Geography of gender; Geographies of disability; Disability in the Global South; Experiences and representations of mental ill health; The inclusion and exclusion ...
Regional Science and Urban Economics 9 (1979) 41-59. 0 North-Holland
AGGLOMERATION* G.J. PAPAGEORGIOU hfch¶aster
UnicersitJ, Hamilton, Ontario, Canadu L8S &,I 1 Received August 1978
In the realm of the humanities four main classes of reasons for agglomeration have been identified [Carter (!977)]: surplus theories, the city as a wall, the city as a temple and the city as a market place. Deductive urban literature on the o:her hand recognizes two main classes of reasons for agglomeration: the city as a centre of r-oduction and the city -.s a Gublic good. Surplus theories relate to the city as a centre of production while the wall ,nd the temple are specific instances of a public good. But there is no deductive analogue to the city as a service centte; and the nature of such reasons for agglomeration is not intuitively clear, as for example. the persistence of periodic markets tends to suggest. This paper attempts to fill the obvious gap. Circumstances are specified undkr which agglomeration is better than dispersion in the following sense. The agglomerated state can support a higher aggregate profit and for every service of the urban economy, a lower price and a higher demand than those attained in the dispersed state. In the absence of output externalities such potential advantages may obtain under decreasing average costs for the economy. This is less demanding than the usual requirement of incrtJsing returns for agglomeration: under the convex portion of decreasing average costs cities are compatible with the idea of general competitive equilibrium.
lntruduction Back in the winter of 1963 in a classroom in Athens the late Constantinos A. Doxiadis askeu a question and, to my subsequent embarrassment, I gave a naively wrong answer. The subject was birth of cities. The setup, a primitive society able to sustain a single artisan and then a second one. The problem for the latter was where to locate. Having just been exposed to the neat ideas of Christaller (1933) I based my answer on distinct market areas. With just a touch of delighted malice this wizard of a teacher replied: ‘The exact opposite will happen - of course. He will locate next door. It is more profitable and this is precisely the reason for cities.. .’ Almost twelve years elapsed. Then, during the summer of 1975, I ws reading Irwin W. Sandberg’s (1975) ‘Two Theorems on a Justification of the Multi-service Regulated Company’. Colourful images of the scene came back and I suddenly realized in all its simplicity the truth that spatial proximity *Research supported by Canada Council Grant S75-0675 and by a Canada Council leave fellowship. I would also like to thank the National Hellenic Research Foundation for providing me with a good working environment.
42
G.J. Papageorgiod,
Agglomeration
creates interdependencies as indeed other types of organization do, for example administrative organization. Sandberg essentially discusses advantages of horizontal and vertical integration. It turns out that integration and agglomeration possess striking abstract similarities. In consequence this essay constitutes reasoning by analogy: Sandberg’s results are modified, extended and re-interpreted to cast some light on aspects of what is probably the most intriguing area of urban study - that of urban externalities. 1
It seems natural to link agglomeration with the existence of certain initial social advantages. Then optimal agglomeration occurs where these advantages are balanced by the disadva&ages created by agglomeration itself.’ Variations on the nature of advantages and disadvantages generate different reasons for cities. Advantages in the urban-economic literature refer to either production or consumption while disadvantages are dominated by transportation costs.’ Whereas production advantages underline the r6le of the city as a centre of production, consumption advantages underline its rcle as a public good. But the city is more than a centre of production and a public good. Since local service has been fundamental to city life throughout its entire history, the cily is also a ser,:ce centre and this r6le has been neglected as an explicit reason for cities. The nature of agglomeration advantages related to the rble of the city as a service centre is the main object of this essay. However, in order to create background for comparisons, we first briefly discuss the city as a centre of production and as a public good. The city as a centre of production In this case the reason for cities is increasing returns: agglomeration should continue until the social advantages of increasing returns are balanced by ‘the social disadvantages of transportation costs created by the agglomeration itself. To begin with a more general point of view we note that increasing returns alone provide a necessary condition for agglomeration. This is discussed by Cole (1972). His macro-analysis has a natural quasi-spatial interpretation leading to the conclusion that agglomeration implies an average degree of increasing returns for the economy. The converse, that ‘The point of balance varies with the type of the optimum. For a discussion of different optima, see Alonso (1971) and Singe11(1974). Here we argue in terms of.the social optimum. ‘An exception to the rule is the paper by Boruchov et al. (1977) where the human propensity to interact is established as a reason for cities: over soxe featureless plain, a balance between the propensity to interact and the disutility of congestion produces equilibrium agglomerations.
G.J. Papageorgiou, Agglomeration
43
increasing returns imply agglomeration, is not necessarily true. Indeed Cole demonstrates that further agglomeration is ascertained only if the average degree of increasing returns exceeds one plus the capital share for the economy. In addition to the requirement of strongly increasing returns for agglomeration, it is also known that a city of optimal size should be characterized by a particular average degree of increasing returns. This average degree of increasmg returns is such that the excess value of the urban product equals the excess of the urban land value over its opportunity cost.3 Variants of this remarkably simple law have been discussed by Mirrlees (1972), Dixit (1973), Starrett (1974) and Arnott (1976). Now what about an optimal agglomeration as the outcome of balance between increasing returns and transportation costs? An answer is provided by Arnott and Stiglitz (1975) who demonstrate that, if there is no locational preference, a circular city is characterized by an excess of the urban land value over its opportunity cost being greater than, equal to, or smaller than one-half the urban transportation costs according to whether there are increasing, constant or decreasing returns to transportation.’ Hence, provided that there is no locational preference at the optimum, the excess value of the urban product is greater than, equal to, or smaller than one-half the urban transportation costs according to whether there are increasing, constant or decreasing returns to transportation. The case of linear transportation costs has been analysed by Starrett (1974) and Hartwick (1976). Since however transportation is actually characterised by increasing returns, it seems that optima1 agglomeration should correspond to an excess value of the urban product being greater than half the value of transportation. The city as a public good In this case the reason for cities is public investment: agglomeration should continue until the social advantages of public investment are balanced by the social disadvantages of transportation costs created by agglomeration itself. There is a symmetry between the city as a centre of production and the city as a public good. In essence the non-convexity generated by increasing returns is now replaced by public investment. When the public good is pure, a city of optimal size should be characterized by a level of public investment being equal to the excess of the urban land value over its opportunity cost.” Thus optimal agglomeration can be sustained by a single, non-distortionary ‘in the case of externalities the excess value c,f urban land is adjusted. ‘Absence of locational preference for an ind:vidual means that his level of utility is invariant over space. ‘Although this is a condition for optimal city size, the quantity of the public good itself may not be optimal. This remark is due to Arnott (1976).
G.J. Papcgeorgiou, Agglomeration
44
tax on land. Variants of this remarkably simple law, the henry George theorem, have been dis,cussed by Flatters, Henderson and Mieszkowski (1974), Stiglitz (1974), Arnott and Stiglitz (1975) and Arnott (1976). If the city is circular, provided that there is no locational preference at the optimum, the level of public investment is greater than, equal to, or smaller than onehalf the urban transportatio:, costs according to whether there are increasing, constant or decreasing returns to transportation. The city as a service centre We now depart from the established framework on two counts: (1) We consider primarily agglomeration of firms, rather than agglomeration of people induced by production or consumption advantages. Firms produce and distribute urban services. (2) Whereas the established strategy is to impose aggregate advantages sufficient for agglomeration and then to enquire on the nature of an optimum, the conditions we impose are not strong enough to ascertain agglomeration. Consequently our task is to describe circumstances under which these imposed conditions create advantages sufficient for agglomeration. In particular we consider the case where agglomeration reduces aggregate production costs. For example transportation savings may account for this phenomenon. This is not as conceptually demanding as the requirement that agglomeration increases productivity, or that there is public investment ub initio. The nature of such agglomeration economies cannot be other than aggregate as the external effects of agglomeration may prove costly for some firms in the economy. On the other hand, at least where location decisions rest with the agents 01 the economy, agglomeration advantages must in principle refer to these agents. Thus, given the distinction between aggregated agglomeration economies aind disaggregated agglomeration advantages, there is nothing at this stage to ascertain that the former lead to the latter. In what follows we describe the geography and economic conditions under which agglomeration economies lead to agglomeration advantages and we discuss the precise nature of such agglomeration advantages. Urban
geography
The whole approach is based upon comparisons between spatial states of the economy. Spatial states differ by the location of firms. The geography of a state is realistic. No restrictions on the complexity of physical structure and form are imposed. In consequence the spatial distributions of population and firms may be represented by any jixe;l three-dimensional surface and by any
G.J. Papageorgiou,
Agglomeration
45
point pattern respectively. Prices are weighted by transportation costs. Hence demand is explicitly affected by the spatial distribution of population and firms.
Condirionsjor
agglomeration
advantages
Firms may engage in any mix of economic activities. However the set of economic activities is restricted by their tendency to agglomerate. The existence of such constraints in reality is demonstrated by the existence of incompatible land-uses: agglomeration advantages do not extend over the entire spectrum of urban services. It is therefore important to determine subsets over which such advantages apply. The entire analysis deals with subsets of urban services exclusively formed by weak gross substitutes. Although this is not the only requirement imposed upon these subsets, it certainly is the most characteristic. In particular the mix of urban services is such that (i) if the mill price for an urban service remains constant then the demand for this service does not decrease because of other possible mill price increases; (ii) it some mill prices increase while the rest remain constant then the demand for some urban service (associated with an unchanged mill price) increases; (iii) if ;here is a change in mill prices then there is an urban service with higher price and lower demand or cite cersu. We conclude the description of circumstances under which agglomerat;on economies imply agglomeration advantages with two observations on the nature of cost and profit: (1) Urban externalities are fully taken into account since cost depends on total output rather than on the output of the firm. The formulation is general enough to allow for the incredibly complex and subtle net of urban interactions. This is important because significantly less remains to be understood of entities that shape urban form after urban externalities have been discounted. (2) There is no explicit mechanism for the determination of profits as indeed of prices. In consequence the same conclusions apply equally well to both market and to planned economies. This is also important because similar urban forms appear to transcend the entire gamut of socio-economic systems.
G.J. Papageorgiou, Agglomeration
46
7 he nature oj agglomeration advantages
state (dispersed state) with given profits, prices and demands, that corresponds to a non-empty set of states (agglomerated states) exhibiting agglomeration economies in respect to the dispersed state. Then, for the kind of world described previously, the following proposition is true? Consider
.any
Proposition 1. lj’ there are agglomeration economies then any agglomerated state can support a higher aggregate projit and, for every service oj the urban economy, a lower price and a higher demand than those attained in the dispersed state.
The aggregate nature of agglomeration economies has already been stressed. Yet the proposition demonstrates that such economies generate advantages specific to the agents of the economy: producers face a higher potential for profit, consumers face the possibility of lower prices for every single urban service, and the demand for every single urban service has also increased. Although strong, these represent potential rather than actual advantages because the attainment of such desirable conditions remains a matter of, speculation. This, in turn, implies the adoption of a somewhat Darwinian principle: if for certain classes of economic activity agglomeration is potentially more successful than dispersion then the former will eventually dominate the latter. This is in sharp contrast with sulIiciently increasing returns or with public investment which generate actual agglomeration advantages and which, therefore, dictate agglomeration within the ideal realm of theory. Equilibrium
and the city
To require non-convexity as a reason for cities is to vitiate general equilibrium theory. This is not to say that equilibrium is impossible under non-convexity because convexity is only a sufftcient condition for equilibrium. Still, the question remains: is non-convexity really necessary for agglomeration? For the city as a centre of prt>duction the answel is affirmative. We now turn to the remaining riles of the city. In contrast to increasing returns tied to a particular technology, public investment can fluctuate. This is especially true in large, ‘open’ economies where questions of fiscal responsibility are not exclusively related to the local level. It follows that, in contrast to the city as a centre of production where increasing returns impose a certain degree of optimal agglomeration, the optimal size of the city as a public good varies with public investment. Moreover, as Arnott (1976) has demonstrated, if both increasing returns and public investment are taken into account then the excess value of the urban 6The validity of the proposition depends on some additional technical requirements which are discussed in the proof,
G.J. Papageorgiou,
Agglomeration
47
product plus the public investment should, at the optimum, equal the excess of the urban land value over its opportunity cost. Clearly therefore public investment may, in principle, extend the size of optimal agglomeration well beyond the region of production non-convexity. In order to examine non-convexity for the city as a service centre, we concentrate once more on the standard issue of the relationship between agglomeration and production technology. We consider the case where the additional structure imposed on aggregate production costs from the agglomeration of a dispersed state can be expressed as an aggregate cost function defined over the dispersed state. Then, for the kind of world described in the preceding section, the following proposition is also true: lj there are no output externalities then decreasing aceruge costs jot every service of‘ the economy imply agglomerarion economies. Proposition
2.
is immediate that, in the absence of output externalities, decreasing average costs for the economy imply potential agglomeration advantage:; described by the first proposition. This is another condition for agglomeration comparable to the ones already discussed. The comparison is summarized in fig. 1. The region OB illustrates Cole’s strongly increasing returns It
i
i
I
I
i
I
I I Fig. 1
I
G.J. Papageorgiou, Agglomeration
48
sufficient fDr agglomeration. The optimum average degree of increasing returns is to be found somewhere within that range, for example at A. If public investment is introduced, an optimal agglomeration may grow to a size corresponding to D. Potential advantages on the other hand are associated with the region OE. In this case potential advantages obtain under more general conditions than actual advantages. It seems that initial stages of urbanization are characterized by both actual and potential advantages whereas, after some threshold of. public investment potential t,as been surpassed and growth still occurs, only potential advantages remain. The region to the right of, C is compatible with general equilibrium. Therefore, if potential advantages are accepted as sufficient for agglomeration, the region CE represents that portion of.decreasing average costs over which cities may attain full competitive equilibrkm. Intuitively, this region corresponds to large cities.
II
This section contains a formal description of circumstances under which agglomeration economies imply agglomeration advantages. A list of notation appears at the end of the article. Spatial structure The urban realm is described as a set of locations restrictions on this landscape other than the following: Assumption
1.
(i) S is closed and connected.
S. There are no
(ii) (S,n) is a metric space.
Hence no realistic urban spatial form has been excluded. Locations are occupied by households or firms. There are m firms in the city. The location of firm j is sjEs for j = 1,. . ., m. The vector s represents the spatial distribution of firms. There are no restrictions on the location of firms in the city. Therefore: Assumption 2.
SE Sm.
The spatial relation between firm j and the other firms is described by a vector of distances between firm j and the other firms: Definition
I.
dj, =
d[sj,
SJ
for
k = 1,. . ., m;
djk E dj,
G.J. Papageorgiou,
Agg/o,merarion
49
A surface unfolding over the city describes the spatial distribution population. This distribution is exogenous: the incessant metamorphosis the city has been suppressed. Assumption 3.
D: S+R+
c.i of
is continuous on S.
Prices There are n urban services provided in the city. The mill price of urban service i is b,Eb where DEB and i=l,..., n. Mill prices are constrained by some upper and lower limits that belong to B. In consequence: Assumption 4.
B is closed rectangular.
Prices are weighted by transportation costs. Thus there is a distinction between the mill price and the price of an urban service. The price of urban service i obtained from firm j is PijEPj where ~j~ P. Intuitively, thk price increases with distance from the location of firm j:’ Assumption 5.
is continuous
(i) Pij at on B x St.
S,
depends on bj and on d[sj,sl].
(ii) pij: B x S’+R+
Demand and supply The local demand per capita for urban service i is pin p. Since prices are spatially distributed, local demand for urban services crucially depends upon the spatial distribution of, firms. Hence a particular distribution of local demand implies a particular distribution of firms: Assumption 6.
p: P+R:
is continuous on P.
The demand for urban service i is qie4 where q E Q_ Demand. refers to the whole urban area. In consequence, it is expressed as the sum of all local demands weighted by the population distribution: Definition 2.
q = js p[b, s, sJD[s,]ds,
‘For urban services with no distance bias, price coincides with mill price.
SO
G.J. Pupageorgiou, Agglomeration
Since S is compact bounded and q:BxS”+R:
and since p and D are naturally
is continuous
on
B x Sm.
bounded, q is also
(1)
Total demand is partitioned between the firms of the city. That portion of total demand for urban service i supplied by firm j is qijE qk The method of allocating total demand among firms is described by the following: Definition 3. For i = 1,. . ., n and j= 1,. . ., m: (i) The matrix A has elements Uij~O such that CUij=l. (ii) qij=aijqi. j Thus C qij = C Qqi = 4iCb9s]W i
(2)
The advantages of agglomeration do not extend over the entire spectrum of. urban services. Hence the ensuing discussion refers only to certain subsets of urban services. Fundamental to identifying these subsets is the following assuiaption which is non-spatial in that it relates to characteristics of demand for a given spatial distribution of firms: Assumption 7. For every (b’,s), (b’,s)EBxS”: (i) if,b1sb2 and b:=bt then qi[b’,s] zZqi[b2,s], i.e., if the mill price for an urban service remains constant
then the demand for this service does not decrease because of other possible mill price increases. (ii) If b’ < b2 and Ef = bf then there exists j such that bf = bjz and, qj[b’, s] Cj cj[b, s*] then economies in respect to (6, s).
5 asserts that the aggregate costs of production for the agglomeare lower than the aggregate costs of production for the dispersed must be attributed to spatial proximity: the only other difference, total demand, is specified in a way that strengthens this
III
This section contains a proof of the proposition that with agglomeration economies, any agglomerated state can support higher aggregate profit and, for every service of the urban economy, lower price and higher demand than those attained in the dispersed state. Lemma 1. 6’ 4 b2.
(i) lj’q[b1,s]~q[b2,s]
then b’q[b’,s]
then
Prof.$n
(i) The argument proceeds by induction. The desired result ?olds for a single urban service by Assumption 7(iii). Suppose now that it nolds for n - 1 urban services and consider the case of n services. Since q[ b’,s] zq[b2,s], 6’ = b2 may or may not be true. Of course 6’ = b2 is consistent with the desired result. If 6’ f b2 then, according to Assumption 7(iii), there *Part (i) of Lemma 1 is adapted from theorem 5, part (i) of Gale and Nikaidb (1965, p. 87).
G.J. Papageorgiou, Agglomeration
52
exists k such that bi c bf. Name k= 1. For if 1,
qi[b:,b:,** b,‘,S]2 qi [b:, bf,. * bi,S] -7
-9
2_qi[b:, bf, *a-3b,Z,S].
(4)
The last inequality is true by virtue of,Assumption 7(i). Let B* be the image of B under the projection (b,, b, ,..., b,)+(b, ,..., b,). Define q*: B* x Sm-+R”+I-las
qY.32,.. .,bwsl = Mb:, bz,.. .,b,,~1,.. .,qJb:, 62,.. ., b,, s])‘. Using (4), q*Cb:, . . ., b,‘J
2 q*Cb;,
. . ., #I,
(5)
and, by induction, (b:,
. . ., b,‘)r(b;,...,b,l).
(ii) If. q[b’,s] >q[b2,s) then b 1 + b2 and therefore (5) holds. If (5) holds with strict inequality for some index then, again by induction,
(b:,...,b,‘)%(b;,...,b;). Suppose now that for i+ 1, qi[b’,s] =qi[b2, s] and that there exists j such that bf = bj”. According to Lemma l(i), b1 5 b2 and by assumption b’ + b*. In consequence b’ < b2. Then, according to Assumption 7(ii) there exists I such that b,’ = bf and q![b’, s] 1 I and starts at u” =O. The second sequence {w’} J 1 and starts at w” = 1. The gist of the argument is an induction which demonstrates that there is a mapping with the required properties (i)-(iii) such that, starting from b’ and b2 respectively,
UWIIt W-1 and
(b[w’]} 1 b[l.].
Fig. 2
The induction proceeds as follows. Assume a b: [0, l] +B continuous and increasing on [0, l] such that properties (i) and (ii) hold. Suppose that for some kh0 and ill,
WI 2 @oh] 1 b[d] 2 b[ l-j,
(8)
qCW1,~15q*CJlS qCbb+l, ~1,
(9) j = 1,. . ., i - 1,
are true and the set ofj that satisfies (10) for i= 1 is empty. Inequalities (8~(10) are true for k=O and i= 1. Let
Because of Assumption 7(i), f ’ [xl 2.f2C~l, D
W
G.J. Papageorgiou,
54
Agglomeration
and
f2CbiC~“11 S f 'CbiCuk13
Sqi[bCvkl,sl S$CAl SqiCbCw”J,s] ~f2Cb~C~31~~f’tb~C~ll~
(12)
by (1l), by (10) and Assumption 7(i), by (9), by (P), by (10) and Assumption 7(i), and by (ll), respectively. Now f’ and fZ are continuous by (1) and decreasing functions of.x by Assumption 7(G). In consequence there exist ijk,tik which are unique and satisfy ‘[bi[$]]
f
=qT[A] =
f2[bi[Gk]]e
(13)
Furthermore; (14) by (12), by (11; and (13), and by (12), respectively. \ Define bi[Xk+‘I= bi[xk] + p(bi[Zk] - bi[xk])I
where XE[O,l] and ~~10, 11. Since p>O, (16)
by (14) and (15) where Wk)=wk+l.Similarly, (17) where vkivk+‘. Combining (8), (16) and (17), and because bi[i?J&b,[tiJ (14),
biCOI 1 biCVk] 2 h,[vk+‘11 b,[wk+ ‘12
b&d] 2 bi[l]
follows immediately. Hence (8) holds for all i. Furthermore,
by (18)
G.J. Papageorgiou,
Agglomeration
by Assumption 7(i) and (18), by Assumption respectively. In a similar manner,
55
7(iii) and (16), and by (13),
so that
Inequalities (18) and (19) demonstrate k. Thus the induction is complete. As {$}+A,
that (8), (9) and (10) told for all
lim bJ_tj“]=-!~~~ (bi[uk’l]-bi[Uk])+ k+m
i
and
lim bi[t”] k-+co
=
b;.
Similarly, lim bi[~~ = b:. k-a
At the limit, qCb’, sl = q*C~l= q[b4, s], by (13). NOW q[b1,s)~q[b2,s] implies b’ zb2 according to Lemma l(i). Therefore q[b’,s] =q[b4, s] implies both b3 2 b4 and b3 s b4. In consequence q[b, s] is one-to-one and b3 = b4. But then limk _.Q)$ = lim, _,to Wk= A. QED.
Theorem. Suppose that (b’,s*) admits agglomeration economies in respect to (b’,s) and that b2E B. If &;rr,[b’,s]>&7r,[b2,s*] and q[b’,s*]
