akai sakata (2007) - complementarity, fiscal decentralization and economic growth

Economics of Governance (2007) 8: 339–362DOI 10.1007/s10101-007-0032-5

O R I G I NA L PA P E R

Complementarity, fiscal decentralizationand economic growth

Nobuo Akai · Yukihiro Nishimura ·Masayo Sakata

Received: 18 November 2004 / Accepted: 7 November 2006 / Published online: 23 February 2007© Springer-Verlag 2007

Abstract Theories of the voluntary provision of public goods and develop-ment economics have clarified that complementarity in the production processis a crucial ingredient to understanding how alternative economic environmentsaffect economic performance. This paper examines how the structures of intra-and inter-regional complementarity affect the relationship between economicgrowth and fiscal decentralization. We provide a theory that describes howfiscal decentralization affects economic growth under various structures of re-gional complementarity. Our empirical analysis, based on a panel data set of

The previous version of the paper was presented at the 59th Congress of the InternationalInstitute of Public Finance (University of Economics in Prague, Prague), the 2003 Fall Meeting ofthe Japanese Economic Association (Meiji University, Tokyo), the 60th Annual Meeting of theJapanese Institute of Public Finance (Kansai University, Osaka), and in seminars at YokohamaNational University and the University of California, Irvine. The authors acknowledge thecomments and discussions by people including Timothy Goodspeed, Kiyoshi Mitsui, MotohiroSato, Etsuro Shioji, Tsunao Okumura, and Craig Parsons. We are also grateful for the commentsby the Editor (Amihai Glazer) and two anonymous referees. The usual disclaimer applies.Nishimura acknowledges the financial support from JSPS (Japan Society for the Promotion ofScience) Postdoctoral Fellowships for Research Abroad.

N. AkaiSchool of Business Administration, University of Hyogo,8-2-1 Gakuen-Nishimachi, Nishiku, Kobe 651-2197, Japan

Y. Nishimura (B)Department of Economics, Yokohama National University and Queen’s University,Dunning Hall, Room 209, Kingston, Ontario K7L 3N6 Canadae-mail: [email protected]

M. SakataDepartment of Politics, Economics and Law, Osaka International University,3-50-1, Sugi, Hirakata, Osaka 573-0192, Japan

340 N. Akai et al.

the fifty states of the United States over the period of 1992–1997, supports ourtheoretical specification of the production function. Also, we observe a hump-shaped relationship between fiscal decentralization and economic growth thatis consistent with our theoretical result. Our analysis also shows that the opti-mal degree of fiscal decentralization conducive to economic growth is higherthan the average of the data in some cases, and hence further decentralizationis recommended for economic growth.

Keywords Complementarity · Fiscal decentralization · Economic growth

JEL classification O40 · H77

1 Introduction

One role of the public sector is to create a public organization to promote eco-nomic growth. It is well known that the average share of central governmentspending in total government expenditure is typically much higher in develo-ping countries than in developed countries. This fact, combined with the argu-ment in many areas of economic theory in favor of decentralized organizations,has recently encouraged the progress of fiscal decentralization in developingcountries. The problem of centralized versus decentralized governmental orga-nization is also a relevant economic issue in developed countries. For example,in Japan, there are discussions on the reform of taxes and intergovernmentalgrants towards a more decentralized system.

A number of studies attempt to quantify the impact of decentralization byrelating some measure of decentralization to economic growth (for example,Davoodi and Zou 1998; Zhang and Zou 1998; Xie, Zou and Davoodi 1999; Ebeland Yilmaz 2002; Lin and Liu 2000; Akai and Sakata 2002). However, underlyingmacroeconomic structures that can explain possible relationship between thesevariables are not fully examined. Theoretical works are scarce. Recent work bySato and Yamashige (2005) introduces a political economy model with a com-plex principal-agent nature of the government to explain the evolution of fiscaldecentralization and economic development. Extending Barro’s (1990) econo-mic growth model with a public good, theoretical models of Davoodi and Zou(1998) and Xie, Zou and Davoodi (1999) demonstrate a theoretically optimaldegree of fiscal decentralization. However, empirical implications of their stu-dies are either not obvious or very difficult to test. This paper presents a theoreti-cal link between fiscal decentralization and economic growth by incorporating amacroeconomic model that crystallizes the economic structure to explain whe-ther fiscal decentralization is conducive to economic growth or not. We alsoempirically examine the validity of our specification of the macroeconomicstructure as well as that of the derived proposition.

A key concept to analyze the mechanism of the effect of fiscal decentra-lization on economic growth is complementarity. Since policy makers havelimited foresight to find the correct policy to promote growth or individualwelfare, there arise successful and unsuccessful jurisdictions, and also, within

Complementarity, fiscal decentralization and economic growth 341

jurisdictions, there are successful and unsuccessful projects. Public services ineach jurisdiction have a spillover effect on the national economy. The ques-tion is how effectively a high quality public service in one jurisdiction canmake up for a poorly performed outcome in another. The less important is thecontribution of successful jurisdictions, the greater the complementarity. Anexample of high complementarity between jurisdictions is that of the formerSoviet Union and the East European countries, where planners located in-dustries across regions to maintain the comparative advantage of each region.An example of low complementarity (high substitutability) is that of China,where public projects have an experimental nature (Montinola et al. 1996).The notion of complementarity was first discussed by Hirshleifer (1983) inthe public good provision problem, extended theoretically and empirically byCornes (1993), Conybeare et al. (1994) and others (see Cornes and Sandler1996). Contributions in monopolistic competition (see Matsuyama 1995 for anoverview) and development economics (see, e.g., Bénabou 1996) emphasize itsimportance on economic development. This paper examines how the structuresof intra-regional complementarity (referred to as local complementarity) andinter-regional complementarity (referred to as global complementarity) affectthe relationship between economic growth and fiscal decentralization.

We consider a simple two-period OLG model where a country consists of Jjurisdictions, which need public services. There is one bureaucrat in the centralgovernment, and one local bureaucrat in every jurisdiction. The outcomes ofpublic services provided by the central and local governments depend on theability of bureaucrats. We show the following. If the degree of local comple-mentarity is lower than global complementarity, the relationship between fiscaldecentralization and expected economic growth is hump-shaped, and there is aunique interior optimum that is consistent with the expected growth maximiza-tion. An intuition is the following. The risk that the bureaucrat in charge of thepublic input may have low productivity can reduce economic growth. The ques-tion is how a bad project is made up with the good project in the aggregationprocess. If the local complementarity is sufficiently lower than global comple-mentarity, risk-sharing between local governments and the central governmentbecomes more desirable.

We then examine these theoretical results using a panel data set of thefifty states of the United States over the period of 1992–1997.1 We use annual

1 Davoodi and Zou (1998) use cross-country data for 46 countries over the period of 1970–1989,in which the cultural, historical, and institutional differences between countries are substan-tial. Given other determinants of growth, they found a negative relationship between fiscaldecentralization and economic growth. However, Davoodi and Zou (1998, p. 254) recognizethat the cross-country variation among developing countries is one of the reasons for the ne-gative relationship. Ebel and Yilmaz (2002, footnote 16) noted that overrepresentation of thedegree of fiscal decentralization in the aggregate figure for developing countries might bethe reason for the negative relationship in Davoodi-Zou’s study. Ebel and Yilmaz also repli-cated Davoodi-Zou’s regression model for the six European countries (the Czech Republic,Estonia, Hungary, Latvia, Lithuania, and Poland) where cultural and historical differences aremuch smaller, in which they derive a positive relationship between fiscal decentralization andeconomic growth. A similar contrast can be found between Xie, Zou and Davoodi (1999)

342 N. Akai et al.

data and the average of the sample period. We first show that our theoreticalspecification of the production function is supported by the regression where wederive statistically significant values of global and local complementarity. Also,we observe a hump-shaped relationship between fiscal decentralization andeconomic growth that is consistent with our theoretical result. An intriguingissue is whether the optimal level of fiscal decentralization derived by oureconometric model is higher or lower than that of the actual data. In some cases,the optimal degree of fiscal decentralization conducive to economic growthis higher than the average of the data, and hence further decentralization isrecommended for economic growth.

To our knowledge, this is the first work which discusses the hump-shapedrelationship between fiscal decentralization and economic growth theoretically,with consistent empirical analyses. Thiessen’s (2003) empirical work shows thehump-shaped relationship between fiscal decentralization and economic growthin the group of high-income OECD countries. The differences between his studyand this paper are as follows. First, we provide a theoretical model that explainsa relationship between fiscal decentralization and economic growth. Second, weestimate the production function consistent with theoretical hypothesis. Andfinally, we examine whether the current level of fiscal decentralization is optimalor not.

The rest of this paper proceeds as follows. In Sect. 2 we provide an economicgrowth model with complementarity in the public expenditure in an overlappinggenerations setting, and examine expected GDP growth in relation to fiscaldecentralization. In Sect. 3 we empirically verify our theoretical result using UScross-section data. Section 4 concludes. The proof of the proposition is given inAppendix A. In Appendix B we provide the data sources.

2 A model with complementarity and economic growth

2.1 Firms and households

We consider an extension of the standard two-period OLG model with manyjurisdictions, adopted in Nishimura (2006). A country consists of J identicaljurisdictions. In each jurisdiction there are many identical firms and households.

Footnote 1 continuedand Akai and Sakata (2002). Xie, Zou and Davoodi (1999), using time series data for the UnitedStates from 1948 to 1994, find that fiscal decentralization may be detrimental to economic growth.Akai and Sakata (2002) point out that Xie, Zou and Davoodi (1999) used the data from a periodof high economic growth in the United States. During the early stages of economic development,a high level of government expenditure was required to provide public investment that generateslarge externalities, so that the analysis that includes such a period overestimates the contributionof the central government. Akai and Sakata (2002) also argue that the estimation has to be donewith data which embody less cultural and historical differences and have similar development stage.Taking account of these factors, Akai and Sakata (2002) find that fiscal decentralization contributesto economic growth. For the same reason as in Akai and Sakata (2002), we use the data from theperiod of 1992–1997.


Firms use capital K, labor L and the public good G to produce the output Y.The production function takes the familiar form studied by Barro (1990) andothers:

Yt = AK1−βt (GtLt)

β (1)

where Yt is the national output, Kt is capital, Lt is labor, and Gt is the publicgood supplied in period t. As specified below, Gt is a function of public ser-vices provided by the central government and the local governments. Examplesinclude health, literacy and the average IQ level of the citizens, which dependon public expenditures such as health and education. Following a convention ingrowth theory, we assume that the public good G serves as a labor-augmentingtechnology.

There are identical households who live for two periods. When agents areyoung, they supply inelastically one unit of labor and receive the competitivewage level wt, which is taxed at the rate τ .2 A fixed fraction is saved out of thepost-tax income in period t,3 which is consumed at the next period when agentsare old without bequest. Following the conventional models, initial capital stockK1 is owned by the L0 persons who are old in period 1.

2.2 Public good

We now introduce Gt. The public projects are undertaken by different bureau-crats. Some policies are favorable for higher economic growth, and others arenot. The level of public good Gt is determined by the local and nation-wide dis-tribution of the public inputs through inter- and intra-jurisdictional spillovers,which we call complementarity. A crucial question for aggregation is whethera good project in the nation pulls the national product up, or a bad one dragsthe nation down. Our concern is captured by the following formulation as anextension of Nishimura (2006).

Each jurisdiction provides a public service, such as infrastructure, education,or regulation policies. There is a continuum of projects in each jurisdiction j.Let gt(j, m), m ∈ [0, 1] ≡ M be project m in jurisdiction j at period t. The publicgood Gt is a function of these public projects.

Tasks are divided between the central bureaucrat and the local bureau-crat. There are two types of bureaucrats who can provide public serviceseither with high (h) or low (l) quality. A proportion α of bureaucrats has high

2 Capital income tax can be incorporated in the model, but such complication is not essential tothe analysis.3 A constant fraction of savings is similar to the conventional Solow-Swan growth model, butit can also be derived from a micro-founded model where households have a logarithmic utilityfunction.

344 N. Akai et al.

ability. When a project m is assigned to the central government, one bureau-crat, who can be type h with probability α, is in charge of public project gt(j, m)

for all j. When a project m is assigned to a local government, one bureau-crat in a local government in each jurisdiction, who can be of type h withprobability α, is in charge of one public service, gt(j, m). The quality of eachpublic service in period t is a random variable, depending on the bureaucrat’sability.

At each period, the local bureaucrats in each jurisdiction and the centralbureaucrats are randomly and independently chosen. At the end of period t,the bureaucrat who is assigned to jurisdiction j’s project m decides on theproject, which turns out to be good or bad in the beginning of the next period(period t + 1) according to the ability of the bureaucrat in charge. Each projectis financed by the tax revenue. Let Tt be the tax revenue per jurisdiction atperiod t, and let Tt(j, m) be the amount of public funds to finance jurisdictionj’s project m at period t. The fiscal budget is balanced:

∫ 10 Tt(j, m)dm = Tt

for all j. Tasks are symmetric, so that an equal proportion of tax revenue isallocated to each project: Tt(j, m) = Tt for all j and m. The bureaucrat in chargeof jurisdiction j’s project m uses the public fund Tt(j, m) to provide the publicservice of the next period, gt+1(j, m). The bureaucrats employ the followinglinear production functions.

gt+1(j, m) = µTt(j, m) if the bureaucrat has a high ability, and

gt+1(j, m) = νTt(j, m) if the bureaucrat has a low ability, with µ > ν > 0. (2)

The parameters µ and ν represent the efficiency of the public service. We assumethat the values of α (the proportion of high ability bureaucrats), µ and ν [theparameters for the production function (2)] are the same for central and localgovernments.4

4 Our formulation is an application of Sah’s (1991) fallibility. Bureaucrats cannot perfectly choosepolicies for a desirable outcome. As a result, the outcome of policies becomes stochastic. A society iscalled ‘more centralized’ or ‘less centralized’ depending on whether there are few or many decision-makers (which is slightly different from our setup). The aggregate performance of all fallible de-cisions is different, depending on the degree of decentralization. One may argue that the centralbureaucrats perform better than the local bureaucrats. On the other hand, there is a counterargu-ment that lower-tier government can respond better to local needs than higher-tier governments.For example, Crewson (1995) used the score of the Armed Forces Qualifying Test (AFQT) as an in-dicator of quality of the workers. He showed that the U.S. federal employees hired during the 1980shave scores of the AFQT that are significantly higher than those of private sector employees, whe-reas there is no significant difference in the state sector. On the other hand, Bardhan (2002) reportsseveral empirical findings in developing countries that public services by municipalities respondbetter to local needs than those by the central government. The focus of our analysis is to exa-mine the economic effects of fiscal decentralization (represented by higher allocation of resourcesto the lower-tier government) without assuming a priori which bureaucrats perform better onaverage.


We then consider the following aggregation process that embodies ourmotivation of the paper:

gt(j) ≡⎛

⎝1∫

0

gt(j, m)−σ dm

⎞

⎠

1−σ

, (3)

Gt =⎛

⎝J∑

j=1

1J

gt(j)−ρ

⎞

⎠

1−ρ

. (4)

(3) represents the public input in jurisdiction j as the composition ofpublic projects within the jurisdiction. The value of σ corresponding toBénabou (1996) local complementarity, represents the complementarity bet-ween the projects within a jurisdiction (among the different projects, or amongthe same project accomplished by different bureaucrats). For example, the ove-rall educational level in a jurisdiction depends on the quality of the educationprovided by national and local schools. A higher σ indicates higher comple-mentarity.5 Intuitively, the unsuccessful local project drags the aggregate per-formance down. Consider, for example, infrastructure in a country where theregional production process is highly complementary. If the quality of one roadin a jurisdiction is poor, it reduces the efficiency of production in the jurisdic-tion.6 On the other hand, substitutability becomes higher when public projectshave an experimental nature, where the level of technology is defined by theadvances made by the most successful jurisdiction (Cornes and Sandler 1996,pp. 54–55).7

(4) represents the national level of the public good as a function of jurisdictio-nal public inputs. The value of ρ represents the degree of global complementaritybetween public services in the terminology of Bénabou (1996).

The production functions (3) and (4) resemble the social composition func-tion by Hirshleifer (1983) and its generalization by Cornes (1993) Cornes andSandler (1996) which discuss complementarity and substitutability in the pu-blic good provision. The same formulation is found in Matsuyama (1995) andBénabou (1996) which explain the mechanism behind economic development.It turns out that the value of σ relative to ρ is central in determining the patternof expected growth in relation to fiscal decentralization.

5 The limiting case of high complementarity, gt(j) = minmgt(j, m) (where σ = ∞), is referred to asthe ‘weakest-link’ by Hirshleifer (1983).6 In the former Soviet Union and the East European countries, industries are located accordingto a comparative advantage of each region, and the industrial linkage has a highly complementarystructure (see also Prud’homme 1995; Montinola et al. 1996).7 The limiting case of high substitutability, gt(j) = maxmgt(j, m) (where σ = −∞), is referred to asthe ‘best-shot’ by Hirshleifer (1983). See also Cremer et al. (1996). See Montinola et al. (1996) forthe case of China.

346 N. Akai et al.

In each jurisdiction, the local government undertakes the projects in theinterval [0, θ ], and the central government undertakes those in [θ ,1]. The valueof θ (hence the proportion undertaken by the local government) is assumed tobe the same in every jurisdiction. Since the projects are symmetric, the value ofθ also indicates the spending share of the local government.8

2.3 Economic growth

Due to the uncertainty in the productivity of the bureaucrats, the growth ofthe national output is stochastic. Let E denote an expectation operator. We can

derive the following result regarding expected GDP growth, E[ln Yt+1

Yt

]:

Proposition Suppose that σ < min {ρ, 0}. Then∂E

[ln

Yt+1Yt

]

∂θ

∣∣∣∣∣θ=0

> 0,

∂E[ln

Yt+1Yt

]

∂θ

∣∣∣∣∣θ=1

< 0 and∂2E

[ln

Yt+1Yt

]

∂θ2 < 0 for all θ ∈ [0, 1]. That is, the relationship

between fiscal decentralization and expected economic growth is hump-shaped:a unique interior solution maximizes expected growth.

The proof of the proposition is given in Appendix A.The point here is the interaction of two effects, which are risk-sharing bet-

ween jurisdictions (represented by ρ) and risk-sharing between central andlocal governments (represented by σ ). When σ (the value of local complemen-tarity) is negative, risk-sharing between the central government and the localgovernment within a jurisdiction is beneficial for expected growth, since goodprojects within a jurisdiction dominate in a local aggregation process (gt(j) in(3)). On the other hand, this vertical risk-sharing brings another risk, namely,a correlated risk across jurisdictions undertaken by the central government. Inthis model, the performance of the central government’s public project is morevolatile than that of the local governments as a whole, since the success/failureof the performance of the public project depends on the ability of the centralbureaucrat. A good project promotes national production as a whole, whereasa bad project drags a country in the opposite direction. On the other hand, ifa project is undertaken at the local level, there is a kind of risk diversificationof the performance across different bureaucrats. This risk diversification bydecentralization is conducive to economic growth when ρ (the value of globalcomplementarity) is negative, and vice versa when ρ > 0 (Nishimura 2006).When σ < ρ, the first effect dominates the second, so that an interior optimummaximizes expected growth, and the relationship between fiscal decentraliza-tion and economic growth is concave.

The case of σ < min{ρ, 0} is only a representative case which is consistentwith our empirical analysis in the next section. Various patterns can emerge,

8 Recall that Tt(j, m) = Tt for all j and m. Then the amount∫

m∈[0,θ]Tt(j, m)dm = Tt∫

m∈[0,θ]dm =θTt is spent by the local bureaucrat.


Table 1 Relationship between fiscal decentralization and expected economic growth

ρ\σ σ < min{ρ, 0} σ = 0 σ > max{ρ, 0}ρ < 0 Hump-shaped, interior optimum Positive and monotonic U-shaped, θ = 1 is optimalρ > 0 Hump-shaped, interior optimum Negative and monotonic U-shaped, θ = 0 is optimal

depending on the values of ρ and σ . For example, when ρ < 0 and σ is closeto 0, the second effect mentioned above dominates, and there is a positivemonotonic relationship between fiscal decentralization and economic growth.With σ being close to 0 and ρ > 0 (global complementarity), there is a negativemonotonic relationship.9 On the other hand, when σ > 0 and sufficiently high,the relationship between fiscal decentralization and economic growth tends tobe U-shaped, since the first effect mentioned above is reversed.10 In order toexamine which possibility applies for a particular country, we need to examinethe values of ρ and σ , i.e., how much higher the degree of local substitutabilityis compared with global substitutability (Table 1).

Theoretical models of Davoodi and Zou (1998) and Xie, Zou and Davoodi(1999) demonstrate that there exists a theoretically optimal degree of fiscaldecentralization to maximize the economic growth rate. Their theoretical modelpresumes the productivity of each government’s spending, which is proportionalto the optimal spending share. However, they did not estimate the productionfunction. On the other hand, our model illuminates the relationship betweenfiscal decentralization and economic growth from alternative complementaritystructures.

3 Empirical analysis with US cross-section data

In this section, in order to empirically verify our theoretical results, we examinethe relationship between fiscal decentralization and economic growth by usingstate-level cross-section data for the United States. As outlined in footnote1, Akai and Sakata (2002) argue that the data used have to be free from the

9 A sufficient condition for positive monotonic relationship( ∂E

[

lnYt+1

Yt

]

∂θ> 0 for all θ

)

is σ = 0

and ρ < 0. A sufficient condition for negative monotonic relationship( ∂E

[

lnYt+1

Yt

]

∂θ< 0 for all θ

)

is σ = 0 and ρ > 0. More generally, for all ρ < 0, there is a critical value of σ , denoted by σ(ρ),where the expected growth is an increasing function of θ for all σ ∈ [σ(ρ), 0]. The σ(ρ) satisfiesρ < σ(ρ) � 0 for all ρ < 0. There is also a value σ(ρ) such that σ(ρ) � 0 for all ρ, where theexpected growth is a decreasing function of θ for all σ ∈ [σ(ρ), 0].

10 The relationship is U-shaped iff∂2E

[

lnYt+1

Yt

]

∂θ2 > 0 for all θ . Sufficient conditions forU-shapedness are σ > max{ρ, 0}, which is complementary to the case of the proposition.

348 N. Akai et al.

early stage of economic development and do not embody cultural and historicaldifferences. Taking account of these points, we use a panel data set of the 50states of United States over the period of 1992–1997.11 We have collected thedata from the USA COUNTIES 1998 (CD-ROM), the Statistical Abstract ofthe United States, and State and Metropolitan Area Data Book 1997–98 (CD-ROM).

3.1 Estimation of the production function

We first estimate the production function (1). Due to the limitations of thedata, we use the values of 1992 to estimate the production function. We add asubscript i to indicate state i, and in this subsection, we omit the subscript t. Thelogarithmic expansion of (1) is:

LGSPi = B + (1 − β) LKi + β (LGi + LLi) , (5)

where LGSPi is the logarithm of the per capita Gross State Product of state i,LKi is the logarithm of the per capita private investment of state i, LLi is thenumber of workers per capita in state i. Each is the value in the year 1992. Thevalue of LGi corresponds to the logarithmic form of the regional productionfunction (4).

We construct LGi in the following way. Let θi be the local expenditureshare in the total budget of the governments (the ratio of local governmentexpenditure to combined state and local government expenditure). Let li(j) bethe contribution by the local bureaucrat and ci(j) be that of a central bureaucratin jurisdiction j, respectively. Then (3) implies:

gi(j) = (li(j)−σ θi + ci(j)−σ (1 − θi))1

−σ .12 (6)

In our empirical specification, li(j) represents the fiscal expenditure of thejurisdictional government j in state i per capita, and ci(j) represents the fiscalexpenditure of the state government per capita in state i divided by the numberof jurisdictions (counties) in state i. Each of them is defined by public expendi-ture minus payroll to the public workers divided by θi and 1 − θi respectively torepresent the expenditure per project. This value of net expenditure capturesthe efficiency of the public service, i.e., the values of µ and ν in (2), since a high

11 Washington DC is not included.12 The local bureaucrat transforms the public projects m ∈ [0, θ ] to

∫m∈[0,θ] gt+1(j, m)−σ dm =

∫m∈[0,θ]{µTt(j, m)}−σ dm = (µTt)

−σ∫

m∈[0,θ]dm = (µTt)−σ θ with probability α, and∫

m∈[0,m] gt+1(j, m)−σ dm =∫m∈[0,θ]{νTt(j, m)}−σ dm = (νTt)

−σ∫

m∈[0,θ]dm = (νTt)−σ θ with pro-

bability 1 − α. The same rearrangement applies for the projects by the central government. Substi-tution of the rearranged terms into (3) derives (6).


value of the net expenditure implies efficiency of the jurisdictional intermediatepublic good.

We then consider the following equation:

LGi = ln

⎛

⎝Ji∑

j=1

1J

pi(j)gi(j)−ρ

⎞

⎠

1−ρ

, (7)

where Ji is the number of the jurisdictions in state i. Since the number ofcounties/provinces in each state differs across states, we have to make someadjustment, where pi(j) is a weighting parameter. If we set the number of J asthe lowest common multiple of the number of the jurisdictions in the fifty states,then, setting pi(j) = J/Ji, the equation becomes equivalent to the logarithmicform of (4). However, such a number is too large for a usual econometricsoftware to accommodate. To circumvent this problem, we set J to 1000, anarbitrary high number, and use the following method. First, arrange the numberssuch that j1 � j2 iff|gi(j1) − jgi(j)/Ji| � |gi(j2) − jgi(j)/Ji|, that is, to assign alower number to a jurisdiction which has a value of gi(j) closer to the averageof gi(j)’s in state i. Let xi ≡ J − [J/Ji]Ji, where [J/Ji] is the Gauss number ofJ/Ji. The weight pi(j) satisfies:

(a) pi(j) = [J/Ji] + 1 if 1 � j � xi,(b) pi(j) = [J/Ji] otherwise.That is, the weighting system, which satisfies jpi(j) = J, is to lower the impact

of the jurisdictions further from the average of the jurisdictional expenditurein the state. This is called Case (i). To test the robustness, we also considerthe polar case where condition (a) above is replaced with pi(j) = [J/Ji] + 1 ifJi − xi + 1 � j � Ji, that is, if gi(j) is further from the average of gi(j)’s in state i.This is called Case (ii).

(6) and (7) are substituted into (5) to estimate the production function.We use the nonlinear least squares method. Our estimation result is shown inTable 2.

The null hypothesis of σ = 0 is rejected with 1% significance level. In addi-tion, we test the null hypothesis of σ − ρ = 0 by Wald test. The χ2 (1) statisticsare 7.937 (P value = 0.005) and 7.866 (P value = 0.005), respectively, so thatthe value of σ − ρ is significantly negative. Therefore, the parameter values of

Table 2 Estimation of the production function

Case (i) Case (ii)

Parameter Estimate t-statistic P-value Parameter Estimate t-statistic P-value

σ −1.19521 −4.54942 [0.000] σ −1.19792 −4.5428 [0.000]ρ −0.39793 −3.45696 [0.001] ρ −0.404898 −3.67224 [0.000]β 0.548132 9.41181 [0.000] β 0.548103 9.41356 [0.000]

350 N. Akai et al.

ρ and σ satisfy the assumptions of the proposition.13 In the next section, weempirically verify our theoretical results.

3.2 Growth regression model

To test our theoretical results, a stylized growth regression model is augmentedwith variables representing θ (the degree of fiscal decentralization). We usea panel data set of the 50 states of United States over the period of 1992–1997. The annual growth rate of Gross State Product (GSP) of state i at time t,�GSPit, is regressed upon, along with other control variables, a measure offiscal decentralization, θit. We assume a quadratic functional form �GSPit =a + b1θit + b2(θit)

2. From the proposition and our estimation of the production

function, the estimate of b1 should be positive(

since∂E

[

lnYt+1

Yt

]

∂θ|θ=0 > 0

)

, and

the sign of b2 should be negative(

since∂2E

[ln

Yt+1Yt

]

∂θ2 < 0 for all θ ∈ [0, 1])

, with

0 < −b1/(2b2) < 1(

since∂E

[ln

Yt+1Yt

]

∂θ

∣∣∣∣θ=1

< 0)

.

The indicator of fiscal decentralization is defined in accordance with thefiscal authority that the central/local government has.14 Following the lite-rature, we adopt two indicators of fiscal decentralization: (1) the local ex-penditure share in the total budget of the governments (the ratio of localgovernment expenditure to combined state and local government expenditure);and (2) the local revenue share in the total budget of the governments (the ratioof local government revenue to combined state and local government revenue).Both can correspond to the value of θ in the theoretical model, but they dif-fer in reality because of the intergovernmental grants. The first indicator canrepresent the local governments’ authority when they have an influence overtheir expenditure, that is, the amount of taxes to be collected and the type ofexpenditure to be made, implicitly assuming that all intergovernmental grantsare non-matching or lump-sum grants. In calculating the expenditure share,the measure includes all grants from other governments. The second indicatorrepresents the local governments’ authority when the government collectingrevenue also has authority over its own revenue, that is, the tax amount to becollected and the type of expenditure to be made, assuming that all intergo-vernmental grants are conditional or matching grants. In calculating the revenueshare, we use the ‘own revenue’ that does not include any grants from other

13 There are two additional remarks. First, the value of B is not reported here, since it is not centralto our study. The values are not significant in both cases. Since this is not a dynamic growth analysis,the implication for the Solow residual is not entirely clear. Second, estimations are done using amore generalized model of LGSPi = B+β1LKi+β2LGi+β3LLi. The conclusion of σ < min{ρ, 0}is robust.14 In Hoxby (1999), for example, the centralized system is defined as a system where the statecollects and allocates revenue among districts (p. 21).


governments. Both indicators are commonly used in research on fiscal decen-tralization, for example, Xie, Zou and Davoodi (1999) and Wallis and Oates(1988).15,16

The other control variables are chosen on the basis of previous empiri-cal studies of economic growth (for example, Barro and Sala-i-Martin 1995;Davoodi and Zou 1998). These variables are annual growth rate of the po-pulation (POP), the logarithm of the initial GSP per capita (GSP), the initialeducation level (EDUC), the initial Liberal vs. Conservative tendencies mea-sured by the initial share of the seats in the state legislature held by Democrats(LIB vs. CON), the initial Gini coefficient (GINI), the initial percentage of pa-tents (PATENTS), and the initial percentage of export in GSP (OPENNESS).Detailed descriptions of the data are given in Appendix B. We use explanatoryvariables measured at each initial fiscal year of �GSPit to deal with potentialendogeneity problems, with the exception of POP.

Our growth equation is the following:

�GSPit = a + b1θit + b2(θit)2 + b3POPit + b4GSPit + b5EDUCit

+ b6LIBvsCONit + b7GINIit + b8PATENTSit

+ b9OPENNESSit + εit (8)

To test robustness, we examine the versions that include time and regionaldummies, and exclude some explanatory variables. We also examine the casewhere the dependent variable is the average growth rate of the sample period.The maximum likelihood method is used for estimation.

3.3 Results

The results are shown in Tables 3 and 4. Tables 3 and 4 present the resultswhen we adopt revenue and expenditure respectively as a fiscal decentralizationindicator.

The first column of estimated results is for a model that includes all controlvariables and a constant term. Regardless of the definition of fiscal decentra-lization, the estimated value of b2 is significantly negative and that of b1 issignificantly positive in both tables, with 0 < −b1/(2b2) < 1. The results areconsistent with the proposition. We will examine the value of −b1/(2b2), theestimated value of the optimal degree of fiscal decentralization, in the nextsection.

The second column shows the regression result with only a fixed time effect.The introduction of a fixed time effect does not affect the results that b2 is

15 Xie, Zou and Davoodi (1999) use the expenditure share as an indicator of fiscal decentrali-zation, while Wallis and Oates (1988) use expenditure and revenue shares as indicators of fiscaldecentralization. See also Akai and Sakata (2002, pp. 97–98), for a recent empirical analysis.16 The result of the production function estimation in Sect. 3.1 is invariant when we use the revenueindicator for θi under several alternative specifications.

352 N. Akai et al.Ta

ble

3R

egre

ssio

nre

sult

sfo

rth

ere

venu

ein

dica

tor

Dep

.ver

:ann

ualg

row

thD

ep.v

er:a

vera

gegr

owth

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Con

stan

t−0

.02

−0.0

01−0

.01

−0.0

3−0

.03

−0.0

05−0

.01

0.01

[1.6

1][0

.11]

[1.5

4][1

.64]

[2.3

6][0

.36]

[1.1

5][0

.55]

Dec

entr

aliz

atio

n0.

150.

120.

140.

120.

140.

100.

120.

24[3

.09]

***

[2.9

4]**

*[3

.84]

***

[2.3

0]**

[3.0

6]**

*[2

.29]

**[3

.05]

***

[2.1

0]**

Dec

entr

aliz

atio

n2−0

.17

−0.1

4−0

.15

−0.1

4−0

.15

−0.1

1−0

.13

−0.2

5[2

.68]

***

[2.4

4]**

[3.1

1]**

*[2

.09]

**[2

.63]

***

[1.9

7]**

[2.5

2]**

[1.5

5]P

OP

0.08

0.06

0.10

0.35

0.34

0.28

0.30

0.11

[0.4

7][0

.48]

[0.8

9][1

.44]

[1.6

0][1

.54]

[1.7

7]*

[0.2

7]G

SP−0

.000

1−0

.000

3−0

.000

20.

0000

0.00

01−0

.000

1−0

.000

2−0

.001

[0.7

7][2

.21]

**[2

.05]

**[0

.25]

[0.4

6][0

.95]

[1.0

6][3

.05]

***

ED

UC

0.08

0.02

0.06

0.15

0.16

0.08

0.08

−0.0

4[1

.35]

[0.3

9][1

.67]

*[1

.99]

**[2

.29]

**[1

.16]

[1.3

7][0

.40]

LIB

vers

usC

ON

−0.0

02−0

.01

–−0

.003

–−0

.01

––

[0.4

3][1

.50]

–[0

.62]

–[1

.37]

––

GIN

I0.

01−0

.02

–−0

.01

–−0

.03

––

[0.3

6][0

.83]

–[0

.16]

–[1

.00]

––

Pat

ents

−0.0

10.

002

–0.

01–

0.02

––

[0.4

1][0

.11]

–[0

.50]

–[0

.75]

––

Ope

nnes

s0.

01−0

.004

–0.

01–

−0.0

03–

–[0

.62]

[0.3

2]–

[0.4

2]–

[0.0

2]–

–L

oglik

elih

ood

988.

411,

045.

761,

044.

2699

5.37

994.

911,

052.

821,

051.

2117

1.11

Opt

imal

valu

eof

fisca

ldec

entr

aliz

atio

n0.

440.

430.

470.

430.

470.

450.

460.

48V

aria

bles

incl

uded

ingr

owth

equa

tion

:R

egio

nalfi

xed

effe

cts

No

No

No

Yes

Yes

Yes

Yes

No

Tim

efix

edef

fect

sN

oY

esY

esN

oN

oY

esY

esN

o

Figu

res

inpa

rent

hese

sar

eth

eab

solu

teva

lues

oft-

stat

isti

cs.A

ster

isks

indi

cate

vari

able

sw

hose

coef

ficie

nts

are

sign

ifica

ntat

the

10%

(*),

5%(*

*)an

d1%

(***

)le

vels

,res

pect

ivel

y.T

hesa

mpl

esi

zeis

300

or50

.Due

tolim

its

onsp

ace,

we

dono

tre

port

the

resu

lts

for

the

esti

mat

edco

effic

ient

sof

the

indi

vidu

aldu

mm

yva

riab

les

inth

eta

ble.

We

pres

ent

fifty

stat

esdi

vide

din

toei

ght

area

sas

follo

ws.

New

Eng

land

=C

onne

ctic

ut,M

aine

,Mas

sach

uset

ts,N

ewH

amps

hire

,Rho

deIs

land

,Ver

mon

t;M

idea

st=

Del

awar

e,M

aryl

and,

New

Jers

ey,N

ewY

ork,

Pen

nsyl

vani

a;G

reat

Lak

es=

Illin

ois,

Indi

ana,

Mic

higa

n,O

hio;

Pla

ins

=Io

wa,

Kan

sas,

Min

neso

ta,M

isso

uri,

Neb

rask

a,N

orth

Dak

ota,

Sout

hD

akot

a,W

isco

nsin

;Sou

thea

st=

Ala

bam

a,A

rkan

sas,

Flo

rida

,Geo

rgia

,Mis

siss

ippi

,Ken

tuck

y,L

ouis

iana

,N

orth

Car

olin

a,O

klah

oma,

Sout

hC

arol

ina,

Tenn

esse

e,V

irgi

nia;

Sout

hwes

t=

Ari

zona

,N

ewM

exic

o,Te

xas;

Roc

kyM

ount

ian

=C

olor

ado,

Idah

o,M

onta

na,

Uta

h,W

yom

ing;

Far

Wes

t=A

lask

a,C

alif

orni

a,H

awai

i,N

evad

a,O

rego

n,W

ashi

ngto

n

Complementarity, fiscal decentralization and economic growth 353Ta

ble

4R

egre

ssio

nre

sult

sfo

rth

eex

pend

itur

ein

dica

tor

Dep

.ver

:ann

ualg

row

thD

ep.v

er:a

vera

gegr

owth

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Con

stan

t−0

.05

−0.0

2−0

.03

−0.0

4−0

.05

−0.0

2−0

.03

−0.0

16[2

.61]

***

[1.2

6][1

.99]

**[2

.16]

**[2

.72]

***

[0.8

7][1

.48]

[0.5

9]D

ecen

tral

izat

ion

0.26

0.21

0.22

0.19

0.20

0.15

0.17

0.36

[3.3

3]**

*[2

.90]

***

[3.1

1]**

*[2

.26]

**[2

.65]

***

[1.9

4]*

[2.4

0]**

[2.8

3]**

*D

ecen

tral

izat

ion2

−0.2

7−0

.22

−0.2

3−0

.19

−0.2

0−0

.15

−0.1

7−0

.34

[2.9

6]**

*[2

.58]

***

[2.7

7]**

*[1

.92]

*[2

.24]

**[1

.64]

*[2

.03]

**[2

.36]

**P

OP

0.03

0.02

0.02

0.22

0.21

0.20

0.19

−0.1

0[0

.17]

[0.1

5][0

.20]

[0.9

2][0

.98]

[1.0

7][1

.11]

[0.2

8]G

SP−0

.000

1−0

.000

2−0

.000

30.

0000

0.00

00−0

.000

2−0

.000

2−0

.001

[0.5

7][2

.15]

**[2

.57]

***

[0.0

1][0

.05]

[1.3

1][1

.81]

*[3

.25]

***

ED

UC

0.07

0.01

0.03

0.13

0.13

0.05

0.05

−0.1

2[1

.27]

[0.2

2][0

.91]

[1.7

1]*

[1.7

7]*

[0.7

7][0

.74]

[1.5

0]L

IBve

rsus

CO

N0.

001

−0.0

03–

0.00

0–

−0.0

04–

–[0

.25]

[0.9

4]–

[0.0

9]–

[0.8

6]–

–G

INI

−0.0

1−0

.03

–−0

.01

–−0

.03

––

[0.2

2][1

.53]

–[0

.31]

–[1

.19]

––

Pat

ents

−0.0

10.

01–

0.00

5–

0.02

––

[0.4

0][0

.27]

–[0

.18]

–[0

.69]

––

Ope

nnes

s0.

004

−0.0

1–

0.00

3–

−0.0

04–

–[0

.27]

[0.5

8]–

[0.2

1]–

[0.3

0]–

–L

ogL

ikel

ihoo

d99

2.63

1,04

7.79

1,04

6.36

998.

0699

7.97

1,05

4.06

1,05

2.90

174.

50O

ptim

alva

lue

offis

cald

ecen

tral

izat

ion

0.48

0.48

0.48

0.50

0.50

0.50

0.50

0.53

Var

iabl

esin

clud

edin

grow

theq

uati

on:

Reg

iona

lfixe

def

fect

sN

oN

oN

oY

esY

esY

esY

esN

oT

ime

fixed

effe

cts

No

Yes

Yes

No

No

Yes

Yes

No

Figu

res

inpa

rent

hese

sar

eth

eab

solu

teva

lues

oft-

stat

isti

cs.A

ster

isks

indi

cate

vari

able

sw

hose

coef

ficie

nts

are

sign

ifica

ntat

the

10%

(*),

5%(*

*)an

d1%

(***

)le

vels

,res

pect

ivel

y.T

hesa

mpl

esi

zeis

300

or50

.Due

tolim

its

onsp

ace,

we

dono

tre

port

the

resu

lts

for

the

esti

mat

edco

effic

ient

sof

the

indi

vidu

aldu

mm

yva

riab

les

inth

eta

ble.

We

pres

ent

fifty

stat

esdi

vide

din

toei

ght

area

sas

follo

ws.

New

Eng

land

=C

onne

ctic

ut,M

aine

,Mas

sach

uset

ts,N

ewH

amps

hire

,Rho

deIs

land

,Ver

mon

t;M

idea

st=

Del

awar

e,M

aryl

and,

New

Jers

ey,N

ewY

ork,

Pen

nsyl

vani

a;G

reat

Lak

es=

Illin

ois,

Indi

ana,

Mic

higa

n,O

hio;

Pla

ins

=Io

wa,

Kan

sas,

Min

neso

ta,M

isso

uri,

Neb

rask

a,N

orth

Dak

ota,

Sout

hD

akot

a,W

isco

nsin

;Sou

thea

st=

Ala

bam

a,A

rkan

sas,

Flo

rida

,Geo

rgia

,Mis

siss

ippi

,Ken

tuck

y,L

ouis

iana

,N

orth

Car

olin

a,O

klah

oma,

Sout

hC

arol

ina,

Tenn

esse

e,V

irgi

nia;

Sout

hwes

t=

Ari

zona

,N

ewM

exic

o,Te

xas;

Roc

kyM

ount

ian

=C

olor

ado,

Idah

o,M

onta

na,

Uta

h,W

yom

ing;

Far

Wes

t=A

lask

a,C

alif

orni

a,H

awai

i,N

evad

a,O

rego

n,W

ashi

ngto

n

354 N. Akai et al.

significantly negative and b1 is significantly positive. To examine the robustnessof the results in the second column, we exclude four control variables, LIBversus CON, GINI, PATENTS and OPENNESS in the third column. Even inthis case, similar results hold. We next introduce a regional fixed effect.17 Wedivide the fifty states into eight different geographical areas (New England,Mideast, Great Lakes, Plains, Southeast, Southwest, Rocky Mountain, and FarWest; see the description in the table for a detailed explanation). The fourthand fifth columns present the results with only a fixed regional effect, and thesixth and seventh columns show the results with both fixed regional and timeeffects. The coefficients of b1 and b2 show the same statistical properties as inthe previous cases.

The eighth column shows the result where the dependent variable is theaverage growth rate of the sample period, and all explanatory variables arethose of 1992. Except for the fact that the significance of b2 in Table 3 is weaker,our result is robust.

Next, turning to other characteristics of coefficient estimates, some conclu-sions emerge from the results in Tables 3 and 4. First, we have controlledthe quality of regional human capital by using two variables (PATENTS andEDUC). EDUC has the expected positive effect, which is statistically significantin several regressions, while the effect of PATENTS is not. Thus we evaluatethe results that a higher level of education enhances economic growth in theregion. Second, the estimated coefficient of GSP is negative and significant inregression models (2), (3), and (8) in Table 3, and (2), (3), (7) and (8) in Table 4,which is consistent with a conventional convergence property.

In this section, we examine our theoretical results with the use of the conven-tional growth regression. Consistent with our theoretical result applied to theestimated parameters of σ and ρ which are statistically significant, we showconvincing evidence of a hump-shaped relationship between fiscal decentrali-zation and economic growth. This result is robust when we use the revenue andexpenditure indicator as a variable of fiscal decentralization.

3.4 Optimal degree of decentralization

In our model, the growth of the GSPit is maximized when θit = −b1/(2b2).The values of −b1/(2b2) are 43–48 % in the case of the revenue indicator, and48–53% in the case of the expenditure indicator. An intriguing issue is whetherthey are higher or lower than the actual data. The average of the data is 0.38in the case of the revenue indicator (that is, on average, thirty eight percent ofthe total government revenue goes to the local governments) and 0.44 in thecase of the expenditure indicator. The hypothesis that the optimal degree of

17 It should be noted that a state fixed effect is highly correlated to fiscal decentralization measurein each state because there is little variation of fiscal decentralization measure across time. Thiscreates a multicolinearity problem, which causes distortion in estimating the real effect of fiscaldecentralization. In order to avoid this problem, we adopt regional dummies, instead of statedummies.


Table 5 Optimality of fiscal decentralization

Model (1) (2) (3) (4) (5) (6) (7) (8)

Revenue indicator 3.451 3.178 4.130 1.961 2.941 2.091 3.089 1.006[0.06]* [0.07]* [0.04]** [0.16] [0.09]* [0.15] [0.08]* [0.32]

Expenditure indicator 2.424 2.051 2.406 1.468 1.970 1.156 1.694 1.711[0.12] [0.15] [0.12] [0.23] [0.16] [0.28] [0.19] [0.19]

Null hypothesis is defined as: optimal value = average value. Figures in parentheses are p-values

fiscal decentralization is significantly higher than the data average is tested bythe statistical significance of the null hypothesis of −b1/(2b2) = the averageof the data, i.e., the existing spending shares (or the revenue shares) for localgovernments have been consistent with growth maximization. We use the Waldstatistics of the null hypothesis.18 Results are shown in Table 5. In the caseof the revenue indicator, the null hypothesis is rejected in case (3) with 5 %significance level, and cases (1), (2), (5), and (7) with 10% significance level. Inthe case of the expenditure indicator, the null hypothesis cannot be rejected.

Our results show that further revenue decentralization is conducive to eco-nomic growth, but the expenditure share is consistent with maximal economicgrowth. Since revenue is less decentralized than expenditure on the average ofthe data, our results also suggest mitigating the divergence of expenditure andrevenue assignments for higher economic growth.

4 Conclusion

The effect of fiscal decentralization on economic growth has been a majorfocus of debate and discussion in the context of recent public reforms. Ithas been suggested that some countries succeed with fiscal decentralizationand others do not. The empirical works regarding whether there is positive ornegative relationship between economic growth and fiscal decentralization arestill controversial. In this paper, we approach this issue from a different anglethan the previous literature. Taking account of complementarity in global andlocal levels, we found that, in general, the relationship between fiscal decen-tralization and economic growth is not linear, so that the economic growthimplication of fiscal decentralization depends on the structure of complemen-tarity. Using data for the United States, we first show that our specification ofthe production function fits well. From our growth regression model, we obser-ved a hump-shaped relationship between fiscal decentralization and economicgrowth, which is consistent with our theoretical result applied to the estimatedparameters. We also examined whether the optimal level of fiscal decentrali-zation derived by the data is higher or lower than that of the actual data. In

18 The Wald statistics = (−b1/(2b2)-the average of the data)2/V̂(−b1/(2b2)), where V̂(−b1/(2b2))

is a root-n consistent estimator of the variance obtained by the delta method (e.g., Greene (1993),p. 297).

356 N. Akai et al.

some cases, the optimal degree of fiscal decentralization conducive to economicgrowth is higher than the average of the data, and hence further decentralizationis recommended for economic growth.

As far as we know, no theoretical work discusses the hump-shaped relation-ship between them with a formal model that identifies the reason. Thiessen’s(2003) empirical work shows the hump-shaped relationship between fiscaldecentralization and economic growth in the group of high-income OECDcountries, but he does not identify the reason for the humped shape. Giventhis background, this paper is the first work which discusses the hump-shapedrelationship between fiscal decentralization and economic growth theoretically,with consistent empirical analyses. We first provided a theoretical model thatexplains a relationship between fiscal decentralization and economic growth,extending a conventional macroeconomic model with central and local govern-mental sectors. Second, we estimated the production function consistent withtheoretical hypothesis. And finally, we examined whether the current level offiscal decentralization is optimal or not. Given that existing theories cannotexplain the hump-shapedness, and that the comprehensive analysis regardingthis issue has not existed yet, our paper has provided persuasive explanationstheoretically and empirically. The macroeconomic implications of fiscal de-centralization depend on the complementarity structures (Table 1). Furtheranalyses with different data sets may complement our analyses.

Appendix A

In order to prove the proposition, we first derive the market clearing conditions.There are two markets in this economy—labor and capital. As in the conventio-nal model, the goods market is complementary to the capital market. First, thelabor market equilibrium condition is met where the competitive firms’ labordemand equals (exogenous) supply of labor. This is equivalent to the followingrelationship between the wage rate wt, the national output Yt and the laborsupply Lt:19

wt = βAK1−βt Gβ

t Lβ−1t = β

Yt

Lt(A.1)

We now show the condition for capital (goods) market equilibrium, followingthe conventional models (e.g., Barro and Sala-i-Martin (1995, Eqs. (3.A.10)–(3.A.12))):Lemma The condition for the capital market equilibrium implies:

Kt+1 = aβ(1 − τ)Yt, (A.2)

where a is a fraction of the net income that is saved.

19 The expression of competitive firms’ capital demand, which determines the interest rate rt , isomitted, since this condition is not used in the following derivation.


Proof of the Lemma The condition for the capital market equilibrium (or thegoods market equilibrium) is where aggregate net investment equals total netincome minus total consumption. Tax revenue is used for public projects, sothat there is no net savings from the government sector. Formally, let Ct andDt denote the consumption of the young and old, respectively, and let rt be theinterest rate. Then:

Kt+1 − Kt = (1 − τ)wtLt + rtKt − CtLt − DtLt−1.

By an assumption of constant fraction of savings, (1− τ)wt −Ct = a(1− τ)wtand Dt = (1 + rt)a(1 − τ)wt−1. The above equation is rearranged to:

Kt+1 = a(1 − τ)wtLt + (1 + rt)(Kt − a(1 − τ)wt−1Lt−1).

At the beginning, initial capital stock K1 is owned by L0, so that D1L0 =(1 + r1)K1 [Barro and Sala-i-Martin (1995, p. 120, footnote 28)]. Since D1L0 =(1+r1)a(1−τ)w0L0, the difference equation above implies Kt+1 = a(1−τ)wtLt.Combining with (A.1), we derive (A.2). ��

Proof of the Proposition Let n (0 � n � J) be the number of successful juris-dictions (those who draw µ). This is a random variable which follows a binomial

distribution with probability y Pr(n|J) ≡(

Jn

)

αn(1 − α)J−n.

Let Tt be the tax revenue per jurisdiction at period t. Recall that tasksand jurisdictions are symmetric, so that an equal proportion of tax revenueis allocated to each project: Tt(j, m) = Tt for all j and m. The local bureau-crat transforms the public projects m ∈ [0, θ ] to

∫m∈[0,θ] gt+1(j, m)−σ dm =

∫m∈[0,θ]{µTt(j, m)}−σ dm = (µTt)

−σ∫

m∈[0,θ]dm = (µTt)−σ θ with probability

α, and∫

m∈[0,m] gt+1(j, m)−σ dm =∫m∈[0,θ]{νTt(j, m)}−σ dm = (νTt)

−σ∫

m∈[0,θ]dm = (νTt)

−σ θ with probability 1 − α. The same rearrangement applies forthe projects undertaken by the central government. Recall that labor incometaxation takes place [so that the tax revenue = τwtLt = τβYt by (A.1)].Recall also that the jurisdictions are symmetric, so that Tt = τβYt/J.Substituting these expressions into (3) and (4), we now describe Gt+1 by theability of the central bureaucrat (µ or ν) and the number of successful juris-dictions (n), as follows. When the central bureaucrat has a high ability (whichyields gt+1(j, m) = µTt(j, m) for m ∈ [θ , 1]) and the number of successful localbureaucrats is n(0 � n � J),

Gt+1 ={

nJ(µ−σ θ + µ−σ (1 − θ))

ρσ + J − n

J(ν−σ θ + µ−σ (1 − θ))

ρσ

} 1−ρ

τβYt

J.

(A.3a)

358 N. Akai et al.

When the central bureaucrat has a low ability (which yields gt+1(j, m)=νTt(j, m)

for m ∈ [θ , 1]) and the number of successful local bureaucrats is n (0 � n � J),

Gt+1 ={

nJ(µ−σ θ + ν−σ (1 − θ))

ρσ + J − n

J(ν−σ θ + ν−σ (1 − θ))

ρσ

} 1−ρ

τβYt

J.

(A.3b)

Notice that any realization of n(0 � n � J) is a-priori possible.Substituting (A.2), into (1),

Yt+1 = A(aβ(1 − τ)Yt)1−β(Gt+1Lt+1)

β .

Substituting (A.3a) and (A.3b) into the above equation, for c = µ, ν and 0 �n � J, we obtain:

lnYt+1

Yt= k + β ln γ c

t+1(n),

where k = ln

{

Aaβ(1 − τ)1−β(ν

τβJ Lt+1

)β}

, and γ ct+1(n)(c = µ, ν, 0 � n � J) is

a random variable such that:

γ ct+1(n) = γ

µ

t+1(n) ≡ ln

(nJι−ρ + J − n

J(θ + ι−σ (1 − θ))

ρσ

) 1−ρ

with probability Pr(µ, n) ≡ α Pr(n|J),

γ ct+1(n) = γ ν

t+1(n) ≡ ln

(nJ(ι−σ θ + (1 − θ))

ρσ + J − n

J

) 1−ρ

with probability Pr(ν, n) ≡ (1 − α) Pr(n|J),

withι ≡ µ

ν> 1.

Taking expectation of the above expression (i.e., multiplying both sides byPr(c, n) and summing up with respect to (c, n)),

∑

c,n

[

lnYt+1

Yt

]

Pr(c, n) = k + β∑

nln γ

µ

t+1(n)α Pr(n|J)

+β∑

nln γ ν

t+1(n)(1 − α) Pr(n|J).

Or, rewriting the expression:

E[

lnYt+1

Yt

]

= k + αβ

−ρEn

[

ln

(nJι−ρ + J − n

J(θ + ι−σ (1 − θ))

ρσ

)]

+ (1 − α)β

−ρEn

[

ln

(nJ(ι−σ θ + (1 − θ))

ρσ + J − n

J

)]

(A.4)


where En is the expectation operator with respect to the binomial distribution

Pr(n|J) =(

Jn

)

αn(1 − α)J−n, i.e., En[ln γ ct+1(n)] ≡ ∑

n ln γ ct+1(n) Pr(n|J) for

c = µ, ν. Taking the derivative with respect to θ :

∂E[ln Yt+1

Yt

]

∂θ= α

β

−σEn

[− J−nJ (ι−σ − 1)(θ + ι−σ (1 − θ))

ρσ

−1

nJ ι−ρ + J−n

J (θ + ι−σ (1 − θ))ρσ

]

+ (1 − α)β

−σEn

[−nJ (1 − ι−σ )(ι−σ θ + (1 − θ))

ρσ

−1

nJ (ι−σ θ + (1 − θ))

ρσ + J−n

J

]

. (A.5)

Evaluated at θ = 0,

∂E[ln Yt+1

Yt

]

∂θ

∣∣∣∣∣∣θ=0

= αβ

−σEn

[− J−nJ (ι−σ − 1)ι−ρ+σ

ι−ρ

]

+ (1 − α)β

−σEn

[−n

J(1 − ι−σ )

]

= β

−σEn

[

−αJ − n

J(ι−σ − 1)ισ − (1 − α)

nJ(1 − ι−σ )

]

= β

−σα(1 − α)(ι−σ − 1)(1 − ισ ) > 0.

Notice that En

[nJ

]= α, ι > 1 and σ < 0.

Also, evaluated at θ=1,

∂E[ln Yt+1

Yt

]

∂θ

∣∣∣∣∣∣θ=1

= β

σEn

[α J−n

J (ι−σ − 1) + (1 − α)nJ (1 − ι−σ )ι−ρ+σ

nJ ι−ρ + J−n

J

]

.

We first prove that this value is negative when ρ is negative. Let

A(n) = α J−nJ (ι−σ −1)+(1−α)n

J (1− ι−σ )ι−ρ+σ and B(n) =(

nJ ι−ρ + J−n

J

)−1.

Then∂E

[ln

Yt+1Yt

]

∂θ

∣∣∣∣θ=1

= βσ{Covn(A(n), B(n)) + En[A(n)]En[B(n)]}, where Cov

denotes covariance. By supposition, both A(n) and B(n) are decreasingfunctions of n,20 so that the covariance is positive. Also, En[A(n)] = α(1 − α)

(ι−σ − 1)(1 − ι−ρ+σ ) > 0 and En[B(n)] > 0, so that∂E

[ln

Yt+1Yt

]

∂θ

∣∣∣∣θ=1

< 0.

20 ∂A(n)∂n = − 1

J (ι−σ − 1)(α + (1 − α)ι−ρ+σ ) < 0 (since ι−σ > 1) and ∂B(n)∂n =

−(

nJ ι−ρ + J−n

J

)−2 1J(ι−ρ − 1

)< 0 (since ι−ρ > 1).

360 N. Akai et al.

Also,

∂2E[ln Yt+1

Yt

]

∂θ2 = −αβ

−σEn

⎡

⎢⎣

(J−n

J (ι−σ − 1)(θ + ι−σ (1 − θ))ρσ

−1)2

(nJ ι−ρ + J−n

J (θ + ι−σ (1 − θ))ρσ

)2

⎤

⎥⎦

− (1 − α)β

−σEn

⎡

⎢⎣

(nJ (1 − ι−σ )(ι−σ θ + (1 − θ))

ρσ

−1)2

(nJ (ι−σ θ + (1 − θ))

ρσ + J−n

J

)2

⎤

⎥⎦

+αβ

−σ

(ρ

σ− 1

)En

[J−n

J (ι−σ − 1)2(θ + ι−σ (1 − θ))ρσ

−2

nJ ι−ρ + J−n

J (θ + ι−σ (1 − θ))ρσ

]

+ (1 − α)β

−σ

(ρ

σ− 1

)En

[nJ (1 − ι−σ )2(ι−σ θ + (1 − θ))

ρσ

−2

nJ (ι−σ θ + (1 − θ))

ρσ + J−n

J

]

.

The supposition of σ < min{ρ, 0} implies that the expression is negative forall θ .

Finally, when ρ > 0, E[ln Yt+1

Yt

]∣∣∣θ=0

> E[ln Yt+1

Yt

]∣∣∣θ=1

(Nishimura 2006).21

To be consistent with∂E

[ln

Yt+1Yt

]

∂θ

∣∣∣∣θ=0

> 0 and∂2E

[ln

Yt+1Yt

]

∂θ2 < 0 for all θ ∈ [0, 1], it

has to be∂E

[ln

Yt+1Yt

]

∂θ

∣∣∣∣θ=1

< 0.

Inspection of (A.5) shows that, when σ = 0,∂E

[ln

Yt+1Yt

]

∂θ� 0(

resp.∂E

[ln

Yt+1Yt

]

∂θ� 0

)

if ρ < 0 (resp. ρ > 0). Such relationship continues to

hold if σ is sufficiently close to 0. Also,∂2E

[ln

Yt+1Yt

]

∂θ2 > 0 for all θ ∈ [0, 1] whenσ > max{ρ, 0} (complementary to the case of the proposition). Combined with

Nishimura’s (2006) result that E[ln Yt+1

Yt

]∣∣∣∣θ=0

≷ E[ln Yt+1

Yt

] ∣∣∣∣θ=1

iff ρ><0, we de-

rive the relationship between fiscal decentralization and economic growth inTable 1.

21 θ = 0 corresponds to the centralized regime, and θ = 1 corresponds to the decentralized regimeof Nishimura (2006). Notice that the sign of ρ in (4) is opposite to Nishimura’s (2006) Eq. (5).


Appendix B Variable definitions, means, and standard deviations

Variable Mean Standard deviation Definition

�GSP 0.0134 0.0094 Average annual growth rate of real GSPper capita over the 1992–1998 period

POP 0.0047 0.0040 The lag of average annul growth rate ofstate population

GSP 24.1878 4.3925 Per capita real GSPEDUC 0.0950 0.0137 Percentage of high school graduates in

total population aged 18–24 yearsLIB versus CON 0.5419 0.1582 The share of seats in state legislature held

by DemocratsGINI 0.0957 0.0233 Income difference between counties

within a statePATENTS 0.0203 0.0290 The state’s share of total US patents

issuedOPENNESS 0.0588 0.0367 Ratio of states’s exports to other countries

and other states to nominal GSPIndicators of fiscaldecentralization

Expenditure indicator 0.4402 0.0730 Ratio of local government expenditure tostate and local government expenditure

Revenue indicator 0.3820 0.0824 Ratio of local government revenue tostate and local government revenue

All variables except for �GSP and POP relate to 1992, 93, 94, 95, 96, 97 each periodSources: USA COUNTIES 1998 (CD-ROM), Statistical Abstract of United States, and States andMetropolitan Area Data Book 1997–98 (CD-ROM)

References

Akai N, Sakata M (2002) Fiscal decentralization contributes to Economic growth: evidence fromstate-level cross-section data for the United States. J Urban Econ 52:93–108

Bardhan P (2002) Decentralization of Governance and Development. J Econ Perspect 16:185–205Barro RJ (1990) Government spending in a simple model of endogenous Growth. J Polit Econ

98:S103-S125Barro RJ, Sala-i-Martin X (1995) Economic Growth. McGraw-Hill, New YorkBénabou R (1996) Heterogeneity, stratification, and growth: macroeconomic implications of com-

munity structure and school finance. Am Econ Rev 86:584–609Conybeare J, Murdoch JC, Sandler T (1994) Alternative collective-goods models of military

alliances: theory and evidence. Econ Inquiry 32:525–542Cornes R (1993) Dyke maintenance and other stories: some neglected types of public goods.

Q J Econ 108:259–271Cornes R, Sandler T (1996) The theory of externalities, public goods. and club goods, 2nd edn.

Cambridge University Press, New YorkCremer J, Estache A, Seabright P (1996) Decentralizing public services: what can we learn from

the theory of the firm? Revue d’Economie Politique 106:37–60Crewson PE (1995) A comparative analysis of public and private sector entrant Quality. Am J Polit

Sci 39:628–639Davoodi H, Zou HF (1998) Fiscal decentralization and economic growth: a cross-country study.

J Urban Econ 43:244–257Ebel RD, Yilmaz S (2002) On the measurement and impact of fiscal decentralization. World bank

policy research working paper no. 2809Greene WH (1993) Econometric Analysis. Macmillan, New York

362 N. Akai et al.

Hirshleifer J (1983) From weakest-link to best-shot: the voluntary provision of public goods. PublChoice 41:371–386

Hoxby M (1999) The productivity of schools and other local public goods producers. J Publ Econ74:1–30

Lin JY, Liu Z (2000) Fiscal decentralization and economic growth in China. Econ Cult Change49:1–21

Matsuyama K (1995) Complementarity and cumulative process in models of monopolistic compe-tition. J Econ Lit 33:701–729

Montinola G, Qian Y, Weingast BR (1996) Federalism, Chinese style: the political basis for economicsuccess. World Polit 48:50–81

Nishimura Y (2006) Human fallibility, complementarity and fiscal decentralization. J Publ EconTheory 8:487–501

Prud’homme R (1995) The dangers of decentralization. World Bank Res Observ 10:201–220Sah RK (1991) Fallibility in human organizations and political systems. J Econ Perspect 5:67–88Sato M, Yamashige S (2005) Decentralization and economic development: an evolutionary

approach. J Publ Econ Theory 7:497–520Thiessen U (2003) Fiscal decentralization and economic growth in high-income OECD countries.

Fiscal Stud 24:237–274Wallis JJ, Oates WE (1988) Decentralization in the public sector: local government. In: Harvey S,

Rosen H (eds) Fiscal federalism: quantitative studies. The University of Chicago Press, Chicagoand London

Xie D, Zou HF, Davoodi H (1999) Fiscal decentralization and economic growth in the UnitedStates. J Urban Econ 45:228–239

Zhang T, Zou HF (1998) Fiscal decentralization, public spending, and economic growth in China.J Publ Econ 67:221–240

akai sakata (2007) - complementarity, fiscal decentralization and economic growth

Documents

economic performance

queens university

yokohamanational university

university of california

university of hyogo

areas of economic theory

osaka international

yokohama national university