A further generalization of the Solow growth model: the role of the public sector

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1 Econocs Letters 68 (2000) locate/ econbase A further generalzaton of the Solow growth odel: the role of the publc sector Oscar Bajo-Rubo* Departaento de Econoıa, Unversdad Publca de Navarra, Paplona, Span Receved 7 Aprl 1999; accepted 4 January 2000 Abstract We develop n ths paper an augented verson of the Solow (1956) [Solow R., A contrbuton to the theory of econoc growth. Quarterly Journal of Econocs 70, 65 94] growth odel, ncludng the role of governent. The odel leads to a non-onotonc relatonshp between the rate of growth of per capta output and governent sze, generalzng prevous results by Barro (1990) [Barro R., Governent spendng n a sple odel of endogenous growth. Journal of Poltcal Econoy 98, S103 S125] to the case n whch returns to scale to prvate factors are not constant Elsever Scence S.A. All rghts reserved. Keywords: Econoc growth; Neoclasscal and augented growth odels; Publc captal; Transfers; Fscal polcy JEL classfcaton: E62; O40 1. Introducton It has becoe wdely accepted that the Solow (1956) growth odel, augented by the ncluson of other productve factors n addton to prvate captal and labor, s able to explan roughly well cross-country dfferences n growth rates of per capta ncoe [see, e.g., Barro and Sala--Martn (1995), Mankw et al. (1992), or Nonnean and Vanhoudt (1996)]. A certan aount of ths eprcal research s addressed to analyze the effects of fscal polcy on growth [whch s surveyed, e.g., n Slerod (1995) or Tanz and Zee (1997)]. The standard result of ths lterature s that of Barro (1991), who fnds a negatve and sgnfcant effect of the level of publc consupton as a percentage of GDP (whch would proxy governent sze), on the growth rate of a cross secton of countres. Ths s justfed on the grounds that a greater governent nterventon would dstort the ncentves systes, so that a hgher governent sze would be assocated wth a *Tel.: ; fax: E-al address: obajo@unavarra.es (O. Bajo-Rubo) / 00/ $ see front atter 2000 Elsever Scence S.A. All rghts reserved. PII: S (00)

2 80 O. Bajo-Rubo / Econocs Letters 68 (2000) lower productvty, and hence a lower growth. However, ths effect dd not appear robust to changes n the condtonng varables n the nfluental study of Levne and Renelt (1992). In addton, and ore portant, t does not see very clear the use of governent consupton as a proxy of the whole publc expendture, snce there would be other coponents ore drectly lnked to growth. On the other hand, ost of the eprcal lterature on fscal polcy and growth s not based on an explct theoretcal fraework, addng only a proxy of the sze of the publc sector (usually, governent consupton), n an ad hoc fashon, to a standard convergence equaton. In fact, fscal polcy nstruents have been eboded n theoretcal odels of growth only fro the pont of vew of endogenous growth. So, Barro (1990) consders publc servces as a productve (flow) nput, and, by takng nto account how the governent fnances those servces, obtans a non-lnear relatonshp between governent sze and growth. Subsequently, Cashn (1995) odfes Barro s odel to nclude nto the producton functon the governent s captal stock (rather than a flow nput), as well as the role of transfer payents. The a of ths paper s to develop an augented verson of the Solow growth odel, ncludng the role of governent. The odel wll lead to a growth equaton n ters of the shares of prvate factors and fscal polcy nstruents, wth a non-onotonc relatonshp between governent sze and growth. As a by-product of the analyss, we wll be able to derve an expresson for the optal governent sze. 2. A odel of fscal polcy and growth The producton functon of our odel wll nclude, together wth prvate nputs, those governent nputs that could a pror be thought to strctly affectng the level of output. One s a reproducble factor, enterng drectly nto the producton functon: publc physcal captal. The other s assued to nfluence ndrectly, va externaltes, the ncentves to accuulaton and growth; followng Cashn (1995), ths nput wll be called transfer payents. The ncluson of transfers ay be justfed snce they would allow to renforce property rghts (on rasng the opportunty cost of crnal actvtes), as well as retrng fro the labor force those people wth a lower level of huan captal (Sala--Martn, 1996, 1997). Hence, we postulate a producton functon such as: g TR u a b1 b 12a 2O b Y 5 K Z 1???Z (AL) 51 S]]DS]D (1) K K where Y denotes output; K s prvate physcal captal, Z (51,...,) are other prvate nputs (such as huan captal, as n Mankw et al. (1992), knowledge captal, as n Nonnean and Vanhoudt (1996), and the lke), L s labor, and A s a labor-augentng factor; fnally, and TR are the governent-provded nputs: publc physcal captal and transfer payents, respectvely. Notce that our forulaton allows for congeston of the publc servces, whch would be rval but non excludable goods: every producer benefts fro the provson of publc nputs but, for a gven level of the latter, the quantty avalable to each producer declnes as other producers rase ther levels of prvate nputs (Barro and Sala--Martn, 1992). In the producton functon above, t s assued that a. g 1u.

3 O. Bajo-Rubo / Econocs Letters 68 (2000) Wrtng the producton functon n per capta ters we have: g TR u a b1 b S]]DS]D...z y 5 Ak z 1 K K (2) where sall letters denote per capta varables, and sall letters wth a bar ndcate per capta varables n effcency unts (.e., for any varable X: x 5 X/L,x 5 X/AL). Notce that the per capta producton functon (2) exhbts decreasng returns to scale n both prvate captal and all prvate nputs, for a gven state of congeston n the use of publc captal and transfers. Ths dffers fro Barro (1990), where prvate captal was subject to constant returns to scale. Next, we turn to the accuulaton equatons. We assue that prvate reproducble factors accuulate accordng to the followng equatons: ~K 5 s (1 2 t)y 2 dk K (3) ~Z 5 s (1 2 t)y 2 dz ; 5 1,..., (4) Z where s and s are the shares of gross nvestent on prvate physcal captal and the other prvate K Z nputs, respectvely, n prvate output, t beng the sze of the publc sector (.e., the share of the publc budget n total output); d s the deprecaton rate (assued to be the sae for all nputs); and a dot over a varable denotes ts te dervatve. In a slar way, publc captal would accuulate accordng to: ~ 5 s ty 2 d (5) where s s now the share of gross publc nvestent n publc output, and the deprecaton rate s agan assued to be the sae for prvate nputs. Fro here, the rates of change n the stocks of the reproducble factors, n effcency ters, would be gven by: ~K g 5] k 2gA2 n K (6) ~Z ] g 5 2g 2 n ; 5 1,..., (7) z Z ~ g] kg 5]] 2gA 2 n A where gx denotes the rate of growth of varable X, and n s the rate of populaton growth (.e., n5g L); n partcular, ga s the rate of techncal progress. By equatng (6), (7), and (8) to zero, we can fnd the ] steady-state values of k, z, and kg; and, assung further that: where s ] strty* tr* 5]]] A TR s the share of transfers n publc output, and astersks denote steady-state values, we can (8) (9)

4 82 O. Bajo-Rubo / Econocs Letters 68 (2000) obtan the (log of the) steady-state per capta output by replacng those values n the steady-state counterpart of Eq. (2): a 1O b 2u a 2 g 2u ln y* 5 ln A 1 gt2]]]]] lnsd 1 g 1 nd1]]]] ln s 1 2 a 2O b 1 2 a 2Ob 0 A A K b1 b g 1]]]]] ln s 1???1]]]]] ln s 1]]]]] ln s Z1 Z a 1Ob 2 g 2u u g 1u 1]]]]] ln s 1]]]]] ln t 1]]]]]] lns1 2 t d (10) TR where A 0 s the ntal value of the technologcal paraeter A,.e., At 5 A 0 e, wth t denotng te. To derve a growth equaton, we ake an approxaton around the steady state (see Mankw et al., 1992 or Barro and Sala--Martn, 1995), so that, n effcency ters, we can wrte: dlny ]] 52 sln y 2 ln y * d1usgtr 2 ga 2 nt d (11) dt s 51 ds A d where l a 2 o b 1u d 1 g 1 n s the speed of convergence. Solvng the dfferental equaton gven by (11), replacng the deternants of the steady state gven by Eq. (10), dvdng by t, and rearrangng, we obtan the fnal expresson for the rate of growth of per capta output: where y 2lt a 1O b 2u (1 2 e ) g 5 (1 2u )g 1]]]] ln A 2]]]]] ln(d 1 g 1 n) t a 2O b y A 0 A 0 a 2 g 2u b1 b 1]]]]] ln s 1]]]]] ln s 1???1]]]]] ln s K Z1 Z g u g 1u 1]]]]] ln s 1]]]]] ln s 1]]]]] ln t TR 6 0 TR a 1O b 2 g 2u 1]]]]]] ln(1 2 t) 2 ln y 1u( g 2 n) (12) 1 2 a 2O b s the ntal per capta output, and sln yt 2 ln y0d gy 5 ]]]]]. t Notce that, n Eq. (12), s and s (51,...,) would denote the shares of gross nvestent on K Z g t A

5 O. Bajo-Rubo / Econocs Letters 68 (2000) prvate nputs n prvate output, and s and str the shares of gross publc nvestent and transfers n publc output, nstead of the shares n total output. Ths allows us to derve explctly fro the odel a non-onotonc relatonshp between the rate of growth of per capta output and the sze of the publc sector, leadng to an nverted U-shaped relatonshp between the two varables. Hgher levels of publc nputs would lead drectly to a hgher growth, but they wll leave a saller quantty of output avalable for the accuulaton of prvate nputs; and the rate of growth of per capta output, together wth ts steady-state level, would be axzed for: g 1u t 5]]] (13) a 1Ob 51 Notce, fnally, that the non-onotonc relatonshp found between the rate of growth of per capta output and the sze of the publc sector, as well as the optal sze of the latter gven by Eq. (13), would be equvalent to the results derved n Barro (1990), beng a generalzaton of the to the case n whch returns to scale to prvate factors are not constant. 3. Conclusons We have developed n ths paper an augented verson of the Solow growth odel, ncludng the role of governent. To ths end, the producton functon has been extended to ncorporate those publc nputs presued to affect strctly the producton process (.e., publc captal and transfer payents). The odel led to a non-onotonc relatonshp between the rate of growth of per capta output and the sze of the publc sector, together wth an expresson for the optal governent sze, generalzng Barro s (1990) prevous results to the case n whch returns to scale to prvate factors are not constant. Fnally, the odel presented n ths paper could be used as a fraework for eprcal analyss, snce ost of the eprcal lterature on fscal polcy and growth s not based on an explct theoretcal odel. References Barro, R., Governent spendng n a sple odel of endogenous growth. Journal of Poltcal Econoy 98, S103 S125. Barro, R., Econoc growth n a cross secton of countres. Quarterly Journal of Econocs 106, Barro, R., Sala--Martn, X., Publc fnance n odels of econoc growth. Revew of Econoc Studes 59, Barro, R., Sala--Martn, X., Econoc Growth, McGraw-Hll, New York. Cashn, P., Governent spendng, taxes, and econoc growth. Internatonal Monetary Fund Staff Papers 42, Levne, R., Renelt, D., A senstvty analyss of cross-country growth regressons. Aercan Econoc Revew 82, Mankw, G., Roer, D., Wel, D., A contrbuton to the eprcs of econoc growth. Quarterly Journal of Econocs 107, Nonnean, W., Vanhoudt, P., A further augentaton of the Solow odel and the eprcs of econoc growth for OECD countres. Quarterly Journal of Econocs 111,

6 84 O. Bajo-Rubo / Econocs Letters 68 (2000) Sala--Martn, X., A postve theory of socal securty. Journal of Econoc Growth 1, Sala--Martn, X., Transfers, socal safety nets, and econoc growth. Internatonal Monetary Fund Staff Papers 44, Slerod, J., What do cross-country studes teach about governent nvolveent, prosperty, and econoc growth? Brookngs Papers on Econoc Actvty 2, Solow, R., A contrbuton to the theory of econoc growth. Quarterly Journal of Econocs 70, Tanz, V., Zee, H., Fscal polcy and long-run growth. Internatonal Monetary Fund Staff Papers 44,