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Regional Redistribution and Growth

Joan Rossell1

Universitat Illes BalearsDept. of Economics

Ctra. Valldemossa, 7.5 km

Palma de Mallorca. Spain.E-mail: joan.rossello@uib.es

october 2001

1 I wish to thank Roland Bnabou, Jordi Gal, the participants at theMacroworkshops at NYU, Columbia University and Harvard University and par-ticularly to Xavier Sala-i-Martin, for their comments and suggestions. I also thanktwo anonymous referees for their comments. The usual disclaimer applies.

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Abstract

In the last twenty years, Interregional Transfers in the form of public infras-

tructures have been used as an instrument to reduce regional disparities inoutput levels. In this paper we provide theoretical foundation on the eectsof such transfers on both regional output and regional welfare disparities. Weconduct our a

hen increasing redistribution areduction in regional output disparities might be compatible with a loss intotal welfare, even in the recipient. This result is due to the fact that suchtransfers force a reallocation of factors of production towards those regionswith lower productivities and this has a negative eect on the growth rate ofthe economy.

JEL: H54, H73, R11, R53.Keywords: Interregional Transfers, Endogenous Growth, Public Investment.Palabras Clave: Transferencias Interregionales, Crecimiento Endgeno, In-versin Pblica

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activity (contributing to avoid the concentration of economic activity) and

that such a policy has not contributed to economic cohesion (which is takenas a measure of welfare) because they observe a polarization in the levels ofregional incomes.

In the following lines we provide two interesting results. First, it is shownthat Interregional Transfers through public investment allow to reduce dier-ences in regional per capita output levels and allow to avoid the concentrationof economic activity in that region with higher productivities. Second, weshow that if the level of redistribution is to be decided according to a criteriabased in the reduction of the regional dierences in output levels, the policymaker could choose a level of redistribution for which agents in the poorerregions could be worse o compared to a policy that xed a lower level ofredistribution. This is due to the fact that redistribution has negative eectson the returns on assets and on the growth rate of the economy that mayoutweigh the increase in equilibrium wages in the poorer regions.

From the second result we derive a policy implication that is quite relevantin those economies that are currently decentralizing their political systems,such as Spain or Italy1 . In these economies there is an important debateon the level of regional redistribution that is to be maintained after decen-tralization. Regional Governments have received many responsibilities onpublic expenditure and they are demanding to control scal revenues, thatstill belong to the Central Governments. The problem is that there are sig-

nicant dierences in regional scal capacities and regions have not reachedany agreement on any criteria for Regional Redistribution. In this paperwe shed some light on the eects of a specic type of interregional transfers,those that take the form of public investment. Regional Redistribution basedon personal transfers is beyond the scope of our analysis because it does notbelong to the framework of analysis of Regional Policy.

We must point out also that we do not introduce political economy consid-erations explicitly. Extending our analysis to a political economy approachwould be quite interesting. This extension would be rather relevant giventhat we obtain that for some values of regional redistribution there are somelosses in aggregate welfare levels. In case that redistribution yields winners

and losers it makes sense to study regional redistribution as the outcome ofa voting process or as the policy that maximizes a social welfare function.However, introducing those considerations in a dynamic set up makes ourmodel cumbersome.

1 The same type of argument applies at the European Union, in which there is a bigdebate over the optimal level of interregional redistribution to be implemented in order toreduce the costs of transition of the new incumbents.

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2 The literature

There is a large literature that analyses the eects of interregional redistribu-tive policies. The literature on Fiscal Federalism has extensively studied thedesign of instruments for Regional Redistribution to correct those distortionsthat are due to the existence of scal externalities (see for instance Flatters,Henderson and Mieszkowski (1974), Wildasin (1983), Dahlby and Wilson(1994), Dahlby (1996) and Burbidge and Myers (1994), among others). Ac-cording to this approach, Interregional Transfers might be used to achievean ecient allocation of public resources. However, in those papers FederalGovernment intervention is assumed to be through personal transfers thataect the agents budget constraints rather than through the provision of

productive public services.Our contribution to the Fiscal Federalism literature is that we use a

model of Endogenous Growth in which a Central Government redistributesresources between regions by providing public services that enter the produc-tion function. This allows us to study the eects of Regional Redistributionon the level of gross product, private capital accumulation, consumption andwelfare. Another contribution to this literature is that we show, contrary towhat expected, that under certain conditions even the regions that receivethe transfers could be worse o in terms of welfare.

Barro (1990) and Barro and Sala-i-Martin (1992, 1995) have included

governments provision of productive public services in an endogenous growthmodel. Their focus is on the relationship among productive governmentservices, the distortionary taxes levied to pay for them and long run growthin a closed economy. Additionally, Clarida (1993) shows that, in small openeconomies, international aid that reduces the rental cost of private capitalemployed in the production of public capital allows to increase the optimalrate of public investment, the equilibrium rate of private investment and thespeed of convergence to the steady state.

Nevertheless, Regional Redistribution is not considered in any of the pre-vious papers. Barro (1990) and Barro and Sala-i-Martin (1992, 1995) con-sider a single region and in Claridas paper the provision of public expen-

diture is nanced through international aid in the form of lump sum grantsthat are exogenous to the model and therefore they do not have any eco-nomic cost for the recipient. Our paper goes one step further. We introduceseveral regions and some of them receive additional public services nancedthrough Interregional Transfers. This establishes a scal link between thedonor and the recipient. We show that redistribution introduces distortionson the donor and that these eects spread to the recipient through scal andprivate capital interregional ows.

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It is important to note that we do not consider the accumulation of public

capital as in Clarida (1993). Instead, we work with productive public servicesthat are non-accumulable. We agree that it is the stock of public capitalthat aects the level of output. However, using this specication wouldyield a model with two sectors for which the transition to the steady state isvery comp

e losses. Consequently,there is not strong empirical evidence on the eects of interregional transferson regional outputs and on the regional growth rates.

In section 3, we present the set up of the model. In section 4, mainresults concerning regional disparities, growth rates and welfare are provided.Finally, in section 5 we present the main conclusions.

3 The Model

We consider a country with several regions and a unique Central Govern-

ment (CG from now on) that provides public services and reallocates publicresources across regions.2 For simplicity, we assume that there are only tworegions, one is poor (P) and the other is rich (R). Dierences between regionsare based on dierences in productivities and the initial levels of assets.

Regions are linked through private capital ows and through the tax

2 A natural extension of our model would be to consider a multi jurisdictional frame-work in which regional authorities may have scal autonomy and may take decisions thatcontradict those taken by the Central Government.

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system because the CG imposes a proportional tax on output ( ) which is

the same in all regions. We use a proportional tax, for simplicity.

3.1 Households

We consider the standard model of the representative, innite-lived agent.The problem for the household in region iis to maximize the overall utility:

Max Ui

Z 1

0et

c1ti 11

Lti dt (1)

s:t: :ati= rt ati+wti cti;

a0i 0; i= R; P;

where cti is consumption per capita and is the rate of time preference.We dene = 1 (with > 0) as the constant inter temporal elasticity ofsubstitution. rtis the market interest rate and wti represents the wage, bothare take as given by households. Lti denotes the family size; as usual, wenormalize the number of adults at time t to unity, therefore Lt= ent, wherenis the population growth rate that we consider equal to zero, for simplicity.

Additionally, we assume that: i) labor is immobile and the labor supplyis inelastic and ii) all agents have the same preferences, independent of theregion of residence; therefore, and are the same for any representativeagent.

Agents hold the quantityatiof real assets in the form of ownership claimson capital. Households oer their capital to domestic rms and rms in theother region. We model private capital mobility by considering a market forinterregional bonds. Therefore

ait= Kit+ b

idt ;

where Kit are domestic claims on capital in region i, and bidt denotes the

interregional bonds demanded by households in region i. a0i is the initial

level of assets, which are assumed to be dierent between regions,a0R > a0P.Domestic and the other regions claims on capital are assumed to be

perfect substitutes as stores of value. In equilibrium, households receive thesame rate of return r on their assets no matter where they nally allocatethem.

Another constraint is imposed in order to rule out the possibility of un-bounded borrowing, although this constraint is also derived from the marketequilibrium

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limt!1

ati eRt

0 (r(v)

n)dv 0: (2)

3.2 Firms

The objective of a representative rm in region i is to maximize after-taxprots

Max ti= (1 )Yti rtKti wti Lti; i= R; P:

We assume that all rms have access to the same technology and that thelevel of output in region i(Yti) follows, as in Barro and Sala-i-Martin (1995):

Yti= AiKti(LtiGti)

1; i= R;P; (3)

where 0< AP, therefore thereare dierences in regional productivities. This assumption is sucient forYtR > YtPto hold before redistribution.

This specication implies constant returns to scale in the private inputs.Therefore, for xed Gti there are diminishing returns to the accumulationof aggregate capital. But if Gti rises along with Kti, then diminishing re-turns will not arise, and the production function presents constant returnsto scale inKtiandGti. With this specication we allow for the possibility ofendogenous growth.

Alternatively, we could consider a technology that explained poverty

traps, which are often used justify interregional transfers. This type of modelwould be really interesting and it would allow us to explain the results inGarcia-Mila and McGuire (1996) regarding the possibility that the amountof interregional transfers in Spain has been too small to have caused any

3 These characteristics may include natural resources with prohibitively transportationcosts, social conditions, immobile factors, endowments of knowledge which are not trans-ferable, etc. In short, all those characteristics that are taken into account by rms whendeciding their location.

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eect. However, even that this kind of model is not dicult to solve for a

single region, when dealing with two regions it becomes extremely dicultto solve. There is a multiplicity of equilibria (stable and unstable) for eachof the regions, and due to the scal ows and private capital mobility it isvery dicult to derive some conclusions on the eects of redistribution.

Another important assumption of the model is that we allow for free andperfect capital mobility between regions. Producers in one region have accessto capital from households in both regions. The level of capital input in theith region is

Kti= Kit + b

ist ;

where Kit is the capital from domestic households in region i, and bist is thesupply of interregional bonds made by producers in region i. As we mentionedin the previous section, we model capital mobility by assuming a market forinterregional bonds.

In equilibrium, rms will pay the same r wherever they hire the capital.The market clearing condition for the interregional bonds is

bRst +bPst = b

Rdt +b

Pdt :

3.3 The Government

In this multi-regional economy there is a CG that provides local public ser-vices (Gti) that are used by rms in the production process.

The CG xes a proportional tax on output at a rate , which is the samein both regions. The national tax revenue collection from all regions at timet (Rt) is entirely devoted to the provision of productive public services (Gt)therefore satisfying the balanced budget constraint

Gt= Rt= (YtR+YtP):

The allocation of resources across regions by the CG must satisfy that

Gt= GtR+ GtP:

The Government applies the following redistribution policy: a shareofthe tax revenues collected in the richer region is devoted to provide additionalpublic productive services to the poorer region. As we mentioned in the rstsection, our goal is not to analyze how is decided, but which are the eectsof dierent levels of redistribution. The amount of public services providedin both regions follow

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GtR=(1 )YtR;

GtP =(YtP+YtR);

with 0 1 and 0 1. The tax rate and the redistributionparameter are constant and exogenous to individual decision-makers.

In the following pages we will show that the decision on is independentof the decision on . This will imply that the government will not have twoinstruments for Regional Policy. We will show that there is an optimal tax

policy that depends on the positive eect of public services and the distortionscaused by the taxes. Therefore, once the optimal tax has been decided (aswell as the aggregate level of expenditure), the problem of the government ishow to allocate public expenditure across regions.

4 Results

In section 4.1 we study the eects of Regional Redistribution (RR from nowon) on regional output disparities. In section 4.2 we present the eects ofredistribution on regional welfare in the economy.

4.1 Regional disparities in output levels

The assumption of free and perfect capital mobility becomes crucial whenanalyzing the eects of RR on regional disparities in output levels. Anotherassumptions that are also crucial in our model are those of immobile workersand the inelastic supply of labor.

Considering those three assumptions, the after-tax prot maximizing con-ditions are dierent in the rich and in the poor region. In the rich region wehave that capital input and labor will be demanded according to the followingconditions:

P M gKR= (1 ) A1

R [(1 )]1

= r;

and

P M gLR= (1 ) (1 ) A1

R KtR [(1 )]1

= wtR;

while in the poor region the conditions will take the form

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P M gKP = (1 ) AP K1tP GtP

1 =r;

and

P M gLP = (1 ) (1 ) AP KtP GtP

1 =wtP;

whereP M gLi, andP MgKi denote the marginal product of labor and capitalrespectively.

The free and perfect capital mobility condition requires that in equilib-rium marginal products of capital in both regions equalize:

(1 ) A1

R [(1 )]1

= (1 ) AP K1tP GtP

1: (4)

which implies that the rate of return to capital is the same in both regionsand, more important, it is constant

r= (1 ) A1

R [(1 )]1

: (5)

From condition (4) we derive the result that capital input, output andpublic services grow at the same rates in both regions. Therefore

Ki =Yi =Gi = ; i= R; P:

We must note that when increasing redistribution we only consider thepossibility of eliminating completely regional output dierences, such thatYtRYtP

1 after the economy is again in equilibrium. If the CG xed a larger

level of redistribution such that YRYP

AP regional redistribution reduces regional

disparities in output levels.Proof: see Appendix 6.1.

Our results derive from the fact that RR reallocates public resourcesfrom the productive region to the less-productive region. Therefore, RRcompensates for dierences in regional productivities. This is precisely whereineciencies come from, because RR motivates a reallocation of factors ofproduction due to the eects on the marginal products of capital.

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Redistribution motivates a reduction in the level of public services in the

rich region, which negatively aects the level of regional output in that region.Additionally, the poor region receives larger amounts of public services, whichhas a positive eect on the regional level of output.

However, there is another eect to be considered. The new level of re-distribution changes the marginal products of capital in both regions thatmotivates the reallocation of factors of production. When redistributionincreases, the marginal product of capital in the rich region decreases andproducers in that region demand lower amounts of private capital input whileproducers in the poor region demand higher amounts of private capital input.These opposite eects explain why the ratio of regional outputs is negativelyrelated to redistribution.

We must note that under the situation of no redistribution, = 0, themodel provides a result that is obvious due to the specication of the pro-duction function: all economic activity concentrates in one region.4 In thiscase, the marginal products of private capital in both regions never equalize,because we assumed AR > AP. Therefore, the type of transfers that weare dealing with prevent the concentration of production in the region withhigher productivity.

Nevertheless, although with = 0 the Marginal Product of Capital isthe highest possible and households get the highest rate of return on theirassets, this does not assure that all agents will be better o in terms of

welfare. Regional equilibrium wages enter the households budget constraintsand they depend on the regional levels of capital inputs. Therefore, due tothe assumption of immobile workers, with = 0 equilibrium wages in thepoor region are zero and this will have a signicant impact on the householdsbudget constraints.

4.2 Welfare analysis

In the previous section we have shown that Interregional Transfers succeedin the goal of reducing regional output disparities. This however does notsay anything about the desirability of such a policy, which depends on the

eects on welfare in each region. In this section we show that the impact onwelfare depends on the eects of redistribution on the sources of income andthe eects on the growth rate.

4 If we want to analyze perfect capital mobility allowing for = 0and avoiding the totalconcentration of production in one region, we should introduce some kind of concavity onthe private capital input. But this does not allow us to get endogenous growth and thetransition to the steady state is very complicated.

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The solution to the problem of the household yields5 that the growth rate

of consumption in both regions, c, follows

c=

:cti

cti=

1

[r ] 8 i = R; P: (6)

Once we take into account the free and perfect capital mobility conditionand given that the interest rate is constant, it is easy to show that theconsumption growth rate is constant also

c=1

(1 ) A

1

R ((1 ))1

8 i = R; P: (7)

Equation (7) says that consumption in both regions grows at the sameconstant rate (cR = cP). c depends both on the tax rate () and on theredistribution parameter (). The consumption growth rate is negativelyrelated to redistribution (@c

@ < 0). By increasing the CG lowers the

amount of public services in the rich region and this aects negatively therate of return on assets (r). This eect also spreads to the recipient because ofthe free and perfect capital mobility condition. We apply the usual conditionthat allows for positive growth.

Given that consumption grows at a constant rate, we can write consump-tion as

cti= c(0)i ect; i= R; P;

wherec(0)i denotes the initial level of consumption in region i .Substituting the expression for consumption in the utility function of the

households (equation 1) we obtain the maximum level of utility attainablein any of the regions

Ui=Z 1

0

c(0)1i e[(1)

c]t 1

1 dt i= R; P:

Given the positive growth and the bounded utility conditions, the maxi-mum level of utility in any of the regions takes the form

Umax i= 1

(1 )

" c(0)1i

c(1 )

1

#; i= R;P; (8)

wherec(0)irepresents the initial level of consumption that satises the transver-sality condition. There is only a c(0), for each region, that satises thiscondition.

5 This result is derived from the First Order Condition that we obtain when we solvethe households problem.

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The next step is to study what is the eect of redistribution on the level

of welfare. Redistribution has two dierent eects. On the one hand there isan eect on the growth rate of consumption (c). On the other hand, thereis an eect on the initial level of consumption.

The derivative of the level of welfare with respect to redistribution israther cumbersome:

@Umax i

@ =

1

c(0)i

1

[ c(1 )]2

[( c(1 ))@c(0)i

@ +

@c@

c(0)i]; i= R; P:

Lemma 2 If there exist a negative relationship between the initial level of

consumption in one region and the level of redistribution, then regional redis-

tribution causes a negative eect on the level of welfare in that region. See

proof in Appendix 6.2

If condition in Lemma 2 is satised then we can assure that redistributionwill have a negative impact on the level of welfare. If this condition holds forthe poor region, this would be a surprising result in the literature because oneexpects that redistribution has a positive eect on the recipient at least. Ifthis condition is not satised redistribution may have a positive or a negativeeect on welfare depending on the sign of the whole term in brackets.

In this paper we will focus our analysis on the conditions for which theremight be a negative relationship between redistribution and the level of wel-

fare in the poor region. We will s how that the impact of redistribution oninitial consumption will depend on its eects on the returns on assets as wellas on the equilibrium wages.

In order to nd these levels of initial consumptions we must solve thedierential equations:

:ati =rati+wti cti; i= R;P:

where6

wtP =r KtP1

and wtR= r KtR

1

:

The previous dierential equations cannot be solved if we do not knowKtP and KtR: In order to nd both variables we will show that consumptionand aggregate capital input levels grow at the same rates.

According to the solution to the problem of the rms, the level of aggre-gate output in the economy can be written as

Yt= A1

R [(1 )]1

Kt:

6 We derive these expressions from the F.O.C to the problem of the rms.

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whereKtdenotes the aggregate level of capital input.

Additionally, the accumulation of aggregate capital in the economy fol-lows7

:

Kt=Yt Ct

or:

Kt=A1

R [(1 )]1

Kt Ct:

We know that consumption grows at a constant rate (c). If we solve theprevious dierential equation8 we derive the result that consumption andcapital input grow at the same rates (c= K)

Ct= Kt [ r

(1 ) c]:

Therefore, capital input levels follow

Kt =KtR+KtP =K(0) ect: (9)

Finally, the free and perfect capital mobility condition provides

KtR

KtP=

(1 )

AR

AP

11

1

: (10)

If we work with equations (9) and (10) we can write equilibrium wages inboth regions as

wtR= r 1

1 + Kt;

wtP =r 1

1

1 + Kt;

where = (1)ARAP

11

1:In these two equations we can observe that theequilibrium wages in both regions depend both on the interest rate and theportion of aggregate capital input that is allocated in each of the regions.

According to the previous considerations, the budget constraint for eachrepresentative household becomes

:atR= r atR+r

1

1 + Kt ctR; (11)

7 We consider there is no depreciation of capital.8 The transversality condition that we imposed on the level of assets also applies to the

aggregate levels of capital input.

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:

atP =r atP +r 1

1

1 + Kt ctP: (12)

The solution to these two equations will allow us to nd the level of initialconsumption that satises the transversality condition.

4.2.1 Welfare in the rich region

The dierential equation in (11) is solved considering the bounded utilitycondition, the condition for positive growth and the transversality condition.The solution provides the result that households in the rich region devotea constant fraction of their assets to consumption and that consumption

depends on the amount of capital input allocated in the rich region

ctR= (r c) atR + r 1

1 + Kt

with r c > 0 due to the bounded utility condition. However, the resultthat we must stress is the fact that capital input, consumption and assetsgrow at the same, constant, rates (c= a= K). This expression, togetherwith previous results, also tells us that all variables in the rich region growat the same rates9

KR

= GR

=YR

=aR

=cR:

The initial level of consumption for which the transversality condition issatised is

c(0)R = (r c) a(0)R + r 1

1 + K(0);

where K(0) the initial level of capital input (which is independent of redis-tribution) and 1+ is the portion of aggregate capital input that is allocatedin the rich region.

Proposition 3 If 1 then regional redistribution causes a negative im-pact on households in the rich region, in terms of welfare. See the proof in

Appendix 6.5.

If this condition is satised the initial level of consumption is negativelyrelated to redistribution. As we stressed in Lemma 2 this is a sucientcondition to show that redistribution has a negative eect on the level of

9 This result is standard in models of Endogenous Growth.

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welfare of households in the rich region. Given the characteristics of our

model this is the result that one could expect for the rich region as the CGincreases redistribution.

Redistribution has three negative eects on households in the rich region.There is a ow of capital input towards the poor region that reduces the rateof return on assets as well as the growth rate of the economy. Moreover, theow of private capital from the rich to the po or region motivates a reductionin the equilibrium wages in the rich region.

We must stress that even if this condition is not satised, as me mentionedin the previous lines, it could still be the case that we found this result. Theproblem is that the condition is rather cumbersome.

4.2.2 Welfare in the poor region

In order to solve the dierential equation that corresponds to the poor region(12), we apply the bounded utility condition and the positive growth ratecondition. We also take into account that consumption and capital input inone region grow at the same rates and that these growth rates are the samein both regions, therefore, cP =cR = aR= K. All these considerationsyield that consumption in the poor region takes the form

ctP= (r c) atP + r (1

)

1

1 + Kt:

The initial level of consumption that satises the transversality conditionin the poorer region follows

c(0)P =a(0)P(r c) + r (1

)

1

1 + K(0)

where 11+ is the portion of aggregate capital input located in the poor region.

Proposition 4 If 1 and if 1

then regional redistribution causes

a negative eect on the level of welfare of households in the poor region. See

the proof in Appendix 6.6

If both conditions are satised then the initial level of consumption isnegatively related to redistribution. The model yields this result becauseredistribution has several opposite eects. On the one hand higher redistri-bution will attract more capital to the poor region and this will positivelyaect the equilibrium wage. On the other hand, higher redistribution reducesreturns on capital as well as the consumption growth rate. However, whenthe level of redistribution is very large the negative eects on the growth rate

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and on the returns to assets outweigh the positive eect on the equilibrium

wage.Another interesting result that is derived from the previous proposition

is that the second condition is the one that measures the impact of redistri-bution on the equilibrium wage. If this condition is satised it occurs thateven the equilibrium wage can decrease when redistribution increases. Wemust remind that there are three elements in the equilibrium wage. Firstthe interest rate, second the portion of aggregate capital input (which de-pends positively on redistribution) invested in the poor region and third theconsumption growth rate.

We must also point out the role that plays in the condition 1

:

If 1

2

this condition is never satised because by assumption

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that maximizes welfare. The government takes into account the distortion of

the tax rate and the positive eect of public services. Therefore, the governdoes not have two dierent instruments of Regional Policy, and the optimaltax is not aected by the allocation of public services across regions.

5 Conclusions

In this paper we have analyzed the eects of Regional Redistribution in amodel in which the Central Government provides public productive servicesto all regions in the economy independently of the amount of taxes collectedin each region. This type of transfers are widely used in several countries

characterized by large dierences in regional productivities, such as Spain,Italy and even in the European Union. The goal of such transfers is to reduceregional disparities in per capita output levels by motivating a reallocationof private capital from the rich to the poor regions.

We have shown that this policy has several interesting eects, beyondthose direct eects that are well known in the literature that analyzes theimpact of public capital in the economy.

First, there is a reallocation of the stocks of private capital that mo-tivates a reduction in regional output disparities. Under free and p erfectcapital mobility conditions, Central Government intervention modies theMarginal Products of Capital in both regions which motivates interregional

ows of capital and an increase in the level of output in the poor region.This intervention introduces some ineciency because capital ows to theless-productive region.

Second, there is a negative eect on the growth rate of the economy. Inour model the rich region is the one that leads the economy due to its largerproductivity of private capital. CG intervention negatively aects the richregions growth rate and this eect spreads to the recipient region throughscal and private capital ows.

Finally, we showed that although higher redistribution reduces regionaloutput disparities, this reduction might be compatible with a decrease in total

welfare. If several conditions are satised, it might be the case that higherredistribution implied lower levels of welfare in the poor region. This isbecause although redistribution has a positive eect on the equilibrium wagein the poor region, it has a negative eect on the rate of return on assets.We concluded also that the negative eect of redistribution on the level ofwelfare in the poor region is related to the contribution of capital inputin regional output. In economies in which the role of capital input is small,redistribution through public productive services with the goal of reallocating

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private capital towards the poor regions could be welfare-worsening for all

households.A further extension of this paper would consider endogeneizing the po-

litical decision on the optimal level of redistribution. In this paper, we haveconsidered that the level of redistribution is xed but not how redistributionis decided.

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6 Appendix

6.1 Proof of Proposition 1.

1. We know that YtP = APKtP(LtPGtP)

1and that GtP = YtP(1 +

YtRYtP

):

2. From the free and perfect capital mobility condition we derive:

YtR

KtR=

YtP

KtP= A

1

R [(1 )]1

:

2 Operating with the previous expressions we see the ratio of regional

outputs, as well as the ratio of regional levels of capital inputs, areconstant and follow

KtR

KtP=

YtR

YtP=

(1 )

AR

AP

11

1

:

The derivative of the ratio of regional outputs with respect to redistri-bution is

@(YtRYtP

)

@ =

1

2

AR

AP

11

+ 1

2;

which is always negative given that (ARAP) > 1:

6.2 Proof of Lemma 2.

1. The consumption growth rate is negatively related to redistribution(@c@ (1 )c: (13)

The derivative of the maximum level of welfare attainable in any of theregions becomes

@Umax i

@ =

1

c(0)i

1

[ c(1 )]2

[( c(1 ))@c(0)i

@ +

@c@

c(0)i]; i= R; P:

The sign of this derivative depends on the sign of the term in brackets.Given the previous conditions we can say that if @c(0)i

@ 0 then the level of

welfare is negatively related to redistribution, that is, @Umax i@

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6.3 Proof of Proposition 3.

According to Lemma 2

if @c(0)i

@ 0then

@Umax i

@ 0 i= R; P: (14)

The derivative of initial consumption with respect to redistribution canbe written as

@c(0)R@

=a(0)R@r

@ [1

1

] +

1

K(0)

1

1 + [

@r

@ + r

1

(1 + )

@

@]:

(15)Given that @r

@

0 and @

@

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