Note on economic geography

economics letters ELSEVIER

Economics Letters 50 (1996) 291-297

Note on economic geography J o o n - M o Yang* Department of Economics, Yonsei University, 134 Shinchon Seodaemungu, Seoul, South Korea Received 13 March 1995; accepted 8 July 1995

Abstract

This paper attempts to establish a phenomenological view on economic geography. We show that the intrinsic

probability structure determines the dynamicsof a city size formation, created by centripetal and centrifugal forces in economic geography. Keywords: Self-sustainingeffects; Congestioneffects; Neighborhoodeffects JEL classification: R12

1. Introduction

This paper analyzes the basic questions concerning urban clustering processes. General equilibrium models, such as those of Debreu (1959) and Ellickson and Zame (1993), study economic geography by giving a special treatment to location, so that they establish a competitive general equilibrium view of economic geography. Even though these approaches adopt well-defined, individual, decision-making procedures and their consequences in equilibrium, empirical applications based on those models may be difficult to handle. Krugman (1991a) develops a model that shows how a country can endogenously become differentiated into an industrialized 'core' and an agricultural 'periphery'. In his model, increasing returns and imperfect competition lead economic agents to cluster, so that they can exploit gains from specialization. From the individual decision-maker's point of view, specialization is rational, given this environment, but how that environment was formed has yet to be answered. To summarize the current literature, we argue that the dynamics of a city's population distribution is composed of an adjustment process created by both centripetal and centrifugal * Correspondence address: Joon-Mo Yang, 3326 Sawtelle BI. #12, Los Angeles, CA 90066, USA. Tel: 310-391-9529; e-mail: [email protected]; or: [email protected]. 0165-1765/96/$12.00 (~) 1996 Elsevier Science S.A. All rights reserved SSDI 0 1 6 5 - 1 7 6 5 ( 9 5 ) 0 0 7 3 7 - 7

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forces in economic geography. Centripetal forces are forces such as a self-sustaining effect and a neighborhood effect, while a centrifugal force is a congestion effect. The self-sustaining effect is such that the population of a city is expected to keep growing in the next period, and the congestion effect, that the population is expected to decrease if the population is above a certain critical point. The neighborhood effect is such that, once the population of the neighborhood of a city exceeds that of the city, then this neighborhood population is expected to dominate the city in the next period. We show that a simple model with a transition probability matrix can adequately explain both centrifugal and centripetal forces and derive the sufficient conditions under which these phenomena can be observed. Our model can be related to the concept of a self-organizing process. The spread of knowledge or the diffusion of technology, the pecuniary externalities, and the entrance and exit from Darwinian competition are created by interactions among agents with heterogeneous characteristics that may be purely random. We argue that these dynamics form a self-organizing process that displays the intrinsic fundamentals of a system (see McCall, 1993, for details). In Section 2 we show that the centripetal and centrifugal forces stem from the underlying logic of an allocation scheme in the city population. In a simulation, we show the emergence of a structure as a self-organizing process. Section 3 discusses further implications of various allocation schemes.

2. An illustration

What we have in mind is a stochastic scheme that allocates people into cities where the spatial distribution is hierarchically organized. 1 People are originally scattered randomly into cities, and they have the opportunity to revise their decisions about where they live. The revision process is not the usual rational individual process, but rather a stochastic process following a hierarchical structure. Suppose that there are m cities and n people to allocate. Pij is the probability that an individual is allocated from city i to city ] each time. The probability depends on where the individual lived at t - 1. Let us assume that, for all individuals, the probability is the same, and what matters is where the individual lived. L - - ( / t , 1 2 , ' " ,lm) is the allocation vector, where li is the size of the population of city i. P,.j is constructed by the hierarchy, which means that Pq satisfies the ultrametric condition:

Pij >~ min{ Pik, Pki } . This condition says that, if city i and city ] are at the same level of the hierarchy, then people in both cities have the same probability of moving to another city at a different level of the hierarchy. For example, those who live in a city among midwest cities have the same probability of moving to Los Angeles. Note that there is no difference between cities in the given system; that is, we have no reason to believe that Los Angeles, for example, should W h a t we m e a n is that the distance m e a s u r e satisfies an ' u l t r a m e t r i c ' condition.

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grow faster than any other city as a result of some attraction. The allocation scheme is such that each individual chooses the city that he/she will move to in the next period, by following a given probability. In the sense that every individual faces the same probability structure, the process that governs the allocation is an exchangeable process. What matters is how many people are allocated into each city, rather than who they are. From the point of view of the stochastic clustering process on the formation of the city population, we show that these effects can be explained by the underlying process of the system. The following propositions show that the centrifugal and centripetal forces are created by a given probability pattern rather than by different characteristics of the agents. In a simulation we illustrate the emergence of the hierarchical intrinsic logic of the system as time goes by and provide a view that the emergence of a metropolitan area is realized at random. Proposition 1. The congestion effects. If lj >i n/(3 - 2P/j), the population o f city ] is expected to decrease in the next period. Proof.

E(AIj(t))

= ~'i li(t)PiJ - ~ . l/(t)Pji - lj(t) = ~' li(t)Pij + lj(t)Pi~ - if' lj(t)Piii#j

iv~j

li(t )

= ~ l~(t)Pij + lj(t)Pjj - lj(t)(1 - Pjj) - lj(t) iv~j

Z l,(t)" 1 + lj(t)(2Pjj- 2) i#j

= n - lj(t) + li(t)(2Pjj - 2)

= n + lj(t)(2Pj/- 3). Therefore, the sufficient condition for the congestion effect is n lj(t) >i 3 - 2Pj~"

[]

Proposition 1 says that, by nature, if the population size of the city is larger than a certain level, it will be expected to decrease. Proposition 2. The self-sustaining effects. If ~_

PiJ ~ 1

1

E

~icjl~

li(t) Pij

'

the population o f city j is expected to grow in the next period.

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Proof. E(Alj(t)) = ~ li(t)p/j - ~ i

lj(t)eji

li(t )

-

i#j

= ~ li(t)p/j + lt(t)Pjj - ~ lj(t)Pti- It(t ) i~"]

i~j

= ~ li(t)p/7 q- lt(t)Pjj-lt(t)(1 - P t t ) - l j ( t ) . i~ t

Therefore, the condition that E(@(t))I> 0 is equivalent to the condition that

PtJ ~ 1 - 1

~ li(t) i~j l-i~ P6 • []

Note that the condition for the self-sustaining effect depends on the relative size of the population at a particular time. If the size is too large, then the congestion effect begins, so the self-sustaining effect is limited by the congestion effect.

Proposition 3. The neighborhood effects. Suppose the allocating scheme is hierarchically constructed and two cities are in the neighborhood at the bottom level of the hierarchy. The condition for the continuing domination of one city (for example, city k) over the other (city l) is that Pkk >~P,

and

P~k >~½(1 + Pk,)"

Proof. E(lk(t + 1) - lt(t + 1)) = ~ li(t)eik i

-- Z i¢k

lk(t)Pki -- ~ li(t)Pit - ~ l,(t)P, i

i~l

= Z li(t)(eik - Pit) - Z Ik(t)eki + Z lt(t)Pti i

i#k

i~l

= (Pkk -- Pk,)l~(t) + (p/k -- P,t)lt(t) - ~ lk(t)ek, + ~ lt(t)P, i~k

(because Phi = P, ,

i~l

Vi ~ k . $)

(2Pkk -- Pk~- 1)(lk(t) -- l,(t)),

ifPkk ~>P , .

Therefore, the condition for continuing dominance is that

Pkk~Pu

and

Pk~>½(I+P~,).

[]

As an illustration of the three propositions given above, let us consider the following simulation. Initially, people are randomly scattered into m cities. Given the set {P/t}, each individual decides whether he/she will move and where he/she will move to. I calculate the city population at the end of the stage, after everyone's decisions are made. The city

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(a)

400~

295

(b)

200~-

40O

Fig. 1. A simulation. (a) One run; (b) 10 runs; (c) 30 runs; (d) 40 runs.

populations should be equal ex ante. However, the small difference in the r a n d o m realization gives a very different result as time goes on. Fig. 1 shows the simulation results. Fig. l(a) is a one-period run, given that, at the previous time, we scattered people randomly into all cities. Most cities have population sizes around 100-150, but there is a trace revealing clustering. After 10 periods, among 25 cities, we can see the emergence of large cities and their neighbors. As we increase the n u m b e r of runs, this clustering process or the emergence of a metropolitan area becomes clear (cf. Figs. l(c) and l(d)). The size of the metropolitan areas did not increase that much after they were formed. We call this the congestion effect. One view of the congestion effect would be that people would find sources of disutility, such as traffic jams and high rental costs, as a result of living in a large population, and they would then leave the city from an individual utility maximization point of view. What if all agents are the same? The answer is obvious. The representative agent model cannot explain the congestion effect because historically we do not observe a sudden m o v e m e n t of all the people. Now suppose heterogeneity exists in the population. The distribution of heterogeneity matters in deciding how many people move out of the city. F r o m Fig. 1 we notice that neighboring cities of a large city cannot exceed the size of the large city. We call this the neighborhood effect. This can be explained by utility equalization across cities, but we can explain this by the nature of the clustering process when we have a hierarchical structure. Let us characterize the equilibrium concept. In equilibrium, the centripetal forces may be equal to the centrifugal forces at the aggregate level, even though there are constant m o v e m e n t s at the micro level.

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Definition. Equilibrium. We define equilibrium as a steady state; that is, an equilibrium is {li(t)} such that

E(Ali(t))=0,

Vt > T

and

~li(t)=n, i

Vi = l . . . .

,m .

We can establish the conditions for the existence of an equilibrium (the usual rank conditions will suffice), but here we need to emphasize the no-equilibrium case. For a city population of constant size, we can imagine the situation where we never attain an equilibrium; that is, we have a constant inflow and outflow of people in a particular city. We illustrate such a situation in the following proposition. Proposition 4. A no-equilibrium allocation scheme. allocation scheme cannot f i n d an equilibrium.

I f Pij = Pji and P , = Pjj,

Proof. By the definition of equilibrium, we have to find lj(t) = lj, ~ I~Pq - ~ ljPj~ - lj = O , i

V]

is)

and

~ lj = n ,

Vt > T,

Vi ~ ] ,

the

V], such that

li>0.

i

With some algebra, and by using the symmetry of P and the fact that Ej Pij -- 1, we have (P~1 - 1)ll + (P22 - 1)12 + " " If P , = P ~ ,

+ (Prom -- 1)lm = 0.

Vi ~ j , the above equation contradicts the condition

li(t ) ---- n ,

Vt .

i

Thus, there is no equilibrium.

[]

Proposition 4 shows the possibility that, under a certain allocation scheme, we can have the constant movement of people forever so that we can envisage the recurrence of increases and decreases in a city's population. From the above propositions, it is necessary to estimate these probabilities to predict the overall trend of city size in addition to the existing micro studies.

3. Discussion

We have shown that we can explain the basic p h e n o m e n a of city formation using a phenomenological approach without mentioning individual decision-making procedures. Furthermore, we argue that this approach is necessary to explain the p h e n o m e n a from the aggregation of interacting heterogeneous agents (see Aoki, 1994, for details). There are two other issues. One is what we can expect from exchangeability and r a n d o m allocation, and the other is how to estimate the probabilities of an allocation scheme. An answer for the first issue may be Zipf's distribution. Zipf's law is a very powerful law for the counting process economy. Its strong empirical relevance was pointed out when Zipf

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introduced the law. Zipf ranked the U.S. metropolitan areas by population, and then plotted rank against population. 2 Even though the economic literature has been aware of this law, a rigorous investigation of its underlying process has not been performed. In other literature, Hill (1970, 1975), Woodroofe and Hill (1975), Hill and Woodroofe (1975), and Chen (1978) show the sufficient and necessary conditions for various types of Zipf's law to hold. One of the assumptions is that the allocation of people to cities in one region is approximately independent of those in other regions. Another is the assumption of a Bose-Einstein allocation of people to cities within regions. Chen (1980) shows that most symmetrical Dirichlet-multinomial models give rise to Zipf's law, and even though both Bose-Einstein and the Maxwell-Boltzman can be categorized as Dirichlet-multinomial distributions, the latter leads to a different result. It appears that other allocation schemes would also yield Zipf's law, but this has not yet been proved. The second issue should be investigated further. We can estimate a prior distribution that generates allocation probabilities or we can calculate the probabilities directly if data on directions of movements are available.

References Aoki, M., 1994, Group dynamics when agents have a finite number of alternatives; Dynamics of a macrovariable with mean field approximation, UCLA Center of Computable Economics Working Paper 13. Chen, W.-C., 1978, On Zipf's law, Dissertation, The University of Michigan. Chen. W.-C., 1980, On the weak form of Zipf's law, Journal of Applied Probability, 611-622. Debreu, G., 1959, Theory of value (Wiley, New York). Ellickson, B. and W. Zame, 1993, Foundations for a competitive theory of economic geography, UCLA Working Paper. Hill, B. M., 1970, Zipf's law and prior distributions for the composition of a population, Journal of the American Statistical Association 65, 1220-1232. Hill, B. M., 1975, A simple general approach to inference about the tail of a distribution, The Annals of Statistics 3, 1163-1174. Hill, B. M. and M. Woodroofe, 1975, Stronger forms of Zipf's law, Journal of the American Statistical Association 70, 212-219. Krugman, P. R., 1991a, Geography and trade (MIT Press, Cambridge, MA). Krugman, P. R., 1991b, Increasing returns and economic geography, Journal of Political Economy 99, 483-499. Krugman, P. R., 1994, The self-organizing economy, Mimeo. McCall, J., 1994, Inequality and efficiency are natural phenomena, Mimeo. Woodroofe, M. and B. Hill, 1975, On Zipf's law, Journal of Applied Probability 12, 425-434. 2 Krugman (1994) also confirms this by plotting the ranks and populations of the 130 largest U.S. metropolitan areas in 1991 with logarithmic scales.