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On the birth and growth of cities:
Regional Science and Urban Economics 22 (1992) 243-258. North-Holland
On the birth and growth of cities: Laissez-faire and planning compared Alex Anas* State University of New York at Buffalo, Amherst NY, USA
Received October 1989, final version received June 1990
In a growing laissez-faire economy, the optimal time when the large cities should be decentralized into new settlements is missed because such timing requires collective migration to the same place by many self-interested agents. New cities are born at stochastically determined times when existing cities are larger than their optimal sizes and unstable. With negligible migration costs, such city births are associated with panic-migrations and related economic shocks. Thus, laissez-faire city systems develop in cycles of booms and busts. The planned timing of settlements smooths these cycles by rounding the tops and raising the troughs in the per capita utility profile over time. Efficient growth requires alternating cycles of laissez-faire and planning. Planning establishes each new city at the optimal time and nurses it through an initial period of instability. As the economy grows large, the duration of the necessary planning periods vanishes asymptotically. In a declining laissez-faire economy, cities are abandoned at deterministic times and at smaller sizes than the sizes at which they would be born during growth. The decline path cannot be obtained by reversing the growth path, and a hysteresis effect is present.
1. Introduction The spatial concentration of e c o n o m i c activity in u r b a n areas has been explained by invoking a variety of factors. A m o n g these are economies of scale in p r o d u c t i o n internal or external to the firms CMills (1967), Dixit (1973)], interindustry linkages [Goldstein and Moses (1975)], the need for social interaction EBeckmann (1976)], expenditure on public g o o d s EStiglitz (1977), A r n o t t and Stigiitz (1979)], business externalities a m o n g firms of the same industry, also k n o w n as localization economies EFujita and O g a w a (1982)] and positive localization externalities which increase the marginal p r o d u c t of labor and o v e r c o m e the heterogeneity in laborers' tastes for location [Anas (1988)]. I n these explanations, limits on concentration arise from the costs of transportation, the costs of transactions a m o n g e c o n o m i c agents or the burden of environmental externalities which increase with concentration. An optimal city size is reached when the marginal gains from Correspondence to: Alex Anas, Department of Economics, 608A O'Brian Hall, SUNY at Buffalo, Amherst, NY 14260, USA. *I would like to thank Richard Arnott, Robert Helsley and Edwin Mills for reading this paper and for detailed discussions on the results. Any shortcomings in the paper are strictly mine.
0166--0462/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
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adding a small increment of economic activity just equal the marginal costs created by that growth [see, for example, Dixit (1973), Tolley (1974) or Arnott (1979]. 1 The presence of an optimal city size for a single city has implications for the distribution of activity among cities in economies with large aggregate populations, as studied by Henderson (1974) and Upton (1981). In these static models, cities are identical in technology, amenities and all other aspects and there is one homogeneous good. Population and malleable capital moves costlessly among cities until returns and utilities across cities are equalized in a long-run equilibrium. At such an equilibrium, in an economy with a single city-type, all cities are of equal and optimal size and a sufficient number of cities emerges to accommodate all activity. The large aggregate economy exhibits constant returns to scale. As aggregate economic activity expands, cities are spawned continuously at the rate of aggregate population growth so that the intercity equilibrium is maintained all along the economic expansion path [Kanemoto (1980, oh. 7), Henderson and Ioannides (1981)]. With different types of cities, due to variations in technology and local resource endowments, there is a size distribution of cities and the number of cities of each type adjusts to maintain equal returns and utilities across cities. For a review of this literature see Henderson (1987). Such models of long-run city size distributions serve well for the comparative statics analysis of city systems with large aggregate populations. This is not fully satisfying because of two troublesome observations. First, many developing nations (being in the early phases of industrialization) are newly experiencing agglomerative forces and have huge portions of their growth concentrated in one big city (the primate city) or, at most, in several big cities. Such nations face the question of whether it is efficient (or even possible) for them to spawn new settlements, and if so with what frequency. It is often observed that such new settlements are beneficial but costly to start because the existing primate city is overgrown and too difficult to decentralize. 2 Second, the large primate city problem is also observed in many developed countries where a host of regional development policies are frequently drafted and sometimes implemented to induce or directly subsidize decentralization. Again, such decentralization is costly even though the primate cities are considered to have grown beyond their optimal sizes. The paper aims for the simplest possible theoretical model which adds 1For a discussion of policy issues stemming from the optimal city size concept see Tolley and Crihfield (1987). 2Despite the enormous costs, a number of developing countries have set up new national capitals partly in response to extreme growth in their large cities (for example, Brasilia in Brazil or Islamabad in Pakistan). Still other countries like Nigeria or Korea have drafted plans for such cities but have either cancelled or postponed them because of the costs.
A. Anas, On the birth and growth of cities
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substance to these observations. In the spirit of the previous city system models, economic differentiation which can lead to various sizes and types of cities is ignored. However, our analysis differs from the previous long-run equilibrium studies by focusing instead on the transient dynamics of an economy which grows from very small (initially having only one city) to very large, spawning new cities as it grows. The process of decline in which settlements disappear successively as population falls, is also examined and found to be very different from that of growth. In particular, the decline path cannot be obtained by retracing the growth path backwards in time. Migration costs are assumed to be z e r o . 3 A closely related paper is by Henderson (1986) who considered the occupation and development of the remote regions of a growing country, by migration from the areas developed earlier. Henderson found that the development of the remote regions occurs later than the optimal time. This result agrees with the finding of this paper, that it is optimal to set up the second city much earlier than the date at which it would emerge under laissez-faire. However, this paper differs from and adds to Henderson's in several respects. First, while Henderson's model is complicated by realistic assumptions and by discussions of government policy, we develop the simplest possible theoretical model uncluttered by complex, yet important, policy issues. Second, we perform a precise stability analysis, tying together the theory of optimal city size with the mathematics of multiple equilibria and bifurcation analysis. Third, we show that periods of laissez-faire and planning must alternate on the optimal growth path and that the length of the required planning periods vanishes as the number of settlements increases. Fourth, unlike Henderson (1986), who examined the case of growth only, we consider decline as well as growth, showing that the decline path cannot be obtained by reversing the growth path and that a hysteresis effect exists. Our simple economy consists of only labor, which grows, and land, which is limited: the former is the source of a positive localization economy, the latter the source of a land use congestion externality. In such a setting, with labor growing exogenously over time, all economic activity is initially concentrated into one city because of agglomeration economies. Such a city grows to its optimal size and beyond, until it becomes feasible for population perturbations under laissez-faire to establish a second city.4 This happens when the first city has become unstable in response to sufficiently small aPositive migration costs would change the results in a predictable way but without negating our essential findings. Several footnotes in the paper comment on the effect of migration costs. *'Perturbations' in this paper are migrations between the cities which occur at random intervals and are of random magnitude. In the tradition of stability analysis for deterministically dynamical systems, such perturbations arc exogenous to the model and can be explained as shocks, errors or deviations from rationality, or as the result of a trial-and-error process through which agents explore the states of the system.
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.4. Anas, On the birth and growth of cities
declines in its population. At such time, it takes a small number of migrants (at the extreme only one) to set up a second settlement which matches the utility of the first and large city. Social optimality requires that the second city be set up much earlier and at a larger initial size, requiring a much larger and planned migration out of the first. Individual agents (in our model, laborers) cannot, in a market economy based on self-interest, antidpate the optimal time when decentralization should occur because the timing of such decentralization requires coordinated action among agents and is thus external to each agent. Under laissez-faire conditions, the second city is either born too late or may not emerge at all if agents in the economy do not experiment with suboptimal and unstable situations. This is so because the first city remains a stable equilibrium for a long time after it exceeds its optimal size. When the second city ultimately emerges, it is always associated with a sudden panic-migration of population out of the first. Under planned optimal growth, the second city must be spawned much earlier than the laissez-faire case, and must be 'nursed' through a period of instability until it reaches a critical size thereafter growing on its own. Under both laissez-faire and planning the sizes of cities pass through stages of growth and decline as does the path of the inhabitants' utilities. The planned optimal timing of settlements smooths these cycles by rounding the peaks and raising the bottoms in the per capita welfare profile over time. As the aggregate population grows very large, the need for planning vanishes and the system is capable of spawning cities at an asymptotically continuous and optimal rate. This cyclical pattern of growth is not reversible in a two-city system declining in total population. One of the two declining cities suddenly vanishes at a smaller size than the size at which the second city is born during growth (a hysteresis effect). The time and city size at which this occurs is determinate. In contrast, under growth, the time at which the second city is formed and the size of the newly formed city are indeterminate and depend on the strength of perturbations at that time. Aspects of the real world, intentionally left out of the analysis, strengthen our conclusion: costs of adjusting capital, sunk public infrastructure and the costs of migration all suggest that primate cities in a laissez-faire economy can grow too large and remain too large for too long. With such costs, a planned settlement policy appears potentially beneficial. Planning must be introduced at regular intervals to 'seed' settlements at the optimal times and to 'nurse' each new settlement to stability. 2. Growth and decline in a two-city laissez-faire system
Consider, for now, an economy which has two a priori identical locations where cities can emerge. A simple example may be an island where the interior is uninhabitable and cities can emerge at either the West End and/or
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the East End. At each end, L acres are available for urban development. Will such a simple economy spawn one city, two identical cities or two cities of different sizes? How will the urban pattern evolve over time as the aggregate population of the economy grows (or declines) smoothly over time? Will the number and relative sizes of the cities (if two) change along such a growth path for the aggregate economy? Suppose that the aggregate economy is the simplest labor economy possible with land the only other input. There are N identical laborers. An undifferentiated good is produced in the urban area(s) and since there is only one good, there is no trade in or out of equilibrium: the inhabitants of each city consume the total product of their city. Cities are linked only by labor migration (assumed to occur quickly and costlessly) as laborers will move between cities so that the utilities of living in either city are equal. Suppose that each laborer derives utility from his output (which he consumes entirely) and the size of his private land lot. Suppose also that a simple form of a localization externality governs production: the output of each laborer rises with the number of laborers in the same city because the presence of other laborers generates innovation in production by means of communication and information exchange among laborers: each city is simply a world of 'several heads working together are better than one working alone'. Land in a city is divided equally among the identical workers and, because the total amount, L, is finite, each worker's land lot is reduced as the city's population increases. Thus, z---f (n) [with f ' ( n ) > 0 and f " ( n ) < 0 ] is a single laborer's output as a function of the number of laborers, n, land per laborer is q---L/n, and utility per laborer (as a strictly concave function of consumption and land) is u=-u(z,q)=u(f(n), L/n)-V(n). It makes sense to assume that f ( 0 ) = 0 , u(z,q)-,O as z--,0 or q--,0. Also, assume that u(f(n),L/n)>O for any n. Thus, u(f(n),L/n)--,O as n--,oo. This simplest of models gives an optimal city size: a maximum occurs where the marginal rate of substitution between consumption and land (or the rent for land, R) just equals the ratio between the marginal land use congestion effect (decreases in land per worker) and the marginal localization benefit (increase in output per worker) from the addition of just one more worker:
q/n - R . uJuq-f,(n )
(1)
Resident-workers in the city own the land collectively as equal shareholders, hence there are no absentee landlords and rental dividend income is endogenous. Migrants to other cities lose their rights to the rent dividend in the city of origin, but become shareholders in the city of destination. Then,
RS.U.E. D
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A. Anas, On the birth and growth of cities
letting y be the income of a worker, utility maximization by each worker can be stated by max,,au(z,q), subject to y = z + R q , where y = f ( n ) + q 2 / n f ' ( n ) : income consists of the worker's output plus the value of his land holding. It is now necessary to set up a dynamic migration process whereby laborers move between the cities. We let an overdot denote a derivative with respect to time. Then, if n and N - n are the two cities' labor forces, the following is a continuous and once~differentiable migration adjustment process: h = f [ V ( n ) -- V ( N - n)];
(2)
with sign(ri) = signEV(n)- V(N--n)] and a f E . ] / d [ . ] >0. The migration process (2) simply says that the population of a city increases when its utility is higher than its rival's, decreases when its utility is lower than its rival's and remains the same otherwise. An equilibrium distribution of population between the two cities occurs when the time derivative of n vanishes, which requires V(n)= V ( N - n ) . 5 In a symmetric equilibrium, each city has exactly half of the total population. Asymmetric equilibria also exist, because it is possible for one city to be large (and thus productive but congested in the use of land) and achieve the same utility as the other which is small (and thus relatively unproductive but also uncongested in the use of land). There are two other possible equilibria in which only one city exists but the time derivative (2) need not vanish. These are the corner solutions (N,0) and (O,N). Fig. 1 shows the loci of all the equilibria for all values of N. The ray OO' is the locus of all symmetric equilibria. The hyperbolic curve AA' is the locus of all asymmetric equilibria and the two axes are the loci of all one-city equilibria. In the figure, the level of population is fixed by a straight line with slope minus one and growth is traced by moving this line away from the origin. Two times the optimal city size is the critical threshold for total population. If N <2n* there are three equilibria: the symmetric one and the two one-city equilibria. If N exceeds this threshold size all five equilibria are possible: the symmetric one, the two one-city equilibria and the two asymmetric equilibria. To see which equilibria are in fact attainable, we have to examine the global system stability of the entire set of equilibria for the two regimes of N. From (2): dfi/dn = ( df /d [" ]) {~ V (x)/ax Ix=~+ a V(x)/ax Ix =(N- n~}.
(3)
For a symmetric equilibrium, the sign of (3) is positive for all N smaller than SBecause we are assuming that migration is costless and instantaneous, the scale of df[']~ d [ . ] will be taken to be large enough to allow us to ignore the actual durations in migrations.
249
A. Anas, On the birth and growth of cities r1
nl
A
o1+02=N / o ,
I
1
__~~ I
rl
-EI~
v(nl) = v(n2)
E2,~
'y V(n 1)
"-
,
0
N'N \\%
II N<2n*
\\
[
N=2n*
n*
N>2n*
~
n2
\,
-\ ,-- f V(n2) Fig. 1. Equilibriain the two-cityeconomy(nl: populationof city one; n2: populationof city two; E~: symmetric two-city equilibrium; E2,E3: one-city equilibria; E4,Es: asymmetric two-city equilibria). 2n*, it is zero when N = 2 n * and it is negative for N larger than 2n*. Therefore, the symmetric equilibrium is unstable when the city system has population less than or equal to the threshold size (see figs. 2a and 2b) and stable when the system has population larger than the threshold size (see fig. 2c). As the system's population just passes through the threshold size (see fig. 2b), then the system bifurcates from the three-equilibrium state to the fiveequilibrium state. Since for economies with below-threshold populations, the symmetric equilibrium is unstable, such 'small' economies will initially concentrate all population in one city. To examine the stability of such one-city equilibria we note from (2) that ri = f[V(O)-- V(N)] <0,
(4) fi = f [ V ( N ) -
V(O)] > O.
A. Anas, On the birth and growth of cities
250
}'1
N < 2n*
I Stable Unst,able~1111
(a) Ea ~ f N E / 1 2 h 1/Stable (b) 5 3
1 2 N = 2n* Unstable ~ ' ~ ~ E1 /2
h
N > 2n* Stable
I
Unstable
Stable
Stable
Stable .
/ II E2
Unstable
n
I
i
A
Stable
Fig. 2. The stability of the equilibria in relation to aggregate population in the economy.
These two conditions mean that one-city equilibria are always stable (see fig. 2c). By the continuity of the migration process given by (2), it follows that since one-city equilibria and the symmetric equilibrium are stable, then the asymmetric equilibria must be unstable (see fig. 2c).
2.1. The growth regime We are now ready to examine the implications of our analysis for the dynamics of urban development. To do so we adopt a monotone growth regime in which the population growth function N(t) is given as a function of time t, begins at time t = 0 from N(0)
251
A. Anas, On the birth and growth of cities
(t)
///~
N(t)
..........n-!'!-'---g-~~ ~ Threshold aggregatesize 2n' n* ~ " Optimal citysize •~ .~ t t V(t) '
1/2 N(t) n(t),decline t, time
.
~
~ t*
, i"
growth t, time
Hysteresis Fig. 3. The time paths of population and utility under growth and decline in the laissez-faire case.
settlers returning to the West End. However (because the locus A A ' approaches the two axes asymptotically), the size of such migrations sufficient to destabilize the West End gets continuously smaller as the city becomes larger and its utility level lower. Eventually, just a few migrants will again move (for r a n d o m reasons) out of the West End to start a small settlement at the East End. By this time, utilities at the West End are far lower than optimum, and the small settlement at the East End immediately achieves higher utility than that which prevails at the West End. When word of this reaches the West End, a panic-migration suddenly ensues and, because of costless migration, the two cities are instantly equalized in size and in utility: the system j u m p s suddenly from a one-city equilibrium to the symmetric two-city equilibrium. 6 Thereafter, the two equal cities grow equally over time and the symmetric division of activity holds. Fig. 3a shows 6The migration is characterized as a 'panic' because of our assumption that it is costless: once it is learned that the new city is viable the adjustment to equilibrium is sudden. If migrations involved cost or lags due to information diffusion, then utilities between the two settlements will be unequal and the system will be out of equilibrium until all of the required population has migrated to the second city.
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A. Anas, On the birth and growth of cities
that even though N(t) is monotonic over time (drawn as linear in the figure), the population of the West End undergoes a cyclical (or unbalanced) transition pattern: the city grows much beyond optimal size by becoming too congested and suffers a gradual decline in utility until it becomes sufficiently unstable in the face of outmigration. At some point (when a sufficiently large outmigration occurs), the West End quickly loses exactly half of its population to the East End and recovers a good part of the lost utility by virtue of reduced land use congestion. Thereafter, the two identical cities continue to decline in utility as they grow equally. After the system 'bifurcates' from the three-equilibrium regime to the fiveequilibrium regime (see figs. 1 and 2c), the one-city equilibrium is initially very robust in the face of migrations. Growth must occur until this robustness deteriorates to the point where a small perturbation is sufficient to start a stable second city going. 7 When this time actually comes, depends on how big the perturbations are. Therefore, in fig. 3b, the time [ is determined by chance events. It is also possible that it never happens, if migrations do not occur. This means that a certain amount of experimenting, randomization or curiosity is necessary for the system to find the preferred state within a laissez-faire environment. Highly regimented or uninnovative societies may become 'stuck' in overconcentrated states not taking advantage of the considerable improvement in utilities which can occur by spawning a second city. 2.2. The decline regime
A monotonic decline process in which the exogenous population growth function N(t) is exactly reversed in time does not result in the exact reversal of the urban development pattern. Figs. 3a and 3b show that, because the symmetric equilibrium remains stable all the way down to the critical size, the declining twin cities reach maximum utility as they become gradually decongested in the use of land. Then, at exactly critical size, the symmetric equilibrium is unstable and all activity must suddenly concentrate into one of the cities (to be selected by chance). This results in a sudden drop in utility as the remaining single city is now larger than optimal in size. We note that whereas in the case of growth, the time at which the second city was born was stochastically determined, in the case of decline the time at which one city is abandoned is deterministic. As population continues to decline, the remaining city becomes gradually decongested to reach maximum utility again. After that point, population and utility decline continuously and 7A measure of this robustness is seen directly from fig. 1 to be proportional to the difference between the population of the city under the stable one-cityequilibrium(N) and the population that city would have if it were in the unstable asymmetric equilibrium. As N grows, the assymetricequilibrium population gets closer to N and thus the margin of stabilitydeteriorates.
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eventually the city vanishes altogether. The important finding here is that there is an hysteresis effect: the transition from one to two cities under growth and the reverse transition from two cities to one under decline do not necessarily occur at the same population level. In the case of decline, the time of transition from two cities to one always occurs at time t*, because after that time the two-city equilibrium is unstable.
3. Optimal growth of the two-city system A related set of questions naturally emerges from our finding that the laissez-faire (or equilibrium) growth path is nonmonotonic and that it may result in inferior welfare levels because of the aggregate economy's tendency to remain in a one-city equilibrium for too long. What is the optimal growth (decline) path which maximizes the discounted present value of utility? What are the planned transition times from one city to two (and from two to one) on these optimal paths? Is the planned symmetric growth of a two-city system from the beginning ever optimal? These questions are resolved by maximizing the discounted present value of the per capita utility path in the economy, while imposing the condition that identical laborers are treated equally at each point in time: the per capita utility on the optimal path (while changing over time) must be the same in all on the existing cities at each point in time. Ignoring any planning costs associated with setting up the second city, this welfare function is t**
max {t** ~ O}
W(t**) = ~ e -~s V(N(s)) ds 0
+ J" e-*sV
ds,
(5)
t**
where t** is the optimal time at which the planned system should switch from the one-city equilibrium to the two-city symmetric equilibrium by establishing the second city. ~ is the social rate of discount. The first-order condition of (5) gives
V(N(t**))= v(N(t~*)),
(6)
which, not surprisingly, says that the second city should be set up when halving the population of the first leaves utility unchanged as reduced congestion in the first city just offsets lost output. This occurs when the first
254
A. Arias, On the birth and growth of cities
v(t V(t*) v(t**)
i Optimalplannedpath ~--Symmetric two~ plannedpath
"', i' "', ,.' "i
~,,~ Laissez-fairepath \~ "~
Fig. 4. Planning and laissez-fairepaths compared.
city's population is between the optimal size and twice the optimal (or the bifurcation threshold) size (see fig. 4). Note that, therefore, the switch time is earlier than the point in time when a symmetric equilibrium is stable. Planners must act at t** to break up the city into two equal smaller cities and must plan to keep the two cities going until they are stabilized at time t*. Thereafter, no further intervention is needed. Thus, the optimal path requires that the second city be established before it is stable and therefore before trial-and-error migrations could establish it under laissez-faire. To achieve this, a temporary substitution of planning for laissez-faire practices is required. Fig. 4 shows the path of per capita utility over time when the system follows this optimal development pattern. This path dominates the symmetric path (also enforceable by planning) in which two equal cities are set up from the beginning and 'nursed' until the two-city equilibrium becomes stable at just beyond critical aggregate population) The optimal path also dominates the laissez-faire paths in which the panic-migration ensues at some randomly determined time. A nice property of the optimal path (unlike the laissez-faire paths) is that it is reversible if the national population were declining. This is seen, by reversing the limits on the two integrals in the social welfare function, (5). 9 It is interesting to note that the inhabitants of the first city have an incentive to subsidize the migrants in order to induce them to move and establish the second city at the socially optimal time. Such subsidies are SThe symmetricplanned path would become optimal if the planning costs of setting up two equal cities from the outset were sufficientlylower than the planning and migration costs of breaking up the first city at the optimal time. 9 To emphasize, this reversibility hinges on maintaining the infinite horizon nature of the problem: we start at t=O with two equal infinitelylarge cities and decline monotonicallyuntil all population vanishesat t = oo.
A. Anas, On the birth and growth of cities
255
wasteful if migration costs are zero. All that is needed is the assurance that the second city will be established as a temporary 'club' (until it becomes stable) at the designated time and place. More than the needed number of migrants will seek to enroll in such a club. If migration is costly, then the inhabitants of the first city can be taxed to raise subsidies which might offset in part the private moving costs of the migrants. A private developer can also help establish the city earlier than the laissez-faire time. As seen in fig. 3b, the birth of the second city under laissez-faire is associated with an upward jump in utility. By announcing that a city will be available, a developer acts as a coordinator and, hence, makes collective migration possible. In return for this, the developer can extract rents from the city inhabitants. If many potential developers are available, they will attempt to undercut each other by proposing to set up the second city at an even earlier date, and hence one which is closer to the optimal one. Returning to the planning problem, it is easy to extend the analysis [by generalizing (5)] to determine the time at which the ruth city ought to be set up in an economy which has a large numer of available sites of L acres each. The social welfare maximization problem now becomes
max W(0 = tltt=O
e-*" V
ds,
(7)
m = 2 tin-1
where the infinite vector f = [ 0 , t 2 . . . . . tm.... ] gives the times at which each successive city should be set up by the planners (tt=0). The first-order conditions result in the following relationship from which the mth city's birthdate is to be derived:
V(~)= v(N(--~)),m>2.
(8)
Fig. 5 shows the way in which laissez-faire and planned utility paths will differ as new cities are spawned one at a time. Cities are spawned at intervals which get shorter with each new city and the planned growth path tends to round out the peaks and raise the bottoms of the laissez-faire utility path. So far, we assumed that cities are set up one at a time. Is it possible to do better by increasing the number of cities more than one at a time? For example, is it better to break up the first city into two equal cities (as was assumed so far) or to wait until it is larger and then break it up into three equal cities? Similarly, in a growing two-city system, is it better to set up a third city according to (8), or is it better to wait until four equal sized cities can be established by the cleavage of each existing city into two? Under our
256
A. Anas, On the birth and growth of cities
V(t)
- -
Optimal Path
......
Laissez-faire Growth Path
V(n )
t2
t3
t4
t5
to t7 t8
LF: period of laissez-faire under growth P:
period of planning under growth
Fig. 5. Welfare effects of city sequences under planning and under laissez-faire in growing and declining economies.
assumptions of costless migration and costless planning, waiting to establish more cities later involves a welfare loss relative to establishing each next city at the earliest possible date. This follows by recalling that it is only efficient to start a new city when existing cities (one or more) are larger than the optimal size and thus when per capita utility is declining with growth. Welfare is maximized by reducing the durations of these stretches in time when utility is declining. Therefore, new cities must be set up as rapidly as possible and this requires setting them up one at a time: the optimal welfare path which maximizes (7) dominates all other welfare paths in which cities are set up more than one at a time. 3.1. An example o f a birthdate sequence
It is worthwhile to consider the following specific example. Suppose that the product of a laborer is given by z = n=. Let each laborer's utility function be of the following constant elasticity of substitution form with the degree of homogeneity equal to one: V(n) = (n- ~# + (L/n) - #)- 1/#,
where 15 is the substitution parameter.
(9)
A. Anas, On the birth and growth of cities
257
This utility function gives an optimal city size n*: n* = (~Lp)1/~1+~)p.
( 1O)
Let the aggregate population grow exponentially at the rate r. Then
N(t.)= N(O)exp(rtm)
(11)
where N(0)
i-L,/Cl+~)[~,m___cT___~_l 1).P_m~B _~_pj '~1/p{1+~)]. (m_
tm=rlnL_~l
(12)
Let ~ = f l = l / 2 and choose L such that n*=100,000 from (10). Also, let population grow at 2% annually (r =0.02) and set N(0)= 50,000. In such an economy, the second city must be set up in 51.74 years when the population of the first city reaches 140,716 (or nearly 41% larger than its optimal size). The third city is set up 27.63 years after the second, when the combined population of the first two cities reaches 244,350 (or nearly 22% above optimal size per city). The fourth city is set up 17.37 years after the third, when the combined population of the first three reaches 346,111 (or nearly 15% above optimal size per city). 260 years from the beginning, there are 90 cities and new cities are spawned at a rate of two per year and city sizes at birth are within 0.56% of the optimal city size. At the risk of appearing simplistic, these numbers demonstrate the cycles that can prevail during optimal economic growth and suggest that the welfare losses from failing to plan new cities at the correct times can be large. 4. Concluding comments
Time lags needed in building public infrastructure, the immobility of capital, the durability of real estate, the costs of migration and the costs of planning new cities are all factors that give large urban concentrations a margin of stability even wider than our analysis suggests. The breaking up of such concentrations is costly, and thus the spawning of new settlements or the transfer of growth to less developed regions remains a valid yet underachieved goal in both developed and developing countries. Societies which like to experiment with new settlements despite these costs are more likely to come close to the optimal path, whereas inflexible societies which reinforce large settlements with government programs and subsidies may become rigidly stuck in inefficient but stable city-system configurations. The
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sizes o f large cities in such s i t u a t i o n s is n e i t h e r the o u t c o m e of c o n c e n t r a t i o n benefits (which have been e x h a u s t e d l o n g ago) n o r the a t t r a c t i o n of ' b r i g h t city lights'. Rather, such sizes m a y be e x p l a i n e d b y insufficient shocks, o r p l a n n i n g o p p o r t u n i t i e s foregone b y p o o r timing. E c o n o m i c efficiency suggests t h a t a bias t o w a r d d e c e n t r a l i z a t i o n c a n offset the stability o f large places.
References Anas, A., 1988, Aggiomeration and taste heterogeneity: Equilibria, stability, welfare and dynamics, Regional Science and Urban Economics 18, %35. Arnott, R.J., 1979, Optimum city size in a spatial economy, Journal of Urban Economics 6, 65-89. Arnott, R.J. and J.E. Stiglitz, 1979, Aggregate land rents, expenditure on public goods, and optimal city size, Quarterly Journal of Economics 93, 471-500. Beckmann, M.J., 1976, Spatial equilibrium in the dispersed city, in: G.J. Papageorgiou, ed., Mathematical land use theory (Lexington Books, Lexington, MA). Dixit, A., 1973, The optimum factory town, Bell Journal of Economics and Management Science 637-651. Fujita, M. and H. Ogawa, 1982, Multiple equilibria and structural transition of non-monocentric configurations, Regional Science and Urban Economics 12, 161-196. Goldstein, G.S. and L.N. Moses, 1975, Interdependence and the location of economic activity, Journal of Urban Economics 2, 63-84. Henderson, J.V., 1974, The sizes and types of cities, American Economic Review 64, 640-656. Henderson, J.V., 1986, The timing of regional development, Journal of Development Economics 23, no. 2, 275-292. Henderson, J.V., 1987, General equilibrium modeling of systems of cities, in: E.S. Mills, ed., Handbook of regional and urban economics, Vol. 2: Urban economics (North-Holland, Amsterdam). Henderson, J.V. and Y. Ioannides, 1981, Aspects of growth in a system of cities, Journal of Urban Economics 10, 11%139. Kanemoto, Y., 1980, Theories of urban externalities (North-Holland, Amsterdam). Mills, E.S., 1967, An aggregative model of resource allocation in a metropolitan area, American Economic Review 57, 19%210. Stiglitz, J.E., 1977, The theory of local public goods, in: M. Feldstein and R. Inman, eds., The economics of public services (Macmillan, New York) 274-333. Tolley, G., 1974, The welfare economics of city bigness, Journal of Urban Economics 1, 324-345. Tolley, G. and J. Crihiieid, 1987, City size and place as policy issues, in: E.S. Mills, ed., Handbook of regional and urban economics, Vol. 2: Urban economics (North-HoUand, Amsterdam). Upton, C., 1981, An equilibrium model of city sizes, Journal of Urban Economics 10, 15-36.
On the birth and growth of cities: Laissez-faire and planning compared Alex Anas* State University of New York at Buffalo, Amherst NY, USA
Received October 1989, final version received June 1990
In a growing laissez-faire economy, the optimal time when the large cities should be decentralized into new settlements is missed because such timing requires collective migration to the same place by many self-interested agents. New cities are born at stochastically determined times when existing cities are larger than their optimal sizes and unstable. With negligible migration costs, such city births are associated with panic-migrations and related economic shocks. Thus, laissez-faire city systems develop in cycles of booms and busts. The planned timing of settlements smooths these cycles by rounding the tops and raising the troughs in the per capita utility profile over time. Efficient growth requires alternating cycles of laissez-faire and planning. Planning establishes each new city at the optimal time and nurses it through an initial period of instability. As the economy grows large, the duration of the necessary planning periods vanishes asymptotically. In a declining laissez-faire economy, cities are abandoned at deterministic times and at smaller sizes than the sizes at which they would be born during growth. The decline path cannot be obtained by reversing the growth path, and a hysteresis effect is present.
1. Introduction The spatial concentration of e c o n o m i c activity in u r b a n areas has been explained by invoking a variety of factors. A m o n g these are economies of scale in p r o d u c t i o n internal or external to the firms CMills (1967), Dixit (1973)], interindustry linkages [Goldstein and Moses (1975)], the need for social interaction EBeckmann (1976)], expenditure on public g o o d s EStiglitz (1977), A r n o t t and Stigiitz (1979)], business externalities a m o n g firms of the same industry, also k n o w n as localization economies EFujita and O g a w a (1982)] and positive localization externalities which increase the marginal p r o d u c t of labor and o v e r c o m e the heterogeneity in laborers' tastes for location [Anas (1988)]. I n these explanations, limits on concentration arise from the costs of transportation, the costs of transactions a m o n g e c o n o m i c agents or the burden of environmental externalities which increase with concentration. An optimal city size is reached when the marginal gains from Correspondence to: Alex Anas, Department of Economics, 608A O'Brian Hall, SUNY at Buffalo, Amherst, NY 14260, USA. *I would like to thank Richard Arnott, Robert Helsley and Edwin Mills for reading this paper and for detailed discussions on the results. Any shortcomings in the paper are strictly mine.
0166--0462/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
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adding a small increment of economic activity just equal the marginal costs created by that growth [see, for example, Dixit (1973), Tolley (1974) or Arnott (1979]. 1 The presence of an optimal city size for a single city has implications for the distribution of activity among cities in economies with large aggregate populations, as studied by Henderson (1974) and Upton (1981). In these static models, cities are identical in technology, amenities and all other aspects and there is one homogeneous good. Population and malleable capital moves costlessly among cities until returns and utilities across cities are equalized in a long-run equilibrium. At such an equilibrium, in an economy with a single city-type, all cities are of equal and optimal size and a sufficient number of cities emerges to accommodate all activity. The large aggregate economy exhibits constant returns to scale. As aggregate economic activity expands, cities are spawned continuously at the rate of aggregate population growth so that the intercity equilibrium is maintained all along the economic expansion path [Kanemoto (1980, oh. 7), Henderson and Ioannides (1981)]. With different types of cities, due to variations in technology and local resource endowments, there is a size distribution of cities and the number of cities of each type adjusts to maintain equal returns and utilities across cities. For a review of this literature see Henderson (1987). Such models of long-run city size distributions serve well for the comparative statics analysis of city systems with large aggregate populations. This is not fully satisfying because of two troublesome observations. First, many developing nations (being in the early phases of industrialization) are newly experiencing agglomerative forces and have huge portions of their growth concentrated in one big city (the primate city) or, at most, in several big cities. Such nations face the question of whether it is efficient (or even possible) for them to spawn new settlements, and if so with what frequency. It is often observed that such new settlements are beneficial but costly to start because the existing primate city is overgrown and too difficult to decentralize. 2 Second, the large primate city problem is also observed in many developed countries where a host of regional development policies are frequently drafted and sometimes implemented to induce or directly subsidize decentralization. Again, such decentralization is costly even though the primate cities are considered to have grown beyond their optimal sizes. The paper aims for the simplest possible theoretical model which adds 1For a discussion of policy issues stemming from the optimal city size concept see Tolley and Crihfield (1987). 2Despite the enormous costs, a number of developing countries have set up new national capitals partly in response to extreme growth in their large cities (for example, Brasilia in Brazil or Islamabad in Pakistan). Still other countries like Nigeria or Korea have drafted plans for such cities but have either cancelled or postponed them because of the costs.
A. Anas, On the birth and growth of cities
245
substance to these observations. In the spirit of the previous city system models, economic differentiation which can lead to various sizes and types of cities is ignored. However, our analysis differs from the previous long-run equilibrium studies by focusing instead on the transient dynamics of an economy which grows from very small (initially having only one city) to very large, spawning new cities as it grows. The process of decline in which settlements disappear successively as population falls, is also examined and found to be very different from that of growth. In particular, the decline path cannot be obtained by retracing the growth path backwards in time. Migration costs are assumed to be z e r o . 3 A closely related paper is by Henderson (1986) who considered the occupation and development of the remote regions of a growing country, by migration from the areas developed earlier. Henderson found that the development of the remote regions occurs later than the optimal time. This result agrees with the finding of this paper, that it is optimal to set up the second city much earlier than the date at which it would emerge under laissez-faire. However, this paper differs from and adds to Henderson's in several respects. First, while Henderson's model is complicated by realistic assumptions and by discussions of government policy, we develop the simplest possible theoretical model uncluttered by complex, yet important, policy issues. Second, we perform a precise stability analysis, tying together the theory of optimal city size with the mathematics of multiple equilibria and bifurcation analysis. Third, we show that periods of laissez-faire and planning must alternate on the optimal growth path and that the length of the required planning periods vanishes as the number of settlements increases. Fourth, unlike Henderson (1986), who examined the case of growth only, we consider decline as well as growth, showing that the decline path cannot be obtained by reversing the growth path and that a hysteresis effect exists. Our simple economy consists of only labor, which grows, and land, which is limited: the former is the source of a positive localization economy, the latter the source of a land use congestion externality. In such a setting, with labor growing exogenously over time, all economic activity is initially concentrated into one city because of agglomeration economies. Such a city grows to its optimal size and beyond, until it becomes feasible for population perturbations under laissez-faire to establish a second city.4 This happens when the first city has become unstable in response to sufficiently small aPositive migration costs would change the results in a predictable way but without negating our essential findings. Several footnotes in the paper comment on the effect of migration costs. *'Perturbations' in this paper are migrations between the cities which occur at random intervals and are of random magnitude. In the tradition of stability analysis for deterministically dynamical systems, such perturbations arc exogenous to the model and can be explained as shocks, errors or deviations from rationality, or as the result of a trial-and-error process through which agents explore the states of the system.
246
.4. Anas, On the birth and growth of cities
declines in its population. At such time, it takes a small number of migrants (at the extreme only one) to set up a second settlement which matches the utility of the first and large city. Social optimality requires that the second city be set up much earlier and at a larger initial size, requiring a much larger and planned migration out of the first. Individual agents (in our model, laborers) cannot, in a market economy based on self-interest, antidpate the optimal time when decentralization should occur because the timing of such decentralization requires coordinated action among agents and is thus external to each agent. Under laissez-faire conditions, the second city is either born too late or may not emerge at all if agents in the economy do not experiment with suboptimal and unstable situations. This is so because the first city remains a stable equilibrium for a long time after it exceeds its optimal size. When the second city ultimately emerges, it is always associated with a sudden panic-migration of population out of the first. Under planned optimal growth, the second city must be spawned much earlier than the laissez-faire case, and must be 'nursed' through a period of instability until it reaches a critical size thereafter growing on its own. Under both laissez-faire and planning the sizes of cities pass through stages of growth and decline as does the path of the inhabitants' utilities. The planned optimal timing of settlements smooths these cycles by rounding the peaks and raising the bottoms in the per capita welfare profile over time. As the aggregate population grows very large, the need for planning vanishes and the system is capable of spawning cities at an asymptotically continuous and optimal rate. This cyclical pattern of growth is not reversible in a two-city system declining in total population. One of the two declining cities suddenly vanishes at a smaller size than the size at which the second city is born during growth (a hysteresis effect). The time and city size at which this occurs is determinate. In contrast, under growth, the time at which the second city is formed and the size of the newly formed city are indeterminate and depend on the strength of perturbations at that time. Aspects of the real world, intentionally left out of the analysis, strengthen our conclusion: costs of adjusting capital, sunk public infrastructure and the costs of migration all suggest that primate cities in a laissez-faire economy can grow too large and remain too large for too long. With such costs, a planned settlement policy appears potentially beneficial. Planning must be introduced at regular intervals to 'seed' settlements at the optimal times and to 'nurse' each new settlement to stability. 2. Growth and decline in a two-city laissez-faire system
Consider, for now, an economy which has two a priori identical locations where cities can emerge. A simple example may be an island where the interior is uninhabitable and cities can emerge at either the West End and/or
A. Anas, On the birth and growth of cities
247
the East End. At each end, L acres are available for urban development. Will such a simple economy spawn one city, two identical cities or two cities of different sizes? How will the urban pattern evolve over time as the aggregate population of the economy grows (or declines) smoothly over time? Will the number and relative sizes of the cities (if two) change along such a growth path for the aggregate economy? Suppose that the aggregate economy is the simplest labor economy possible with land the only other input. There are N identical laborers. An undifferentiated good is produced in the urban area(s) and since there is only one good, there is no trade in or out of equilibrium: the inhabitants of each city consume the total product of their city. Cities are linked only by labor migration (assumed to occur quickly and costlessly) as laborers will move between cities so that the utilities of living in either city are equal. Suppose that each laborer derives utility from his output (which he consumes entirely) and the size of his private land lot. Suppose also that a simple form of a localization externality governs production: the output of each laborer rises with the number of laborers in the same city because the presence of other laborers generates innovation in production by means of communication and information exchange among laborers: each city is simply a world of 'several heads working together are better than one working alone'. Land in a city is divided equally among the identical workers and, because the total amount, L, is finite, each worker's land lot is reduced as the city's population increases. Thus, z---f (n) [with f ' ( n ) > 0 and f " ( n ) < 0 ] is a single laborer's output as a function of the number of laborers, n, land per laborer is q---L/n, and utility per laborer (as a strictly concave function of consumption and land) is u=-u(z,q)=u(f(n), L/n)-V(n). It makes sense to assume that f ( 0 ) = 0 , u(z,q)-,O as z--,0 or q--,0. Also, assume that u(f(n),L/n)>O for any n. Thus, u(f(n),L/n)--,O as n--,oo. This simplest of models gives an optimal city size: a maximum occurs where the marginal rate of substitution between consumption and land (or the rent for land, R) just equals the ratio between the marginal land use congestion effect (decreases in land per worker) and the marginal localization benefit (increase in output per worker) from the addition of just one more worker:
q/n - R . uJuq-f,(n )
(1)
Resident-workers in the city own the land collectively as equal shareholders, hence there are no absentee landlords and rental dividend income is endogenous. Migrants to other cities lose their rights to the rent dividend in the city of origin, but become shareholders in the city of destination. Then,
RS.U.E. D
248
A. Anas, On the birth and growth of cities
letting y be the income of a worker, utility maximization by each worker can be stated by max,,au(z,q), subject to y = z + R q , where y = f ( n ) + q 2 / n f ' ( n ) : income consists of the worker's output plus the value of his land holding. It is now necessary to set up a dynamic migration process whereby laborers move between the cities. We let an overdot denote a derivative with respect to time. Then, if n and N - n are the two cities' labor forces, the following is a continuous and once~differentiable migration adjustment process: h = f [ V ( n ) -- V ( N - n)];
(2)
with sign(ri) = signEV(n)- V(N--n)] and a f E . ] / d [ . ] >0. The migration process (2) simply says that the population of a city increases when its utility is higher than its rival's, decreases when its utility is lower than its rival's and remains the same otherwise. An equilibrium distribution of population between the two cities occurs when the time derivative of n vanishes, which requires V(n)= V ( N - n ) . 5 In a symmetric equilibrium, each city has exactly half of the total population. Asymmetric equilibria also exist, because it is possible for one city to be large (and thus productive but congested in the use of land) and achieve the same utility as the other which is small (and thus relatively unproductive but also uncongested in the use of land). There are two other possible equilibria in which only one city exists but the time derivative (2) need not vanish. These are the corner solutions (N,0) and (O,N). Fig. 1 shows the loci of all the equilibria for all values of N. The ray OO' is the locus of all symmetric equilibria. The hyperbolic curve AA' is the locus of all asymmetric equilibria and the two axes are the loci of all one-city equilibria. In the figure, the level of population is fixed by a straight line with slope minus one and growth is traced by moving this line away from the origin. Two times the optimal city size is the critical threshold for total population. If N <2n* there are three equilibria: the symmetric one and the two one-city equilibria. If N exceeds this threshold size all five equilibria are possible: the symmetric one, the two one-city equilibria and the two asymmetric equilibria. To see which equilibria are in fact attainable, we have to examine the global system stability of the entire set of equilibria for the two regimes of N. From (2): dfi/dn = ( df /d [" ]) {~ V (x)/ax Ix=~+ a V(x)/ax Ix =(N- n~}.
(3)
For a symmetric equilibrium, the sign of (3) is positive for all N smaller than SBecause we are assuming that migration is costless and instantaneous, the scale of df[']~ d [ . ] will be taken to be large enough to allow us to ignore the actual durations in migrations.
249
A. Anas, On the birth and growth of cities r1
nl
A
o1+02=N / o ,
I
1
__~~ I
rl
-EI~
v(nl) = v(n2)
E2,~
'y V(n 1)
"-
,
0
N'N \\%
II N<2n*
\\
[
N=2n*
n*
N>2n*
~
n2
\,
-\ ,-- f V(n2) Fig. 1. Equilibriain the two-cityeconomy(nl: populationof city one; n2: populationof city two; E~: symmetric two-city equilibrium; E2,E3: one-city equilibria; E4,Es: asymmetric two-city equilibria). 2n*, it is zero when N = 2 n * and it is negative for N larger than 2n*. Therefore, the symmetric equilibrium is unstable when the city system has population less than or equal to the threshold size (see figs. 2a and 2b) and stable when the system has population larger than the threshold size (see fig. 2c). As the system's population just passes through the threshold size (see fig. 2b), then the system bifurcates from the three-equilibrium state to the fiveequilibrium state. Since for economies with below-threshold populations, the symmetric equilibrium is unstable, such 'small' economies will initially concentrate all population in one city. To examine the stability of such one-city equilibria we note from (2) that ri = f[V(O)-- V(N)] <0,
(4) fi = f [ V ( N ) -
V(O)] > O.
A. Anas, On the birth and growth of cities
250
}'1
N < 2n*
I Stable Unst,able~1111
(a) Ea ~ f N E / 1 2 h 1/Stable (b) 5 3
1 2 N = 2n* Unstable ~ ' ~ ~ E1 /2
h
N > 2n* Stable
I
Unstable
Stable
Stable
Stable .
/ II E2
Unstable
n
I
i
A
Stable
Fig. 2. The stability of the equilibria in relation to aggregate population in the economy.
These two conditions mean that one-city equilibria are always stable (see fig. 2c). By the continuity of the migration process given by (2), it follows that since one-city equilibria and the symmetric equilibrium are stable, then the asymmetric equilibria must be unstable (see fig. 2c).
2.1. The growth regime We are now ready to examine the implications of our analysis for the dynamics of urban development. To do so we adopt a monotone growth regime in which the population growth function N(t) is given as a function of time t, begins at time t = 0 from N(0)
251
A. Anas, On the birth and growth of cities
(t)
///~
N(t)
..........n-!'!-'---g-~~ ~ Threshold aggregatesize 2n' n* ~ " Optimal citysize •~ .~ t t V(t) '
1/2 N(t) n(t),decline t, time
.
~
~ t*
, i"
growth t, time
Hysteresis Fig. 3. The time paths of population and utility under growth and decline in the laissez-faire case.
settlers returning to the West End. However (because the locus A A ' approaches the two axes asymptotically), the size of such migrations sufficient to destabilize the West End gets continuously smaller as the city becomes larger and its utility level lower. Eventually, just a few migrants will again move (for r a n d o m reasons) out of the West End to start a small settlement at the East End. By this time, utilities at the West End are far lower than optimum, and the small settlement at the East End immediately achieves higher utility than that which prevails at the West End. When word of this reaches the West End, a panic-migration suddenly ensues and, because of costless migration, the two cities are instantly equalized in size and in utility: the system j u m p s suddenly from a one-city equilibrium to the symmetric two-city equilibrium. 6 Thereafter, the two equal cities grow equally over time and the symmetric division of activity holds. Fig. 3a shows 6The migration is characterized as a 'panic' because of our assumption that it is costless: once it is learned that the new city is viable the adjustment to equilibrium is sudden. If migrations involved cost or lags due to information diffusion, then utilities between the two settlements will be unequal and the system will be out of equilibrium until all of the required population has migrated to the second city.
252
A. Anas, On the birth and growth of cities
that even though N(t) is monotonic over time (drawn as linear in the figure), the population of the West End undergoes a cyclical (or unbalanced) transition pattern: the city grows much beyond optimal size by becoming too congested and suffers a gradual decline in utility until it becomes sufficiently unstable in the face of outmigration. At some point (when a sufficiently large outmigration occurs), the West End quickly loses exactly half of its population to the East End and recovers a good part of the lost utility by virtue of reduced land use congestion. Thereafter, the two identical cities continue to decline in utility as they grow equally. After the system 'bifurcates' from the three-equilibrium regime to the fiveequilibrium regime (see figs. 1 and 2c), the one-city equilibrium is initially very robust in the face of migrations. Growth must occur until this robustness deteriorates to the point where a small perturbation is sufficient to start a stable second city going. 7 When this time actually comes, depends on how big the perturbations are. Therefore, in fig. 3b, the time [ is determined by chance events. It is also possible that it never happens, if migrations do not occur. This means that a certain amount of experimenting, randomization or curiosity is necessary for the system to find the preferred state within a laissez-faire environment. Highly regimented or uninnovative societies may become 'stuck' in overconcentrated states not taking advantage of the considerable improvement in utilities which can occur by spawning a second city. 2.2. The decline regime
A monotonic decline process in which the exogenous population growth function N(t) is exactly reversed in time does not result in the exact reversal of the urban development pattern. Figs. 3a and 3b show that, because the symmetric equilibrium remains stable all the way down to the critical size, the declining twin cities reach maximum utility as they become gradually decongested in the use of land. Then, at exactly critical size, the symmetric equilibrium is unstable and all activity must suddenly concentrate into one of the cities (to be selected by chance). This results in a sudden drop in utility as the remaining single city is now larger than optimal in size. We note that whereas in the case of growth, the time at which the second city was born was stochastically determined, in the case of decline the time at which one city is abandoned is deterministic. As population continues to decline, the remaining city becomes gradually decongested to reach maximum utility again. After that point, population and utility decline continuously and 7A measure of this robustness is seen directly from fig. 1 to be proportional to the difference between the population of the city under the stable one-cityequilibrium(N) and the population that city would have if it were in the unstable asymmetric equilibrium. As N grows, the assymetricequilibrium population gets closer to N and thus the margin of stabilitydeteriorates.
A. Anas, On the birth and growth of cities
253
eventually the city vanishes altogether. The important finding here is that there is an hysteresis effect: the transition from one to two cities under growth and the reverse transition from two cities to one under decline do not necessarily occur at the same population level. In the case of decline, the time of transition from two cities to one always occurs at time t*, because after that time the two-city equilibrium is unstable.
3. Optimal growth of the two-city system A related set of questions naturally emerges from our finding that the laissez-faire (or equilibrium) growth path is nonmonotonic and that it may result in inferior welfare levels because of the aggregate economy's tendency to remain in a one-city equilibrium for too long. What is the optimal growth (decline) path which maximizes the discounted present value of utility? What are the planned transition times from one city to two (and from two to one) on these optimal paths? Is the planned symmetric growth of a two-city system from the beginning ever optimal? These questions are resolved by maximizing the discounted present value of the per capita utility path in the economy, while imposing the condition that identical laborers are treated equally at each point in time: the per capita utility on the optimal path (while changing over time) must be the same in all on the existing cities at each point in time. Ignoring any planning costs associated with setting up the second city, this welfare function is t**
max {t** ~ O}
W(t**) = ~ e -~s V(N(s)) ds 0
+ J" e-*sV
ds,
(5)
t**
where t** is the optimal time at which the planned system should switch from the one-city equilibrium to the two-city symmetric equilibrium by establishing the second city. ~ is the social rate of discount. The first-order condition of (5) gives
V(N(t**))= v(N(t~*)),
(6)
which, not surprisingly, says that the second city should be set up when halving the population of the first leaves utility unchanged as reduced congestion in the first city just offsets lost output. This occurs when the first
254
A. Arias, On the birth and growth of cities
v(t V(t*) v(t**)
i Optimalplannedpath ~--Symmetric two~ plannedpath
"', i' "', ,.' "i
~,,~ Laissez-fairepath \~ "~
Fig. 4. Planning and laissez-fairepaths compared.
city's population is between the optimal size and twice the optimal (or the bifurcation threshold) size (see fig. 4). Note that, therefore, the switch time is earlier than the point in time when a symmetric equilibrium is stable. Planners must act at t** to break up the city into two equal smaller cities and must plan to keep the two cities going until they are stabilized at time t*. Thereafter, no further intervention is needed. Thus, the optimal path requires that the second city be established before it is stable and therefore before trial-and-error migrations could establish it under laissez-faire. To achieve this, a temporary substitution of planning for laissez-faire practices is required. Fig. 4 shows the path of per capita utility over time when the system follows this optimal development pattern. This path dominates the symmetric path (also enforceable by planning) in which two equal cities are set up from the beginning and 'nursed' until the two-city equilibrium becomes stable at just beyond critical aggregate population) The optimal path also dominates the laissez-faire paths in which the panic-migration ensues at some randomly determined time. A nice property of the optimal path (unlike the laissez-faire paths) is that it is reversible if the national population were declining. This is seen, by reversing the limits on the two integrals in the social welfare function, (5). 9 It is interesting to note that the inhabitants of the first city have an incentive to subsidize the migrants in order to induce them to move and establish the second city at the socially optimal time. Such subsidies are SThe symmetricplanned path would become optimal if the planning costs of setting up two equal cities from the outset were sufficientlylower than the planning and migration costs of breaking up the first city at the optimal time. 9 To emphasize, this reversibility hinges on maintaining the infinite horizon nature of the problem: we start at t=O with two equal infinitelylarge cities and decline monotonicallyuntil all population vanishesat t = oo.
A. Anas, On the birth and growth of cities
255
wasteful if migration costs are zero. All that is needed is the assurance that the second city will be established as a temporary 'club' (until it becomes stable) at the designated time and place. More than the needed number of migrants will seek to enroll in such a club. If migration is costly, then the inhabitants of the first city can be taxed to raise subsidies which might offset in part the private moving costs of the migrants. A private developer can also help establish the city earlier than the laissez-faire time. As seen in fig. 3b, the birth of the second city under laissez-faire is associated with an upward jump in utility. By announcing that a city will be available, a developer acts as a coordinator and, hence, makes collective migration possible. In return for this, the developer can extract rents from the city inhabitants. If many potential developers are available, they will attempt to undercut each other by proposing to set up the second city at an even earlier date, and hence one which is closer to the optimal one. Returning to the planning problem, it is easy to extend the analysis [by generalizing (5)] to determine the time at which the ruth city ought to be set up in an economy which has a large numer of available sites of L acres each. The social welfare maximization problem now becomes
max W(0 = tltt=O
e-*" V
ds,
(7)
m = 2 tin-1
where the infinite vector f = [ 0 , t 2 . . . . . tm.... ] gives the times at which each successive city should be set up by the planners (tt=0). The first-order conditions result in the following relationship from which the mth city's birthdate is to be derived:
V(~)= v(N(--~)),m>2.
(8)
Fig. 5 shows the way in which laissez-faire and planned utility paths will differ as new cities are spawned one at a time. Cities are spawned at intervals which get shorter with each new city and the planned growth path tends to round out the peaks and raise the bottoms of the laissez-faire utility path. So far, we assumed that cities are set up one at a time. Is it possible to do better by increasing the number of cities more than one at a time? For example, is it better to break up the first city into two equal cities (as was assumed so far) or to wait until it is larger and then break it up into three equal cities? Similarly, in a growing two-city system, is it better to set up a third city according to (8), or is it better to wait until four equal sized cities can be established by the cleavage of each existing city into two? Under our
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V(t)
- -
Optimal Path
......
Laissez-faire Growth Path
V(n )
t2
t3
t4
t5
to t7 t8
LF: period of laissez-faire under growth P:
period of planning under growth
Fig. 5. Welfare effects of city sequences under planning and under laissez-faire in growing and declining economies.
assumptions of costless migration and costless planning, waiting to establish more cities later involves a welfare loss relative to establishing each next city at the earliest possible date. This follows by recalling that it is only efficient to start a new city when existing cities (one or more) are larger than the optimal size and thus when per capita utility is declining with growth. Welfare is maximized by reducing the durations of these stretches in time when utility is declining. Therefore, new cities must be set up as rapidly as possible and this requires setting them up one at a time: the optimal welfare path which maximizes (7) dominates all other welfare paths in which cities are set up more than one at a time. 3.1. An example o f a birthdate sequence
It is worthwhile to consider the following specific example. Suppose that the product of a laborer is given by z = n=. Let each laborer's utility function be of the following constant elasticity of substitution form with the degree of homogeneity equal to one: V(n) = (n- ~# + (L/n) - #)- 1/#,
where 15 is the substitution parameter.
(9)
A. Anas, On the birth and growth of cities
257
This utility function gives an optimal city size n*: n* = (~Lp)1/~1+~)p.
( 1O)
Let the aggregate population grow exponentially at the rate r. Then
N(t.)= N(O)exp(rtm)
(11)
where N(0)
i-L,/Cl+~)[~,m___cT___~_l 1).P_m~B _~_pj '~1/p{1+~)]. (m_
tm=rlnL_~l
(12)
Let ~ = f l = l / 2 and choose L such that n*=100,000 from (10). Also, let population grow at 2% annually (r =0.02) and set N(0)= 50,000. In such an economy, the second city must be set up in 51.74 years when the population of the first city reaches 140,716 (or nearly 41% larger than its optimal size). The third city is set up 27.63 years after the second, when the combined population of the first two cities reaches 244,350 (or nearly 22% above optimal size per city). The fourth city is set up 17.37 years after the third, when the combined population of the first three reaches 346,111 (or nearly 15% above optimal size per city). 260 years from the beginning, there are 90 cities and new cities are spawned at a rate of two per year and city sizes at birth are within 0.56% of the optimal city size. At the risk of appearing simplistic, these numbers demonstrate the cycles that can prevail during optimal economic growth and suggest that the welfare losses from failing to plan new cities at the correct times can be large. 4. Concluding comments
Time lags needed in building public infrastructure, the immobility of capital, the durability of real estate, the costs of migration and the costs of planning new cities are all factors that give large urban concentrations a margin of stability even wider than our analysis suggests. The breaking up of such concentrations is costly, and thus the spawning of new settlements or the transfer of growth to less developed regions remains a valid yet underachieved goal in both developed and developing countries. Societies which like to experiment with new settlements despite these costs are more likely to come close to the optimal path, whereas inflexible societies which reinforce large settlements with government programs and subsidies may become rigidly stuck in inefficient but stable city-system configurations. The
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sizes o f large cities in such s i t u a t i o n s is n e i t h e r the o u t c o m e of c o n c e n t r a t i o n benefits (which have been e x h a u s t e d l o n g ago) n o r the a t t r a c t i o n of ' b r i g h t city lights'. Rather, such sizes m a y be e x p l a i n e d b y insufficient shocks, o r p l a n n i n g o p p o r t u n i t i e s foregone b y p o o r timing. E c o n o m i c efficiency suggests t h a t a bias t o w a r d d e c e n t r a l i z a t i o n c a n offset the stability o f large places.
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