spatial solow

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Augmenting the Solow-Romer Model with

Spatial Externalities: An Application to theBrazilian Case.

Waldery Rodrigues J unior

Institute of Applied Economic Research,Directorate of Urban and Regional Studies,E-mail: [email protected]

Pedro Henrique Melo Albuquerque*Camilo Rey LauretoMarina Garcia PenaRafael Dantas GuimaraesGilberto Rezende de Almeida J unior

Institute of Applied Economic Research,Directorate of Urban and Regional Studies

Abstract: The main objective of this paper is to show how the resultsobtained with traditional economic growth models will change with theintroduction of spatial elements. Specically we show how the Arrow-Romer externalities coupled with spatial externalities will change theresults obtained with the use of the traditional Solow-Swan Model foreconomic growth. We use a spatial econometric specication based onErtur and Koch (2007) and Koch (2010) and applied the model tothe analysis of per capita income for the 558 micro regions in Brazil.The results show that spatial externalities are important. Using variousways for constructing the spatial weigh matrix we also identify whatare the most important municipalities in terms of the elasticity of percapita income for Sao Paulo with respect to the rates of saving of itsneighbors. Explicit formulae for this Spatial version of the Solow modelare provided throughout the paper.

Keywords: Spatial Solow, Spatial Econometrics

Reference to this paper should be made as follows: Rodrigues J unioret. al. (2010) Augmenting the Solow-Romer Model with SpatialExternalities: An Application to the Brazilian Case., 32 o EncontroBrasileiro de Econometria .

JEL: C21, R15

*Professor at Departament of Management - University of BrasiliaCopyright c 2010. Sociedade Brasileira de Econometria


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Augmenting the Solow-Romer Model with Spatial Externalities 3

the parameter 0 < 1. This is assumed to be identical for all countries.However the net effect os spatial externalities is dependent on the how

connected one region is to its neighbors and this is represented by thefrictions elements wij .

3 Spatial Dependence and Economic Growth

In order to estimate the parameters of the model one needs to necessarily quantifythe neighbors inuence (in a quantitative way). This is usuallly done through theuse of the contiguity matrix W (1)n n also known as the neighborhood matrix ouweight matrix.

Given a set of n polygons ( P 1 , . . . , P n ) we construct the matrix W(1)n n where

each element (friction) wij represents a measure of spatial proximity betweenpolygons P i and P j .

This measure can be evaluated via the following criteria:

wij = 1 if the centroid of P i is at a given (non zero) distance of polygon P j ;otherwise wij = 0.

wij = 1 if P i shares at least a common side with P j ; otherwise wij = 0(Rook).

wij = 1 if P i shares at least a common point with P j ; otherwise wij = 0(Queen).

wij = l ijl i where lij s the length of the common frontier between the polygonsP i and P j and lij is the perimeter of polygon P i .

The diagonal matrix is equal to zero by denition. Since it is used in buildingspatial indicators it is common to apply a normalization process in order for thesum of the elements for any given row to be equal to 1.

We can generalize the denition of matrix W (1) for high degree orders, i.e., itis possible to dene if P i and P j are neighbors and P i is a neighbor of P k then P jand P k are also neighbors (transitivity property).

Another idea related to continguity matrix is to take for each i = j , wij =t ijt i

where t ij can represent the number of links (phone calls, for example) for i to jand t i =

jt ij is the number of calls having i as the starting point.

The choice of matrix to be used in spatial analysis is not prone to a certaindegree of arbitrary selection. An analytical proposal is to choose the matrix of neighborhood that maximizes a given measure, say the likelihood.

Cliff-Ord (1982) generalize the matrix W via a functional relation between therelative size of the common frontier between the two polygons adjusted by the

inverse of the distance between the two observations, i.e., wij =bijd ij

where bij isthe common frontier between regions i and j and , are adjusting parameters.

These weights may represent a measure of potential interaction between units iand j . For example, the may have a direct relationship with the spatial interaction(element) as wij = 1d ij or wij = e

d ij or else in a more complex way it may use


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4 Rodrigues J unior et. al.

other measures of distance such as the Manhattan, Minkowsky or even stastisticalmeasure named Mahalanobis distance.

Tipcally the vectors of parameters( , ) are given apriori (Example: = 2reecting gravity) instead of the case where they would be given jointly with othermodel parameters.

Clearly if these parameters are jointly estimated the objective function willhave a high degree of non-linearity, which led to a complex otimization problem.

Figure 1 Figure 1

Example: The gue below is represented by the following weight matrix of therst order-Queen:

W (1) =

0 1 0 0 01 0 1 0 00 1 0 1 10 0 1 0 10 0 1 1 0

Note that we still need to use the usual normalization (in order to obtain theproperty that the sum of every row is one).

4 Spatial Solow Model

Given the type of technological interdependence implied by the model thelocation (countries) cannot be analyzed in isolation. Instead they should take intoconsideration the inuence of neighbors and the role played by the matrix W. Wecan rewrite the function for A i (t) in matrix form:

A = + k + WA

where A is a N 1 vector of logarithms of the technology level in aggregateterms, k is a N 1 vector of logarithm of the physical capital per worker, W isthe N N contiguity matrix . Solving for A led us with:

A = ( I W ) 1 + (I W ) 1k (2)


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Rewriting the above equation and considering | | < 1 we obtain:

A i (t ) = 11 (t)ki (t)n

j = ik j

r =1 r

w( r )

ij (3)

Note that the level of technology depends on its own physical capital per workerand the level of physical capital per worker of its neighbors.

Rewriting the above equations and using the equation for the productionfunction we obtain:

yi (t ) = 1

1 (t)ku iii (t)n

j = iku ijj (t ) (4)

where the terms

u ii = + 1 +

r =1

r w( r )ii

and

u ij =

r =1

r w( r )ij

are the new elemente in the spatial model. Here the terms of friction wij are theelements of row i and column j of matrix W when using power r . Finally,

yi =Y i (t )L i (t )

is the usual level of output per worker.This model implies spatial heterogeneity in the parameters of the production

function. Note that for the special case where there is no capital externalityregarding the physical capital , i.e., = 0, we are left with u ii = and u ij = 0.This gives the standard production funciton of the Cobb-Douglas type.

5 Elasticities

We can evaluate the elasticity of income per worker in country i with respect tothe physical capital. When a country i increases its own physical capital per worker

it gets a social return of u ii . This is augmented to u ii +N

j = iu ij = + 1 , if the

countries simultaneously decide to increase their stocks of capital per worker.In order to assure local convergence and avoid explosive or endogenous growth,

we assume diminishing social returns: +

1 < 1As in the Solow model, we assume that a constant fraction of the output, s i ,is saved and that work grows exogenously at a rate of n i for a country i . We alsoassume an annual rate of physcal capital depreciation for all countries denotedby . The evolution of the output per worker in country i is given by the Solowfundamental dynamic equation:

ki (t) = s i yi (t) (n i )ki (t ) (5)


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where the line over the variable denotes its derivative with respect to time.Since the output per worker is characterized by diminishing returns, equation

5 implies that the physical capital-to-output ratio of country i , for i = 1 , . . . , N isconstant and converges to a balanced growth dened by k ik i = g or also

k iy i

=s i

n i + g+ . In other words:

ki = 1

(1 )(1 u ii ) (t) s in i + g + 1

1 u iiN

j = ik

u ij1 u ii

j (t) (6)

Since production technology is characterized by externalities between countries,we can observe that physical capital per worker in steady state depends not onlyon usual technology and preference parameters, but also on the level of physicalcapital per worker in neighbor countries.

In order to determine the equation that describes the real income per worker

in a country, we rewrite the production function in matrix form:

y = A + k

Substituting A in equation 2, and multiplying both sides by ( I W ) weobtain:

y = + ( + )k W k + W y

Rewriting the equation for the economy i and introducing the logarithm of theoutput capital ratio in the steady state, we obtain the real income per worker incountry i in steady state:

log [yi (t )] =1

1 log [(t )] + +

1 log [s i ] +

1 log (n i + g + )

1 N

j = iwij log [s i ] + 1

N

j = iwij log (n j + g + ) + (1 )1

N

j = iwij log yj (t)

This Solow model - spatially augmented - has the same qualitativeinterpretations as in the traditional Solow model, especially concerning theinuence of the internal savings rate and the population growth rate on real incomeper worker in steady state.

First, the real income per worker in steady state for a country dependspositively on its own savings rate and negatively on its own population growth

rate. Second, it can also be shown that the real income per worker in a countrydepends positively on the savings rates of neighbor countries and negatively ontheir population growth rates.

Actually, even though the sign of the coefficient for the neighbor countriessavings rate is negative, each of these savings rates ( log [s j ]) positively inuencesits own real income per worker in steady state ( log yj (t ) , which positivelyinuences the real income per worker in steady state for country i throughexternalities and global spatial technological interdependence.
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We then have,

k

i( t )

k i ( t )

k i (t )> 0

provided that u ii > 1.Externalities on physical capital and technological interdependence only delay

the decrease in the marginal productivity of physical capital. Therefore, theconvergence result is still valid under the hypothesis + 1 < 1, in opposition tothe endogenous growth models, where the marginal productivity of physical capitalis constant.

Furthermore, this model allows for quantitative predictions about theconvergence speed to the steady state. As in the literature, the transition dynamicscan be quantied by means of a log-linearization of equation 9 around the steadystate, for i = 1 , . . . , N :

dlog [k i ( t )]dt = g (1 u ii )(n i + g + ) [log (ki (t )) log (k

i )]

+N

j = iu ij (n i + g + ) log (kj (t )) log kj

We obtain a system of linear differential equations whose solution is quitecomplicated. However, considering the following relations between the countriesrelative to their steady states:

log (ki (t)) log (ki ) = log (k j ) log kj

log (yi (t)) log (y

i ) = j [log (yi (t )) log (y

i )]With convergence speed equal to:

dlog [yi (t)]dt

= g i {log [yi (t )] log [yi ]}

where:

i =

N

j =1u ij 1 j (n j g + )

N

j =1u ij 1 j

N

j =1

1j

(n j + g + )

These hypotheses state that the difference between the country i and its ownsteady state is proportional to the correspondent differential for country j .Therefore, if j = 1, countries i and j are at the same distance from their

steady states.If j > 1 (alternatively j < 1) then country i is farther (or alternatively

closer) from country j s steady state.The relative difference between the countries with respect to their steady states

affects the convergence speed.
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Actually, i j =u ij (n j + g + )

2j> 0 and the convergence speed is high, if the

country i is far from its own steady state.Furthermore, the convergence speed is high if country j is close to its steady

state.Therefore, there is a strong form of heterogeneity in this model, since the

convergence speed in country i is a function of the parameters wij representing thedistance to neighbor countries, from their own steady states.

When there are no externalities on physical capital ( = 0), heteregeneity of theconvergence speed is reduced to the traditional Solow model: i = (1 )(n i +g + )

In this sense, we have the same relation between the externalities on physicalcapital and the heterogeneity we obtained with the production function.

The solution for log (yi (t)), subtracted from log (yi (0)) - the real income perworker in some initial moment - on both sides, we obtain:

log (yi (t )) log (yi (0)) = gt (1 e i t ) 1 1

i (1 e i t )log (yi (0)) + (1 e i t )log (yi )

(10)

The model predicts convergence, since the growth of real income per worker is anegative function of the initial level of income per worker, but only after controllingfor the determinants of the steady state.

More specically, the growth rate of real income per worker depends positivelyon its own savings rate and negatively on its own population growth rate.

Furthermore, it also depends on the same variables on the neighbor countries,due to technological interdependence. We can see that the growth rate is higherwhen the initial level of income per worker is higher and when the growth rate of neghbor countries is higher.

Finally, it can be seen that the growth rate of a country depends on thegrowth rate of its neighbors, weighted by the speed of convergence and the spatialneighboring conditions.

7 Analysis of Data for Brazil

Following the empirical literature on growth, we have used data from IPEADATA,which contains information on real income, investments and population (amongother variables) for a great number of Brazilian cities.

Specically for this work, we have used data from 558 micro-regions for Brazil.We measure n as the average growth rate of the active population (aged 15 to

64). We also compute the number of workers.Real income per worker is measured by dividing total income by the numberof workers. Also, the savings rate is denoted by s .

The matrix W dened before is the weight matrix commonly used in spatialeconometric models of spatial interdependence between countries or regions(Anselin (1988); Anselin and Bera (1998);Anselin (2006)).

More precisely, each country is linked to a set of neighbor countries by meansof a purely spatial model introduced by W .
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The elements wii of the main diagonal are conventionally equal to zero, whereasthe elements wij indicate with which intensity country i is spatially connected to

country j .In order to normalyze the external infulence on each country, the weights of the matrix are standartized so that the elements of the rows sum to 1.

For variable x , this transformation means that the expression W x , calledspatially lagged variable, is simply the weighted average of neighbor observations.

It is important to stress that the elements wij must be exogenous to the model.In this sense, we consider the purely geographical concept of queen neighborhood,according to which two countries are neighbors if their borderlines share at leastone point, which is indeed strictly exogenous.

8 Econometric Model Specication

In this section, we aim at assessing the impact of savings, population growth andlocation on per capita real income.Taking equation 10, we can see that the real income per worker along the

balanced growth path, at a certaint moment ( t = 0, for the sake of simplicity), is:

log Y iL i = 0 + 1 logs i + 2 log (n i + g + ) + 1N

j = iwij logs j +

2N

j = iwij log (n j + g + ) +

N

j = iwij log

Y jL j + i

(11)

Where 11 log(0) = 0 + i for i = 1 , . . . , N with 0 a constant, i astochastic term and the term (0) reects not only technology, but also resource,climate and other endowments, and thus can differ from country to country.

We also assume that g + = 0 .05, as is common in the literature of MRW(1992) and Romer (1989). Finally, we have the following theoretical limitations forthe coefficients:

1 = 2 = +

1 and

2 = 1 =

1

= (1 )1 Also,

Y = X + W X + y +

y = W y + X +
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where,

N (0, 2

I)In this model, the vector y contains n observations of the dependent variable

of interest, X is the delimitation matrix that contains the covariates that explainthe event y and W is the matrix of known spatial weights.

Parameter is responsible for measuring the spatial dependence of thedapendent variable y and its respactive neighbors.

The vector of parameters reects the inuence of the covariates on thedependent variable (does not include the spatial component of neighboring).

The model is called mixedmodel because it combines the standard regressionwith a spatial component which is the spatially lagged variable Y .

Thus, the model tries to explain the dependent variable of the region yi , withthe value of the dependent variable from its neighbors and with covariates for its

own region.We can estimate these parameters through numeric optimization or, morecommonly, through concentrated likelihood. In both cases, rst it is necessary tobuild a likelihood function (joint distribution of vector y)

Note that:

y = W y + X +

y W y = X +

(I W )y = X +

(I W )y X =

Since the distribution is known, we can use the Jacobian transformation inorder to obtain the joint distribution of y. For that, we need the Jacobian:

J =

y= |I W |

so, the joint dostribution function of y is:

f (y) = f ( )

y

f (y) = f (I W )y X

y

f (y) = (2 )n2 | 2I |

12 exp 12 (I W )y X

T

(2I) 1 (I W )y X |I W |

The equation above represents the joint distribution of y and also the likelihoodof the parameters = ( , T )T .

Taking logarithms, we obtain the log-likelihood function:


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l(,, 2) = n2 log (2 ) 12 log |

2I | 12 (I W )y X T

(2I) 1 (I W )y X + log |I W |

To proceed with the concentrated likelihood, we need to dene whichparameters are nuisance. In this case, the parameters of disturbance are and

2 . The next step is to obtain the maximum likelihood estimators of nuisanceparameters that should be a function of the parameter of interest . For thenuisance parameter we have:

l(,,

2) = 12 2 [ X ]T (I W )y X = 0

[ X ]T

(I W )y X = 0 X T (I W )y + X T X = 0

X T X = X T (I W )y = 0

= ( X T X ) 1X T (I W )y

= ( X T X ) 1X T y (X T X ) 1X T W y

= O

L

Where O

is the estimator of the parameter vector of the model y = X O

+

Oby the method of ordinary least squares and

Lis the OLS estimator of the

model W y = X L

+L

. Similarly, we estimate the parameter 2 :

2 l(,,

2) = n2 2 +1

2 4 (I W )y X T

(I W )y X = 0

2 =( I W )y X

T

( I W )y X

n

2 =y W y X

T

y W y X

n ,como = O

L

temos:

2 =y W y X (

O

L)

T

y W y X ( O

L

)

n

2 =y X

O W y X

L

T

y X O

W y X L

n

2 =

O

L

T

O

L

n

Where O

is the residue of the estimated model y = X O

+O

and L

is the

residue of the estimated model W y = X L

+L

.


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Figure 2 Figure 2

Table 1 Spatial Auto Regressive ModelParameter Coefficient Standard

Errort Stat P-value

Intercept 5.446541 0.293504 18.556929 0.000000LOG SI 0.238113 0.024377 9.768142 0.000000LOG X 0.892737 0.145027 6.155638 0.000000

W LOG SI -0.022716 0.033231 -0.683595 0.494231W LOG X 1.708643 0.181219 9.428597 0.000000

Table 2 Spatial Error Model

Parameter Coefficient StandardError

t Stat P-value

Intercept 7.817188 0.314011 24.894615 0.000000LOG SI 0.239212 0.022426 10.666562 0.000000LOG X 1.167668 0.128494 9.087370 0.000000

W LOG SI -0.124858 0.061760 -2.021655 0.043212W LOG X 0.736193 0.213608 3.446463 0.000568

Considering the results from the SAR, as for the spatial lag parameter, weobtain = 0 .687192 with standard error equal to 0.023957 and p-value of 0.000000.These results imply = 0 .573379, = 0 .160321 and = 0 .031999.

Considering the results from the SEM, as for the spatial error lag parameter, weobtain = 0 .877103 with standard error equal to 0.019836 and p-value of 0.000000.These results imply = 0 .808547, = 0 .068422 and = 0 .124614.

The results from the OLS imply = 0 .633265, = 0 .137914 and = 0 .052577.Then there is evidence that spatial dependence is signicant, this way, the usual

Solow model cannot be used for that data set.Our model predicts that the savings rate and population growth have larger

effects on real income per worker because of physical capital and technologicalinterdependence externalities.


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References

L. Anselin. Spatial Econometrics: Methods and Models , Studies in OperationalRegional Science, 1988.

L. Anselin. How (Not) to Lie with Spatial Statistics , American Journal of Preventive Medicine, 2, 2006.

L. Anselin and A. Bera, Spatial dependence in linear regression models with an introduction to spatial econometrics , New York: Marcel Dekker, 1998.

A. Cliff and J. Ord, Spatial Processes Models and Applications , London: Pion, 1981.

W. Koch and C. Ertur Growth, technological interdependence and spatial externalities: theory and evidence. , J Appl Econom, 2006.

Instituto de Pesquisa Econ omica e Aplicada (IPEA), IpeaGEO 1.0.4 ,http://www.ipea.gov.br/ipeageo/, 2010.

Mankiw, N. Gregory and Romer, David and Weil, David N., A Contribution to theEmpirics of Economic Growth , The Quarterly Journal of Economics, 2, 1992.

D. Romer Staggered Price Setting with Endogenous Frequency of Adjustment. ,Economics Working Papers, 1989.


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11 Annex I: Map of Income-Savings Rate Elasticities

Here we present maps of the Income-Savings Rate Elasticities for the 9Metropolitan Regions of Brazil for the SAR, SEM and OLS.

Figure 3 Micro region for Belem. Figure 4 Micro region for Fortaleza.

Figure 5 Micro region for Recife. Figure 6 Micro region for Salvador.


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Figure 11 Micro region for Porto Alegre. Figure 12 Micro region for Belem.

Figure 13 Micro region for Porto Alegre.


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Figure 27 Micro region for BeloHorizonte.

Figure 28 Micro region for Rio deJaneiro.


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Figure 29 Micro region for S ao Paulo. Figure 30 Micro region for Curitiba.


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