The rest of the paper s organsed as follows. Secton 2 presents the economc envronment. Secton 3 solves for optmal polcy. Sectons 4 and 5 study, respectvely, long-run propertes and transtonal dynamcs. Secton 6 concludes. 2. The economy and decentralzed equlbrum Consder a decentralzed closed economy populated by prvate agents (households and frms) and a government. Households purchase goods for consumpton and save n the form of captal. There are 5 1, 2,..., households. Frms produce goods wth the use of captal only, and make rental payments to the 9 owners of captal. For smplcty, there s a sngle frm. The government mposes taxes on households captal ncome to fnance publc producton servces, publc 10 consumpton servces and transfer payments. We assume nfnte horzons and perfect foresght. Dfferent households can start wth dfferent ntal captal stocks. Then, at each pont n tme, household s ndexed by ts own captal stock, k, relatve to the average captal stock, k (where k ; K/ and K ;o51k s the aggregate captal stock). Ths ndex,.e. (k 2 k ), can be thought of as beng a measure of wealth nequalty The role of economc polcy It s convenent to start wth the role of economc polcy. The government receves tax revenues urk from each household, where 0 #u, 1 s the tax rate on captal ncome, r s the market return to captal, and k s household s begnnng-of-perod captal stock. The government uses ts total tax revenues, o51urk, to fnance the provson of aggregate publc producton servces, G, aggregate publc consumpton servces, H, and transfer payments to households, o51d, where d s a transfer payment receved by each household. Thus, at each 11 nstant of tme, the government budget constrant s: 8 The ssue of redstrbuton, allocaton and votng s stll open (see Mueller (1989, chapters 23, 24)). 9 For notatonal smplcty, we do not nclude labor. Our results do not change, f households supply nelastcally ther labor servces. However, our results do change, f households make labor/ lesure choces (see e.g. Alesna and Rodrk (1994)). For detals, see Park and Phlppopoulos (1998). 10 Thus, followng Barro (1990), the engne of long-term growth s publc producton servces. In Glomm and Ravkumar (1992) and Perott (1993), the engne s human captal. See also Fernandez and Rogerson (1995, 1996) for human captal and redstrbuton. 11 For smplcty, the government budget s balanced and there s no publc debt.

6 520 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) G 1 H 1Od 5Ourk. (1) A share 0, b, 1 of total tax revenues s used to fnance productve government expendtures (.e. publc producton servces) and a share 0, 1 2 b, 1s used to fnance non-productve government expendtures (.e. publc consumpton servces plus transfer payments). Then, (1) s decomposed to (2a) and (2b): G 5 bourk 51 H 1Od 5 (1 2 b)ourk We now model the transfer payment d receved by each household from the government. We assume that d s a lnear functon of relatve captal ownershp, 12 (k 2 k ). Thus, each household receves: (2a) (2b) d 5 d(k 2 k ) (3) where d $ 0 s a polcy parameter. Eq. (3) mples that the government subsdses those households who are endowed less than the average. That s, f k. k, household s a recever. If k, k, household s a donor. Ths s a commonly used polcy rule Households Household maxmzes ntertemporal utlty: ` 2rt E[u(c, H)]e dt (4a) 0 where c s household s prvate consumpton, H s aggregate publc consumpton servces (whch are a pure publc good), and the parameter r. 0 s the rate of tme preference. The functon u(.) s ncreasng and concave n ts two arguments, and also satsfes the Inada condtons. For smplcty, we assume that u(.) s addtvely separable and logarthmc. Thus, 12 For smlar state-contngent lnear rules, see e.g. Aghon and Howtt (1998, chapter 9). Here, as n most of the lterature, we take redstrbuton as gven wthout justfyng t. If we do not wsh to ntroduce captal market mperfectons, rent-seekng actvtes, etc., then we could justfy redstrbuton by assumng that an ndex of nequalty provdes drect dsutlty to households, or that ths ndex exerts a negatve producton externalty to prvate frms. Snce ths would also be ad hoc and would not affect our man results, we smply prefer to take redstrbuton as gven. Also note that redstrbuton can take many forms n addton to transfers (e.g. publc educaton, progressve ncome taxaton, mnmum wages). See Benabou (1996) and Atknson (1999).

7 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) u(c, H) 5 log c 1 nlog H (4b) where the parameter n $ 0 measures the weght gven to publc consumpton 13 relatve to prvate consumpton. Households can save n the form of captal. When household rents out k to frms, t receves an after-tax captal ncome (1 2u )rk. It also gets a proft share, p. Furthermore, t receves a transfer payment d from the government. Usng Eq. (3) for d, the household s budget constrant s: ~ c 1 k 5 (1 2u )rk 1 p 1 d(k 2 k ) (5) where a dot over a varable denotes tme dervatve, and the ntal captal stock for 14 each s gven. Households take prces, tax rates and publc goods as gven. The control varables are c and k, so that the frst-order condtons for utlty maxmzaton 15 are Eq. (5) as well as the Euler equaton: 1 c~ 5 c F(1 2u )r 2 r 2 ds1 2 ] DG (6) where the last term on the rght-hand sde captures moral hazard behavor. That s, drect captal-contngent redstrbutve transfers dstort prvate decsons, n the sense that nvestors have an ncentve to be poor so as to receve transfers (see also below) The frm Followng Barro and Sala--Martn (1995, pp ) and Alesna and Rodrk (1994), technology at the frm s level takes a Cobb Douglas form. Thus, the sngle frm faces the producton functon: Y 5 G 12a K a (7) where 0, a, 1 s a parameter, G s aggregate publc producton servces (whch are a pure publc good) and K s the aggregate captal stock (recall that there s a sngle frm). At any pont of tme, the frm maxmzes profts, p: 12a a p 5 G K 2 rk. 13 We assume no congeston. Also note that (4b) allows for a balanced growth path (see below). 14 In Persson and Tabelln (1994), u 5 d n Eq. (5). 15 The necessary condtons (5) and (6) are completed wth the addton of the transversalty 2rt condton lm[u c(.)k e ] 5 0. It s known that there exsts a unque soluton to ths problem. The other t ` argument n the utlty functon must also be bounded. Ths s taken as gven by households. See below.

8 522 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) The frm takes prces and publc goods as gven. Ths s a statc problem. The control varable s K, so that the frst-order condton for proft maxmzaton s smply: r 5 ag 12a K a21 whch equates the rate of return to the margnal product of captal. (8) 2.4. Decentralzed equlbrum ( for any feasble economc polcy) We now solve for a decentralzed equlbrum (DE). Ths s for any feasble economc polcy. We choose to express the DE n terms of the tax rate, u, and the share of total tax revenues used to fnance publc producton servces, b. As Eqs. (1), (2a) and (2b) show, the choce of u and b completely characterzes economc polcy because only two of the four polcy nstruments (G, H, u, b) can be ndependently set. Usng (2a) n (8), the return to captal s: 1/a (12a)/a r 5 a (bu ) ; D(u, b) (9) whch s the return that drves prvate agents consumpton/ savng decsons n a DE. In the presence of producton externaltes, ths return s smaller than the realzed one (see footnote 17). Usng (9) n (2a), (2b) and (7), the economy-wde publc producton servces 16 (G), publc consumpton servces (H) and output (Y) are respectvely: 1/a G 5 (abu ) K (10a) H 5 (1 2 b)ud(u, b)k 2Od(k 2 k ) 51 (10b) (12a)/a Y 5 (abu ) K (10c) 51 where k, and hence K ;o k, have been chosen by prvate agents who gnored externaltes. We now calculate aggregate profts, p ;o51p. Snce realzed captal pay- ments and profts exhaust total output, we have Y 5 RK 1 p, where R s the realzed return to aggregate captal, K. But from (10c), R ; Y/ K 5 (12a)/a 17 (abu ). Thus, p 5 0 (10d) 16 Eq. (10c) shows that our model s a varant of Rebelo s (1991) Ak model. Thus, at aggregate level, output s lnear n captal. 17 Snce 0, a, 1 and u. 0, R. r. Thus, n the presence of producton externaltes, the economywde return exceeds the return perceved by prvate agents n (9).

9 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) e. at an economy-wde level, all realzed ncome goes to captal. We summarze ths secton. We have solved for a decentralzed equlbrum (DE). Ths s for any feasble economc polcy. The latter s fully descrbed by the tax rate, u, and the share of tax revenues used to fnance publc producton servces, b. In ths equlbrum: () prvate decsons maxmze households utlty and frms profts (ths s summarzed by (5), (6), (9) and (10d)). () The government budget constrant s satsfed (ths s summarzed by (10a) and (10b)). () All markets clear. 19 The next secton wll endogenze polcy, u and b. Before we do so, we pont out two results for gven u and b. Frst, usng (9) nto (6) mples that the economy s growth rate s G ; [(1 2u )D(u, b) 2 r 2 d], ˆ where ˆ c d ; d(1 2 1/). 0 denotes the effectve redstrbutve parameter. The effect of ˆ d on growth s negatve. As we argued above, when prvate agents nternalse the provson of redstrbutve transfers, moral hazard behavor reduces economc growth. Observe that when there s a large number of households (.e. `), the effectve redstrbutve parameter ncreases, so that free-rdng ncentves and hence the moral hazard problem become worse. On the other hand, when there s only one household (.e. 5 1), the effectve redstrbutve parameter becomes zero and there are no moral hazard problems. Secondly, G / u 5 (1 2 a 2 u )D(u, c b)/au. Thus, G c / u. 0, f 0,u, 1 2 a, 1; on the other hand, G c / u, 0, f 1 2 a,u, 1. In other words, when polcy s exogenous, the relaton between the economy s growth rate and the captal tax rate s an nverse U-curve. At low tax rates, 0,u, 1 2 a, 1, an ncrease n the tax rate ncreases growth. At hgh tax rates, 1 2 a,u, 1, the growth rate declnes wth the tax rate. Ths result s smlar to that n e.g. Barro (1990) and Alesna and Rodrk (1994) Optmal polcy and poltco-economc equlbrum We now endogenze economc polcy, u and b. To do so, we suppose that 21 economc polcy s chosen by the th ndvdual voter. The th ndvdual voter 18 Ths s because the model s a varant of Rebelo s (1991) Ak model. Thus, there s nothng left for profts or labor (see also Barro and Sala--Martn (1995, pp )). 19 d s exogenous. Alternatvely, d could be chosen optmally. See below. 20 Therefore, when nequalty leads to redstrbuton to the less endowed, ths has negatve ncentve effects at both the gvng end (redstrbuton requres hgher taxes whch dscourage nvestment) and the recevng end (moral hazard problems dscourage effort). On the other hand, when the nstrument of redstrbuton s not drect transfers but say publc educaton, then redstrbuton can be growthenhancng. Therefore, our results do not necessarly mean the abolshment of the welfare state. See Atknson (1999) for these ssues. 21 Alternatvely, economc polcy can be chosen by a benevolent government. See Mueller (1989, chapters 18 and 22) for selfsh voters, altrustc voters and a benevolent government. Here, we wll solve for optmal polcy under selfsh voters and a benevolent government.

10 524 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) behaves selfshly and ratonally, n the sense of maxmzng hs own utlty functon (see Eqs. (4a) and (4b)) subject to hs optmal decson rules n a DE (see Eqs. (5) and (6) n combnaton wth (9) and (10d)) and the moton of the economy s publc goods n a DE (see Eqs. (10a) and (10b)). The current-value 22 Hamltonan, H, oftheth ndvdual voter s: S H ; log c 1 nlog (1 2 b)ud(u, b)k 2Od(k 2 k ) s s ˆ 1 l c (1 2u )D(u, b) 2 r 2 d d 51 1 g (1 2u )D(u, b)k 1 d(k 2 k ) 2 c where g and l are the multplers assocated wth (5) and (6), respectvely. Recall that K 1/a (12a)/a ˆ 1 k ; ], K ;Ok, D(u, b) ; a (bu ) and d ; d 1 2 ]. 51 Snce all ndvduals voters move smultaneously, the th ndvdual voter takes the decsons and transfer payments of all other ndvduals voters as gven. That j j s, the th ndvdual voter takes k and d, where j ±, as gven. The frst-order 23 condtons for u, b, c, l, g, k are respectvely: n(1 2 b)ksd(u, b) 1uD u(u, b) d ]]]]]]]]] 1 (l c 1 g k )(12u s )D u(u, b) 2D(u, b) d (1 2 b)ud(u, b)k 2Od(k 2 k ) (11a) nuk s(1 2 b)d b(u, b) 2D(u, b) d ]]]]]]]]] 1 (l c 1 g k )(1 2u )D b(u, b) 5 0 (11b) (1 2 b)ud(u, b)k 2Od(k 2 k ) 51 1 ~ ˆ l 5 rl 2] 2l [(1 2u )D(u, b) 2 r 2 d] 1 g (11c) c ˆ ~c 5 c [(1 2u )D(u, b) 2 r 2 d] (11d) d D S D 22 We use (10d) and also assume that proft shares across ndvduals, p, are zero. 23 The problem of the benevolent government s lke the problem of the th voter except that when the government chooses u, b, c, l, g, k, then (a) t maxmzes the unweghted sum of all ndvduals welfare functons subject to all ndvduals optmal decson rules n a DE, and (b) t nternalzes the j effect of ts choces on d, where j ±. That s, snce j d k j j 1 d 5 d(k 2 k ),] 5 d] 5 d ]. k k See below for comparson and detals.

11 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) ~k 5 (1 2u )D(u, b)k 1 d(k 2 k ) 2 c (11e) s n (1 2 b)ud(u, b) 1 ˆ d ~g 5 rg 2]]]]]]]]] 2g [(1 2u )D(u, b) 2 d]. ˆ (1 2 b)ud(u, b)k 2Od(k 2 k ) 51 d (11f) These necessary condtons are completed wth the addton of the transversalty condton: f(1 2u )D(u, b) g2 r 2 ˆ d, r (11g) 24,25 whch follows from (11d) and ensures utlty s bounded Propertes of economc polcy along the optmal path Eqs. (11a) (11f) mply the followng sx propertes of optmal polcy. 26 Frstly, (11a) mples (1 2 a 2u ), 0. That s, the optmal tax rate, u, s wthn the subset 0, 1 2 a,u, 1. The optmal tax rate s hgher than 1 2 a (where 1 2 a s the productvty of publc producton servces) because the government provdes n addton to publc producton servces publc consumpton servces and transfer payments. Ths mples that when polcy s chosen, we can only be on the downward-slopng part of the growth rate tax rate relaton (compare t wth the case n whch polcy s exogenous at the end of the prevous secton). Ths s an ntutve result: tax polcy cannot be optmal when a hgher tax rate can ncrease growth (whch happens when 0,u, 1 2 a). 27 Secondly, Eq. (11b) mples (1 2 a 2 b), 0. That s, for the same reason as 24 Captal cannot grow faster than consumpton n steady state. Hence, the utlty from publc consumpton servces s also bounded f (11g) s satsfed. 25 The utlty functon and the constrants are contnuous and bounded, the utlty functon s strctly concave n the controls (c, u, b), and the constrants are lnear n c and strctly concave n u and b. Further, snce the utlty functon and the constrants are both jontly concave n the controls and the state varable (k ), the necessary condtons (11a) (11g) are also suffcent for optmalty. 26 Snce (1 2 a)d(u, b) (12 a 2u )D(u, b) D (u, b) 5 ]]]].0, s(1 2u )D (u, b) 2D(u, b) d5]]]]] u u au au must be negatve for (11a) to hold. Hence, (1 2 a 2u ), Snce (1 2 a)d(u, b) (12 a 2 b)d(u, b) D (u, b) 5 ]]]].0, s(1 2 b)d (u, b) 2D(u, b) d5]]]] b b ab ab must be negatve for (11b) to hold. Hence, (1 2 a 2 b), 0.

12 526 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) above for u, the optmal share of total tax revenues used to fnance publc producton servces, b, s wthn the subset 0, 1 2 a, b, 1. Thrdly, Eqs. (11a) and (11b) combned mply that n each tme perod: (1 2u ) (12 a 2 b) ]]]] 5 ]]]],0. (12) (1 2 a 2u ) (12 a)(1 2 b) Ths atemporal condton mples that the two polcy nstruments, u and b, move n opposte drectons n each tme perod. Intutvely, when the government allocates a larger share of tax revenues to publc producton servces (.e. b ncreases), t can afford a lower tax rate (.e. u decreases), because publc producton servces stmulate prvate nvestment and hence ncrease the tax base. That s, u and b are substtutes along the optmal path. Fourthly, n the specal case n whch publc consumpton servces provde no utlty,.e. n 5 0, Eq. (11a) mples (1 2 a 2u ) 5 0,.e. u a n each tme 28 perod. Ths s the popular tax rate of Barro (1990) and Barro and Sala--Martn (1992, 1995). In other words, when n 5 0, the optmal tax rate s constant over tme and equals the productvty of publc servces, (1 2 a), all the tme. Also note that when n 5 0 and u a, (12) mples b 5 1 n all tme perods,.e. when publc consumpton servces offer no utlty, t s optmal to use all tax revenues to fnance publc producton servces only. Also note that a constant u and a constant b mply a constant captal return, D(u, b). Then, Eq. (11d) mples that the economy has no transtonal growth dynamcs (see also Alesna and Rodrk (1994)). Therefore, when n 5 0, the necessary condtons for optmalty mply that the economy adjusts mmedately to ts steady state, balanced growth path. Ffthly, when publc consumpton servces do not offer any utlty (.e. n 5 0) and also government expendtures do not provde any producton externaltes to frms (.e. a 5 1), then u 5 0 n all tme perods. That s, the optmal tax rate s zero all the tme. Ths mples zero tax revenues and zero transfer payments. It also mples that the equlbrum return to captal s not bounded below (.e. D(u, b) 5 0), and hence long-term growth s not optmal (see (11d)). Therefore, the government fnds t optmal to redstrbute ncome from the rch to the poor only when t also provdes publc producton servces; t s the latter that generates long-term growth. Sxthly, for gven aggregate values of consumpton, captal and ther shadow prces, total dfferentaton of (11a) mples that the tax rate, u, ncreases wth (k 2 k ). Thus, ndvduals voters wth captal endowments below (resp. above) the average captal stock prefer hgh (resp. low) tax rates. In other words, those 28 When n 5 0, (11a) requres ether (lc 1 lk) 5 0, or D(u, b) 5 0, or (1 2 a 2u ) 5 0. The frst possblty,.e. (lc 1 lk) 5 0, cannot occur, because ths contradcts the dynamc Eqs. (11c) (11f). The second possblty,.e. D(u, b) 5 0, cannot occur ether, when the return to captal s postve. So only the last possblty remans.

13 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) who are relatvely lttle captal-endowed prefer hgher redstrbuton. Then, f the medan ndvdual voter s less endowed than the average, votng majorty leads to hgher taxes. In turn, snce the growth rate s negatvely affected by the tax rate along the optmal path, t follows that ntal nequalty hurts growth. Ths s as n e.g. Persson and Tabelln (1994), Alesna and Rodrk (1994) and Benabou (1996). However, our model s more general because: (a) the government provdes explct redstrbutve transfers as well as publc (producton and consumpton) servces; (b) we have a moral hazard problem; (c) we have transtonal dynamcs A symmetrc equlbrum (.e. ex-post homogenety) It wll be useful for what follows n the next sectons, to consder the specal case n whch all ndvduals voters are alke ex post. Ths s a symmetrc ash 29 equlbrum. Ths equlbrum concdes wth the decson that t s optmal for the th ndvdual voter, who (n ths specal case) s the one wth the representatve wealth. That s, the wnner of the votng process s the representatve ndvdual. Apparently, n a symmetrc equlbrum, all ndvduals own ex post the same amount of captal, so that ntal nequalty cannot affect equlbrum outcomes and no actual transfers take place n equlbrum. Ths s not very restrctve. We have already shown how nequalty affects polcy and growth along the optmal path. The crtcal feature of redstrbuton s the antcpaton of transfers of wealth as opposed to actual transfers of wealth. And ths has been already captured n our model by moral hazard behavor. As Benabou (1996) ponts out, the fght over the pe does not necessarly lead to hgher transfers, just to hgher dstortons. Therefore, we nvoke the symmetrcty condtons k ; k, c ; c, l ; l, and g ; g nto the optmalty condtons (11a) (11f). Then, as Appendx A shows, f we use the transformatons z ; c/k and c ; gk, the dynamcs of (11a) (11f) are equvalent to the dynamcs of (13a) (13d) below, whch consttute a four-equaton dynamc system n z, u, b, c. Ths transformaton reduces the dynamc dmensonalty of the model and facltates analytcal tractablty. Thus, the dynamcs of a symmetrc equlbrum are summarzed by: ˆ ~z 5 (z 2 r 2 d )z (13a) ~u 5 A(u )B(u, b)g(u, b, c) (13b) ~b 5 E(u, b)b(u, b)g(u, b, c) (13c) 29 Symmetrc equlbra are not only nterestng n themselves, but they can also help us to study the propertes of non-symmetrc equlbra whch are more complcated algebracally (see below). Ths s a rather usual methodology (see e.g. Persson and Tabelln (1992)). ot that most of poltco-economy models also use ex post homogenety. However, see Fernandez and Rogerson (1995) and Greenwood and Jovanonc (1990) for ex post heterogenety.

14 528 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) ˆ ˆd c~ 5 (r 1 d 2 z)c 2 ˆ n 1 1 ]]]]] (1 2 b)ud(u, b) S where, a(1 2 b)(1 2 a 2 b) A(u ) ; 2 a(1 2 a 2u ). 0, E(u, b) ; ]]]]]], 0, (1 2u ) F 21 n 1 2u 2 a(2 2 a) (12 a)(1 2 b)(1 2 a 2 b) ˆ n ;], B(u, b) ; ]]]]] 2 ]]]]]]]], u b(1 2u ) (1 1 ˆ n )(1 2 a 2u )D(u, b) G(u, b, c) ; r 1]]]]]]]] n d(1 ˆ 2 a 2u )[nˆ 2 (1 2 b)ucd(u, b)] 1 ]]]]]]]]]]. n(1 2 b)u D G (13d) In the above analyss, economc polcy was chosen by ndvduals voters who 30 played ash vs-a-vs each other. If, nstead, economc polcy s chosen by a benevolent government (see footnote 23), the dynamcs of a symmetrc equlbrum are summarzed by the three-equaton system: ~z 5 (z 2 r 2 ˆ d )z (14a) ~u 5 A(u )B(u, b)m(u, b) (14b) ~b 5 E(u, b)b(u, b)m(u, b) (14c) where M(u, b) ; r 1 ((1 1 n)(1 2 a 2u )D(u, b)/n). That s, we now have a three-equaton dynamc system n z, u, b only. Also, nspecton of Eqs. (14b) and (14c) reveals that the two polcy nstruments, u and b, are ndependent of the state of the economy (the nherted captal stock) and redstrbuton polcy (the parameter d ). In other words, when economc polcy s chosen by a benevolent government, the government fnds t optmal to set ts polcy nstruments ndependently of economc fundamentals. Ths s the least dstortng polcy. Ths s dfferent from the case n whch economc polcy s chosen by selfsh ndvduals voters; see (13a) (13d) above, where u and b do depend on economc fundamentals. 30 ote that the redstrbuton polcy coeffcent d s exogenous. That s, we do not ask what s the optmal degree of redstrbuton. If d were endogenous, we would have one more optmalty condton n (11) above. Ths would complcate the algebra a bt. More mportantly, t would not allow us to study how commonly used redstrbuton polces, lke n (3) above, affect the economy. It can be easly shown that n the specal case of symmetrc equlbra, the optmal value of d s ndetermnate n the sense that the relevant optmalty condton vanshes and so we are left agan wth (13a) (13d). Ths s not surprsng; when there are no wealth dfferences n equlbrum, redstrbuton polcy s rrelevant n terms of effcency. We therefore treat d as a parameter.

15 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) Long-run equlbrum Ths secton studes the steady state of a symmetrc equlbrum n Eqs. (13a) (13d). That s, we focus on a long-run equlbrum n whch all ndvduals own ex post the same amount of captal. Ths choce of a steady state follows naturally from the assumpton that all ndvduals have the same rate of tme preference. By contrast, heterogeneous rates of tme preference would lead to a 31 long-run equlbrum n whch only patent agents hold captal. We wll look for a long-run symmetrc equlbrum n whch the economy grows at a constant postve rate (known as balanced growth path, BGP) and economc polcy does not change. Specfcally, we solve for the followng long-run equlbrum: () consumpton, c, and captal, k, grow at the same constant postve rate. Ths mples z ; c/k s a constant, or z~ 5 0 n Eq. (13a). () The polcy nstruments do not change. Thus, u~ 5 b~ 5 0 n Eqs. (13b) and (13c). () The socal value of the captal stock (c ; gk) grows at a constant negatve rate. As ~ shown below, for a well defned long-run soluton to exst, t must be that c n ~ (13d) s negatve (see also Eq. (15f)). The ntuton behnd a negatve long-run c s as follows. In standard models of long-run (.e. endogenous) growth, equlbrum returns to captal should be decreasng n order to mantan postve captal accumulaton over tme. Snce our model also ncludes redstrbutve transfers, whch requre hgher captal accumulaton to fnance them, we need a stronger 32 condton. The rest of ths subsecton solves for z, u, b, c, (tldes above varables denote the correspondng long-run values). We start wth (13a). Settng t equal to zero, we smply have: ˆ z 5 r 1 d. 0 (15a) so that the long-run consumpton-to-captal rato, z, equals the dscount factor, r. 0, plus the effectve redstrbutve parameter, ˆ d ; d(1 2 1/). 0, where d. 0 s the redstrbuton polcy parameter and. 1 s the number of ndvduals. We contnue wth (13b) and (13c). For u~ 5 b~ 5 0 to hold, we must have G(u, b, c ) 5 0, where G(u, b, c) has been defned n (13a) (13d). Suffcent condtons for G(u, b, c ) 5 0 are: 31 See also Yano (1981) and Bewley (1982). If we use endogenous dscount rates, they become dentcal to all agents n the long-run, and thereby agents own the same amount of captal (see Epsten (1987) and Benhabb et al. (1988)). 32 We have expermented wth varous steady states. For nstance, a steady state n whch z~ 5 u~ 5 b~ 5 ~c 5 0 does not gve a well-defned soluton (ths becomes obvous from (15f)). Techncally, n steady state, all varables should be equal to a constant (zero s only one of the possble constant varables).

16 530 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) and (1 1 ˆ n )(1 2 a 2 u )D(u, b ) r 1]]]]]]]] 50 n (15b) ˆn 2 (1 2 b )ucd(u, b ) 5 0. (15c) The atemporal condton (12) whch lnks the two polcy nstruments, u and b, must also hold n the long run. Thus, n steady state, we have: (1 2 u ) (12 a 2 b ) ]]]] 5 ]]]]]. (15d) (1 2 a 2 u ) (12 a)(1 2 b ) Let us take stock. Eqs. (15b) (15d) can be solved for the long-run values u, b and c. otce that (15b) and (15d) consttute a two-equaton system n u and b only. So the man task s to solve (15b) and (15d) for u and b. Appendx B shows: Proposton 1. If the parameter values satsfy the followng condton: (12a)/a (12a)/a (1 1 ˆ n )a (1 2 a) r,]]]]]]]]] n (C.1) there exsts a unque optmal long-run tax rate, u, where 0, 1 2 a, u, 1, and a unque optmal long-run allocaton of tax revenues between productve and non-productve government expendtures, b, where 0, 1 2 a, b, 1. In turn, ths polcy supports a unque balanced growth path (BGP) along whch long-run consumpton and captal grow at the same unque postve constant rate. ote that n (C.1) a low r, a low n, oralow help exstence. That s, exstence obtans more easly when economc agents care about the future (.e. when r s low), when they care more about prvate consumpton than publc consumpton servces (.e. when n s low), or when the number of economc agents s relatvely small so that free rdng problems are less acute (.e. s small). These are ntutve results. In turn, usng the above resultng solutons for u and b nto (15c), we get a unque c,.e. ˆn c 5 ]]]]].0 (15e) (1 2 b )ud(u, b ) and usng (15a) and the solutons for u and b nto (13d), we get: S D ~ ˆd c 52nˆ 1 1 ]]]]],0 (15f) (1 2 b )ud(u, b )

17 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) whch verfes that the long-run socal value of the economy s captal stock grows at a constant negatve rate. When economc polcy s nstead chosen by a benevolent government (see Eqs. (14a) (14c) above), the steady state condtons are smply z~ 5 u~ 5 b~ 5 0. Then, z s as n (15a), whle u and b follow from (15b) and (15d) as before, but now (snce M(u, b ) 5 0) we have n nstead of ˆ n n (15b). Then, f we compare the case n whch polcy s chosen by a benevolent government to the case n whch polcy s chosen by ndvduals voters, and snce n. ˆ n ; n/, comparatve statc exercses n (15b) and (15d) mply that the long-run tax rate gets hgher, whle the long-run share of tax revenues used to fnance publc producton servces gets lower, when we move from the benevolent case to the ndvdual voter case. Then, (11d) mples that long-run growth s hgher n the benevolent government case. 5. Dynamc stablty We now move on to study local stablty propertes around steady state. Ideally, we would lke to lnearze (11a) (11f) around (15a) (15f). However, due to heterogenety along the optmal path, ths s not a tractable problem. Snce we cannot explctly study the transtonal dynamcs of the economy under both ex ante and ex post heterogenety, we wll study the transtonal dynamcs n the specal symmetrc case n whch there s a representatve ndvdual voter ex post, and then gve an nformal argument for the general asymmetrc case n whch ndvduals voters can dffer both ex ante and ex post. We therefore lnearze (13a) (13d) around (15a) (15f). The dynamcs are approxmated by the lnear system: ~z r 1 ˆ d z ~u 0 AB(V12 V 2) AB(V62 V 7) 2 ABV3 u 5 ~b 0 EB(V12 V 2) EB(V62 V 7) 2 EBV3 b ~c 2 c V V 0 c (16) where the elements of the Jacoban evaluated at steady state are n Appendx C. The characterstc equaton of the Jacoban n (16) s: s ˆ 2 b(r 1 d 2 b ) b 2 BA(V f 2 V ) 1 E(V 2 V ) gb 1fBV (EV 1 AV g (17) where b s an egenvalue of the Jacoban evaluated at steady state. Eq. (17) mples that there are four egenvalues, denoted as b 1, b 2, b 3, b 4, where b15 0, b 5 r 1 ˆ 2 d. 0 and the values of b 3, b4 are the soluton of the quadratc n the parenthess n (17). Observe four thngs. Frstly, one of the egenvalues s zero. Thus, the matrx n d

18 532 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) (16) s sngular. In our case, a sngular Jacoban of the endogenous varables means that the dmensonalty of the system s not actually four, but three. Ths happens because the two polcy nstruments, u and b, are proportonal to each other n each tme perod (see the atemporal condton (12) above). Thus, the dynamcs of u are also the dynamcs of b, and vce versa. Secondly, and gven the prevous pont, snce there are four non-predetermned or jump varables z, u, b, c determnacy (.e. a unque equlbrum path to the steady state) requres the remanng two egenvalues, b and b, to be postve (.e. unstable). If one of b, b4 s negatve (.e. stable), there s ndetermnacy. That s, f b 3,orb 4, or both, are negatve, there exsts a contnuum of equlbrum paths assocated wth a gven ntal condton and the same steady state. Thrdly, n the specal case n whch there s no redstrbutve polcy,.e. d 5 ˆ d ; 0, the system n (16) s reduced to a three-dmensonal one n z, u, b, and the characterstc equaton n (17) s reduced to b(r 2 b )(b 2 r) 5 0 so that b15 0 and b25 b35 r. 0. That s, wthout redstrbutve polcy, there s always determnacy. Fourthly, when polcy s chosen by a benevolent government so that the economy evolves over tme accordng to (14a) (14c), the characterstc equaton s b(r 1 ˆ d 2 b ) sb 2 BAM f u 1 EMbgd5 0 so that the three egenvalues are b 5 0, b 5 r 1 ˆ d. 0 and (1 1 n)(1 2 a 2 u )D(u, b ) b 5 ]]]]]]]].0. 3 n That s, agan there s determnacy. Let us now go back to the general case n (17), where there s redstrbutve polcy and economc polcy s chosen by ndvduals voters. As we sad above, determnacy depends on the sgn of b and b. From (17), the sum of b and b s: ˆndB(u, ˆ b )(1 2 a 2 u ) BA(V f 12 V 2) 1 E(V62 V 7) g5 r 1]]]]]] nu(1 2 b ) F G 2 (1 2 a 2 u ) (12 a 2 b ) 3]]]] 2 ]]]]. (18) u b(1 2 u ) If (18) s negatve, at least one of b 3, b4 s negatve, so that there s ndetermnacy. Observe that B(u, b ), 0 s a necessary condton for ndetermnacy. The crtcal value of the redstrbutve polcy parameter, d. 0, that makes (18) negatve s: 2 2 ru(1 2 b ) d.]]]]]]]]]]]]]]]].0 (19) 2 (1 2 a 2 u ) (12 a 2 b ) (1 2 a 2 u )B(u, b ) F]]]] 2 ]]]] G( 2 1) u b(1 2 u ) 33 Here we follow the termnology of Benhabb and Perl (1994).

19 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) whch s a suffcent condton for ndetermnacy. Therefore, we summarze: Proposton 2. Under the condton n Proposton 1, the unque steady state s locally ndetermnate (resp. determnate), when condton (19) holds (resp. does not hold). That s, when the redstrbutve polcy parameter d s hgh enough, there are multple (nfnte) paths of economc polcy and growth, all of whch are assocated wth a gven ntal condton and wth convergence to the same steady state. On the other hand, when economc polcy s chosen by a benevolent government, or when there s no redstrbutve polcy, the steady state s determnate. What does Proposton 2 mean? It means that the antcpaton of suffcently large redstrbutve transfers opens the door for multplcty. That s, there are many possble equlbrum paths for economc polcy, consumpton and captal accumulaton, each of whch s consstent wth a gven ntal condton and wth convergence to the same, unque steady state. Any of these dfferent transton paths can be obtaned as a self-fulfllng prophecy. Ths result mples that, n the presence of moral hazard behavor due to redstrbuton, ndvduals who start wth smlar endowments may consume and save at completely dfferent rates over tme, even f they eventually converge to the same long-run rate of consumpton and captal (although not to the same long-run level of consumpton and captal). In other words, the antcpaton of redstrbuton can tself generate ncome nequalty along the optmal path. We fnally dscuss the general case n whch ndvduals are heterogeneous not only ex ante but also ex post. As we sad above, although we cannot explctly study the transtonal dynamcs of ths general case, we can use Proposton 2 above, to gve an nformal argument. It s obvous that when there s ex post 35 heterogenety, the dmensonalty of the dynamc system (11a) (11f) ncreases. Hence, stablty cannot become easer when we move from the specal case n whch ndvduals are heterogenous ex ante to the general case n whch 34 Here s a numercal example that shows the plausblty of our analyss. Assume the followng parameter values: the productvty of prvate captal vs-a-vs ` publc producton servces n (7) s a 5 0.9, the relatve weght gven to publc consumpton servces relatve to prvate consumpton n (4b) s n 5 0.3, the dscount factor s r and the number of ndvduals s 5 5. Then, the soluton of (15b) and (15d) gves the long-run values of the tax rate and the share of total tax revenues used to fnance publc producton servces, whch are u and b 5 0.3, respectvely (note that 0, 1 2 a,u, 1 and 0, 1 2 a, b, 1 as t should be). In turn, B(u, b ) 520.9, 0, whch s negatve (ths s necessary for ndetermnacy). Then, the crtcal value that gves ndetermnacy n (19) s d ote that these numbers can become more realstc (especally the dscount factor r whch s too hgh) f we add more structure (.e. parameters) to the model. However, we prefer to keep structure and notaton at a mnmum. 35 In partcular, Eqs. (11c) (11f) appear tmes,.e. one for each ndvdual 5 1, 2,....

20 534 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) ndvduals are heterogenous both ex ante and ex post. Then, Proposton 2 mples that the possblty of ndetermnacy under a hgh enough d ncreases once ndvduals dffer both ex ante and ex post. 6. Conclusons and extensons Ths paper has studed optmal fscal polcy n a growng economy when the government mposes dstortng taxes to fnance publc producton servces, publc consumpton servces and state-contngent redstrbutve transfers. Focusng on effcency and dynamcs, we gave a more realstc vew of the effects of optmal (second-best) redstrbutve and allocatve fscal polces on economc growth. We close wth three lmtatons and possble extensons. A frst lmtaton s that we have not taken a stand as to why redstrbuton occurs. We just took t as gven. Although ths s a rather standard way (see also Atknson (1999)), we fully understand that our framework does not ncorporate any of the contngences for whch the welfare state s ntended to exst (socal conflcts, unnsured uncertanty, nvoluntary unemployment, etc) whch could ratonalze redstrbutve polcy. Secondly, t would be nterestng to add human captal and publc educaton, and also assume that households have dfferent human captal endowments (see Glomm and Ravkumar (1992), Perott (1993) and Fernandez and Rogerson (1995, 1996)). Thrdly, we have solved for commtment equlbra. That s, we assumed that optmzng economc agents can commt themselves to future polces. Whle ths s the type of equlbra that most of the lterature on endogenous growth has focused on, we realze that solvng for tme-consstent equlbra (e.g. Markovperfect equlbra) would be more realstc. We leave these extensons for future work. Acknowledgements We would lke to thank the edtor and an anonymous referee for constructve crtcsms and suggestons. We also thank J. Malley, T. Moutos, K. shmura, H. Polemarchaks, S. Prce and E. Tsakalotos for dscussons and comments. Ths paper was completed when the second author was vstng Queen Mary and Westfeld College, Unversty of London. He s grateful for the hosptalty. All errors are ours. Appendx A. From Eqs. (11a) (11f) to Eqs. (13a) (13d). If we mpose the symmetrcty condtons, take the logarthms of (11a) and (11b), dfferentate wth respect to tme, use (11c), (11d), (11e) and (11f) for the

21 H. Park, A. Phlppopoulos / Journal of Publc Economcs 87 (2003) rates of growth of l, c, k and g, and also use the defnton c ; gk, then we get Eqs. (13b) and (13c). Eq. (13a) follows easly f we dfferentate z ; c/k wth respect to tme and then use (11d) and (11e). Fnally, Eq. (13d) follows f we dfferentate c ; gk wth respect to tme and then use (11e), (11f) and the defnton z ; c/k. Appendx B. Proof of Proposton 1 Consder (15b) and (15d) n the man text and rewrtten here for convenence: (1 1 ˆ n )(1 2 a 2 u )D(u, b ) r 1]]]]]]]] 50 (15b) n (1 2 u ) (12 a 2 b ) ]]]] 5 ]]]]] (15d) (1 2 a 2 u ) (12 a)(1 2 b ) whch are two equatons n u and b. Recall that 0, 1 2 a, u, 1 and 0, 1 2 a, b, 1. Total dfferentaton of (15b) mples: b 2 b[(1 2 a)(1 2 a 2 u ) 2 au] ] 5 ]]]]]]]]],0 (A.1) u u(1 2 a)(1 2 a 2 u ) and total dfferentaton of (15d) mples: b 2 b ] 5 ]],0. (A.2) u u Comparson of (A.1) and (A.2) mples that the slope n (A.1) s always less than the slope n (A.2),.e. (15b) s always steeper than (15d). Therefore, f (15b) and (15d) ntersect, they can ntersect only once. But we also have to check whether they actually ntersect. To do so, t s convenent to see what happens when u approaches ts two boundares, (1 2 a) and 1. As u (1 2 a), (15d) mples b 1, whle as u 1, ths equaton mples b 1 2 a. ow, snce (15b) s always steeper than (15d), for (15b) and (15d) to ntersect, t must be that when b 1 2 a, (15b) mples a value of u less than 1. Indeed, t can be seen that when b 1 2 a, and f we assume (12a)/a (12a)/a (1 1 ˆ n )a (1 2 a) r 2 ]]]]]]]]],0, n then u 1 cannot be a soluton to (15b). Specfcally, f ths condton holds (whch s (C.1) n Proposton 1), the propertes of the functons (n partcular, that (1 2 a 2 u )D(u, b ) s decreasng n u ) mply that u must be less than 1.