Papers Reg. Sci. 81, 49–62 (2002)

c RSAI 2002


Capital cities: When do they stop growing?
Kristof Dascher Viadrina University, Postbox 1786, D-15207 Frankfurt (Oder), Germany (e-mail: [email protected]) Received: 6 June 2000 / Accepted: 27 November 2000

Abstract. This article is an attempt to explain a capital city’s size. We assume away explanations such as exploitation of the capital city’s hinterland. Instead, we emphasise the role of the localisation of government activity (i.e., administration or legislation) in the capital city for both the capital city economy and the hinterland economy. We assume in the model that larger regions benefit from agglomeration economies. We discuss the interaction of those agglomeration economies with an agglomeration diseconomy specific to the capital city. Under certain conditions, a stable population distribution between the capital city and its hinterland emerges where neither region captures the entire population. We also analyse the comparative statics properties of this stable equilibrium. JEL classification: H41, R53 Key words: Capital city, public goods, urban growth

1 Introduction As a stylised fact, capital cities are much larger than other cities: Paris and London clearly overshadow their respective rivals, Marseille and Birmingham. More often than not, it seems that being the largest city coincides with being the capital city. To be sure, capital cities that are big when compared with other cities in their country might be big because big cities become capitals. Also, capital cities that are small when compared with other cities in their country might be small because sometimes small cities are chosen to be capitals, too.
I am grateful for the discussions with Friedel Bolle and Hermann Ribhegge, as well as for referee comments. Any remaining errors are nevertheless my own. A previous version of the paper circulated as How Do Capital Cities Trade?

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Despite their size when chosen, all capitals could possibly benefit from their capital city function, an endeavour that this article attempts to address.1 Various different mechanisms suggest why capital cities could grow more than other cities. For example, capitals can grow because benefits from public goods decline with distance from the capital. Protection from violence in Colombia, for example, may be had more easily in, than around, Bogota.2 Alternatively, capitals may simply grow because they are “parasitic”, importing goods from the hinterland without providing much in return. As Jacobs (1985, p. 231) puts it: “capital cities thrive on transactions of decline. When a city’s principal function is being a capital ... it is obvious that the more transfer payments, subsidies, grants, military contracts and promotion of international advanced-backward trade, the greater the work and the prosperity in the capital city ... Behind its busyness at ruling, a capital city of a nation or an empire, vivacious to the last, at length reveals itself as being a surprisingly inert, backward and pitiable place.” However, in this article neither access to a national public good nor redistribution from hinterland to capital play any role. Instead the role of the localisation of production of capital city output (such as administration or legislation) in the capital city is the emphasis here. We assume that local demand drives local service variety. Building on a “monopolistic competition cum love of variety”model, a larger population in the capital increases the local variety of services: a familiar agglomeration effect. Running counter to this, increasing population in the capital may decrease the local real wage; this is the model’s congestion effect, which results from governmental impact on the capital’s labour market. As we will show in the model, the capital city function plays a stronger role in small rather than large capital cities. Government demand is successful in raising local real wages in small capital cities, but the same is not true in large capital cities. So increasing the capital’s population dilutes the beneficial effect government service demand has on the real wage. The basic structure of the model is borrowed from the literature of monopolistic competition as in, say, Fujita et al. (1999) or Matsuyama (1995). This structure is augmented by the provision of a public good with properties specific to a capital city. Section 2 introduces the assumptions. Section 3 solves for short-run general equilibrium prices and quantities, holding constant the regional distribution of population. Section 4 examines the long-run adjustment of the population. Under certain conditions – to be specified in the model – the interaction of the agglomeration effect and deglomeration produces a stable equilibrium, where neither the capital nor the hinterland capture the entire pop1 See Dascher (2000a) for empirical evidence that is consistent with the idea that capitals indeed grow faster precisely because of their capital city function. 2 See Ades and Glaeser (1994), and Moomaw and Shatter (1996) on how corruption, instability and other factors determine the size of a country’s largest city.

Capital cities: When do they stop growing?

51

ulation. We will discuss some of the comparative statics properties of this stable “interior” equilibrium. Conclusions follow.

2 The framework Let the model’s country be inhabited by L¯ households. These households live in either of two regions, West and East. Normalising L¯ to 1, there are LW households in W (est) and LE = (1 − LW ) households in E (ast). Three important assumptions describe the capital city function. First, the national government produces an amount, G, of a pure national public good. Second, household consumption of the public good does not depend on where households live. And third, in the model there is only one location where the national public good is produced. This assumption probably does well at capturing the localisation of national legislation, national jurisdiction, and national administration in a nation’s capital city. Specifically, in what follows, West is the capital city region while East is the peripheral region or hinterland. We assume that parameters for preferences and technologies are identical in both regions. Whenever possible, we drop the regional index r, where r = W , E . Apart from the public good, two other types of goods enter household utility. First, households consume a manufactured good Z . Second, households consume n different services xi , where i = 1, . . . , n. All these n services are aggregated into a composite service C with the help of the following CES-function:  σ
n  σ−1 σ−1 σ xi where σ > 1 (1) C ≡ i =1

A well-known illustration of the CES-function’s role is when prices of all services are equal. Because of the utility function’s symmetry, consumption of any σ service will be equal too. Then (1) becomes C = n σ−1 x , which can be rewrit1 ten as C = (nx )n σ−1 . Given that prices of all services are equal, nx may be considered as a proxy of expenditures on services. Consider then an increase of n accompanied by a decrease of x that serves to keep expenditure nx constant. Clearly, here, consuming a larger range of services while reducing consumption of each individual service makes the household better off. This strong valuation of variety is why (1) is often also referred to as reflecting the household’s love for variety. The representative household Cobb-Douglas utility function U (C , Z , G) given in (2) depends on the consumption of the composite service, C , of the manufacture, Z , and of the level of the public good G provided by government: U (C , Z , G) = C δ Z 1−δ G

where

0<δ<1

(2)

The household chooses the optimal levels of services xi and manufacture Z , but has no influence on G. The quantity of the public good simply enters utility as a positive externality. Each household inelastically supplies one unit of labour.

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Household income, then, equals labour income because in long-run equilibrium firms’ profits are zero (see below). Transportation costs within regions are always zero. But three particular assumptions specify the tradeability of the three goods across regions. First, by nature the public good G is tradable at no cost. Second, the services xi (i = 1, . . . , n) are non-tradable, again by nature. Third, the manufacture Z is assumed to be tradable without transportation cost. Prices are pi for service i (i = 1, . . . , n) and pZ for the manufacture. Households are not the only entities to have a taste for services. Government also uses services as inputs in the production of the public good, G, as specified by the CES production function:  n  σ  σ−1 σ−1 G= gi σ where σ > 1 (3) i =1

and where gi is the level of service i used in public good production. Obviously, services are inputs in public good production much the same way as they are inputs into the composite service, C , consumed by households (see Equation (1)). Since services as inputs into public good production are not tradable, they must be supplied by the capital region West. According to (3), government relies on the same services for public good production as households for consumption purposes. Examples of this overlap are numerous. Services by restaurants and hotels, newspapers and TV, accounting and consulting firms, lobbyists, and economists are typically offered to government as well as households. Government’s total revenues T comprise tax revenues in West T W and East E T . The tax rate on nominal income is exogenous and in both regions is equal to t, with 0 ≤ t ≤ 1.3 In equilibrium, labour income is the households’ only income, so that total revenues T are: T = T E + T W = t(LE wE + LW wW )

(4)

After having specificed objectives and constraints, we next turn to the agents’ behavioural functions. Each household maximises utility (2) for given net income w(1 − t) by choosing optimal levels of services and manufacture. This optimisation may be decomposed into a two-stage procedure. At the first stage we derive the optimal expenditures that fall onto manufacture on the one hand and on services on the other. Given the Cobb-Douglas specification in (2), and given net income w(1 − t), each household spends δ(1 − t)w on services as a group and (1 − δ)(1 − t)w on the manufacture. Household demand for the manufacture is then simply equal to (1 − δ)(1 − t)w pZ . At the second stage we derive household demand functions for every service. Given the CES-function in (1), and given total household expenditure on all services equal to δ(1 − t)w, it is straightforward to see that total private sector demand for service i by Lr households in region r(r = W , E ) is: 3 See Ades and Glaeser (1994) for a model where tax rates differ across regions. But here we want to exclude exploitation of the hinterland as a source of migration.

Capital cities: When do they stop growing?

xid ,r

53

(p r )−σ δ(1 − t)wr Lr = i (P r )1−σ

1  1−σ  r n  Pr ≡  (pjr )1−σ 

where

j =1

and

i = 1, . . . , n r

(5)

As is standard in the literature on monopolistic competition, we assume that n r is large enough to warrant that any change in a single service‘s price pi is too small to affect the price index P (e.g., see Fujita et al. 1999). Hence, the price elasticity of service demand just equals σ. We assume that government maximises the output of the public good G by choosing service inputs gid (i = 1, . . . , n W ) subject to its fiscal constraint n W W d d i =1 pi gi = T . This yields government input demand gi :

−σ T piW = 1−σ W P

gid



i = 1, . . . , n W

(6)

where P W is defined as in (5). Government’s service demand function, too, has a price elasticity of σ. Note once more that government only uses inputs provided by the capital region West, never those produced in the periphery region East. Aggregate demand for service i in region W is then equal to xid ,W + gid with elasticity of σ. But aggregate demand for service i in region E simply equals private sector demand xid ,E . Within each region, the market structure in the services sector is “monopolistic competition”. Each firm operates under the same increasing returns technology: (7) Li = α + βxi where α, β > 0 and (i = 1, . . . , n) and where Li is firm i ’s labour input. If the nominal wage is w, the cost function simply is wLi and marginal costs are βw. The price corresponding to the monopolist’s optimal output choice then becomes: pi =

σ βw σ−1

(i = 1, . . . , n)

(8)

All service suppliers have identical technologies, face identical price elasticities and buy labour in the same competitive labour market. Each firm will then choose its output such that the resulting output price does not vary across firms within one region: pi = p. Using this symmetry and (5), total private sector demand xid ,r (r = W , E ) for service i simplifies as follows: xid ,r =

δ(1 − t)Lr wr nr pr

(9)

while government demand for service i becomes: gid =

T . nW pW

(10)

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As in the literature, we assume that local variety increases until each monopolist’s profit becomes zero. In this zero profit long-run equilibrium, each firm’s output x is equal to: α x¯ = (σ − 1) (11) β Finally, the manufacture Z is produced by many different firms under conditions of perfect competition. Each firm employs the simple constant returns technology given in (12): Z = LZ (12) LZ is the amount of labour employed in the Z -industry. Labour is assumed to be completely mobile across sectors so that the wage rate w cannot vary by sector. Firms in the Z -sector maximise profits so that at each positive output level supplied marginal revenue must equal marginal cost, i.e., pZ = w. At this price the manufacturing sector’s aggregate supply Z s is perfectly elastic. Any demand for Z will be satisfied. For convenience and later reference, Table 1 summarises the behavioural equations of the model by sector. In the Table, all behavioural equations are indexed by region. Table 1. Behavioural equations of the model Sector

Aggregate behavioural function

LW Households in West Z -Industry in West n W Service Suppliers in West LE Households in East Z -Industry in East n E Service Suppliers in East 1 Service Supplier in West LW Households in West

LW LdZ ,W = Z s,W n W Ldx ,W = n W (α + β x¯ ) LE LdZ ,W = Z s,E n E Ldx ,E = n E (α + β x¯ ) x¯  δ(1 − t)LW wW n W p W

Government

T

Service Supply East Service Demand East

1 Service Supplier in East LE Households in East

x¯  δ(1 − t)LE wE n E p E

Z-Supply West + East Z-Demand West + East

Z-Industry in East and West LW + LE Households

(Z s,W + Z s,E )  (1 − δ)(1 − t)(LW wW + LE wE ) pZ

Labour Supply West Labour Demand West Labour Supply East Labour Demand East Service Supply West Service Demand West



nW pW

3 Short-run general equilibrium The model consists of n W service markets in region West, n E service markets in region East and two regional labour markets. Finally, the manufactured good is sold and bought on a single interregional market. Only in this market do agents from West and East interact. Besides, equilibrium conditions are identical for service markets within a region. This leaves us with the equilibrium conditions for the two representative service markets in West and East each, for the two

Capital cities: When do they stop growing?

55

Table 2. Sectoral identities by region West: Households in West Service Suppliers in West Z-Industry in West Government

(1 − t)wW LW = pZ Z d ,W + n W p W x d ,W p W x¯ = wW Ldx ,W pZ Z s,W = wW LdZ ,W (LW wW + LE wE )t = n W p W g d ,W

East: Households in East Service Suppliers in East Z-Industry in East

(1 − t)wE LE = pZ Z d ,E + n E p E x d ,E p E x¯ = wE Ldx ,E pZ Z s,E = wE LEZ

regional labour markets in West and East – and for the Z -market. Only five independent equations remain. Making use of the model’s zero-profit-properties, Table 2 summarises the sectoral identities. Neither firms in the manufacturing sector (because of constant returns to scale) nor monopolists in the services sector (because of unrestricted entry in long-run equilibrium) make positive profits.4 Service sector firms completely use up revenues from service sales to pay their workforce. Likewise, firms in the manufacturing sector use up revenues to cover their wage costs. Aggregating the identities in Table 2 by region highlights the consequences of government’s using inputs only from region West. In (13), all sectoral identities in East are consolidated. (We also have to use the fact that there are n E service markets in East).   wE LE − n E Ldx ,E − LdZ ,E



+ n E p E x¯ − x d ,E

+ pZ Z s,E − Z d ,E − tLE wE ≡ 0

(13)

In general equilibrium the labour market in East as well as the service market in East have to be in equilibrium. However, this equality of regional suppy to regional demand does not extend to the Z -market. In the Z -market, it is national supply and demand that must be equated. Hence, in general equilibrium only the first two on the left-hand side of Equation (13) disappear, terms in brackets

leaving pZ Z s,E − Z d ,E −tLE wE ≡ 0. As households pay taxes (t > 0), Z s,E > Z d ,E . East’s production of Z exceeds East’s demand for Z . This reflects the real transaction behind taxation of the peripheral region East. East can pay its taxes only by exporting the only tradable good in the economy – which is Z . Total taxes in East T E equal the value of East’s Z -exports to the capital city region West. A similar story can be told for region West. Aggregation of regional identities in West gives West’s regional constraint (14): 4 As a potential source of confusion, there are two long-run equilibrium concepts in the model. The first refers to a situation with no more entry into the services sector. The second, in Sect. 4, will refer to a situation with no more household migration across regions.

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K. Dascher

  wW LW − n W Ldx ,W − LdZ ,W



+ n W p W x¯ − x d ,W − g d ,W

+ pZ Z s,W − Z d ,W + tLE wE ≡ 0 (14)

(Here we use the fact that there are n W service markets in West). By applying a similar discussion as above, we conclude that for positive taxes, the capital region West will produce less of the Z -good than its households actually consume.5 The presence of government clearly gives rise to regional specialisation. Jointly, (13) and (14) give Walras’ law. Only four independent equilibrium equations remain. In what follows, the interregional market for the manufacture is ignored, with the manufacture as num´eraire: pZ ≡ 1. Equilibrium in the representative service market in West and East, respectively, holds if: x¯

=

x¯

=

δ(1 − t)LW wW + T nW pW δ(1 − t)LE wE nE pE

(15) (16)

West’s labour market and East’s labour market are in equilibrium if: = n W (α + β x¯ ) + LdZ ,W

LW E

L

= n (α + β x¯ ) + E

LdZ ,E

(17) (18)

respectively. Since regional interaction is limited, equilibrium solutions are straightforwardly realisable for the short-run, i.e., for a given allocation of labour to the two regions. Solutions for n W und n E are readily obtained from Equations (15) and (16), by substituting the solutions for x¯ and p in (11) and (8): nW

=

nE

=

δ(1 − t)LW wW + T ασwW δ(1 − t)LE ασ

(19) (20)

Given the solutions for regional varieties, we are left to seek solutions for the remaining two endogenous variables. Inserting (19) and (20) into the labour market equilibrium conditions (17) and (18) yields: LW LE

T + LdZ ,W wW = δ (1 − t) LE + LdZ ,E = δ(1 − t)LW +

(21) (22)

We look at labour market equilibrium in East first. Because in East labour employed in the services sector does not exhaust total available labour, there will always be a positive quantity of labour left to be employed in the manufacturing sector. Rearranging (22) shows: LdZ ,E = LE (1−δ(1−t)) > 0. This implies wE = 1. 5 If taxes are zero, each region would produce precisely as much as it needs to meet its own demands. After all, no government would then exist, so that no need arises to export a tradable good in order to pay taxes.

Capital cities: When do they stop growing?

57

Next we look at labour market equilibrium in West. In West, it may be not possible to bring about labour market equilibrium via quantity adjustments of Z . This can be seen when inspecting (21). There, δ(1 − t)LW equals labour employed in the services sector because of households’ demand for services. Then LW (1 − δ(1 − t)) is the maximum quantity of labour available to expand services production further. We cannot a priori say whether this is sufficient in order to  satisfy extra labour demand induced by government demand for services, i.e., T wW , even if production of Z were already reduced to zero. If LW is “too small”, West’s labour market may well be in disequilibrium unless wW increases.6 We can summarise these issues as follows. As long as Z -firms operate in West, wW = 1. If, however, wW exceeds one, no Z -firm is willing to produce. These two aspects may be condensed into a single statement of complementary slackness: (wW −1)Z W = 0. Apparently, the smaller a capital city’s labour supply is relative to government-induced labour demand, the more likely is its labour market to be equilibrated by price adjustment instead of quantity adjustment. Such switching occurs where wW = 1 and Z = 0. Inserting these conditions into (21) gives the critical allocation L˜ W : L˜ W =

t 1 − δ(1 − t)

where

0 ≤ L˜ W ≤ 1

(23)

For LW > L˜ W we have labour market equilibrium in West at wW = 1 and Z > 0. Conversely, for LW < L˜ W we have labour market equilibrium in West at wW > 1 and Z = 0. In the latter case we can also calculate the wage necessary to bring about equilibrium. Inserting LdZ ,W = 0 into (21), West’s market wage becomes: wW =

tLE (1 − t)(1 − δ)LW

for

LW ≤ L˜ W

(24)

From (24), as LW approaches L˜ W from below, wW falls. If LW is equal to or larger than L˜ W , then wW = 1. Hence, as the capital city’s population grows, its wage – expressed in units of Z – falls. At the same time, according to (8), the wage expressed in units of any service w p is constant. So a larger population unambiguously reduces the capital city households real wage. This is the model’s congestion effect, as mentioned in the introduction. It arises from the impact the national government has on the capital city’s labour market. We emphasise that no such congestion effect is present in East. Early on one might suspect that the presence of government in the capital city should drive up the capital city’s wage. But in fact this only happens if the capital city’s labour market is small. If it is large, then government demand for services instead implies the crowding out of the manufacturing industry. Since the Z -industry’s labour demand is perfectly elastic, no wage increase is needed to induce such crowding out. 6

I owe this important insight to a referee of an earlier version of this article.

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K. Dascher

4 General equilibrium in the long-run Thus far the regional distribution of the population LW has been held fixed. By now allowing households to migrate, we explore the long-run properties of the model. In particular, we look for stable long-run equilibria that are not corner equilibria. Under certain circumstances a stable long-run equilibrium in (0, 1) may emerge. In having identified a mechanism of agglomeration - as well as one of deglomeration - we first analyse how utility in East and utility in West develop as LW changes. The ratio of utility in West to utility in East, v ≡ V W V E , is found by applying the utility function in (2) to both regions. Hence,  v=

nW nE

δσ   σ−1

 δ  d ,W  W 1−δ x d ,W LW Z L   d ,E E d ,E L LE x Z

(25)

Naturally, the level of the national public good G provided in the capital plays no role in the welfare comparison. From Table 1, we see that equilibrium aggregate demands for the manufacture in West and East are (1 − δ)(1 − t)LW wW and (1 − δ)(1 − t)LE wE , respectively. From (16), equilibrium aggregate demand for a single service in East is x d ,E = x¯ . Finally, equilibrium aggregate demand for a single service in West is x¯ times the share of private sector  demand for services in total demand for services, i.e., x¯ times δ(1 − t)LW wW (δ(1 − t)LW wW + T ). These expressions, as well as (19) and (20), are inserted into (25). Simplifying gives:   δ δ(1 − t)LW wW + T σ−1 W 1−δ v= w (26) δ(1 − t)LE wW Depending on whether LW is below or above L˜ W , we have to distinguish between the two cases discussed above. In the first case, LW ≤ L˜ W , implying Z s,W = 0. Inserting (4) and (24) into (26) gives: 

LW 1 v= E L δ(1 − t)

δ  σ−1 

LE t W L (1 − t)(1 − δ)

1−δ for

LW ≤ L˜ W

(27)

In a diagram with v on its y-axis and LW on its x-axis, (27) gives the section of the v-curve for values of LW smaller than L˜ W . Suppose that LW falls. Then the first factor in (27) decreases, reflecting increasing agglomeration economies due to more service variety in East as well as decreasing agglomeration economies due to less service variety in West. However, the second factor in (27) increases as LW falls, thus representing the increase in West’s real wage due to the capital city function. The net effect is not clear but depends on the weights (that is, exponents)  of the two factors. In what follows we assume that δ (σ − 1) < 1 − δ.7 This requires that either δ or 1/(σ − 1) be sufficiently small. These parameters reflect 7 As one referee has pointed out, this condition is similar to the no-black-hole-condition in Fujita, Krugman and Venables (1999, p. 58).

Capital cities: When do they stop growing?

59

the weight of services in the utility function (2). Thus, as LW falls the loss of agglomeration economies is assumed to weigh less than the parallel increase in real income, thereby creating a net positive impact of a shrinking population. Put differently, v is a decreasing function in LW for LW ≤ L˜ W . In the second case, alternatively, LW ≥ L˜ W so that wW = 1. Hence, the part of the v-curve corresponding to those allocations LW larger than L˜ W , is given by:   δ δ(1 − t)LW + t σ−1 v= for LW ≥ L˜ W (28) δ(1 − t)LE Clearly, for LW ≥ L˜ W , v is an increasing function in LW . v, given jointly by (27) and (28), has the following three properties. (i) v is continuous in the open interval (0, 1). (ii) As LW tends to either zero or one, v approaches +∞. (iii) Finally, v is strictly decreasing for LW ≤ L˜ W and strictly increasing for LW ≥ L˜ W . This behaviour merely reflects the relative strength of agglomeration vs. deglomeration economies: seen from the capital city’s perspective, the congestion effect dominates agglomeration economies to the left of L˜ W . Vice versa, agglomeration economies dominate the congestion effect to the right of L˜ W . By property (iii), v must have a minimum at L˜ W . We can evaluate v(L˜ W ) as:  v(L˜ W ) =

t δ(1 − t)2 (1 − δ)

δ  σ−1

(29)

Migration costs being absent, households will migrate to the region with higher utility. Suppose first that v(L˜ W ) is larger than one. In that case, no interior equilibrium can exist. For all possible population splits relative utility is larger than one. All households in East migrate to the West, giving rise to a stable corner equilibrium in West. Alternatively, suppose that v(L˜ W ) is smaller than one. From properties (i) and (ii) we infer that v must equal one for some LW , say Lˆ W , between 0 and L˜ W . Then Lˆ W is a stable equilibrium. Also, a second but unstable equilibrium exists to the right of L˜ W . Note that whether v(L˜ W ) is larger or smaller than one depends on whether t is larger or smaller than δ(1−t)2 (1−δ). A higher national tax rate apparently decreases the likelihood of observing a stable equilibrium. The preceding discussion is illustrated in Fig. 1, where parameter values are σ = 3, δ = 0.5 and t = 0.1. This gives L˜ W ≈ 0.18 and Lˆ W ≈ 0.10, as drawn in Fig. 1. In the stable equilibrium Lˆ W , the capital region West is smaller than the periphery East. Hence, agglomeration economies in West are smaller than agglomeration economies in East too. But this is offset by a higher real wage in the capital. Note, too, that in the stable equilibrium, the capital specialises completely in services; every household in the capital is employed in the service sector.8 8 See Dascher (2000b) for evidence that capital cities do specialise in the production of services, i.e., in the production of non-tradable goods.

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K. Dascher

3.5 3

v(Lw)

2.5 2 1.5 1 0.5 0 0

0.2

0.4

0.6

0.8

1

1.2

Lw Fig. 1. Existence of Interior Equilibria (σ = 3, δ = 0.5, t = 0.1)

If interior equilibria exist, then L˜ W has to lie to the left of 1/2. The reason for this is easy to see. By definition of L˜ W , wW = 1 must be true. Wages do not differ across regions. Given this, a difference in utility can only come from differences in agglomeration economies in the service  sector. These in turn are related to the regional economy’s size. For v = V W V E to be smaller than one, then, East surely has to be larger than West. So v(L˜ W ) < 1 implies L˜ W < 0.5. A fortiori, Lˆ W as the single stable interior equilibrium to the left of L˜ W also has to be smaller than 0.5. We may conclude that a stable capital is always smaller than the periphery. We turn finally to comparative statics. As t increases, the v-curve shifts upwards. Hence the population of the capital associated with the stable interior equilibrium grows. With further increases of t, all of the v-curve will eventually come to lie above the (v = 1)-graph. The stable interior equilibrium disappears altogether. Here the peripheral region empties completely. So when do capital cities stop growing?, to revert back to this article’s title. Given our framework, we may group capital cities as follows. First, there are stable small capitals (where LW = Lˆ W ), which have stopped growing early. And second, there are stable large capitals (where LW = 1), which have only stopped growing after capturing the entire population.9 More precisely, if t > δ(1 − t)2 (1 − δ) then only stable large capitals can be observed. In the alternative scenario, i.e., where t < δ(1 − t)2 (1 − δ), the equilibrium capital city population depends on the initial population split, as is apparent from Fig. 1. Here either the stable large equilibrium or the stable small equilibrium will emerge as the long-run equilibrium. Surely the transition between the two scenarios is reminiscent of models of critical mass. A small increase of t in our model may have a large impact on the equilibrium population split. Note, finally, that by assuming that a small fraction of the population is completely immobile we would prevent a “black hole”, even, in the case where the corner equilibrium at LW = 1 is the only stable 9

Of course, in few countries does the capital comprise the total population.

Capital cities: When do they stop growing?

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equilibrium. From this perspective, Buenos Aires or Montevideo are capitals that effectively have reached this corner equilibrium, attracting all but those immobile households. To interpret the comparative statics results from the perspective of the federal state also seems tempting: Higher tax rates imply a larger share of federal revenues in total government revenues if income taxes are a source of revenue only to the federal budget (and holding other things equal). Higher tax rates on income may indicate a more centralised economy. If these revenues are spent exclusively on non-tradables produced in the capital city, then in our model an increasing tax rate generates a growing capital city. So in this special case less federalism actually means larger capitals.

5 Conclusions By using a framework of monopolistic competition and love-of-variety, we assume that larger regions benefit from economies of agglomeration. So households in capital cities, too, benefit from a growing population. However, we show that households in capital cities may also suffer from a growing population. In small capital cities, the presence of the national government drives up real wages through its induced extra demand for labour. But in large capital cities, this real wage advantage is smaller (if present at all). Hence, a capital city’s growth entails a loss of purchasing power for capital city households. Economies and diseconomies of agglomeration overlap. Given our framework, we show that the interaction of economies and diseconomies of agglomeration can lead to the existence of a stable interior equilibrium, i.e., where neither region captures the economy’s entire population. This equilibrium’s existence becomes all the more likely, the smaller revenues and expenditures of the national government are. We also have a result on regional division of labour between the capital city and its hinterland. A stable small capital city completely specialises in nontradable services, while its hinterland specialises in the production of tradables. But the extent of capital city specialisation declines with increasing capital city size. A stable large capital city produces tradables and non-tradables (while the hinterland as a region ceases to exist). Finally, we look at the model’s comparative statics properties. Higher national taxes make capital cities grow. Using this, we also speculate on the link between federalism and capital city size. Under restrictive assumptions, larger tax revenues and expenditures of the central government may be interpreted as more centralism. But then in the model federalism and capital city size are negatively related.

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