Agglomeration and demographic change

Journal of Urban Economics 74 (2013) 1–11

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Journal of Urban Economics www.elsevier.com/locate/jue

Agglomeration and demographic change Theresa Grafeneder-Weissteiner a,⇑, Klaus Prettner b a b

University of Vienna, Department of Economics, Hohenstaufengasse 9, 1010 Vienna, Austria Harvard University, Center for Population and Development Studies, 9 Bow Street, Cambridge, MA 02138, USA

a r t i c l e

i n f o

Article history: Received 12 November 2010 Revised 13 August 2012 Available online 26 September 2012 JEL classification: F15 J10 C61 R12

a b s t r a c t This article investigates common consequences of demographic change and economic integration for the spatial location of economic activity. In doing so, it provides a unified framework that introduces an overlapping generation structure into a New Economic Geography model. Whether integration leads to agglomeration crucially hinges on the demographic properties of economies. While population aging strengthens concentration tendencies, population growth acts as a dispersion force. This is consistent with the stylized relationships between demography and urbanization found in the data and thus allows us to assess the possibility of agglomeration in various demographic scenarios. Ó 2012 Elsevier Inc. All rights reserved.

Keywords: Agglomeration Constructed capital model Population aging Population growth

1. Introduction Worldwide economic integration during recent decades has led to renewed interest by economists and policy makers in the location of economic activity (Fujita and Thisse, 2002; World Bank, 2009). Within the European Union, for example, regional cohesion policies are at the top of the political agenda. At the same time, as illustrated by a special report on aging in The Economist (2009), demographic change has created serious economic challenges for industrialized countries. In particular, declining fertility has caused upward shifts in the mean age in most countries while simultaneously reducing population growth rates (see Eurostat, 2009; United Nations, 2007). Until now, agglomeration and demographic change have mostly been analyzed independently of each other, with New Economic Geography (NEG) frameworks addressing the impact of deeper economic integration on the spatial concentration of productive factors and overlapping generation (OLG) models investigating the effects of demography on macroeconomic aggregates. This division is unfortunate for several reasons. First – from a theoretical point of view – both demographic change and international integration crucially affect demand patterns, which are decisive

for the spatial distribution of economic activity. Second, empirical evidence suggests the existence of a close relationship between urbanization and demographic transition on the one hand (see Zhang, 2002; Sato and Yamamoto, 2005) and economic development, international integration and demographic change on the other hand (see Durlauf et al., 2005 for an overview of related empirical studies). Explaining agglomeration processes without accounting for demographic developments thus seems to miss a fundamental point. This is also indicated by Fig. 1, which illustrates the relationship between agglomeration and demography, both from a time series and a cross section perspective. The first two diagrams show the evolution of the share of the population living in cities (U), population growth (n), and the share of the population older than 65 (s65) between 1965 and 2010, for the aggregate world economy. While urbanization and the share of the population older than 65 have been steadily increasing, population growth has been slowing down. The results for 189 countries in 2010, displayed in diagrams 3 and 4, are consistent with the time series data: countries with faster population growth tend to feature less urbanization (see also Bloom et al., 1998, for African countries), while the converse holds true for countries with a larger share of the population older than 65.1

⇑ Corresponding author. Fax: +43 1 4277 9374. E-mail addresses: [email protected] (T. GrafenederWeissteiner), [email protected] (K. Prettner). 0094-1190/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jue.2012.09.001

1 The simple correlation coefficients are 0.3170 in the former and 0.5023 in the latter case, both being statistically significant at the 1% level.

2

T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

Fig. 1. Urbanization and demography. Note: The first two diagrams show the evolution of the share of the population living in cities (denoted by U and represented by the solid lines), population growth (denoted by n and represented by the dotted line) and the share of the population older than 65 (denoted by s65 and represented by the dashed line) between 1965 and 2010 for the aggregate world economy. Diagram 3 displays the relationship between U and n for 189 countries in 2010 and diagram 4 shows the relationship between U and s65 for the same set of countries. The data were obtained from World Bank (2012).

Despite this evidence, analysis of the interrelations between demographic change and agglomeration has not attracted much attention so far. To our knowledge, there are three articles that try to address them.2 Gaigné and Thisse (2009) provide a theoretical explanation for the empirical finding of Chen and Rosenthal (2008) that older individuals tend to agglomerate in places with attractive consumer amenities, while younger individuals prefer cities with high quality business environments. In their analysis Gaigné and Thisse (2009) assume that only young people work, while the older generation is retired. Young individuals prefer to work in cities with a thriving business environment and are willing to incur the associated congestion costs, while retirees primarily care about consumption opportunities and tend to settle in places, where these are abundant. Sato and Yamamoto (2005) approach the topic from a historical perspective and aim to explain demographic transition, urbanization and increases in wages over the course of industrialization within one framework. They assume that rural production processes are subject to diminishing returns of labor because of the limited production factor of land, while urban production processes allow for increasing returns due to agglomeration economies. However, city inhabitants have to incur time costs related to commuting. Because costs of having children are measured according to time spent for child rearing, and because city inhabitants have less available time due to commuting costs, their fertility rates are lower. An exogenous rise in population size caused by decreasing child mortality leads to higher wages in the city and lower wages in the countryside. The consequence of this is migration towards cities and a subsequent decrease in a country’s overall fertility rate. This in turn generates a rise in wages and, together with the decrease in child mortality, a population growth rate following an inverted U-shape with respect to time.

2 Furthermore, in the public health literature, the aging of urban populations has gained attention since it raises serious challenges for health policy and city planning alike (see Phillipson, 2004; Beard and Petitot, 2010, for an overview)

Zhang (2002) assumes that there is a dual economy consisting of urban and rural areas. Production and human capital accumulation are more efficient in the urban area, with the latter leading to lower fertility among the urban population compared to the rural population. Migrating from rural to urban areas is costly and therefore more likely to occur if the rural population has a higher income that allows them to pay the associated costs. Altogether, this structure leads to a situation, where urbanization happens gradually over time, driving fertility and population growth down while fostering economic development. The findings of Zhang (2002), Sato and Yamamoto (2005) and Gaigné and Thisse (2009) are consistent with the stylized facts on agglomeration and demography presented above. However, when determining the relative strengths of agglomeration and dispersion forces, these frameworks do not elaborate on the channels by which declining population growth and economic integration might interact with each other. Furthermore, all of these studies have to assume that there are initial differences between two regions in order for agglomeration to take place. By contrast, we assume initially identical regions and focus exactly on the interrelations between agglomeration processes, economic integration and demographic change. This also allows us to illustrate the channels by which declining fertility and population aging emerge as causal explanatory factors for agglomeration. In particular, our framework merges two well-established modeling strands, the NEG on the one hand and life-cycle savings models on the other, to account for the interactions between economic integration and demographic change. In doing so, we provide a unified framework that captures the stylized relationships in the data and allows us to theoretically analyze the linkage between demography, economic integration and the spatial location of economic activity. The NEG literature initiated by Krugman (1991) has provided new insights on how to explain the spatial distribution of industries. These models are characterized by catastrophic agglomeration, meaning that beyond a certain threshold level of economic integration, industrial activity completely concentrates itself in

T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

one region. In particular, demand-linked circular causality effects destabilize the symmetric equilibrium with an equal division of productive factors, turning the core-periphery outcome into a stable equilibrium. However, the richness of agglomeration features in these early models has reduced their analytical tractability. Therefore, in his seminal contribution, Baldwin (1999) introduced the constructed capital framework with interregional labor and capital immobility. The spatial distribution of industrial activity in his model is based on interregional differences in capital accumulation rates being the result of the dynamically optimal savings decisions of forward-looking agents. A higher capital rental rate in the home region increases home relative to foreign savings. Catastrophic agglomeration of capital occurs due to the following demand-linked circular causality effect: a higher capital stock raises capital income and therefore consumption expenditures, leading to an increase in the capital rental rate and subsequently to further capital accumulation. Because neoclassical growth models in the spirit of Ramsey (1928) associate capital accumulation with medium-run growth, Baldwin (1999) describes the economy in the region that is accumulating capital as a growth pole, whereas the other region is referred to as a growth sink. This explanation illustrates how economic integration can lead to the emergence of ‘‘rust’’ and ‘‘boom belts’’. The Ramsey (1928) framework of one single, infinitely lived representative agent, on which the constructed capital model’s saving features rely, does not account for demographic change. We therefore generalize the approach of Baldwin (1999) by introducing the possibility of death, in addition to allowing for changes in population size. In so doing, we adopt the overlapping generations structure of Buiter (1988), where heterogeneity among individuals is caused by their dates of birth. While still following the lines of intertemporally optimizing agents, this results in a more comprehensive model that allows for demographic change and nests the constructed capital set-up as a special case. The findings presented in this paper clearly reveal the importance of demographic structures for demand and saving patterns and thus for agglomeration processes. We first show that both population growth and the turnover of generations act as dispersion forces by weakening the demand-linked circular causality effect of the constructed capital model. Population growth dampens the expenditure increase due to more capital because it means that more resources must be dedicated to maintaining the capital stock instead of to consumption. The turnover of generations, on the other hand, is stabilizing because the distributional effects due to birth and death, i.e., the replacement of older individuals with higher consumption expenditures by newborns with lower consumption expenditures, are more severe in the region with the higher capital stock. This implies that the expenditure increase associated with a rise in the capital stock is reduced. Population aging – as represented by declines in fertility – slows down the turnover of generations and thus promotes agglomerative tendencies. This destabilizing effect is strengthened by the decrease in the population growth rate resulting from lower birth rates. In contrast, declines in the mortality rate weaken agglomeration processes because the stabilizing impact of the associated higher population growth rate is stronger than the destabilizing effect of the reduced turnover due to lower mortality. The remainder of the paper is organized as follows. Section 2 presents the structure of the model and derives optimal aggregate savings behavior and equilibrium capital rental rates for both model regions. Section 3 verifies the existence of a symmetric equilibrium and – by analyzing its stability properties – establishes the link between agglomeration and demographic change. Finally, Section 4 summarizes the findings and draws conclusions for economic policy.

3

2. The model This section describes how we integrate the overlapping generation structure of Buiter (1988) into the constructed capital framework of Baldwin (1999). 2.1. Basic structure and underlying assumptions The model consists of two symmetric regions or countries3 with identical production technologies, preferences of individuals, labor endowments and demographic structures. Each region has three economic sectors (agriculture, manufacturing and investment) with two immobile factors (labor L and capital K) at its disposal. The homogeneous agricultural good, which is also the numéraire and denoted as z, is produced under perfect competition and with constant returns to labor. It can be traded between the two regions without any cost. Manufacturing firms are modeled as in the monopolistic competition framework of Dixit and Stiglitz (1977) and therefore produce varieties, m, with one variety-specific unit of capital as fixed input and labor as the variable production factor. Because one variety requires exactly one unit of capital as a fixed input, a continuum of varieties i 2 (0, K] is manufactured at home, whereas a continuum of varieties j 2(0, K⁄] is produced in the foreign region. In contrast to agricultural goods, trade of manufactured goods involves iceberg transport costs such that u P 1 units of the differentiated good have to be shipped to sell one unit abroad (see Samuelson, 1952). In the Walrasian investment sector, capital, i.e., machines, are produced using labor as the only input and wages are paid out of the individuals’ savings. Following Baldwin (1999), a share d > 0 of the capital stock depreciates at each instant. As far as the demographic structure is concerned, we closely follow Buiter (1988) who introduced population growth into the Blanchard (1985) model of perpetual youth. We assume that at each instant in time, s 2 ½0; 1Þ, a large cohort consisting of new individuals is born. The size of this cohort is given by bNðsÞ, where NðsÞ is population size at time s and b P 0 is the constant birth rate. Newborns receive no bequests and thus start their lives without any wealth. Each individual’s time of death is stochastic with an exponential probability density function parameterized by the constant instantaneous mortality rate l P 0.4 Normalizing initial population Nð0Þ to one, the size of the cohort born at t 0 at a certain instant s is Nðt 0 ; sÞ ¼ bebt0 els .5 Consequently, total population size at time s is given by

NðsÞ ¼

Z s 1

Nðt 0 ; sÞdt 0 ¼

Z s

bebt0 els dt 0 ¼ eðblÞs ;

ð1Þ

1

where we denote its growth rate as n  b  l. Because there is no heterogeneity between members of the same cohort, each cohort can be described by one representative individual, who inelastically supplies her efficiency units of labor on the labor market with perfect mobility across sectors but immobility between regions. Finally, as in Yaari (1965), individuals can insure against the risk of dying with positive assets by buying actuarial notes, which are canceled upon death, from a fair life insurance company. 3 We use an asterisk to indicate foreign variables. If further distinction is needed we additionally use H for home and F for foreign variables. In particular, the superscript F denotes that a good was produced in the foreign region, whereas the asterisk indicates that it is consumed in the foreign region. 4 Due to the law of large numbers, the individual probability of death is equal to the fraction of individuals who die at each instant. 5 In what follows, the first time index of a variable will refer to the birth date, whereas the second will indicate a certain instant in time. Details of these and other derivations can be found in the Supplemental Guide to Calculations, available upon request.

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T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

2.2. Individual consumption behavior Preferences over the agricultural good and a CES composite of the manufacturing varieties are Cobb-Douglas.6 The representative individual of cohort t 0 chooses the consumption of agricultural goods, cz ðt 0 ; sÞ, varieties produced at home, cHm ði; t 0 ; sÞ, and varieties produced abroad, cFm ðj; t 0 ; sÞ, to maximize her expected lifetime utility

Uðt0 ; t 0 Þ ¼

Z

1

t0

h i a eðqþlÞðst0 Þ ln ðcz ðt0 ; sÞÞ1a ðcagg m ðt 0 ; sÞÞ ds;

ð2Þ

where q > 0 is the rate of pure time preference, 0 < a < 1 is the manufacturing share of consumption and

cagg m ðt 0 ;

sÞ 

"Z

KðsÞ 0



cHm ði; t0 ;

sÞ

rr1

di þ

Z

K  ðsÞ

0

#rr1  F r1 r cm ðj; t 0 ; sÞ dj

þ

Z

Z

ð3Þ

KðsÞ

0

pHm ði; sÞcHm ði; t0 ; sÞdi

K  ðsÞ

pFm;u ðj;

0

s

ÞcFm ðj; t 0 ;

sÞdj:

Here, pz ðsÞ is the price of the agricultural good, pHm ði; sÞ is the price of a manufactured variety produced at home and pFm;u ðj; sÞ is the price of a manufactured variety produced abroad, with the subscript u indicating dependence on transport costs. Solving the individual’s utility optimization problem yields the following demand functions for the agricultural good and for each of the manufactured varieties:

cz ðt 0 ; sÞ ¼

ð1  aÞeðt0 ; sÞ ; pz ðsÞ

cHm ði; t 0 ; sÞ ¼ hR KðsÞ 0

cFm ðj; t 0 ; sÞ ¼ hR KðsÞ 0

ð4Þ

aeðt0 ; sÞðpHm ði; sÞÞr i; R K  ðsÞ ðpHm ði; sÞÞ1r di þ 0 ðpFm;u ðj; sÞÞ1r dj aeðt0 ; sÞðpFm;u ðj; sÞÞr ðpHm ði; sÞÞ1r di þ

R K  ðsÞ 0

i; ðpFm;u ðj; sÞÞ1r dj

ð5Þ

ð6Þ

as well as the consumption Euler equation for the representative individual of cohort t 0 ,

_ 0 ; sÞ eðt pðsÞ ¼  d  q: eðt 0 ; sÞ ai wðsÞ

Due to its demographic structure, our model set-up does not feature only one single representative individual. Instead, economy-wide laws of motion of capital and expenditures have to be obtained. For this purpose, we first apply the following aggregation rules:

Z

t

kðt 0 ; tÞNðt0 ; tÞdt0 ;

ð8Þ

eðt0 ; tÞNðt 0 ; tÞdt 0 ;

ð9Þ

1

where wðsÞ denotes the wage per efficiency unit of labor, l refers to the efficiency units of labor supplied by an individual, pðsÞ is the capital rental rate, kðt 0 ; sÞ is the individual capital stock and eðt 0 ; sÞ are individual consumption expenditures defined as

eðt 0 ; sÞ  pz ðsÞcz ðt0 ; sÞ þ

2.3. Aggregation

KðtÞ 

represents consumption of the CES composite, with r > 1 denoting the elasticity of substitution between varieties. Individual savings, defined as income minus consumption expenditures, are converted into capital in the investment sector with a time-independent, exogenous labor input coefficient of ai . The wealth constraint of a representative individual7 can thus be written as

_ 0 ; sÞ ¼ wðsÞl þ pðsÞkðt 0 ; sÞ  eðt0 ; sÞ þ lkðt 0 ; sÞ  dkðt 0 ; sÞ; kðt wðsÞai

As first shown by Yaari (1965), the representative individual’s Euler equation with fully insured lifetime uncertainty is identical to the Euler equation when no lifetime uncertainty exists, i.e., individual saving behavior is not influenced by the mortality rate and, moreover, does not differ across generations.

ð7Þ

6 The following discussion refers to the home region but, due to symmetry, the corresponding equations also hold in the foreign region. 7 The particular law of motion of capital given in Eq. (3) is based on the full insurance result of Yaari (1965). Since all individuals only hold their wealth in the form of actuarial notes, the market rate of return on capital pðsÞ=ai wðsÞ  d has to be augmented by the mortality rate l to obtain the fair rate on actuarial notes (see Yaari, 1965).

EðtÞ 

Z

t

1

where KðtÞ is the aggregate capital stock, EðtÞ denotes aggregate consumption expenditures8 and equivalent equations hold in the foreign region. For each population aggregate variable XðtÞ, the corresponding quantity per capita is defined by ~xðtÞ ¼ XðtÞent (see Buiter, 1988). Using this notational convention, we can finally derive the laws of motion of per capita expenditures ~eðtÞ and per capita capital ~ kðtÞ. The ‘‘per capita’’ Euler equation is given by

  ~ ~e_ ðtÞ kðtÞ pðtÞ  d  q  bðq þ lÞai wðtÞ ¼ ~eðtÞ ~eðtÞ wðtÞai _ 0 ; tÞ ~eðtÞ  eðt; tÞ eðt b ; ¼ ~eðtÞ eðt0 ; tÞ

ð10Þ ð11Þ

where eðt; tÞ are the consumption expenditures of newborns; an analogous equation holds in the foreign region. The per capita Euler equation (economy-wide perspective) differs from the individual ~ Euler equation (cohort perspective) by b eðtÞeðt;tÞ . This turnover cor~eðtÞ 9 rection term captures distributional effects due to birth and death. In particular, consumption expenditure growth is the same for all generations but expenditure levels are age dependent. Older individuals are wealthier and therefore have higher consumption expenditure levels than their younger counterparts. Because, at each point in time, a fraction l of dying wealthier individuals is replaced by a fraction b of newborns with no capital holdings, per capita consumption expenditure growth is smaller than individual consumption expenditure growth. Increases in both birth and mortality rates decrease per capita consumption expenditure growth by strengthening this turnover effect, whereas in the representative agent case, i.e., b ¼ l ¼ 0, the turnover effect completely disappears. Finally, the law of motion of the per capita capital stock is given by

  ~l ~eðtÞ pðtÞ ~_ ~ kðtÞ ¼  þ  d  b þ l kðtÞ: ai wðtÞai wðtÞai

ð12Þ

Compared to the law of motion of individual capital, l only augments the rental rate via its effect on the population growth rate, n ¼ b  l. This captures two facts: First, lkðtÞ in Eq. (3) does not represent aggregate capital accumulation but rather a transfer – via the life insurance company – from individuals who died to those who survived, within a given cohort (see Heijdra and van der Ploeg, 2002, chapter 16). Second, faster population growth 8 The aggregate efficiency units of labor LðtÞ are equal to the individual supply of efficiency units of labor l multiplied by the population size. 9 For the turnover correction term in case of a constant population, i.e. b ¼ l, see chapter 16 in Heijdra and van der Ploeg (2002).

5

T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

obviously decreases per capita capital accumulation. In sharp contrast to the per capita Euler equation, birth and mortality rates affect the law of motion of per capita capital in the opposite way. 2.4. Profit maximization Profit maximization in the manufacturing and agricultural sectors closely follows Baldwin (1999). By choice of units, the input coefficient for labor in the production of the perfectly competitive agricultural good can be set to one. Perfect labor mobility across sectors implies, then, that the equilibrium wage rate in the economy is pinned down by the price of the agricultural good. Since free trade between the home and foreign regions equalizes this price, wages also equalize between the two regions as long as each of them produces some agricultural output.10 Finally, choosing the agricultural good as numéraire leads to11

w ¼ w ¼ 1:

ð13Þ

In the monopolistically competitive manufacturing sector firms face increasing returns to scale production technology, with an associated cost function

p þ wcY m ðiÞ;

ð14Þ

where c is the unit input coefficient for efficiency units of labor, Y m ðiÞ is total output of one manufacturing good producer and the capital rental rate p represents the fixed cost. Profit maximization yields the standard result that prices are given by

pHm ðiÞ ¼

r wc; r1

pFm;u ðiÞ ¼

ð15Þ

r wcu; r1

ð16Þ

and therefore equal a constant mark-up over marginal cost. Moreover, mill pricing is optimal, meaning that any differences in prices between the two regions is due solely to transport costs (see Baldwin et al., 2003). Because we have variety specificity of capital and free entry into the manufacturing sector, driving pure profits down to zero, the capital rental rate is equivalent to the Ricardian surplus, i.e., the operating profit of each manufacturing firm. Using optimal prices as shown in Eqs. (15) and (16) and redefining global quantities and regional share variables gives operating profits and thus capital rental rates as12

p¼





hE ð1  hE Þ/ þ hK þ /ð1  hK Þ /hK þ 1  hK |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

!

aEW ; rK W

ð17Þ



 1  hE hE / þ 1  hK þ /hK /ð1  hK Þ þ hK |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

3. The impact of demography on agglomeration To assess the impact of demographic change on agglomeration processes, we will analyze the stability properties of the symmetric equilibrium more thoroughly. If it turns out that this steady state is unstable, then any slight perturbation will lead us away from an equal distribution of capital and expenditures, thus resulting in the agglomeration of economic activity. 3.1. Equilibrium analysis The dynamics of our model are fully described in the following ~ and k ~ , whose four-dimensional system with the variables ~e; ~ e ; k expressions were derived in Section 2.3 and are given by:

" ~e_ ¼

Bias

p ¼

measure the extent to which a region’s sales per variety differ from world average sales per variety. Additionally, they capture the impact that capital and expenditure shifting have on profits.13 At the symmetric equilibrium, shifting expenditures towards the home economy (dhE > 0) raises p and lowers p because it increases the home market size. A higher expenditure share therefore supports agglomeration of capital at home because capital accumulates, where the rental rate is higher and decumulates in the other region. Production shifting14 to home (dhK > 0), on the other hand, has the opposite impact because it increases competition in the home market. Suppose now that the two regions are in a symmetric equilibrium and capital stocks are slightly perturbed. If this perturbation raises relative profitability in the region with increased capital share, the equilibrium is unstable and agglomeration sets in. Whether catastrophic agglomeration of capital occurs is therefore determined by the relative strengths of the two effects described above. On the one hand, the local competition effect directly decreases the capital rental rate. On the other hand, the higher expenditure share due to the raised income associated with a higher capital share, i.e., the demand-linked circular causality effect, indirectly increases it. Since the relative size of the expenditure and the production shifting effects depends on the level of trade openness /, the degree of economic integration is crucial for agglomeration processes. In particular, Baldwin (1999) shows that, in a situation with initially prohibitive trade costs, agglomeration processes set in as soon as economic integration reaches a certain threshold level. The crucial question to be investigated in the following sections of this paper is whether and how such agglomerative tendencies depend on the economies’ demographic structures.

~ þ /k ~ k

!

aEW ; rK W

ð18Þ

" ~e_  ¼

Bias

where /  u1r is a measure of openness between the two regions; / ¼ 0 indicates prohibitive trade barriers and / ¼ 1 indicates free trade. World expenditures are defined as EW  E þ E and the world capital stock is K W  K þ K  , with hK and hE referring to the home shares of these quantities. As expected, these capital rental rates are identical to those derived in Baldwin (1999), since the introduction of demographic structures does not change the production side of the economy.  In analogy to Baldwin (1999), the terms labeled as Bias and Bias 10 This can be shown to hold if a, the manufacturing share of consumption, is not too large (see Baldwin, 1999), which will be assumed from now on. 11 For the sake of clarity, we will ignore time arguments from this point forward. 12 Note that a ¼ a and r ¼ r due to symmetry between regions.

~_ ¼ k

~e

"

~_  ¼ k

"

þ

~e / ~ ~ /k þ k

!



#

a ~  d  q ~e  bðq þ lÞai k; ai r

# !  ~e/ a ~ ; þ  d  q ~e  bðq þ lÞai k ~ þ /k ~ /k ~ ~ þ k ai r k ~e

~e / þ ~ ~ ~ ~  /k þ k k þ /k ~e

!



ð20Þ

#

~ a ~ þ l  ~e ; dbþl k ai ai ai r

!  #  ~ ~e/ a ~ þ l  ~e : dbþl k þ ~ ~ ~ ~   ai ai k þ /k /k þ k ai r ~e

ð19Þ

ð21Þ

ð22Þ

13 Since capital is immobile between regions, the term ‘‘capital shifting’’ as frequently used in the NEG literature might be misleading. It should, however, only represent an exogenous perturbation of the home capital share (and similarly of the home expenditure share, in the case of expenditure shifting). 14 Since the number of varieties in the home region is equal to the capital stock at home, capital accumulation in one region is tantamount to firm creation.

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T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

Here, the equilibrium wage rate is equal to one in both regions and the rental rates from Eqs. (17) and (18)15 have been incorporated. The demographic parameters b and l affect the system via the turnover correction terms in the per capita Euler equations as well as the population growth rate n ¼ b  l in the laws of motion of per capita capital. Setting b ¼ l ¼ 0, i.e., considering the case of an infinitely lived representative agent, reduces the laws of motion to the ones obtained by Baldwin (1999). Our framework thus nests the constructed capital model as a special case. ~¼k ~ into the Inserting the symmetric outcome ~ e ¼ ~e and k above system indeed reveals that it is a steady state with the equilibrium values given by

1.0

 pffiffiffiffi pffiffiffiffi ~  ~e ¼ lr ðd  lÞ rðA þ BÞ þ b ðd þ q  2lÞr þ 2ðl þ qÞa þ A r ; 2ððd  lÞr þ baÞððb þ d þ qÞr  ðl þ qÞaÞ ð23Þ

0.4

  pffiffiffiffi ~ ~l ðqr  rAÞðr  aÞ þ 2raðb  lÞ þ drðr þ aÞ k ¼ ; 2ai ððd  lÞr þ baÞððb þ d þ qÞr  ðl þ qÞaÞ

0.8

0.6

0.2 KKstable KKunstable

ð24Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi where A  rðd þ qÞ2 þ 4bðl þ qÞa and B  ðd þ qÞ r.16 Fig. 2 illustrates this steady state by plotting the model’s scissor diagram, which shows the two key equilibrium conditions of the constructed capital model, EE and KK (see, e.g., Baldwin et al., 2003). While the EE schedule gives the equilibrium home expenditure share for a given hK , the KK line plots the equilibrium home capital share for any hE .17 Obviously, any intersection of the EE and KK schedules represents a steady state of our model. Similarly to the scissor diagram in Baldwin (1999), the symmetric outcome hE ¼ hK ¼ 1=2 is an equilibrium. Depending on the chosen parameter values,18 this steady state is either stable (KK stable schedule) or unstable (KK unstable schedule).19 In the latter case, starting with any hE > 1=2; KK unstable gives the corresponding equilibrium capital share. The EE schedule then tells us the implied equilibrium expenditure share, which is greater than the initial hE . As indicated by the arrows in the figure, the system thus diverges from the symmetric equilibrium, i.e., ongoing agglomeration processes toward the core-periphery outcome set in.20 In what follows, we will analyze in depth how the demographic parameters b and l affect these agglomeration tendencies by investigating their impact upon the stability properties of the symmetric equilibrium. 3.2. Formal stability analysis

EE

0.0

0.48

0.49

0.50

0.51

0.52

Fig. 2. Scissor diagram.

where the four symmetric 2  2 sub-matrices J i for i ¼ 1; . . . ; 4 are given in appendix A (for this classical approach to stability analysis see, e.g., the appendix on mathematical methods in Barro and Salai-Martin, 2004). Solving the characteristic equation yields the following four eigenvalues (see Appendix A)

pffiffiffiffiffiffiffiffiffiffi 1 ðr1  rad1 Þ; 2 pffiffiffiffiffiffiffiffiffiffi 1 eig2 ¼ ðr1 þ rad1 Þ; 2 pffiffiffiffiffiffiffiffiffiffi 1 ðr  rad2 Þ; eig3 ¼ 2 pffiffiffiffi 2 2ð/ þ 1Þ r pffiffiffiffiffiffiffiffiffiffi 1 eig4 ¼ ðr þ rad2 Þ; 2 pffiffiffiffi 2 2ð/ þ 1Þ r

eig1 ¼

where A r 1  pffiffiffiffi  b  d þ l;

r

To analyze the stability properties of the symmetric equilibrium we first linearized the non-linear dynamic system given in Eqs. (19)–(22) around the symmetric equilibrium values (23) and (24), and then evaluated the eigenvalues of the corresponding 4  4 Jacobian matrix

J sym ¼



J1

J2

J3

J4

 ;

ð25Þ

 2 ðr  aÞ ðA þ BÞ2 þ 4bðl þ qÞa A ; rad1  pffiffiffiffi þ b þ d  l þ ra r

pffiffiffiffi   r 2  3/A þ A þ bð/ þ 1Þ2 þ l  d 2/2 þ / þ 1 þ /ðlð/ þ 2Þ  /q þ qÞ r; 



pffiffiffiffi2 rad2  Að/  1Þ þ bð/ þ 1Þ2 þ lð/ þ 1Þ2 þ dð/  1Þ þ /ð/ þ 3Þq r

 ð/ þ 1Þðð/ þ 1Þr þ ð/  1ÞaÞ 4bðl þ qÞað/ þ 1Þ2 þ ð/  1Þ2 ðA þ BÞ2 þ :

a

15 ~ Note that we have rewritten the rental rates as functions of the variables ~e; ~e ; k ~ . Moreover, due to the assumption of symmetric regions, we have that and k ~l ¼ ~l ; b ¼ b ; l ¼ l ; a ¼ a ; d ¼ d and q ¼ q . i i 16 These and most of the following results were derived with Mathematica. The corresponding files are available from the authors upon request. Also note that attention is restricted to the economically meaningful solution pair, i.e. where consumption and capital are positive for plausible parameter values. 17 These schedules were derived by setting Eqs. (19)–(22) equal to zero, rewriting the resulting system of equations in share and world quantities and, finally, reducing it to two equations in hK and hE . 18 Fig. 2 is plotted for l ¼ 0:0001; b ¼ 0:0001; d ¼ 0:05; q ¼ 0:015; a ¼ 0:3; r ¼ 4; ai ¼ 1 and / ¼ 0:95 ðKK stable Þ or / ¼ 0:98 ðKK unstable Þ. 19 The EE schedule remains the same for both parameter choices. 20 A similar argument applies for the case, where the initial hE < 1=2.

By investigating the signs and nature of these eigenvalues, it is possible to fully characterize the system’s local dynamics around the symmetric equilibrium. First, it is easily established that all of the four eigenvalues are real because both rad1 and rad2 are nonnegative for all possible parameter values.21 Convergence to or divergence from the symmetric equilibrium is thus monotonic. Moreover, as there are two jump variables ~e and ~e , saddle path stability prevails if and only if there are two negative eigenvalues. If fewer than two eigenvalues 21 Recall the parameter ranges l P 0; b P 0; d > 0; r > 1; q > 0; 0 < a < 1 and 0 6 / 6 1, which also imply that A > 0 and B > 0. In particular, note that r > a.

7

T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

0.010

0.10 0.008

0.25 0.002

0.006

0.008

0.08 0.2 0.004

0.1

0.006

0.000

0.06

0.15

0.004

0.04 0.05

0.002

0.02 0.00

0.000 0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.0

0.2

0.4

0.6

0.8

1.0

µ

Fig. 3. Contour plot of eigenvalue 3.

Fig. 4. Contour plot of eigenvalue 1.

are negative, then the system is locally unstable. Evaluating the signs of the eigenvalues turns out to be rather complicated. However, despite the model’s complexity, it is possible to come up with analytical findings even here. In particular, as shown in detail in Appendix B, for b P l, eigenvalues 2 and 4 are nonnegative while eigenvalue 1 is nonpositive. Thus, the stability properties crucially hinge on the sign of eigenvalue 3. If it is negative, then the system is saddle path stable. On the other hand, for parameter ranges within which eigenvalue 3 is positive, the symmetric equilibrium is unstable and agglomeration processes set in. For b < l, only two eigenvalues have unambiguous signs, i.e., eigenvalues 2 and 4 are once again nonnegative. Therefore, with a shrinking population, agglomeration processes will set in for parameter ranges in which at least either eigenvalue 1 or eigenvalue 3 is positive. To be able to assess the impact of demography on agglomeration, we thus focus our analysis on eigenvalues 1 and 3. Investigating them more thoroughly indeed reveals that both of them switch signs depending on the demographic parameters b and l. This is illustrated in Figs. 3 and 4, which plot the contour lines of these eigenvalues for different birth and mortality rates.22 In particular, the zero contour line divides the parameter space into stability and instability regions. As far as the case of population shrinking, i.e., b < l, is concerned, note moreover that for all combinations of b and l for which eigenvalue 3 switches sign, eigenvalue 1 is still negative. We can thus conclude that the possibility of agglomeration crucially hinges on the demographic properties of economies, i.e., there exist some combinations of birth and mortality rates for which the symmetric equilibrium is stable and others for which it is unstable. The relevance of demographic structures for agglomeration processes becomes even clearer when investigating critical levels of trade openness /break . This threshold value identifies the degrees of openness at which eigenvalue 3 switches its sign and therefore the stability properties of the symmetric equilibrium change. They are given by23 22 Figs. 3 and 4 are plotted for d ¼ 0:05; q ¼ 0:015; a ¼ 0:3; r ¼ 4 and / ¼ 0:98. Note that we used different ranges of l and b to focus on the parameter region, where eigenvalue 3 switches sign. 23 We derived them by solving eig3ð/break ; d; q; a; r; b; lÞ ¼ 0 for /break . Setting b ¼ l ¼ 0 reduces them to the break points obtained by Baldwin (1999).

pffiffiffiffiffiffiffiffi T 1  2T 2 ; T3 pffiffiffiffiffiffiffiffi T 1 þ 2T 2 ¼ ; T3

/break1 ¼

ð26Þ

/break2

ð27Þ

where

pffiffiffiffi T 1  4a2 bðl þ qÞ þ a rðb þ d  lÞðB  AÞ þ rBðA þ BÞ; T 2  2a½2a2 bðl þ qÞð2d2 þ ðb  lÞ2 þ 2ðb þ 2d þ lÞq þ 3q2 Þr pffiffiffiffi  4a2 b rAð2b þ d  2l  qÞðl þ qÞ þ aðb2 Aðd  4l  3qÞ  2bðA þ BÞðd  2l  qÞðl þ qÞ þ ðA þ BÞðd þ qÞðl þ qÞ2 Þr3=2 þ 16a3 b2 ðl þ qÞ2  bð4BðA þ BÞðl þ qÞ þ abðd2 þ 8l2 þ 20lq þ 11q2 þ 2dð2l þ qÞÞÞr2 ; pffiffiffiffi T 3  4a2 bðl þ qÞ þ a rAðd þ qÞ þ rBðA þ BÞ þ arð2b þ d þ qÞ  ð2l  d þ qÞ: For all / 2 ð/break1 ; /break2 Þ, eigenvalue 3 is positive and agglomeration processes therefore set in.24 Recognizing that both critical values depend on mortality and birth rates confirms the crucial role of demography for the stability properties of the symmetric equilibrium. Fig. 525 illustrates this observation by plotting these levels as the boundaries of the shaded instability region for the case of a constant population size, i.e., b ¼ l. In addition to showing that the instability region varies with different demographic structures, the figure also already indicates one direction of the impact. The next sections of this paper are dedicated to investigating this issue more thoroughly by identifying the channels by which demography affects the spatial distribution of economic activity. In particular, recall from Eqs. (19)–(22) that the demographic parameters b and l affect our dynamic system via the turnover correction terms as well as the 24 This is, of course, only true for parameter ranges yielding real values for /break1 and /break2 . 25 Fig. 5 and all figures in the next section are plotted for our most plausible choice of parameter values: d ¼ 0:05, implying that capital fully depreciates after 20 years, q ¼ 0:015 (see Auerbach and Kotlikoff, 1987), a ¼ 0:3 (see Baldwin, 1999; Krugman, 1991) and r ¼ 4 (see Bosker and Garretsen, 2007; Brakman et al., 2005; Krugman, 1991; Krugman and Venables, 1995; Martin and Ottaviano, 1999; Puga, 1999).

8

T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

particularly through the effect of production shifting on expenditure shifting, i.e., dhE =dhK . As n increases, the expenditure increase that comes with the rise in income associated with a higher capital stock gets weaker, dampening the demand-linked circular causality and thus making the instability set smaller. This is because, with population growth, similarly to the impact of depreciation on agglomeration in Baldwin (1999), more resources must be dedicated to maintenance of the per capita capital stock instead of to consumption.

1.00

0.98

0.96

0.94

0.92

3.4. The turnover effect

0.0001

0.0002

0.0003

Fig. 5. /break1 (dashed) and /break2 (solid) as a function of

µ 0.0004

l ¼ b.

population growth rate. The latter effect can be isolated by setting b ¼ 0 while still allowing for nonzero mortality rates. The turnover effect, on the other hand, is best understood by considering the case of constant population size ðb ¼ lÞ. Only after having investigated these two channels separately we can proceed with an analysis of the general case, where we allow for both changes in population age structure and population size. 3.3. The population growth effect Setting b ¼ 0 while still allowing for nonzero mortality rates eliminates the turnover correction terms in the per capita Euler equations, (19) and (20), while still maintaining the population growth rate n in the laws of motion of per capita capital, given in Eqs. (21) and (22).26 We can therefore isolate the effects of changes in population size on agglomeration processes. In particular, setting b ¼ 0 simplifies the critical levels of trade openness to

/break1 ¼

rðq þ dÞ  aðq  nÞ ; rðq þ dÞ þ aðq  nÞ

/break2 ¼ 1:

ð28Þ ð29Þ

Clearly, a higher population growth rate n increases /break1 and thus reduces the parameter set within which the symmetric equilibrium is unstable and agglomeration processes set in. Population growth thus acts as a dispersion force, promoting an equal distribution of economic activity between regions. The above analysis identifies the channel by which demographic change, solely capturing changes in population size, impacts agglomeration processes. To clarify the economic mechanism underlying this first relationship between demography and agglomeration, we rely on Baldwin (1999), who showed that the formal stability analysis pursued in Section 3.2 yields the same results as a more informal method of checking the stability of the symmetric equilibrium. This informal method involves investigating how an exogenous perturbation of the home share of capital, hK , influences the profitability of home-based firms relative to foreign-based firms. A positive impact implies instability as even more firms would locate themselves in the home region, i.e., capital accumulation would set in. In Baldwin (1999), there are two ways by which production shifting influences the relative profitability of home-based firms. These are the demand-linked circular causality effect, @ P=@hE dhE =dhK , and the local competition effect, @ P=@hK , (see Section 2.4). Because demographic structures do not affect the production side of the economy and therefore operating profits, they can only exert their influence via the demand-linked circular causality, 26

In particular, n ¼ l in this case.

While the preceding section has worked out how population growth influences agglomeration, this section tries to identify and explain the impact of the turnover correction terms that show up in the per capita Euler equations. To isolate their effect we set b ¼ l such that population size is constant. In doing so, we eliminated any potential effect of population growth. Changes in the demographic parameters b and l can thus influence agglomeration only via the turnover terms. In particular, these correction terms increase with both the birth rate and mortality rate (see Section 2.3). Recall from Section 3.2 that demographic structures can only influence the stability properties of the symmetric equilibrium via eigenvalue 3. Fig. 6 thus plots this eigenvalue as a function of the level of trade openness / for three different birth rates. The graph indicates that higher birth rates27 stabilize the symmetric equilibrium. Fig. 5 confirms this finding by showing that /break1 increases and /break2 decreases in b; this shrinkage of the instability region is equivalently represented by the downward shift of eigenvalue 3. Consequently, the range of / within which eigenvalue 3 is positive and thus agglomeration processes set in is maximal for the representative agent setting of Baldwin (1999), i.e., for b ¼ l ¼ 0. Accounting for the possibility of birth and death by introducing an overlapping generation structure thus considerably weakens agglomeration tendencies.28 The above analysis identifies the second channel by which demographic change impacts agglomeration processes. Clearly, the turnover correction terms resulting from positive birth and mortality rates act as another dispersion force strengthening the stability of the symmetric equilibrium. To understand the economic mechanism of this turnover effect, we return to the informal stability analysis of Baldwin (1999). Similarly to population growth, turnover correction terms can only impact agglomeration via the demand-linked circular causality. In particular, they again weaken the effect of production shifting on expenditure shifting, i.e., dhE =dhK . This occurs because the exogenous rise in the home capital share increases the wealth and thus the expenditure levels of individuals in the home region, relative to foreign-based individuals. The negative distributional effects resulting from birth and death on aggregate expenditures, i.e., the replacement of individuals by newborns whose consumption expenditures are lower because they have zero wealth levels (see Section 2.3), are thus more pronounced in the home region. This, in turn, dampens the increase in the home expenditure share caused by production shifting and thus the demand-linked circular causality. Note that this dispersion force crucially depends on the heterogeneity of individuals with respect to their expenditure and wealth levels: 27 In this section, we will discuss the effects of changes in b. Note, however, that by varying b we also change l by the same amount such that the population size remains constant. 28 This finding is confirmed by extensive numerical simulations of eigenvalue 3. In particular, using MATLAB (the corresponding files are available from the authors upon request) we checked the sign of eig3jb¼l>0  eig3jb¼l¼0 on a narrow parameter grid w i t h v a l u e s o f q 2 ½0; 0:5; d 2 ½0; 0:5; r 2 ½2; 8; a 2 ½0:1; 0:9; / 2 ½0; 1 a n d b 2 ½0; 0:2. This difference is always negative.

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T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

1.00

0.0002

0.96

0.97

0.98

0.99

1.00

0.98

0.000

0.0002 0.002

0.96

0.004

0.0004 0.0006

0.94

0.0008

0.006

0.92 0.0010 Fig. 6. Eigenvalue 3 as a function of / for b ¼ 0 (dashed line), b ¼ 0:0002 (solid line) and b ¼ 0:0004 (dotted line).

0.90

as long as the consumption expenditures of newborns are smaller than average per capita consumption expenditures, i.e., as long as full bequests do not prevail,29 the turnover effect is active and the associated distributional effects due to birth and death work against agglomeration. Interestingly, the turnover effect is even strong enough to make agglomeration a highly implausible outcome. This is illustrated in Fig. 7, which plots the contour lines of eigenvalue 3 for varying b and /. The graph nicely shows the ‘‘smallness’’ of the instability region characterized by all the parameter combinations inside the zero contour line. Only in the case of very low birth and mortality rates it is possible to find critical values of economic integration within which the symmetric equilibrium becomes unstable and agglomeration processes set in. With a constant population size, our model thus predicts the symmetric outcome to be predominant even in the presence of high economic integration. In particular, we can establish that for b ¼ l > 0:00028, corresponding to a life expectancy of less than approximately 3500 years, there exists no level of trade costs such that the symmetric equilibrium is unstable (i.e., the downward shift in Fig. 6 is such that eigenvalue 3 does not cross the horizontal axis anymore, where it would become positive).30 With reasonable demographic structures, deeper economic integration therefore does not necessarily result in spatial concentration of economic activity. This is in sharp contrast to other NEG models and implies that the agglomeration induced growth findings of Baldwin (1999) primarily apply to the very special case of infinitely lived individuals. Finally, Figs. 5–7 do not only show that the instability region shrinks in the birth rate; they also reveal that, for low but positive birth and mortality rates, the instability set is non-monotone in /. Agglomeration processes thus only set in for an intermediate range of trade costs and the symmetric equilibrium is stable again for sufficiently high levels of economic integration. The reason for this is that the relative strength of the agglomeration force in the constructed capital model of Baldwin (1999) is highest for an intermediate level of trade costs. Fig. 6 shows that this does not have any consequences for the instability set in the case of b ¼ l ¼ 0. For b ¼ l > 0, however, it qualitatively changes the stability properties of the symmetric equilibrium because the introduction of birth and death decreases eigenvalue 3 for all levels of trade costs.

0.88

29 In particular, only a bequest scheme that exactly equalizes the expenditure levels of newborns and average per capita consumption eliminates the turnover effect. Other schemes would only dampen it by reducing heterogeneity with respect to expenditure levels. 30 These calculations are based on our most plausible choice of parameter values d ¼ 0:05; q ¼ 0:015; a ¼ 0:3 and r ¼ 4. We also performed similar simulations for other parameter ranges, in particular for the parameter choice q ¼ d ¼ 0:1; a ¼ 0:3 and r ¼ 2, which follows Baldwin (1999). In this case, the symmetric equilibrium is always stable as soon as b ¼ l > 0:00395. This implies that in the set-up of Baldwin (1999), a life expectancy of less than approximately 250 years prevents any agglomerative tendencies.

0.008

0.01 0.86 0.000

0.001

0.002

0.003

0.004

0.005

Fig. 7. Contour plot of eigenvalue 3.

3.5. Agglomeration processes in aging societies So far, we have identified the two channels by which demographic structures impact agglomeration processes. This allows us to assess the effects of varying birth and mortality rates in a setting with nonconstant population size. In doing so, we will also answer how population aging influences the spatial distribution of economic activity. With nonconstant population size, i.e., b – l, varying the demographic parameters changes the age structure of the population as well as its growth rate. In particular, declines in the birth rate imply both population aging and a lower population growth rate, whereas declines in the mortality rate leave the mean age unchanged31 and only increase population growth (see Preston et al., 2001). To assess agglomeration tendencies in aging societies, we will thus focus on the case of declining birth rates. Figs. 8 and 9 plot eigenvalue 3 as a function of trade openness for different values of b and l.32 Eigenvalue 3 decreases with the birth rate and increases with the mortality rate.33 Only for sufficiently low b or sufficiently high l values can we find levels of economic integration for which eigenvalue 3 turns positive and the symmetric equilibrium becomes unstable. The destabilizing impact of decreasing birth rates is easily explained when recalling the turnover and population growth channel discussed in the previous sections. Obviously, lower birth rates decrease both the turnover correction terms (see Section 2.3, particularly Eq. (10)) and the population growth rate, thus dampening their stabilizing effects on agglomeration. To put it another way,

31 This can easily be shown by noting that the proportion of the population at age t  t0 is given by Nðt0 ; tÞ=NðtÞ ¼ bebðtt0 Þ which is independent of l. Intuitively, a lower mortality rate contemporaneously leads to individual aging and faster population growth. At the aggregate level, these two forces exactly cancel each other out. 32 Figs. 8 and 9 are again plotted for d ¼ 0:05; q ¼ 0:015; a ¼ 0:3 and r ¼ 4. For Fig. 8, we moreover set l ¼ 0:001 and for Fig. 9, we set b ¼ 0:001. 33 These findings are once again confirmed by extensive numerical simulations for eigenvalue 3. In particular, using MATLAB (the corresponding files are available from the authors upon request) we checked the sign of eig3jb>0;l>0  eig3jb¼0;l>0 as well as eig3jb>;l>0  eig3jb>0;l¼0 on a narrow parameter grid with values of q 2 ½0; 0:5; d 2 ½0; 0:5; r 2 ½2; 8; a 2 ½0:1; 0:9; / 2 ½0; 1; b 2 ½0; 0:2 and l 2 ½0; 0:2. The former difference is always negative, while the latter is always positive.

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T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

0.96

0.97

0.98

0.99

1.00

0.0005 0.0010 0.0015 0.0020 0.0025 Fig. 8. Eigenvalue 3 as a function of / for b ¼ 0:0001 (solid Line), b ¼ 0:001 (dotted line) and b ¼ 0:002 (dashed line).

0.88

0.90

0.92

0.94

0.96

0.98

1.00

0.002

0.004

0.006

0.008

Fig. 9. Eigenvalue 3 as a function of / for line) and l ¼ 0:0001 (dashed line).

l ¼ 0:05 (solid line), l ¼ 0:02 (dotted

population aging strengthens agglomeration processes. This is similar to the case of a constant population size considered in the previous section, where contemporaneous declines in b and l, which also raise the mean age of the population, increase the instability region as well. However, also note that, when b – l, agglomeration processes are still implausible if we assume realistic demographic structures, e.g., for a mortality rate of l ¼ 0:0125 resulting in a life expectancy of 80 years, spatial concentration of economic activity does not take place for b > 0:00051.34 A birth rate below such a value would clearly be at odds with reality. Turning to the impact of the mortality rate, the analysis becomes slightly more complicated. While lower mortality – similarly to decreasing birth rates – dampens the stabilizing turnover effect, it strengthens the dispersion force due to population growth. The preceding analysis, however, shows that lower mortality rates decrease eigenvalue 3 and thus make the symmetric equilibrium more stable. We can therefore conclude that the effects via the population growth-based channel are stronger than those of the turnover channel for the case of declining mortality rates.

as slower population growth, strengthen concentration tendencies. Decreases in mortality, which have no effect on the population age structure but increase the population growth rate, weaken agglomeration processes. Consistent with the data presented in the introduction, we can thus conclude that population aging fosters agglomeration, while population growth acts as a dispersion force. Our model framework is suited to assessing the possibility of agglomeration in various demographic scenarios. The one with a low fertility rate, implying an old age structure and slow population growth, is most relevant for industrialized countries like Japan and the members of the European Union. In such a situation, agglomeration tendencies are stronger than in emerging markets, where birth rates are higher, making the mean age lower and population growth faster. However, the above results indicate that this might change with decreasing fertility also in these countries. Moreover, for economies, where both birth rates and mortality rates are in decline, concentration processes are strengthened as long as population growth rates do not increase. This is typically the case in developed countries, where birth rates decrease faster than mortality rates. Finally, we want to stress that this article only represents a first step toward a more comprehensive understanding of the interrelations between demographic change, economic integration and agglomeration. Allowing for age-dependent mortality would further enrich our notion of demography and thereby yield additional insights with respect to its effects on agglomeration. Moreover, recognizing the tight link between the spatial distribution of economic activity and economic growth, it is worth investigating how demographic structures influence regional growth rates. Because demography has been shown to crucially affect agglomeration, we also suspect an important impact on the relationship between growth and concentration tendencies. To study this requires a NEG framework that allows for both demographic change and endogenous growth, a task that is at the top of our research agenda. Acknowledgments We thank Dalkhat Ediev, Ingrid Kubin, Alexia Prskawetz, Gerhard Orosel, Michael Rauscher, Gerhard Sorger, Jens Südekum, Vladimir Veliov, Matthias Wrede, Stefan Wrzaczek, three anonymous referees and the conference participants at the 2009 RIEF Doctoral Meeting, the 2009 Annual Meeting of the Austrian Economic Association, the 2009 ERSA Congress and the 2009 Annual Congress of the Verein für Socialpolitik as well as the seminar participants at the Institute for Advanced Studies, University of Vienna, Vienna University of Economics and Business, and the Vienna Institute for International Economics Studies, for their helpful comments and suggestions. The paper was prepared within the ‘‘Agglomeration processes in aging societies’’ research project, which has been funded by the Vienna Science and Technology Fund (WWTF) through project MA07-002. Appendix A. Intermediate results for the stability analysis

4. Concluding remarks We introduced demography into the NEG by generalizing the constructed capital model to allow for an overlapping generation structure that features both population aging and varying population size. This framework enables us to study the effects of demographic change on the spatial location of economic activity. Decreases in the birth rate, which lead to population aging as well 34

A birth rate of b ¼ 0:00051 implies 0.00051 children per individual. The other parameters were again set to d ¼ 0:05; q ¼ 0:015; a ¼ 0:3; r ¼ 4 for this calculation.

The Jacobian matrix J sym , which was evaluated at the symmetric equilibrium and is given in Eq. (25), has the following entries J i for i ¼ 1; . . . ; 4:

J1 ¼

1 pffiffiffiffi 2ð/ þ 1Þ r 0

B J2 ¼ @





/ðA þ BÞ

ai ð/2 þ1ÞðAþBÞ2 4ð/þ1Þ2 a 2

i /ðAþBÞ  a2ð/þ1Þ 2 a



Að/ þ 2Þ  B/ /ðA þ BÞ Að/ þ 2Þ  B/

; 1

2

i /ðAþBÞ  bai ðl þ qÞ  a2ð/þ1Þ 2 a



ai ð/2 þ1ÞðAþBÞ2 4ð/þ1Þ2 a

 bai ðl þ qÞ

C A;

T. Grafeneder-Weissteiner, K. Prettner / Journal of Urban Economics 74 (2013) 1–11

J3 ¼

1 ai ð/ þ 1Þr

0 B J4 ¼ @

A/þ





a  ð/ þ 1Þr /a ; /a a  ð/ þ 1Þr

pffiffiffi

rðbð/þ1Þ2 dð/2 þ/þ1Þþlð/þ1Þ2 þ/qÞ pffiffiffi ð/þ1Þ2 r

/ðAþBÞ  ð/þ1Þ 2 pffiffiffi r

Appendix C. Supplementary material 1

/ðAþBÞ  ð/þ1Þ 2 pffiffiffi r C A: pffiffiffi A/þ rðbð/þ1Þ2 dð/2 þ/þ1Þþlð/þ1Þ2 þ/qÞ 2 pffiffiffi ð/þ1Þ r

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with the parameter clusters A  rðd þ qÞ2 þ 4bðl þ qÞa as well as pffiffiffiffi B  ðd þ qÞ r. Appendix B. The signs of the eigenvalues When investigating the signs of the four eigenvalues, we distinguished between the three cases of constant population ðb ¼ lÞ, growing population ðb > lÞ and shrinking population ðb < lÞ. In the case of a constant population size, we set b ¼ l. By inserting the expression for A, it turns out that r 1 > 0. We can thus immediately conclude that eigenvalue 2 is positive. To discover the sign of eigenvalue 1, we compared r1 with the corresponding part under the radical, i.e., rad1 . The square of the former was smaller than the latter, implying that eigenvalue 1 is always negative. It remained to investigate the signs of eigenvalues 3 and 4. Again we first checked whether r 2 is nonnegative. By inserting the expression for A, r2 can be rewritten as

pffiffiffiffi   pffiffiffiffi r 2 ¼  rd 2/2 þ / þ 1 þ rð1  /Þ/q |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} term2

term1

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ð1 þ 3/Þ rðd þ qÞ2 þ 4lðl þ qÞa : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ðB:1Þ

term3

All three terms are increasing in q; a and l but react differently to changes in /; d and r. To show that r 2 is nevertheless nonnegative for all parameter values, we set q; a and l equal to zero resulting in the ‘‘worst’’, i.e., most negative, outcome with respect to these parameters. Because, even in this case, it was established that r 2 is nonnegative for the whole feasible parameter space, the fourth eigenvalue is definitely positive. This shows that, for a constant population size, stability crucially hinges on eigenvalue 3. In the case of positive population growth, eigenvalue 3 retains its decisiveness. This becomes clear when checking the signs of the remaining eigenvalues for b > l. Note that now the sign of r 1 becomes ambiguous.35 As far as eigenvalue 1 is concerned, it is surely nonpositive as long as r 1 < 0. In the case of r 1 > 0, we can, however, show that r21 < rad1 which implies that eigenvalue 1 never becomes positive. The last inequality, moreover, yields that eigenvalue 2 is always nonnegative.36 Finally, turning to the sign of eigenvalue 4, note first that r 2 again does not have an unambiguous sign.37 For r2 P 0, eigenvalue pffiffiffiffiffiffiffiffiffiffi4 is surely nonnegative. It can, however, be shown that r 2 þ rad2 > 0 even for r 2 < 0. Hence eigenvalue 4 is nonnegative for all possible parameter ranges. In summary, we again have two positive and one negative eigenvalue in the case of a growing population. Thus, as with a constant population, the symmetric equilibrium becomes unstable for parameter values that yield a positive eigenvalue 3. In the case of a shrinking population, i.e., l > b, eigenvalues 2 and 4 have unambiguous signs. Note first, that r1 > 0, which immediately proves the nonnegativity of eigenvalue 2. Similarly, it can be shown that r2 > 0, implying that eigenvalue 4 is always nonnegative as well. Thus, in the case of a shrinking population, agglomeration processes will set in for parameter ranges for which at least either eigenvalue 1 or eigenvalue 3 is positive.

35 36 37

In particular, r 1 becomes negative for sufficiently high b. For r 1 > 0 this follows trivially, r 21 < rad1 also shows it for r 1 < 0. In particular, r 2 becomes negative for sufficiently high b.

11

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