EQUILIBRIUM AND ECONOMIC GROWTH : SPATIAL

ECONOMETRIC MODELS AND SIMULATIONS

Bernard Fingleton

Department of Land Economy, University of Cambridge, Cambridge CB39EP U.K.

E-mail: bf100@cam.ac.uk

JEL categorizations : C2,O1,O2,O3,O4,R0,R1

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ABSTRACT. Neoclassical theory assumes diminishing returns to capital and spatially constant

exogenously determined technological progress, although it is questionable whether these are realistic

assumptions for modeling manufacturing productivity growth variations across EU regions. In contrast, the

model developed in the paper assumes increasing returns and spatially varying technical progress, and is

linked to endogenous growth theory and particularly to ‘new economic geography’ theory. Simulations,

involving 178 EU regions, show that productivity levels and growth rates are higher in all EU regions when

the Objective 1 regions have faster output growth. This also reduces inequalities in levels of technology.

Allowing the core regions to grow faster has a similar effect of raising productivity growth rates across the

EU, although inequality increases. The simulations are thus seen as an attempt to develop a type of

‘computable geographical equilibrium’ model which, as suggested by Fujita, Krugman and

Venables(1999), is the way theoretical economic geography needs to evolve in order to become a

predictive discipline.

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1. INTRODUCTION

Somewhat paradoxically, the starting point for the model described in this paper

is neoclassical theory (Solow 1956), although the assumptions we make are essentially

the converse of basic neoclassical assumptions. Basic neoclassical growth theory assumes

constant returns to scale (or diminishing returns to capital) and an exogenously

determined spatially uniform rate of technical progress, assumptions which are untenable

or at least debatable when one examines the empirical evidence (see for example

Fingleton and McCombie, 1998).

Basic neoclassical theory assumes that the well-established empirical fact, that

initially low technology regions tend to faster productivity growth, is due to diminishing

returns to capital. Put simply, regions starting from a position of a relatively high capital-

labour ratio grow relatively slowly, and vice versa, so that low technology regions catch-

up and convergence on a common steady-state. Hence at its simplest, neoclassical

growth theory implies the elimination of differences between capital-labour ratios and

productivity levels as regions converge. More recently, in response to the evident gap

between this theoretical prediction and empirical reality, neoclassical theory has been

extended to accommodate the existence of regionally differentiated steady-states, leading

to conditional convergence. In conditional convergence, convergence is to region or

country-specific steady-states rather than to a single steady-state, and this does improve

the empirical performance of the model (Barro and Sala-i-Martin 1995). None the less,

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theoretical objections remain. The reason why diminishing returns is in doubt is because

of theory which argues to the contrary, and because of empirical evidence.

Empirical evidence supporting increasing returns is found in the urban and

regional economics literature which has for a long time made reference to internal

economies of scale and to external ones, namely urbanization and localization economies

(Armstrong and Taylor 1985), so that increasing returns have become ‘almost an article

of faith of regional economists’ (Fingleton and McCombie,1998). A focal point for much

of this literature is the dynamic Verdoorn Law (Verdoorn, 1949), which is an empirical

relationship positively associating manufacturing labor productivity growth to

manufacturing output growth, which was brought to a wide audience by Kaldor(1966) in

particular, who stressed its importance for understanding the determinants of economic

growth. The ratio of productivity and output growth was evidently thought of as a

measure of returns to scale, although it is not possible to make this interpretation on the

basis of the underlying equation system initially developed by Verdoorn (Rowthorn,

1979, McCombie and Thirlwall, 1994). It is possible to relate the Verdoorn coefficient1

and the degree of returns to scale in the Cobb-Douglas production function which is not

homogeneous of degree one, although the value typically estimated for the coefficient has

meant that the degree of returns to scale as measured by the homogeneity parameter is

unrealistic. This led to a more appropriate interpretation (Kaldor, 1972) of the coefficient

as also containing the effects of induced technical progress and induced investment. It is

with this background that estimates of the dynamic Verdoorn Law using regional data by

Bernat(1996), Fingleton and McCombie(1998), Leon-Ledesma(2000), and Pons-Novell

and Viladecans-Marsal(1999) are generally seen as consistent with the existence of

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increasing returns, and Harris and Lau(1998) come to the same conclusion as a result of

estimating long-run cointegration vectors. The Verdoorn Law is thus a reasonable starting

point for empirical models that are attempting to capture increasing returns, such as those

estimated in this paper. However, it is also true that data which are consistent with the

Verdoorn Law also are consistent with neoclassical explanations of economic growth.

Fingleton and McCombie(1998), for example, fit spatial econometric versions of a Solow

growth model which allow an interpretation of diminishing returns with more rapid

productivity growth in those regions with capital-labor ratios that are initially lower than

their equilibrium values. They find it is not possible to discriminate between the opposing

interpretations purely by data analysis, since the two models are not nested and are

derived from different underlying assumptions. A similar open-ended interpretation is

provided by Cheshire and Carbonaro(1995), who argue that ‘beta convergence’ is only a

signal that data are not inconsistent with neoclassical theory, rather than being a direct

test of diminishing returns to capital. It is thus necessary to turn to the power of

theoretical arguments in order to more strongly argue the case for increasing returns.

Theoretical arguments pointing to the presence of increasing returns are

embodied in endogenous growth theory (Aghion and Howitt, 1998), based on early

insights2 that technical progress was not exogenous but was itself determined by new

capital and that this impacted on the returns to scale. These insights led to the modern

endogenous growth theory stimulated by the work of Romer(1986) and Lucas(1988)

involving non-diminishing social returns to investment rather than exogenous technical

progress. In Romer(1986), factor accumulation arises from individual firms’ decisions to

invest in R&D which are based on temporary monopolistic market power, the expected

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gains and market cost of additional R&D, and imperfect copyright and patent laws

causing knowledge spillovers. Knowledge spillovers compensate for private diminishing

returns to investment in knowledge creation so that the stock of knowledge capital in the

economy as a whole is raised. Lucas(1988) envisages a spillover mechanism involving

educational improvements by individuals motivated by the prospect of higher wages. In

the paper we attempt to capture the theoretical arguments of endogenous growth theory in

our specification by terms representing the spillover of technical progress across regions.

Most recently, increasing returns has reasserted its importance as a central

concept in economic theory via ‘new economic geography’ (Krugman 1991a,b, Krugman

and Venables 1995, Ottaviano and Puga 1998, Puga and Venables, 1997,1999, Fujita,

Krugman and Venables, 1999). The revival of increasing returns as a valuable analytical

concept by proponents of new economic geography has been influenced by a non-

neoclassical perception of how the economy operates, most notably provided by the work

of Kaldor(1957,1966,1970,1972). As Krugman(1991a), states, ‘we live in an economy

closer to Kaldor’s vision of a dynamic world driven by cumulative processes than to the

standard constant returns model’. This vision embraces the notion of the dynamic

Verdoorn Law, and cumulative causation mechanisms (Myrdal 1957) which have been

formalized as structural models by Dixon and Thirlwall (1975a,b) (see also McCombie

and Thirlwall 1994, Fingleton 2000a, 2000b). By using the Dixit and Stiglitz(1977)

model of monopolistic competition, a way has been found to link economic geography

more closely to micro-level theoretical foundations as is typical of much mainstream

economic theory. However there has been little or no testing of new economic

geography assumptions or predicted outcomes that are based on the assumptions. The

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paper argues that our model, built around the productivity-output growth nexus and

embellished to capture causes of varying technical progress, is not inconsistent with what

one might expect as an empirical manifestation of new economic geography theory, and

it therefore is seen as a way of confronting our version of new economic geography with

data.

It should be recognized from the above that Verdoorn’s Law appears to be

consistent with different theoretical positions or with different underlying technical

relationships. In order to illustrate how increasing returns are compatible with the

Verdoorn Law, we choose to commence in Section 2 with the derivation from the Cobb-

Douglas production function (Black, 1962), although this is not without its critics (see

McCombie and Thirlwall 1994, Leon-Ledesma, 2000). It is possible to show on this basis

that, given appropriate assumptions, increasing returns to scale as understood from the

Cobb-Douglas perspective can be inferred from the Verdoorn law estimation. In Section

3 we develop our model specification to also accommodate spillovers which per se

would, via the arguments of endogenous growth theory, also be consistent with

increasing returns. In Section 4 we show that the empirical specification is consistent

with ‘new economic geography’ theory, thus providing a basis for the preferred model in

contemporary theoretical developments. Section 5 concerns model estimation, Section 6

gives expressions for the equilibria which are implied by the model, and section 7

illustrates these by simulating various scenarios in the context of EU regional

development. Section 8 concludes.

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2. THE COBB-DOUGLAS PRODUCTION FUNCTION AND THE DYNAMIC

VERDOORN LAW

The basic single equation specification of the dynamic Verdoorn Law, given as

equation (1) below, holds that there is a linear relationship between the exponential

growth rates of labour productivity (p) and output (q). In equation (1) the coefficient m0

is the autonomous rate of productivity growth and m1 is the Verdoorn coefficient, for

which a value of about 0.5 is usually found when this specification is fitted to data on

manufacturing productivity growth and output growth. This implies that a one

percentage point increase in output growth induces an increase in the growth of

employment of about one-half of one percentage point and an equivalent increase in the

growth of productivity. Thus as the scale of activity expands, there is a less than

proportionate increase in employment because of the labour saving effect of productivity

growth due to increasing returns to scale. For example, if as the scale of activity grows

workers are more able to specialize in activities more suited to their talents, output per

worker will increase3. The error term ξξξξ reflects the other effects on p which in this initial

specification are assumed to behave as random shocks.

(1) p = m0 + m1q + ξξξξ

Actually, it is apparent that equation (1) should also include an additional term,

since the growth of capital (k) will also enhance productivity growth. This becomes

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evident assuming that the underlying static model is the conventional Cobb-Douglas

loglinear production function

Q = A0exp(λλλλt)KααααEββββ

in which λλλλ is the growth of total factor productivity, Q, K and E are output, capital and

employment levels respectively, and αααα and ββββ are elasticities. It then follows that a

version of the dynamic Verdoorn Law (equation 2) which includes the additional variable

k can be derived directly from this production function. Moreover, we can show, using a

reasonably acceptable restriction imposed on this revised dynamic Verdoorn Law, that

the often-replicated estimate of the Verdoorn coefficient (m1) of around 0.5 resulting

from fitting equation (1) is consistent with the presence of increasing returns. To see this,

we first take (natural) logs of the production function and differentiate with respect to

time. The resulting expression is then rearranged to give equation (2) in which

productivity growth (p) is now a linear function of both output growth (q) and of capital

stock growth (k).

(2) p = λλλλ/ββββ + ((ββββ - 1)/ ββββ)q + (αααα/ββββ)k + u.

The restriction on (3) mentioned in the previous paragraph is that k = q, in other

words that the capital – output ratio is constant. This assumption is a necessary result of

a common limitation of regional analysis, the absence of comprehensive and reliable

capital growth data. However, we do have evidence to suggest that capital growth will be

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about the same as output growth, so we use this assumed equality of q and k to arrive at

an equation that can be estimated using pan-European data4. With this restriction, k

drops out of the equation and we are left with a specification

(3) p = λλλλ/ββββ + ((αααα + ββββ - 1)/ ββββ)q + u

and if m1 = ((αααα + ββββ - 1)/ ββββ) > 0, as is normally the case, then (αααα + ββββ) > 1 and we have

increasing returns.

Assume that q and k are unequal, so that k = γγγγq, and γγγγ > 1 consistent with capital

growing faster than output. It then follows that (γγγγ αααα + ββββ -1)/ ββββ = m1 and also that

ββββ =( γγγγ αααα -1)/(m1-1), and making assumptions about αααα and m1 we can find the values of γγγγ

that are consistent with the presence of increasing returns. Hence, assume αααα = 0.333, and

m1 = 0.5, then γγγγ < 2 is consistent with increasing returns. So on this basis capital would

have to be growing twice as fast as output for the Verdoorn coefficient (m1 = 0.5) not to

be consistent with the presence of increasing returns. Fingleton and McCombie(1998)

estimate an output elasticity with respect to capital of 0.709. Harris and Lau(1998)

estimate output elasticities by sector and UK region, and the across-region mean of their

weighted averages (calculated across sectors) is 0.753. Assuming the latter for αααα gives γγγγ

< 1.17 for increasing returns, although assuming a higher value for the Verdoorn

coefficient allows a higher γγγγ.

An alternative approach, which has the advantage that the resulting estimating

equation is easy to compute because it does not need capital stock estimates, is to

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commence from the more general constant elasticity of substitution production function.

Thus we assume

Q = A[δδδδK1/µµµµ +(1-δδδδ) E1/µµµµ ]υυυυµµµµ

in which δδδδ is the distribution parameter, µµµµ is a substitution parameter so that the elasticity

of substitution is µµµµ/(µµµµ-1), and υυυυ is the homogeneity parameter (υυυυ = 1 for constant returns to

scale). From this it is possible (see Dixon and Thirlwall, 1975b) to derive the relationship

ln P = a + µµµµ/(µµµµ-1) ln(W/E) + c lnQ

in which W/E is earnings per worker in real terms and a and c are coefficients which are

functions of the above parameters (this reduces to a dynamic form with productivity

growth rates dependent on the growth of real wages and output, which can be compared

with equation 2). Evidently this derivation has a basis in assumptions allowing an

‘equivalence relationship’ for the elasticity of substitution, so that it equals the ratio of the

proportionate change in productivity to the proportionate change in the wage rate,

assumptions which may not hold in all circumstances (Harcourt, 1972). Although the

elasticity of substitution is not constrained a priori to one, in fact, using regional cross-

sectional levels data, Dixon and Thirlwall (1975b) find that the estimated elasticity of

substitution tends not to be significantly different from 1, pointing towards the Cobb-

Douglas form as an approximation on empirical grounds, and on the whole they find

increasing returns to scale (υυυυ > 1) for UK manufacturing sectors.

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3. ENDOGENOUS GROWTH THEORY AND SPILLOVERS

Having introduced increasing returns, we now turn to endogenous technical

progress. We assume that the rate of growth of technical progress, represented by λλλλ,

depends on spillovers, on the diffusion of technology, and on the level of human capital

within regions, and thus equation (3) is extended below to make these effects explicit.

Two forms of technology spillover are envisaged, that occurring as a result of within-

region technical change, and that occurring as a result of extra-regional technical change

perhaps in ‘neighbouring’ regions or in more remote high technology regions. In either

case, the underlying spillover process is the same, with non-internalized technical change

becoming an externality captured by other firms and individuals, as propounded in the

literature of endogenous growth theory cited above. To see how this impacts our model

specification, assume that technical change is proportionate to capital accumulation (in the

form of the growth of capital per worker). Assume also that the growth of capital per

worker (k – e, where e is the growth of employment) is equal to the growth of productivity

(p = q – e), an assumption that follows from our ‘stylized fact’ that k = q. It then follows

that

(4) λλλλ = λλλλ♦ + φφφφp +κκκκWp

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In (4), λλλλ is proportionate to p and, via the matrix product Wp, λλλλ is also a function

of capital accumulation occurring within the particular set of ‘neighbours’ for each

region. Here W is a square matrix5 with n2 cells defining the interaction between n

regions, with the cell (i,j) of the W♦♦♦♦ matrix (ie W before it is standardized) given by

(5) W♦♦♦♦ij = Qj0

ααααdij-γγγγ

in which Qj0 is the output level in economy j at time 0 and dij is the great circle distance

between the centres of regions i and j. The coefficients αααα and γγγγ are chosen a priori rather

than estimated6 with each taking the value 2. This is thus a broad measure of interaction

that encompasses technology level since Q = PE. We standardize W so that its has row

totals equal to 1, in which case each element i of Wp is the weighted average of the other

regions with weights proportional to the level of technology of their economies and their

distance from i.

The term λλλλ♦ in equation (4) depends on other characteristics of each region,

notably internal socio-economic conditions that determine the extent to which

innovations are adopted. We divide these into two, the initial level of technology, denoted

by G below, and the level of human capital (s) within each region. The argument is that

the lower the level of technology the more a region will benefit from adopting new

technology. An advanced region, on the other hand, will see little or no benefit since the

technology will already be in use. In addition, Governments and the EU policy

instruments designed to promote innovation adoption will also have an effect, being

stronger the lower the technology level and diminishing as the level of technology rises.

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We represent this process by equation (6),

(6) λλλλ♦ = ππππG + δδδδs

in which variable Gi = (Pi*-Pi)/Pi

* represents the start-of-period (and therefore

exogenous) technology gap between region i and the leading technology region (*), with

initial technology levels proxied by the initial productivity levels Pi and Pi*. The

implication is that ππππ > 0.

The level of human capital is assumed to be a function of peripherality (l), since

peripheral regions are sparsely populated and culturally distinct from more central

regions, and an increasing function of the level of urbanization (u). In equation (7) we

therefore anticipate that θθθθ < 0 and ΓΓΓΓ > 0.

(7) s = εεεε + θθθθl + ΓΓΓΓu

If we combine the various influences as a single equation the outcome is

p = δδδδεεεε////(ββββ - φφφφ) + δδδδθθθθl////(ββββ - φφφφ) + δδδδΓΓΓΓu////(ββββ - φφφφ) + ππππG////(ββββ - φφφφ) + κκκκWp////(ββββ - φφφφ)

+ (αααα + ββββ - 1)q////(ββββ - φφφφ) + ξξξξ

and representing the combinations of parameters by the hybrids ρρρρ and b0,…,b4 we obtain

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(8) p = ρρρρWp + b0 + b1l + b2u + b3G + b4q + ξξξξ

It is possible to show that, under certain assumptions, this specification is equal to

a well known version of endogenous growth theory. Hence, from the starting point of

equation(8), we can work back, imposing and relaxing various assumptions, to obtain an

underlying static model equal to a very simple form of endogenous growth model known

as the AK model (see Aghion and Howitt, 1998). Assume that λ♦♦♦♦ = 0 and that κ = 0 since

this simplifies the analysis. We also revoke the earlier assumption that k = q, so that our

start point is

p = (φφφφ (k-e))/ββββ + ((ββββ - 1)/ ββββ)q + (αααα/ββββ)k + ξξξξ

Ignoring ξξξξ, integrating with respect to time and rearranging, this leads to

Q = A(K/E)φφφφ Kαααα Eββββ

Assume that φφφφ + αααα = 1, which equates to constant social returns to capital, and

diminishing private returns with αααα + ββββ = 1, it then follows that

Q = AK

In this AK model, the presence of the spillover offsets the diminishing returns and causes

output to increase proportional to capital.

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As explained by Aghion and Howitt(1998), the kind of rationale underpinning the

simple AK model has been a stimulus for further theoretical development (for example

Romer, 1987, 1990). Implications of this include links between the scale of the economy

and growth, and divergence in incomes per capita across regions or countries. Also, the

theory extends to incorporate imperfect competition, a development which is also the

hallmark of new economic geography theory as outlined below.

4. THE PREFERRED SPECIFICATION AND NEW ECONOMIC GEOGRAPHY

THEORY

In this section we show that the preferred model can be obtained on the basis of

assumptions underpinning ‘new economic geography’ theory (Fujita, Krugman and

Venables 1999, Krugman 1991). In order to demonstrate this we adapt a version of this

genre of theory as it is presented by Ciccone and Hall (1996). We again commence with a

Cobb-Douglas production function, but this is specific to production of a final

manufactured good

(9) Q = (MββββI(1-ββββ))αααα

in which Q is the amount of final good produced, M is the amount of labour used directly

to make Q measured in labour efficiency units, and I is the amount of (immobile)

intermediate products. The production is of Q per unit area, and therefore as the level of

output rises, one might anticipate that there are diminishing returns because of the effects

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of increasing congestion. The parameter αααα < 1 defines diminishing returns due to the

congestion effects. The ββββ ( < 1) parameter determines the importance of I to the amount

of final production. Recent developments in new economic geography are largely based

on the assumption of a constant elasticity of substitution production function (Fujita,

Krugman and Venables 1999, Dixit and Stiglitz 1977). This allows one to avoid the

unrealistic assumptions of perfect competition, and yet impose market structure

assumptions at the micro-level, namely monopolistic competition, which allow increasing

returns to scale internal to the individual firm. We thus assume a constant elasticity of

substitution sub-production function for I , so that

I = [∩∩∩∩♦♦♦♦ i(t)1/µµµµ dt]µµµµ

There are x available varieties of intermediate good and i(t) represents the amount of

variety type t in the assumed continuum of varieties. The µµµµ parameter (> 1), which is

equal to the profit maximising price to marginal cost ratio, controls the importance of

variety, and as µµµµ increases there is less substitution among the differentiated goods and

more monopolistic power available to producers of intermediate products. Given µµµµ, it is

possible to deduce the level of intermediate goods input at the zero profit point, which it

turns out is a constant across varieties, and also intermediate good productivity. The

level of output of intermediate goods i depends on wages w, µµµµ and labour efficiency units

i + s used to produce i . Production costs are w(i + s) and revenues are prices charged

times output (wµµµµi), and with intermediate firms freely entering the market until revenues

just equal costs the level of output i is s /(µµµµ-1). The latter is constant across varieties of

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intermediate goods and it therefore follows from the constant elasticity of substitution

sub-production function that I = xµµµµi, and since this requires xi , it is then the case that

intermediate good productivity is xµµµµ - 1 so that there is a positive relation between the

number of intermediate goods varieties an area has and productivity (Ciccone and Hall,

1996). Also as a consequence of the constant elasticity of substitution production

function within the Dixit-Stiglitz theory, an increase in the scale of production in an area

increases the variety of intermediate goods. This follows from the fact that we have

constant intermediate goods inputs, so that the number of varieties (x) can be obtained as

a result of dividing the total amount of labour efficiency units employed producing

intermediate goods, (1- ββββ)N, by the common amount of labour efficiency units used to

produce a variety (i + s), and this indicates that the number of varieties is proportional to

the total labour efficiency units N (used both for final and intermediate goods

production). The term (1- ββββ)N is based on the equilibrium allocation of labour efficiency

units to final goods production as a consequence of equation (9), which shows that the

share of Q accruing to M is ααααββββ, so that wM = ααααββββQ, while wN = ααααQ since the share not

going to labour is (1-αααα)Q. Hence M/N = ββββ. Given x it is possible to calculate I and this

combined with M in equation (9) leads to the following relationship between final good

production and total labour efficiency units,

Q = φφφφNγγγγ

in which φφφφ is a function of other constants and γγγγ is the elasticity with

γγγγ = αααα[1+(1-ββββ)(µµµµ-1)]

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From this we see that

(10) ln(Q/N) = ln(φφφφ)/γγγγ + [(γγγγ-1)/ γγγγ]ln(Q)

Assuming that the same relationship deduced for the micro-scale holds for regions, so

that k′′′′Q = φφφφ(k′′′′N)γγγγ, with vector k′′′′ representing scaling factors applied to unit areas. It then

follows that

(11) ln(Q/N) = ln(φφφφ)/γγγγ + [(γγγγ-1)/ γγγγ]ln(k′′′′Q)

indicating that as we go to regions with higher levels of output, we see higher

productivity levels. However, with large regions, it may be the case that the micro-level

increasing returns are not very evident. Alternatively, if it is assumed that k′′′′Q = k′′′′φφφφNγγγγ, in

other words the coefficient on k′′′′ is equal to 1 rather than γγγγ > 1, then the connection

between productivity level and output level is equation (10) not (11). If this is the case

we should see little relationship between the levels of regional productivity and regional

output, and this does appear to be a finding in a number of empirical analyses (see

Fingleton and McCombie, 1998). However, as it turns out, the time constant variable k′′′′

makes no difference to the analysis pursued below, which focuses on change over time

rather than levels, and hence k′′′′ is omitted from the equations that follow.

In addition, since M = ββββN, it is apparent that

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ln(Q/M) = ln(φφφφ)/γγγγ + [(γγγγ-1)/ γγγγ]ln(Q) – ln(ββββ)

Assume labour input in efficiency units at time t is

Mt = EtAt = Et A0 eλλλλt

in which Et is the level of final product employment, At is the efficiency level at time t

and λλλλ is the rate of technical change. This means that

ln(Q/E) = ln(φφφφ)/γγγγ + [(γγγγ-1)/ γγγγ]ln(Q) – ln(ββββ) + ln(A0) + λλλλt

Following the arguments of section 3, and anticipating the subsequent empirical

evidence, we now include additional effects in the specification. Assume that the rate of

technical progress is a function of urbanisation (u), peripherality (l), the level of

technology gap (G), and the rate of productivity growth in ‘neighbouring’ regions (Wp).

The choice of these variables reflects an assumption that core urban regions inherently

adopt or create innovations at a faster rate because of the larger demographic scale of

such regions. Alternatively, since labour efficiency is commonly attributed to schooling,

it is possible that the technical progress is a function of urban/rural and core/periphery

schooling differences. In addition we again assume that low technology regions benefit

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more from adopting new technology and that policy instruments also have a bigger effect

in raising the technical progress rate of the lower technology regions. With regard to Wp,

the assumption is that if, for whatever reason, firms in ‘neighbouring’ regions have rapid

productivity growth, this will stimulate technical progress as a response, hence

λλλλ = b0 + b1l + b2u + b3G + ρρρρWp

We view the effect of Wp to be a consequence of pure external economies involving

knowledge spillovers. Differentiating (18) with respect to time, and adding an error term,

we again obtain the reduced form encountered earlier as equation (8), thus

(12) p = ρρρρWp + b0 + b1l + b2u + b3G + b4q + ξξξξ

in which p is the exponential growth rate of final good productivity and q is the

exponential growth rate of final good output. It is apparent that in equation (12) b4 would

normally be expected to take a value greater than 0 with increasing returns. Equivalently,

as noted by Ciccone and Hall(1996), the elasticity γγγγ will tend to exceed 1. It does

depend partly on the decreasing returns (αααα) to the level of input of M and I to final

production, which represents the effects of congestion as the activity in the area

increases. However this one would expect to be offset by the advantages provided by

production in an area offering a larger variety of intermediate goods. While intermediate

goods need to be relevant (ββββ < 1) there is no possible offsetting of the congestion effect

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unless they are also imperfect substitutes for each other (µµµµ > 1 and is large enough) so

that a larger variety becomes relevant.

To summarise, the theory outlined above leads to a specification in which

productivity growth is positively related to output growth because the latter is associated

with an increase in the variety of inputs as the density of activity increases, and the

effects of increased variety more than offsets congestion effects also associated with the

increasing density. We have also hypothesised that productivity growth is stimulated by

the rate of technical progress which is faster in urban regions and core regions, on

productivity growth in ‘neighbouring’ regions reflecting pure external economies, and on

the diffusion of technology from high technology to low technology regions. Thus, the

preferred model combines concepts taken from endogenous growth theory with new

economic geography theory, the intention being to devise a theoretically sustainable

model that can also be tested against real world data.

5. MODEL ESTIMATION

This section provides empirical support for (a version of) the reduced form seen

to be consistent with the above theory7. The model is fitted to data for 178 NUTS 2

regions of the EU, with each region’s average annual growth rate for manufacturing

output (q) and productivity (p) calculated for the period 1975-1995. The initial level of

productivity gap (G) is as at 1975, and urbanization (u) is treated as a dummy variable,

23

while peripherality (l) is measured as distance from Luxembourg. Details of these data

and of the NUTS 2 regional system are given in the Appendix. The estimated coefficients

of equation (12) given in Table 1 indicate that each of the variables is highly statistically

significant with appropriate signs on the coefficient estimates. The significant coefficient

to standard error ratios suggest that elimination of either Wp or any of q, u, l and G would

result in a mispecified model, so it is not possible to simplify equation (12) by

eliminating any of the variables. For instance, omitting the spatial autoregressive term

Wp, and fitting via OLS the resulting regression8 induces significant residual

autocorrelation, which could be taken as a sign of mispecification error.

There are also indications that adding9 extra variables to model (12) would not

produce significant improvements in goodness of fit. One indication is via the model (12)

diagnostics given in Table 2. The LM error test finds no spatial pattern in the residuals.

If there were significant omitted variables10 it is likely that these would show up as

spatially autocorrelated residuals, since it is probable that an omitted variable would be

spatially autocorrelated and would manifest itself as a residual pattern. One possible

source of additional variables are the spatial lags of the exogenous variables q, u, l and G,

namely Wq, Wu, Wl, and WG , on the premise that the boundaries between regions11 are

easily transgressed allowing the effect of a variable to spill over to neighboring regions.

However, attempts to add the exogenous lags12 to the model produces no significant

improvement in fit.

The same final model, equation (12), results from simplifying a more complex

model. The more complex model treats p as a function of exogenous variables u, l, q, G,

but also includes their spatial lags Wl, Wq, WG, and Wu, together with the endogenous

24

lag Wp. This specification encompasses equation (12) and we are able to reduce back to

equation (12) by eliminating insignificant effects. Hence reducing the more complex

model by eliminating the exogenous lags produces no significant loss of fit, but it is not

possible to eliminate the endogenous lag Wp or the regressors u, l, q, and G. It is

noteworthy that in the more complex model the endogenous lag remains significant in the

presence of the exogenous lags, pointing to a real endogenous effect rather than it simply

proxying for significant exogenous lags that might have been omitted from equation (12).

Table 1,2 near here

While equation (12) appears resistant to change, there is a source of

misspecification which suggests modification. The diagnostics in Table 2 indicate

(random coefficients) heteroscedasticity using a linear specification comprising a

constant and the single (dominant) variable l and also using all four regressors together.

We therefore re-estimate the model to allow for heteroscedasticity as a function of a

core-periphery dichotomy13. The groupwise heteroscedastic estimation results obtained

using SPACESTAT (Anselin, 1999) are given in Table 1. It is apparent that these differ

only slightly from the initial model (12) estimates, although the difference in likelihoods

indicates that the peripheral regions have significantly more variable productivity growth

than core regions. Table 1 also contains the results of Bootstrap estimation (see Anselin,

1988), thus avoiding assumptions of normality and homoscedasticity, and this produces

parameter estimates and standard errors which are quite similar to both sets of ML

estimates. While evidently we could use any of the three sets of parameter estimates

25

given in Table 1 with only marginal impact on subsequent simulations, those obtained via

groupwise heteroscedastic estimation are considered optimal, providing the closest fit to

the data. These are therefore the basis of the empirical simulations described below.

6. CONVERGENCE WITH SPATIAL EFFECTS – SIMULATION ANALYSIS

In order to obtain the expression for equilibrium implied by the preferred model,

it is convenient to represent equation (12), plus heteroscedasticity, by the more compact

equation

p = ρρρρWp + Xb + ξξξξ

ξξξξ~ N(0, σσσσi2I)

in which σσσσi2 denotes error variance conditional on whether region i is in the core or

periphery, X is the n by p♦ matrix of regressors and b is the p♦ by 1 vector of

coefficients. Thus,

(I - ρρρρW)p = Xb + ξξξξ

E(p) = (I - ρρρρW)-1Xb

26

Assume a steady-state exists, and define R as the vector of productivity level

ratios (R = U –G and U is a vector of 1s) then at steady-state the proportional rate of

growth of R, R•/R, equals zero. Since R•/R = E(p – p*), in which p* is the productivity

leader’s productivity growth rate, then

(13) E(p – p*) = (I - ρρρρW)-1Xb – (I - ρρρρW)-1X*b = 0

which can be re-expressed as

(14) E(p – p*) = (I - ρρρρW)-1(X- X*)b = 0

In equations (13) and (14), each row of X* is equal to the productivity leader’s

row of X. The aim is to obtain an expression for Ge and hence the steady-state vector Re,

and this is achieved by removing variable G from X, the result being matrix X♦ which has

one less column, with the equivalent vector of (reduced) coefficients represented by b♦ .

Variable G is placed in the (n x 1) matrix X′′′′′′′′, and the coefficient corresponding to G is

represented by b′′′′′′′′. We can now write

(I - ρρρρW)-1X♦ b♦ + (I - ρρρρW)-1X″″″″b″″″″ – (I - ρρρρW)-1X*b = 0

and rearrange and simplify to obtain

X″″″″ = Ge = (X*b – X♦ b♦ )(b″″″″)-1

27

(15) Re = U - (X*b – X♦ b♦ )(b″″″″)-1

in which Re denotes the steady-state vector of productivity level ratios.

Equation (15) shows that if b″″″″ = 0, no steady-state solution exists, so the presence

of a significant catch-up effect is a necessary condition for the model to converge. Note

also that the steady-state otherwise depends on the regressors, and that the endogenous

spillover (ρρρρWp ) does not form part of the expression for steady-state. This might appear

surprising given the significance attributed to ρρρρ in the empirical analysis. The reason is

that at equilibrium p is constant across regions, and since Wp is the weighted average of

this constant it too is a constant and thus cannot affect interregional productivity level

ratio differences at equilibrium. However, the out-of-equilibrium dynamics are affected

by the endogenous lag, as we show subsequently in our simulations. Also of course, Wp

has a hidden effect since its presence precludes bias in the estimates of the parameters in

equation (15).

In fact it is also possible to calculate the equilibrium productivity ratio vector Re

using the iteration given by equations (16a) to (16e). The advantage of the iterative

approach is that it allows us to obtain a picture of disequilibrium dynamics which might

occur as the result of assuming gradually diminishing catch-up and spatial interaction

effects as the system evolves to steady-state, and it enables the leading productivity

region to change and allows us to introduce stochastic disturbances14. In the equations,

the estimated coefficients ρρρρ and b are combined with the matrix W and the matrix of

regressors Xt, starting at iteration t = 1 using the observed values of l, u, G and q. The

28

subscript indicates that (part of) matrix Xt is updated in each cycle. Thus, column v of Xt

changes with t, since Gt = 1-Pt/Pt*= 1- Rt

changes as the productivity levels Pt and Pt*

change with expected productivity growth rates E(pt) and E(pt*). The other variables

remain fixed since they are assumed not to depend on E(pt).

(16a) E(pt) = (I - ρρρρW)-1Xtb

(16b) Pt+1 = Ptexp(E(pt))

(16c) P*t+1 = P*

texp(E(p*t))

(16d) Gt+1 = 1 – (Pt+1/ P*t+1)

(16e) Xt+1,v = Gt+1

Note however that from equation (5), W also depends on P. We therefore endogenise W

so that

W♦♦♦♦ij,t+1 = Ej,t+1

αPj,t+1α/dij

γ

Wij,t+1 = W♦♦♦♦ij,t+1 / ΣΣΣΣj W♦♦♦♦

ij,t+1

For simplicity it is assumed that E is constant across regions and grows at a constant rate.

This assumption means that the active component determining the development of W is

29

the evolution of P. Also, as E(p) becomes constant across regions as they converge to the

steady state, W also becomes constant.

7. SIMULATING THE EU STEADY-STATE

In this section we explore the distributional consequences of different

assumptions regarding the growth of output (q). The growth of output (q) is assumed to

be exogenous, depending on factors outside the model. The output growth distribution

has implications for the distribution of technology in the steady-state. If, apart from

catch-up, productivity growth is determined solely by the exogenous variable q, then if q

is spatially constant all regions converge to the productivity level of the leading region.

This is apparent from the expression from equation (15), since in this case X*b – X♦ b♦ =

0 and thus Re = U. This will not be the case if differences between regions’ output

growth are assumed to exist. This is illustrated in a ‘laboratory’ simulation comprising

an imaginary set of 10 regions with assumed values15 for X,W and b and with ρ = 0.6.

The dynamics of the imaginary landscape are illustrated by Figure 1 which plots the

productivity level ratios against iteration number. Endogenising W produces a similar

outcome. However, we have seen with our data analysis that in the real world p depends

on several exogenous variables. This has the consequence that for some regions faster

than average output growth may compensate for the negative impact of peripherality and

a rural location, allowing technology levels to move closer to the technology leadership

than would otherwise be the case. In the simulation that follows, the other exogenous

variables (l, u) are held at their historical levels to isolate the effect of different q

30

assumptions. These are combined with the parameter estimates of the space lag model

with groupwise heteroscedasticity given in Table 1, which, since they are not

endogenously determined in the model, are also assumed to hold constant to isolate the q

effect. With regard to the coefficient on q, this may not be unreal, given the empirical

evidence in the literature relating to the Verdoorn Law per se (already cited above) in

which estimates in the vicinity of 0.5 have been found using different sets of data.

It is important to appreciate how changing q for some regions impacts p in other

regions. It is useful to observe that since E(p) = (I - ρρρρW)-1Xb, assuming ρρρρ < 1, it is

also true, as pointed out for instance by Kelejian and Prucha(1998), that

(17) E(p) = (∑ρρρρi W i)Xb

where the summation is from i = 0 to ∞, W0 = I, W2 is the matrix product of W and W,

W3 is the matrix product of W2 and W and in general Wi is the matrix product of Wi-1 and

W . We re-express this as

(18) E(p) = Xb + ρρρρ W Xb + ρρρρ2 W 2Xb + ρρρρ3 W 3Xb + ρρρρ4 W 4Xb......

with the W matrix powers generally defining different remote inter-regional connections

to those of W, so that E(p) at any one location depends on q and thus X to infinite spatial

lags.

31

We first assume that q will be spatially constant. This assumption follows as a

possible consequence of the development of a highly integrated single European

economy. We characterize economic integration by very low barriers and enhance market

penetration. In this scenario, regions experiencing low demand growth benefit from the

growth of demand in other regions, and vice versa, so that there is no marked spatial

variation in the growth of output across regions. The set of regions behave collectively as

a single region. Consequently, output growth in each region is set equal to 5% per annum.

Combined with the parameter estimates from Table 1 in each iteration, the outcome is

Figure 2. Well before 100 iterations, productivity growth has become almost equal

across regions and thus the ratios of productivity levels (R) tend to the horizontal. Figure

2 summarizes the dynamics in terms of mean, maximum and minimum R, distinguishing

between the total set of regions and the Objective 1 regions (which have the highest level

of financial assistance from the European Union). We see from Figure 2 an unequal

steady-state with the Objective 1 regions having a lower mean, although overall the level

of inequality reduces as the dynamics evolve towards the steady-state. This is evident

from the progression of the Gini coefficient16 for the total set of regions (the 0.05 line in

Figure 5). The steady-state is also characterized by highly significant spatial polarization,

and on the basis of the evidence from the standardized17 values of Moran’s I (the 0.05

line in Figure 6), regions form clusters so that the pattern deviates considerably from

what one would expect from a random allocation18.

We now use our model to look at the impact of increasing demand in the

Objective 1 regions in order to see if faster output ameliorates the steady-state

distribution. The range of alternative growth rates considered is from 6% to 10%, while

32

output growth in the unassisted regions is held constant at 5%. A summary of the

dynamics is given by Figures 3 and 4, corresponding to Objective 1 region output growth

rates of 8% and 10%. Figure 3 shows that faster Objective 1 output growth makes the

mean R for Objective 1 regions approach the overall mean, and the lesser inequality in

the distribution is reflected in the smaller range compared to that of Figure 2. Figure 4

shows Objective 1 mean R surpassing the total mean R as equilibrium is approached, an

Objective 1 region with maximum R, and a non-Objective 1 region with minimum R.

Figure 5 shows that in terms of inequality as measured by the Gini coefficient,

assuming Objective 1 output grows faster than in the unassisted regions reduces

inequality. However the most equal distribution, with a Gini coefficient equal to 0.05371

in the steady-state, occurs when we assume that OBJECTIVE 1 output grows at 8%,

faster Objective 1 growth increases inequality as non-Objective 1 regions are left behind.

Figure 6 shows that increasing the output growth in the assisted regions progressively

lowers, by a very small amount, standardized values of Moran’s I for the steady-state

productivity ratios, but the patterns remain very significantly different from random.

Figure 1,2,3,4,5,6,7,8 near here

While faster growth, up to a point, in the assisted regions tends to reduce

inequality, it could be argued that the cost of improved regional equity is too high and the

outcome only marginally better than doing nothing. However this would be to ignore the

other consequences of unilaterally boosting output growth in the Objective 1 regions.

Figure 7 shows that increasing output growth in the Objective 1 regions alone boosts

33

productivity growth in the unassisted regions also, so that the level of technology is

higher in both assisted regions and in unassisted regions, as anticipated by equations (17)

and (18). Figure 7 compares productivity levels and growth for two sample regions,

Greater London and Ireland, the latter being Objective 1, assuming first the same output

growth (5% p.a.) across the EU regions, and then assuming faster output growth (10%

p.a.) in the Objective 1 regions. With equal output growth, Greater London achieves a

level of productivity that is permanently higher than that of Ireland. This implies

permanent regional aid to promote social and economic cohesion. With output growth

boosted in the assisted regions, both regions have higher productivity levels at any point

in time at steady-state. The steeper lines of Figure 7 show that at equilibrium productivity

grows faster in both regions than it does without additional output growth in the

Objective 1 regions. The comparative closeness of the steeper lines reflects more similar

levels of productivity (in fact Ireland has a slightly higher manufacturing productivity

level at any time in the steady-state). Similar comparisons can be made for other regions

and for other rates of growth.

Assume the reverse, so that output growth in the unassisted regions is

permanently higher than in the assisted regions. This may be a more likely outcome than

equal output growth since manufacturing production could benefit from being located

close to markets and suppliers in the core regions of the EU which are in general not

assisted by strong regional policy. Figure 8 shows the consequences for the two sample

regions, with faster growth in the core region spilling over. Permanently faster output

growth in the unassisted regions speeds up productivity growth in both assisted and

unassisted regions. Figure 8 shows that allowing faster core output growth (10% versus

34

5%) steepens the lines considerably, indicating permanently faster productivity growth in

equilibrium. The upward shift of the lines reflects the higher technology levels. It is also

evident that boosting core growth increases inequality so that Greater London’s

technology level is even higher and this is represented by the wider separation of the

equilibrium paths. Likewise Figure 5 illustrates the increase in inequality across all EU

regions (the –0.05 line), as measured by the Gini coefficient, and Figure 6 shows that

spatial clustering in the steady-state (again indicated by the –0.05 line) is at a level very

similar to that which obtains from a policy of boosting output growth in the assisted

regions. Hence, faster output growth in core regions does benefit productivity levels and

growth rates in the assisted regions, but in this simulation increases inequality and does

not reduce spatial clustering.

What this information implies for policy involves a weighing up of costs and

benefits. The benefits of ‘allowing’ core regions to grow faster are that the Objective 1

regions also gain because of spatial interaction effects, and in any case faster core output

growth may be a ‘natural’ outcome of market forces that is difficult to change. The

disadvantage is that evidently such a policy increases inequality, and even though, under

this model, it produces faster productivity growth and higher levels, it is unlikely to be

popular in assisted regions. Such a policy would also result in permanent but

demonstrably ineffective regional assistance that would be difficult to justify. On the

other hand, the main benefit of an alternative policy of boosting output growth in the

Objective 1 regions is enhanced productivity growth in both assisted and unassisted

regions and reduced inequality and therefore reducing costs of regional assistance. Apart

from convincing policy makers that this analysis is correct, the main disadvantages would

35

seem to stem from the practical and political difficulties associated with finding

appropriate and effective policy instruments. This is a difficult problem, since boosting

output growth implies a renewed emphasis on the demand side rather than the supply side

policies which have tended to be the dominant feature of EU regional policy.

8. CONCLUSIONS

The future, long term, economic prospects of regions is an important issue for

many governments and other agencies concerned with human progress and welfare. It is

very difficult to know what will happen in the future, but we can say something about

what might happen from our understanding of historical trends, using hopefully

reasonable assumptions about relationships between variables over the longer term. This

is the context of this paper, which endeavours to fit a model to data for the regions of the

EU, and then say something about what this implies long-term.

The basic assumption of the paper is that we can estimate a model from current

data, and that the model has some validity because it replicates the observed data with

some reasonable accuracy, and also because it embodies or represents some fundamental

theoretical forces which have logical coherence and support from the body of theoretical

work. As it turns out, the empirical model we fit has its origins in the Verdoorn Law

which has been shown to be reasonably good at replicating the real world pattern of

manufacturing productivity growth simply as a function of manufacturing output growth.

However we move away from the simple productivity-output growth nexus that this

entails to also include explicit terms for the determinants of technical progress, and this

36

fits rather better. We find that productivity growth in the EU is also faster in core rather

than peripheral regions, in urban rather than rural regions, in low technology regions and

in regions 'surrounded' by regions with fast growing productivity.

Moreover we are able to show in this paper that the preferred model is consistent

with the new developments in theory which have appeared in recent years, notably new

economic geography, and we see this as adding to its appeal. This extra theoretical

coherence is attractive, since one can view the model as an empirically testable version

of new economic geography, going some small way to help bridge the gap between what,

thus far, has been a largely abstract body of theory and empirical reality. This bridge

building exercise is very much on the research agenda mapped out for new economic

geography in the final, ‘The Way Forward’, chapter of Fujita, Krugman and Venables

(1999). Hence, the model attempts to incorporate the forces such as spatial interaction

and regional policy which are the type of empirical reality largely assumed away in the

extant body of new theory, with the result that modified or constrained simulation

dynamics occur which, as it turns out, mimic the conditional convergence produced, for

different reasons, by models with a neoclassical, diminishing returns, theoretical basis.

The outcome is convergence to a differentiated steady state in which consistent level of

development differences between regions persist. Clearly, in this set up, there exists a

critical point, equal to a zero catch-up parameter, about which dynamics bifurcate from

convergent to divergent regions. However, there is no attempt to model sudden

catastrophic change, as in Casetti(1981a,b) (see Vining, 1982 and Clark, Gertler and

Whiteman, 1986, for a critical view of this), or punctuated equilibria as critical points are

passed and regions move from one equilibrium configuration to another, as in Fujita,

37

Krugman and Venables (1999). Evidently the simulation exercise falls more in the realm

of ‘quantification’ as described by Fujita, Krugman and Venables (1999), meaning a

‘theoretically consistent model whose parameters are based on some mix of data and

assumptions, so that realistic simulation exercises can be carried out’, in other words

‘computable geographical equilibrium models’. This kind of modeling is acknowledged

to be difficult, but the payoff would be a major advance in the progress of theoretical

economic geography as a predictive discipline, capable of evaluating how an economy’s

spatial structure might be influenced by hypothetical events, such as sudden changes in

policy or simulated environmental shocks. Hopefully this paper leads the way towards

this goal.

Acknowledgement

I wish to thank the Editors and referees for their most helpful comments and suggestions

relating to the earlier version of this paper. The paper was initially presented at the 39th

European Congress of the European Regional Science Association, held in Dublin,

Ireland, in August 1999. I am grateful to the organizers for making that possible.

38

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42

Table 1: Maximum Likelihood Groupwise Heteroscedastic Estimates

VARIABLE COEFF S.D. z-value Wp [0.642234]

(0.729364) 0.650123

[0.0893122] (0.169942) 0.128873

[7.190886] (4.291834) 5.044684

CONSTANT [-0.0192694] (-0.021555) -0.019942

[0.00397344] (0.0053057) 0.00419611

[-4.849564] (-4.06268) -4.752487

q [0.495972] (0.486726) 0.444352

[0.0601994] (0.0595534) 0.0614415

[8.238824] (8.172928) 7.232119

G [0.0642047] (0.064448) 0.066450

[0.00844908] (0.0080925) 0.00767723

[7.599010] (7.964042) 8.655467

l [-1.4195E-05] (-1.478E-05) -1.4311E-05

[2.59694E-06] (2.6035E-06) 2.48029E-06

[-5.465922] (-5.67777) -5.769887

u [0.00834755] (0.0084096) 0.00929659

[0.00303875] (0.0031395) 0.00279149

[2.747032] (2.67865) 3.330328

Summary regions 178 Variables 6 R2 [0.4619]

(0.4830) 0.4947

Sq corrn. [0.5164] (0.4697) 0.5447

Likelihood [522.212] 526.622

AIC [-1032.42] -1041.24

SC [-1013.33] -1022.15

σ2

periphery core

[0.00016279] 0.000205032 0.000101419

The figures in () are the results of 999 bootstrap replications. The figures in [] are the results of fitting the spatial lag model via ML

43

Table 2: Diagnostics for the Spatial Lag Model

TEST DEGREES OF FREEDOM

VALUE PROB.

HETEROSKEDASTICITY Breusch-Pagan 1 62.897068 <0.000001 Spatial B-P test 1 62.941813 <0.000001 Breusch-Pagan 4 69.360529 <0.000001 Spatial B-P test 4 69.416944 <0.000001

SPATIAL DEPENDENCE Likelihood Ratio (Lag) 1 26.036358 <0.000001

Lagrange Multiplier (error) 1 0.024267 0.876207

44

APPENDIX

THE DATA

Peripherality (l) is represented by great-circle distance from Luxembourg. Urbanization

(u) has two levels, urban (above 500 people per sq. km) and rural. The growth of

manufacturing productivity (p) and manufacturing output (q) are exponential growth rates

for the period 1975-95. Variables p, q and G are calculated from levels series of

manufacturing (which includes energy) employment and constant 1985 prices

manufacturing ‘output’ (Gross Value Added) available in Cambridge Econometrics’

European Regional Databank. The data used are a more or less contiguous subset of 178

regions of the EU covering 13 member states. Thus the data are for NUTS 2 regions of

Germany ( including West Berlin and excluding the Eastern Lander), France (excluding

Départements d’Outre-Mer such as Gaudeloupe), Italy, Belgium, Luxembourg, the UK,

Ireland, Denmark, Greece, Spain (excluding the North African territories of Ceuta y

Melilla), Portugal (excluding non-continental and overseas territories such as the Azores

and Madeira) and Austria (which only joined the EU in 1995).

Although varying considerably in size, the NUTS 2 region, defined by Eurostat, is

the most appropriate unit for modeling and analysis, being sufficiently small, in most

cases, to capture sub-national variations, and it is the unit adopted by the EU to define

Objective 1 regions for Structural Funds purposes. It is also the level at which a

comprehensive data set is provided by Eurostat. In contrast, data problems would be

more acute were the modeling based on the smaller NUTS 3 regions. In cases where no

45

NUTS 2 region is defined, the unit chosen is at the NUTS 1 level, for example Hamburg,

Athens and Madrid.

The Eurostat REGIO database is the prime source of raw data for the database,

which consists of comprehensive series of economic and demographic indicators at the

NUTS 2 level and above. In order to achieve this, it has been necessary to reconstruct

data missing from the source. Where an incomplete series exists at NUTS 2 levels,

interpolation methods have been used which fill gaps in the series from complete series

available for aggregates of NUTS 2 regions. The totals of regions containing interpolated

values are constrained to sum to known totals at higher levels of the spatial hierarchy. In

this way, a detailed series has been built up which is consistent with the higher order

regional values available in published statistics.

The Eurostat regional accounts (REGIO) employment series are defined by

establishment, and these are the usual source of the employment series in the European

databank. However, in the case of Italy, the total employment series was provided by

Irpet, with employment equal to working population minus unemployment, and

employment by sector based on sharing total employment according to the pattern in the

REGIO series. The employment series for the Netherlands refer to employment by

residence, and were provided by Nederlands Economisch Instituut. The employment

data for Austria refer to employees rather than total employment.

Finally, since Groningen and Flevoland have anomalous manufacturing GVA

and GVA per worker values (largely due to fluctuations in gas production in Groningen

and possibly also due to commuting), in these cases the Dutch national averages were

used. In the case of Hamburg, it is apparent that commuting may also be a distorting

46

influence because of the (NUTS 1) region’s small spatial extent. Hence a more

appropriate definition of the city was used, the travel to work area (ROR05) which

comprises the NUTS 1 region of Hamburg and the surrounding Kreise that qualify as part

of the functional urban area. For example, in 1990, the Hamburg TTWA had a population

of 2.9m people, compared with 1.6m people for the NUTS 1 region.

THE FIGURE 1 SIMULATION DATA The W matrix 0.00000 0.00013 0.02368 0.04210 0.06578 0.09472 0.12893 0.26312 0.21313 0.16840 0.00004 0.00000 0.00603 0.04286 0.06698 0.09645 0.13127 0.26791 0.21700 0.17146 0.00307 0.01230 0.00000 0.04919 0.01922 0.01230 0.15065 0.30746 0.24904 0.196770.00290 0.01162 0.02614 0.00000 0.00090 0.10456 0.14231 0.29044 0.23525 0.185880.00291 0.01162 0.02615 0.00046 0.00000 0.10460 0.14238 0.29056 0.23536 0.185960.00073 0.00129 0.02611 0.04642 0.07253 0.00000 0.14215 0.29011 0.23499 0.185670.00075 0.00134 0.02713 0.04824 0.07537 0.10853 0.00000 0.30148 0.24420 0.192950.00567 0.02270 0.05106 0.09078 0.14184 0.20426 0.06950 0.00000 0.05106 0.363120.00581 0.02324 0.05229 0.09296 0.14526 0.20917 0.28470 0.14526 0.00000 0.041320.00035 0.00315 0.02838 0.05046 0.07884 0.11352 0.15452 0.31535 0.25543 0.00000Initial productivity levels 1 2 3 4 5 6 7 10 9 8

The X matrix

1 0.01 0.9 1 0.03 0.8 1 0.05 0.7 1 0.03 0.6 1 0.07 0.5 1 0.08 0.4 1 0.04 0.3 1 0.09 0.0 1 0.02 0.1

47

1 0.01 0.2 The b matrix 0.00001

0.2 0.2

1 The slope coefficient of the simple linear equation linking productivity growth and output growth.

2 The dynamic Verdoorn Law can also be viewed as an early form of endogenous growth theory (Aghion

and Howitt, 1998) when represented as a linear technical progress function relating output growth to the

investment rate (Kaldor, 1957 Dixon and Thirlwall,1975a,b).

3 This is a simplification of the Verdoorn effect as it is commonly presented, which as we noted earlier

argues that the increase in productivity may also be due to a higher rate of induced investment and induced

technical progress.

4 The empirical basis for omitting k is the ‘stylized fact’ that capital stock growth and output growth are

approximately the same in most developed economies. This assumption is supported by the results of

empirical tests. For example McCombie and Thirlwall(1994) found that regressing k on q for a sample of

developed countries produced a regression slope coefficient that was not significantly different from unity.

While data on capital stock growth per se do not exist for the 178 EU (NUTS 2) regions used in this study,

as an alternative to the above restriction, one might consider proxying k by the average share of real

(gross) equipment investment in GDP, or more accurately by the net investment-output ratio and assume

that the capital-output ratio is constant ( McCombie and Thirlwall, 1994). Unfortunately these data do not

exist either. 5 Clearly there is considerable scope for alternative specifications of the W matrix, and it should be

appreciated that the model and diagnostics relating to the empirical fit of the model are based on W as

defined here.

6 Ideally we should like to estimate these, but that would involve a highly complex estimation process

which is rarely, if ever, attempted. It is believed that these assumed coefficients have a more realistic basis

than others which might have been adopted. For example, assuming α = 0 corresponds to a pure distance

effect, while γ = 0 is a pure ‘size’ effect, neither of which appears reasonable in the context. The data

48

analysis shows a high level of significance attributed to the spatial effect using the assume coefficients,

suggesting that they have been reasonably appropriately chosen. However it is the author’s experience that

in these circumstances slightly different coefficient assumptions do not radically alter the results obtained.

7 It is assumed that q is exogenous. In fact since p = q - e, a minor problem is introduced by the presence of

q on either side of the equation. This is best seen via the simplified relation p = a0 + a1q which is entirely

equivalent to e = -a0 + (1-a1)q, so there is no real consequence for the parameter estimates although the

latter specification provides the correct R2.

8 The standardised value so Moran’s I equals 5.89 which exceeds the critical value of 1.96 in the N(0,1)

distribution, and the LM error dependence statistic is equal to 20.09 which is highly significant when

referred to the chi-squared distribution with one degree of freedom.

9 See Florax and Folmer (1992) and Florax, Folmer and Rey (1998) for a discussion of model selection

strategies in spatial econometrics.

10 Such as industrial structure (ie the share of manufacturing in total employment) as implied by

Baumol(1967), or the growth of real wages.

11 The NUTS 2 regions are on the whole formal administrative units rather than self-contained functional

economic regions.

12 Space limitation mean that it is not possible to provide full details of the estimation of alternative

specifications mentioned here, but details are given in Fingleton(2000a) .

13 Dichotomised with core regions defined as those less than 500km from Luxembourg, with peripheral

regions at least 500km from Luxembourg.

14 In fact the present paper is restricted to exploring the deterministic steady-state. Stochastic dynamics are

studied in Fingleton (1999c, 2000a).

15 Given in the Appendix

16 A coefficient equal to zero indicates complete equality, with each region having the same productivity

level ratio. The coefficient approaches 1 as the ratios tend to zero (apart from the productivity leader). In

practice the maximum value depends on the number of cases.

17 Using as an approximation the randomisation moments (Cliff and Ord 1981). Tiefelsdorf and

Boots(1995) provide the exact distribution.

49

18 Note that the W matrix used to calculate Moran’s I is not as in the simulation, but is a symmetric matrix

in which Wij = 1/dij where dij is the great circle distance between regions i and j.