- Email: [email protected]
Efficiency in the urban economy
Regional and Urban Economics 4 (1974) 41-56. © North-Holland Publishing Company
EFFICIENCY IN THE URBAN ECOHOMY A general equilibrium framework*
Alan ABOUCHAR University of Toronto, Toronto, Canada R~eived June 1973
1. Introduction While economists often talk in terms of efficiencies or economies of city size or urban agglomeration, there is little agreement on the content of these terms. Just what do we mean by efficiencies of scale in the urba~ context ? This paper presents an analytical framework for the definition and analysis of urban efficiencies. Although no new measurements of urban efficiency are offered, we will consider the empirical literature within this theoretical framework paying special concern to the internal consistency of empirical investigations and to the relevance of this theory to what the authors have attempted to measure. Finally, we sketch some of the implications of our model for benefit measurements.
2. A model of the urb~, economy
2.1. The problem On its most general p'~ane, an agglomeration economy is a portmanteau expression which means tilat, for a host of reasons, welfare per capita is higher in a large city than in a small one. These reasons include production externalities of firms when indivisibilities require a large market for individual producing units; the same sort ofindivisibilities in the consumption sector, requiring a large market for the provision of a wide variety of amenities; higher incomes per capita owing to the greater labour productivity arising in greater proximity to work and increased use of capital or large scale production, made possible by the larger market; cost reductions in the Provisions of public sector services; and, finally, the intangibles resulting from the larger urban complex, such as the *This study was undertaken as part of a research project of the Institute for Policy Analysis of the University of Toronto. Helpful comments ~,y an anonymous referee are gratefully acknowledged. Errors and conclusions are the author's responsibility alone, how~ver.
42
A. Abouclw.r, Efficiency in the urban economy
improved quality of work, better newspapers, and so on. Some writers intend only subsets of these effects, such as the unit cost behavior in the provisi3n of public services. Whey the net result of all these factors is to lower welfare it is a diseconomy. Obviously, it is difficult to make operational so broad a concept of agglomeration effects. Empirical research has, therefore, concentrated on some of the subconcepts, such as the relationship between unit cost and volun e of individual public sector services Most of the s~udies, which have been based primarily on United States data, have been statistical cross section studies of urban expenditures by type function and by total, sometim¢:s adjusting Oor income and sometimes not, depending on what is the main focus of the investigation. Since most of this literature ha~ been reviewed and its methodolcgy appraise6 by others [see Bird (1970), Hirsch (1968) and Musgrave (1968)], we will make no attempt to treat it in anything like e/haustive fashion, simply referring to it as the need arises. Much of ~.he investigati,,~n of urban expenditures has been criticized for ,c~ffuseness and uncertainty about what is being measured and about the implications ofusing alternative formulations. It has been asked many times whether such studies of expenditure determinants measure demand or supply factors. As Musgrave (1968, p. 573) has put it: "The 'service' function combines a mixture of supply and demand factors which need to •be disentangled if the really interesting insights into the fiscal behavior of communities are to be obtained. To some extent, this might be accomplished by separating demand variables such as income and demographic factors, from supply variables, such as climatic conditions or wage rates. To illustrate, the cost of a ,range of service levels for snow removal (i.e. absence of snow on streets) will depend on factors such as snowfall, density, topography, wages and so forth. This is the co. ~.function p~rt of the problem. Service levels provided will then be a function of these costs and of demand factors, such as i~comes, tastes, and other prices. The gener~ expenditure function is unsatisfact3ry both bo~ause the dependent variable is poorly defined (i.e.~ in terms, of expenditure rather than service levels) and because the independent variables combine cost and demand factors without due separation. While certain variables m . y ~ on both sides of the picture (density reduces the cost of firefighting per house, but raises demand by increasing the cost of conflagration) a better separation is h~dly Im insoluble problem."
The same or similar point has been made by Breton (1965) and a number of others. The dissatisfaction with past research has undoubtedly stemmed in part from the different purposes to which different ~,bservers would like to put it. For example, straightforward expenditure determinant studies may be appropriate as a predictive device, but inadequate as a normative measure. But there is another diversity of purpose to contend with in analyzing the urban economy - there is not one object of measurement, but several. Analyqis must help in framing policy with respect to three quite dif~erent issues: (I) determination of optimal size for new towns; (2) consolidation of already existing neighbouring urban complexes; and (3) growth of already existing cities. Analysis relevant to each of these
43
.4. Abouchar, Efficiency in the urban economy
requires a different approach. They are all important today, the last being perhaps, the mest critical. The following presentation, therefore, will start with the last. The necessary modifications for the others will follow.
2.2. Diagrammatic analysisfor urban growth 2.2.1. The product market The formulation of ap urban economic model is too complex to be dealt with satisfactorily by a single expenditure determinants model. Even for a single service, there are, first of all, interrelationships between city size and demand, at the same time that demand may vary with price for any given city size. Fig. I shows market demand and cost curves for an urban public service. For the sake of definiteness, we will talk about garbage collection. The vertical axis shows the price per unit and the horizontal axis, the number of pickups per unit of time, year, say. The demand curve, DDo, slopes from northwest to southe a s t - at lower prices people will want to have more collections per year. Thus,
P~ L
!
Q,
Q.
I~
LRAC 0 pII
'"
'
"
'
---
DD 0
i
0
i
II
II
I
I
I
I
~
OJ (~0
~
0"
Pickups per Year O Fig. 1. Hyp,:,thetical market demand and supply of garbaFe collection (city size, No).
44
A. Abouchar, Ej~ciency in the urban economj
as the demand curve is drawn, if price is reduced from P' to pw, the quantity demanded per year will rise by 25 % from Q' to Q'. The demand curve DDo relates price to quantity holding constant everything else including the size of the city, whic:~ is here No. More pickups are demanded a~ lower unit prices for essentially the same reason that more of any normal good is always demanded when the price is reduced, i"he specific factors conditioning demand here have the foUowing logic: with more frequent collections people will not have to crush their refuse as intensiwly, objectionable odours wiU lessen, fly attraction will fall, and so on. For this city of size ~,'o we will also have a long-run average cost function L R A C This curve tells us what the lowest cost is per unit at different levels of activity. Being a long-run curve each point represents the most efficient cost which might be attained if we could start from scratch in each case with city s~e fixed, r"or example, the lower cost which is secured in moving from Q' to Qm might be b~:sed on a revised land use pattern with the c:'.y dump moved in closer, and lower labour and vehicle costs per pickup- the reduced driving distance offsets over the range of demand but nothing criti~,d is implied- the demand intersection at price Po and quantity Qo could easily occur over the rising range ¢x in an interval of constancy. The LRAC is taken as the supply schedule and ~ ill be referred to as $S in the future. The demand and ~upply analysis just conducted for city with size No can be repeated for other city sizes. Fig. 2 shows the relevant curves for six different c,ity sizes with equilibrium prices denoted P~. The supply-~emand pairs represent succe.,;sive rightward displacements, ~vhich means tha~: people are willing to pay more per unit for the same quantity of annual pickups as city size increases, and also that it costs more per unit for the same quantity of pickups per year for larger city sizes. The rightward sluft in demand occurs for two reasons. First, individuals' utility functions change and they are wilting to pay more for each pickup since greater density imposes additional health hazards. In the second place, the fact of a larger city implies a larger number of people who would be willing to pay more at any given level of colle~ion activity. (In the ~onsolidation analysis to be discussed below, by contrast, only the second of these factors enters, since the layout of the aggregated metropolitan region remains the same.) The demand curve of fig. 1 and the family of curves of fig. 2 are based on a give~ i n , m e level and we qhould expect that an increase in income will shift the demand curve for any city size to the fight, intersecting the supply schedule a; ~ r~int to the southeast of its present intersection at price P. "[h~ rightward shift in the supply schedules results from increased prices for inp~lts, primarily l~nd. Wages probably also rise somewhat, and final costs may higher since elimination must be more thorough. Several different treatments .have been proposed for the input price variation which depends on city size. Breton (1965) has clarified the issue by drawing the distinction between input
A. Abouchar, E/ficiency in the urban economy
45
price variation and input quantity variation as sc arces of final cost variation. Musgrave (1968, p. 573) has argued that input p:ic~ varietion should be eliminated in expenditure studies to try to gain an idea about ',,;ervice levels'. But this approach would leave us ill-equipped to deal with c,ne of the major questions 8t issue when the focus of study is policy with respect to existing cities: input. price variation accompanying different city size is precisely the way to measure society's needs and resources, and, therefore, the welfare gain or loss to be associated with transition from one si=e to anothel. And for a policy concerned with influencing the growth of existing urban complexes, we do want to observe
aDo Q, =I n_
DO2
0,
Dr,.]
DO4 SS5 DD 5
0 Pickups per year, O
Fig. 2. Market demand and supply curves for cities of varying size N,.
the consequences of input price variation since the very heart of the issue is the cost and benefit of changing from an existing city size to a different one, and the currently existing prices better measure the costs and benefits than would production functions based on the constant prices of some given city size. We may formalize the foregoing as follows. First, the demand function for city size No can be written Qo = fo(Po). In general, for city of size Nj it may be written Qj - f~(Qa). In all normal cases we will have Q, + 1 > Q, i.e., demand will shift to the right with increasing city size. If demand sniffs by a constant percentage, i.e. if the quantity associated with an increase in city size is proportional to the relative size difference, we will have Qt = [1 -l'oe(Na/No)]fo(Qo).
46
A. Abouchar, Efficiency in the urban economy
In like manner, the generalized supply function may be written AC, = A(Q,).
,,~s the supply and demand functions shift to the right, the equilibrium intersection P~ will shift. The series of intersections will be termed a Dynamic Product Equilibrium (DPE) curve. Recalling, moreover, that there will be a difl~rent family of demand curves for each income level, we will require a DPE curve for each income. Fig. 3 shows
Q. E .Q "5 D"
OPEl
DPE"
i
- -
0
NO
N1
i
ii
!
,
N2
m
N3
City Size N i
Fig. 3. Dynamic prodt~ct eqt~ilibrimn curve (equilibriurn price vs. city size).
two such curves, denoted DPE' and DPE". (DPE' corresponds to the s~tuation of fig. 2 and the equ:~ibrium prices/~l have, accordingly, been relabeled/F on fig. 3. DPE" relates to a higher income level since the equilibrium price for higher income is lower, with a higher quantiq, than that of the lower income level, as indicated earlier.) The DPE curve is a Cescriptive rather than an optimizing device. It merely traces out the supply-demand equifiorium prices for different size cities. Optimization requires that this product ma~'lcet equilibrium curve be evaluated with respect to the benefit~, which are associated with different size cities, to which we now turn.
47
A. Abouchar, Efficiency in the urban economy
2.2.2. Demandfor city size (amenities) Since cities perform different functiors (seaports, manufacturing cities, agricultural centres, and so en) it is first necessary to de:ermine the function or nature of the city bein~ analyzed. We will consider cities oriented to commerce and fight industry. Their amer:it~ies will vary with their size- parks, newspapers, .,,ocial life, etc. In general it is to be expected that for any given size, the willingness to spend on municipal services is positively associated with income. Larger cities may be expected to have greater environmental diversity and advantages than smaller cities; individuals would be willing to pay more for these advantages out of any given income. The extra expenditure on the various municipal services that a consumer is willing to undertake to live in a larger city may be regarded as a price - the price ofriving in the larger city. A group of expenditure-income curves is shown in fig. 4. Annual expenditure
J
EE4
j:
3
j j
EE EE
~mb
m
,,m
Q. al
0 Q. (u 0 12 v
e. :1
'0 o x w
.
.
.
.
.
o
i
i
mill
_
7
Income ( Dollars Per Capita)
Fig. 4. Per capita expenditure-income curves foz varying city size.
48
A. Abouchar, Efficiency in ,he urban economy
is shown on the vertical axis, and income on the horizontal.. The curves are labeled EEl, the subscript indk'.ating that the curve pertains to city size N~. The expenditure-income function can then be related to city size by drawing a perpendicular at any income level, say ~, and plotting in fig. 5 the points of its intersection with the f ~ i l y of exj~enditure curves. This diagram shows per capita e~:penditu:e on the vertical :~xis and city size on the horizontal. The curve derived in this way is labeled EY. This curve will be upward sloping, under our assumptions, ff people were unwilling to pay more to live in larger cities, the curves in fig. 4 would be displaced downward with higher subscript i, and the El? curve would slope dow:award. We would expect, normally, that the EEl would move upwards, though at a decreasing rate, and eventually either fall or remain constar~. In the last named situation the El? curve will flatten out. If the E£~curves were to start to fall, the El?curve would start to decline as well.
S~
t11 .,.,.
O "D
,,m,
°,m
Q.
O eL
"O ¢= O Q.
DASPE
I(
UJ
Ni City Size N i
Fig. 5. Product-equilibrium, expenditure-income equilibrium, and determination of efficient city size.
,4. Abouchar, Efficiency in the urban economy
49
We have considered the expenditare city-size curve for income of Y. Such a curve can be constracted in like roamer for any income. The curve deri,~ed for > ~'wou~d lie above the EYcurve in fig. 5. We can now compare the amenities or city size dems~td, in the form of the expenditure-income function, with ~ cost curve. We will assume that the demand for each service, as analyzed in sec~lon 2.2.1 is independent of the demand for omers, as seems reasonable. The I ~ E curves are given in terms of prices per unit of service and must be converted into expenditure terms. To do this, we multiply P~ by the equilibrium quantity and divide by city size to determine the per capita equilibrium expenditure on this service. The same is done for all other services. The resulting curves are then summed vertically to give a Per Capita Dynamic All Services Product Equilibrium Curve for a given income level. This curve is labeled D A~PE in fig. 5. Following cons,derations of the relationship between the DPE curves for different income levels, as shown in fig. 3 and explained earlier, the DASPE curve for a higher income would lie to the left ofthe DASPE curve in fig. 5. Also shown in fig. 5 is the expenditure-income function tbr the same income level, T. The optimal city size is shown by the intersection of the two curves at N~. Thus, ~he optimal city size is determined by four factors: (i) a faz~y of demand functions for individual services for varying city sizes (one family for each income level); (ii) a long-run average cost function for each cky size; (iii) the expenditure-income function, one for each city s~ze; (iv) per capital personal income. There are some obvious similarities between the graphical analysis just concluded and the Hicks-Hansen elaboration of the Keynesian system. Thus, the dynamic all services product equilibrium curve (DC) is the counterpart of the investment-savings function which defines a series of equilibria in the real economy. DASPE tells us the equilibrium annual expenditure which equates supply and demand for urban services in any given city size. The E Y curve is the founterpart of the Hicks-Hansen liquidity preference function and may also be interr zeted as a kind of financial equilibrium curve. It defines a series of equilibria in people's apportionment of their budgets between expenditure on urban services and other items. We assume that people have some idea of what is available in different size cities which serve the respective functions (commerce and fight industry, in this case) and how much they would want to have left after urban service expenditures to enjoy being in that city. If per capita income is ~, the population in this region will prefer cities of size ,¥1 for their commercial centers. Sizes of other kinds of cities would be determined in like manner- people will
50
A. Abouchar, Efficiency in the urban economy
prefer agricultural supply cities of some given size, seaport another, heavy industrial cev.ters another, and ~o on. ~ To carry Hicks-Hansen analogy further, just as an increase in the money supply will shift the L M curve ~to the right), an increase in per capita income will shift the EY curve (to the le£~, however). Since the income level is different, the DASPE curve will also move to the left. z Both movements lead to a smaller optimal city size. If E Y m o v e s more than DASPE, the final equilibrium expenditure under the higher income in the smaller city will be greater than that under the lower income in the bigger city. This means that with great wealth, people will avail themselves of the luxury of smaller cities (where the 'physical' market supply-demand equilibrium occurs at a higher price). People will spend more per capita on urban services and also pay more per unit of urban service; but at the same time they will spend relatively less of their income on urban services (see the slope ofthe EE curves in fig. 4). They will thus have a larger share income unoommitted to the provision of urban services with which they can enjoy more benefits offered in the smaller city, which has fewer benefits to offer, than they c.ouild enjoy in the larger city, which has more to offer. A reduction in per capita income in the region will shift to the right, defining an equilibrium at greater city size. The logic is similar to that just described. 2.3. /~alysis for new towns policy
Where new towns policy is concerned ~ e do wish to abstract from the input price, variation which has come to characterize relative factor costs in actually observed cities and to concentrate instead only on the scale economies of the underlying technological processes, l'he changes that must be introduced into the analysis of ti=e preceding section relate first to the individual product supply curves and thence to the DPE and DASPE curves. We assume that land can be bought at constant unit price in the region in which the new city is to be founded, ,~o that the differential effects of city growth over time have not come to be reflected in land values. In this case, doubling the ~izeof a city dump will 0 .a[ ~, l~and cost. Similarly, labour costs can be estimate6 at a given consta-~ ",,~ce,, the assumption that the workers who o:me to this ~ity come from the ' aargin ,n many different cities and, within the city size rar~ge being contemplated, ~ ay number can be attracted at that price. The physical inputs underlying the cost curves ol fig. 2 are now valued at these unit prices to calculate a single LILAC curve for each service. The family of 'That cities of any given function in any given reckon are not, in fact, all the same size is not to deny this theory. The cities existing at any point in time have started at different times, have different special characteristics (even within the stone broadly defined type), and so on. Ovr analysis is l,~tended as a logical abstraction of the forces conditioning optimal city size. 2The dynamic product equilibrium curve for any product will be displaced downwards the less, the lower is the income elasticity of demand for the service in question (the less does the family of demand curves ir fig. 3 shift to the right with rising income) and the greater is the elasticity of the individual cost ~ r v e s in fig. 3.
A. Abouchar, Efficiency in the urban economy
51
demand curves remains as before. The series of intersections with the LP,A C and the demand curve family traces out a static product equilibrium curve, one for each service, relating equilibrium prices to city size. The observation for each city size is then multiplied by the quantity provided and divided by the city size to get a per capita annual expenditure for that service. These are summed over all services to yield a per capita static all services product equilibrium curve (SASPE). Confrontation of SASPE with the expenditure-income function, as before, then defines the optimal city size.
2.4. Ar,alysis for consolidation policy Under what conditions should contiguous municipali~:ies unite ? The decision should take into account the cost reduction that may be achieved through consolida~ion. This ~ usually a managerial economy. Another main source of scale e c o n o m y - price concessions - is not so straightforward; the quantity discounts offered by a supplier working with forty local communities, say, cannot be considered as observations on a statistical supply function which would then be simply extrapolated to a larger community since the existing pricing pattern may reflect a currently existing discriminating sales pattern in which the higher average prices to some will offset the lower prices to others, overhead being apportioned, in part, inversely to the strength of the buyer. For the same reason we could not even assume that the purchases after consolidation could all be made at the price now granted the biggest user. The main source of scale economy, then, would be management. Pecuniary economies are less certain. Other sources of cost variation such as the changing input prices or factor cost variation considered earlier would not arise in the consolidation analysis-land and labour costs in the various municipalities within the same urban region would be much the same before and after consolidation. Consolidation of forty municipalities into a metropolitan district of one million would not witness the same stresses and price changes that would accomF~ny the growth of a ~ity from 25 thousand to one million. The diagrammatic soluticn for consolidation is illustrated in fig. 6. The product-equilibrium curve ~'ould be developed in the following way. A longrun average cost curve for a single municipality is given by $So in fig. 6. A demand curve for this municir ~lity is shown as DDo. Consolidation of two such units would spread the managerial bu~dep, somewhat and would permit a d~wnward shift in the cost curve to SSo. For the two municipalities taken together a new demand curve must be introduced. It will be a simple horizontal summation of the existing curves. This will intersect the SS'o curve at a lower unit cost and higher output. This process continues, with the equilibrium price falling as long as there are economies causing the SS curve to shift downwards. Eventually, management may start to yield diseconomies, causing the SS curves to start moving upwards and raising the intersection with the successively rightward shifting demand curves.
52
A. Abouchar. Efficiency in th~ urban economy
,
0
SS 0 maw
m
e e n m e
m
m
m e
abe
a
m
~
I I I I | I
.
.
.
.
.
.
.
0o
i
o
. . . . . . . . . . . . . .
8
Quantity per year D
p
m
m
m
aam
.,m
~,. '
-
I
I|
0 Quar,tity per year
i
I
I
I
III
Gill
1
II
I
I
I I
I_]11
I
28
Fig. 6. lllustration of effects of consolidation of two municipalities.
III
A. Abouchar, E.Ofciency in the urban economy
53
3. Empirical mmlysis of efficiency The question of what approach should be followed to determine empirically the relations described in sect. 2 is rather straightforward although any actual attempt naturally entails difficult data problems. Many of the studies which have been conducted in the past constitute appropriate formulations of some of these relations, and in this section we will evaluate them in this light. A few preliminary words are in order, however. For some services, such as police or fire protection, recourse is frequently made to expenditure to measure output. Criticism is encountered that this approach confuses demand and supply factors, as noted earlier. From the point of view of our analysis, this is unobjectionable, provided the expenditure by the public authority is in some direct way related to the final consumer through e.g. an annual property tax, or an earmarked city sales tax, for in this case any expenditure observation can be construed as an intersection of the demand and supply curves relevant to a given city size. A downward sloping demand curve can be imagined, and it woul,~ be interpreted as showing the amount that each additional person would be willing to pay for another unit of service. The intersection is the efficient product eqt, ilibrium point for a city of that size, and the series would then constitute an efficient product equilibrium curve just as would one derived from natural physicai indicators. It could then be summed directly for the derivation of an all services ~roduct equilibrium curve into which services measured in physical until would also enter, but only after a transformation to an annual per capita expenditure basis. How may the past studies be grouped within our classificatory scheme ? Most of them shed fight 0n the various modifications of a product equilibrium curve, divided about evenly between new towns and growth policy. While consolidation is the ostensible framework for many analyses, we feel that the studies done under this rubric really apply to new towns policy. Little has been done which could shed light on the expenditure-income relationship. Exan~ples of studies which illuminate various aspects of the dynamic p,'oduct equilibrium curve (in which input prices change with changing city size) for individual services include the recent sewage disposal study by Downing (1969) or the fire protection study by Will (1965) in which the fire protection expenditures per capita may be interpreted as city-size-related, rightward-shifting, supplydemand intersections. The gas study by Lomax (1951) also seems to be relevant to determination of the dynamic product equilibrium curve; it acknowledges the regionally related input price variability and attempts to avoid its complications by studying the cost relationships for three 0ifferent regions. Studies which relate to the question of whetuer and what size new cities should be include police, fire and refuse collection studies by Hirsch (1959, 1965, 1968)St. Louis municipalities- and the Schmandt-Stephens (1960) police protection
54
A. Abouchar, Efficiency in the urban economy
study- Milwaukee county municipalities- althc .~gh many of them ostensibly relate to consolidation policy. The observations in these studies are made on municipal units within fairly uniform urban regions with boundary lines of individual municipalities being, on the whole, rather artificial. Therefore, it would seem reasonable to view determinants of land values or wages as being fairly uniform. That is to say, we would not expect relative land prices (ceteris paribus) in two adjacent communities of size 5,000 and 50,000 respectively to show the same differences that two isolated ml.nic~~alities of these sizes would show. The two adjacent cities both reflect the sa~ne l,Lctors and, but for historical accident, would probably constitute a single city of 55,000. In this case, differences in land prices would relate to different use pa~terns rather than to differences in demands for homogeneous l~nd in two isolated ci:ies of different size. If this view is accepted, investigations of thi~ kind really tell us, up to a constant, what the average cost of functions would be starting from scratch. That is, they yield information on the static production equilibrium curve. They do not :ell us exactly, since, in a new situation, different relative prices might prevail. When starting a city from caw acreage, land prices would be lower than •those in the heavily urban areas of these studies, and the labour/land price ratio might differ. Finally, not~'~g apvears to have been done from which inferences could be made regarding tee expenditure-income function. Moreover, it is not certain just how one should proceed if he wanted to study this relationship. We could not regard the total expenditure at ~liiferent city sizes ~ a series of productequilibrium points without making some heroic assumptions. We could have to assume that everything that has developed in the past is as it should be. Even a small knowledge of the realities of the budgetary process precludes this view. 4. Benefit measurement under size change
The change in observed consumption in moving from a small to a large city is the result of two effects - a price effect and a demand sh~t effect which depends on city size. The point may be most easily pictured diagrammatically by reference to the analysis for new towns policy. Fig. 7 represents a supply curve for the output of some service. Three demand curves are Shown: DDo, DD[ and DDI. The left-most curve represents the demand for each of two cities of size No. The curve DDo is the sum ofthese two demand curves and represents the amount that both cities taken separately would demand. This curve cuts the supply curve at a lower price than the equilibrium price at which each city would supply the goods. However, for a city of size ~/~ --- 2No the demand curve would not be a simple doubling of the individual demand curves, which would be DD'o; rather it shifts further to the right to DDI t'or reasons already noted in the anal,,,sis of sect. 2.2.1. Thus, to measure tl: ~ increase in we]Lfareof the larger city, which attaches directly to the increased level of service, we must think in terms of DDo
tl. Abouchar, Efficiency in the urban economy
55
Sp
Fig. 7. Price effect and vity size, effect on change in consumption of an urban service.
and DD'o and relate a consumer's surplus to these curves. This then ~s given by the shaded area ABEC. The further consumption resulting from the higher level of demand necessitated by the larger city does not represent any benefit to the consumer- it is a cost of living in the larger city which presumably does have other benefits which could be measured. This last result is applicable for new towns or growth policy rather than consolidation. If an urban development authority is considering the construction oI~'either two cities of size No each, or ~ single city of size Art = 2No, this is the relevant analysis; similarl5', if consideri,lg growth from size No to N1. In these two cases the demand of the larger city will be greater than the sum of the d~emands of the two cities each half its size since different population densities are expected. However, when consolidatic,n is at issue, the demand after consolidation should be equal to the sum o:" the demands before consolidation since nothing but organization is changed. References Bird, R., 1970, The determinants of state and local expenditures, A review of U.S. studies, Appendix B, in: The growth of government spending in Canada (Canadian Tax Foundation). Breton, A., 1965, Scale effects in local and metropolitan government expenditures, Land Economics, 371. Downing, P.B., 1969, The economics of urban sewage disposal (Frederick Praeger, New York). Hirsch, W.Z., 1959, Expenditure implications of L~e~xopolitan growth and consolidation, Review of Economics and Statistics 41.
56
,4. Abouchar, Efficiency in the urban' economy
Hirsch, W.Z., 1965, Cost functions of an urban government service: Refuse collection, Review of Economics and Statistics 47. Ifirsch, W.Z., 1968, The supply of urban public services, in: H.S. Perloff ~md L. Wingo, Jr., Issues in uxban economics, 477-527. Lomax, K.S., 1951, Cost curves for gas supply, Bulletin of the Oxford Institute of Statistics 13, 243-246. Musgrave, R.A., 1968, The urban public economy, Discussion of Part Ill, in: H.S, Perloi~"and L. Wingo, Jr., Issues in urban economics, 567-574. Schmandt, H.J. and G.R. Stephens, 1960, Measuring municipal output, National Tax Journal. Will, R.E., 1965, Scale economics and urba, service requirements, Yale ..EconomicEssays 5.
EFFICIENCY IN THE URBAN ECOHOMY A general equilibrium framework*
Alan ABOUCHAR University of Toronto, Toronto, Canada R~eived June 1973
1. Introduction While economists often talk in terms of efficiencies or economies of city size or urban agglomeration, there is little agreement on the content of these terms. Just what do we mean by efficiencies of scale in the urba~ context ? This paper presents an analytical framework for the definition and analysis of urban efficiencies. Although no new measurements of urban efficiency are offered, we will consider the empirical literature within this theoretical framework paying special concern to the internal consistency of empirical investigations and to the relevance of this theory to what the authors have attempted to measure. Finally, we sketch some of the implications of our model for benefit measurements.
2. A model of the urb~, economy
2.1. The problem On its most general p'~ane, an agglomeration economy is a portmanteau expression which means tilat, for a host of reasons, welfare per capita is higher in a large city than in a small one. These reasons include production externalities of firms when indivisibilities require a large market for individual producing units; the same sort ofindivisibilities in the consumption sector, requiring a large market for the provision of a wide variety of amenities; higher incomes per capita owing to the greater labour productivity arising in greater proximity to work and increased use of capital or large scale production, made possible by the larger market; cost reductions in the Provisions of public sector services; and, finally, the intangibles resulting from the larger urban complex, such as the *This study was undertaken as part of a research project of the Institute for Policy Analysis of the University of Toronto. Helpful comments ~,y an anonymous referee are gratefully acknowledged. Errors and conclusions are the author's responsibility alone, how~ver.
42
A. Abouclw.r, Efficiency in the urban economy
improved quality of work, better newspapers, and so on. Some writers intend only subsets of these effects, such as the unit cost behavior in the provisi3n of public services. Whey the net result of all these factors is to lower welfare it is a diseconomy. Obviously, it is difficult to make operational so broad a concept of agglomeration effects. Empirical research has, therefore, concentrated on some of the subconcepts, such as the relationship between unit cost and volun e of individual public sector services Most of the s~udies, which have been based primarily on United States data, have been statistical cross section studies of urban expenditures by type function and by total, sometim¢:s adjusting Oor income and sometimes not, depending on what is the main focus of the investigation. Since most of this literature ha~ been reviewed and its methodolcgy appraise6 by others [see Bird (1970), Hirsch (1968) and Musgrave (1968)], we will make no attempt to treat it in anything like e/haustive fashion, simply referring to it as the need arises. Much of ~.he investigati,,~n of urban expenditures has been criticized for ,c~ffuseness and uncertainty about what is being measured and about the implications ofusing alternative formulations. It has been asked many times whether such studies of expenditure determinants measure demand or supply factors. As Musgrave (1968, p. 573) has put it: "The 'service' function combines a mixture of supply and demand factors which need to •be disentangled if the really interesting insights into the fiscal behavior of communities are to be obtained. To some extent, this might be accomplished by separating demand variables such as income and demographic factors, from supply variables, such as climatic conditions or wage rates. To illustrate, the cost of a ,range of service levels for snow removal (i.e. absence of snow on streets) will depend on factors such as snowfall, density, topography, wages and so forth. This is the co. ~.function p~rt of the problem. Service levels provided will then be a function of these costs and of demand factors, such as i~comes, tastes, and other prices. The gener~ expenditure function is unsatisfact3ry both bo~ause the dependent variable is poorly defined (i.e.~ in terms, of expenditure rather than service levels) and because the independent variables combine cost and demand factors without due separation. While certain variables m . y ~ on both sides of the picture (density reduces the cost of firefighting per house, but raises demand by increasing the cost of conflagration) a better separation is h~dly Im insoluble problem."
The same or similar point has been made by Breton (1965) and a number of others. The dissatisfaction with past research has undoubtedly stemmed in part from the different purposes to which different ~,bservers would like to put it. For example, straightforward expenditure determinant studies may be appropriate as a predictive device, but inadequate as a normative measure. But there is another diversity of purpose to contend with in analyzing the urban economy - there is not one object of measurement, but several. Analyqis must help in framing policy with respect to three quite dif~erent issues: (I) determination of optimal size for new towns; (2) consolidation of already existing neighbouring urban complexes; and (3) growth of already existing cities. Analysis relevant to each of these
43
.4. Abouchar, Efficiency in the urban economy
requires a different approach. They are all important today, the last being perhaps, the mest critical. The following presentation, therefore, will start with the last. The necessary modifications for the others will follow.
2.2. Diagrammatic analysisfor urban growth 2.2.1. The product market The formulation of ap urban economic model is too complex to be dealt with satisfactorily by a single expenditure determinants model. Even for a single service, there are, first of all, interrelationships between city size and demand, at the same time that demand may vary with price for any given city size. Fig. I shows market demand and cost curves for an urban public service. For the sake of definiteness, we will talk about garbage collection. The vertical axis shows the price per unit and the horizontal axis, the number of pickups per unit of time, year, say. The demand curve, DDo, slopes from northwest to southe a s t - at lower prices people will want to have more collections per year. Thus,
P~ L
!
Q,
Q.
I~
LRAC 0 pII
'"
'
"
'
---
DD 0
i
0
i
II
II
I
I
I
I
~
OJ (~0
~
0"
Pickups per Year O Fig. 1. Hyp,:,thetical market demand and supply of garbaFe collection (city size, No).
44
A. Abouchar, Ej~ciency in the urban economj
as the demand curve is drawn, if price is reduced from P' to pw, the quantity demanded per year will rise by 25 % from Q' to Q'. The demand curve DDo relates price to quantity holding constant everything else including the size of the city, whic:~ is here No. More pickups are demanded a~ lower unit prices for essentially the same reason that more of any normal good is always demanded when the price is reduced, i"he specific factors conditioning demand here have the foUowing logic: with more frequent collections people will not have to crush their refuse as intensiwly, objectionable odours wiU lessen, fly attraction will fall, and so on. For this city of size ~,'o we will also have a long-run average cost function L R A C This curve tells us what the lowest cost is per unit at different levels of activity. Being a long-run curve each point represents the most efficient cost which might be attained if we could start from scratch in each case with city s~e fixed, r"or example, the lower cost which is secured in moving from Q' to Qm might be b~:sed on a revised land use pattern with the c:'.y dump moved in closer, and lower labour and vehicle costs per pickup- the reduced driving distance offsets over the range of demand but nothing criti~,d is implied- the demand intersection at price Po and quantity Qo could easily occur over the rising range ¢x in an interval of constancy. The LRAC is taken as the supply schedule and ~ ill be referred to as $S in the future. The demand and ~upply analysis just conducted for city with size No can be repeated for other city sizes. Fig. 2 shows the relevant curves for six different c,ity sizes with equilibrium prices denoted P~. The supply-~emand pairs represent succe.,;sive rightward displacements, ~vhich means tha~: people are willing to pay more per unit for the same quantity of annual pickups as city size increases, and also that it costs more per unit for the same quantity of pickups per year for larger city sizes. The rightward sluft in demand occurs for two reasons. First, individuals' utility functions change and they are wilting to pay more for each pickup since greater density imposes additional health hazards. In the second place, the fact of a larger city implies a larger number of people who would be willing to pay more at any given level of colle~ion activity. (In the ~onsolidation analysis to be discussed below, by contrast, only the second of these factors enters, since the layout of the aggregated metropolitan region remains the same.) The demand curve of fig. 1 and the family of curves of fig. 2 are based on a give~ i n , m e level and we qhould expect that an increase in income will shift the demand curve for any city size to the fight, intersecting the supply schedule a; ~ r~int to the southeast of its present intersection at price P. "[h~ rightward shift in the supply schedules results from increased prices for inp~lts, primarily l~nd. Wages probably also rise somewhat, and final costs may higher since elimination must be more thorough. Several different treatments .have been proposed for the input price variation which depends on city size. Breton (1965) has clarified the issue by drawing the distinction between input
A. Abouchar, E/ficiency in the urban economy
45
price variation and input quantity variation as sc arces of final cost variation. Musgrave (1968, p. 573) has argued that input p:ic~ varietion should be eliminated in expenditure studies to try to gain an idea about ',,;ervice levels'. But this approach would leave us ill-equipped to deal with c,ne of the major questions 8t issue when the focus of study is policy with respect to existing cities: input. price variation accompanying different city size is precisely the way to measure society's needs and resources, and, therefore, the welfare gain or loss to be associated with transition from one si=e to anothel. And for a policy concerned with influencing the growth of existing urban complexes, we do want to observe
aDo Q, =I n_
DO2
0,
Dr,.]
DO4 SS5 DD 5
0 Pickups per year, O
Fig. 2. Market demand and supply curves for cities of varying size N,.
the consequences of input price variation since the very heart of the issue is the cost and benefit of changing from an existing city size to a different one, and the currently existing prices better measure the costs and benefits than would production functions based on the constant prices of some given city size. We may formalize the foregoing as follows. First, the demand function for city size No can be written Qo = fo(Po). In general, for city of size Nj it may be written Qj - f~(Qa). In all normal cases we will have Q, + 1 > Q, i.e., demand will shift to the right with increasing city size. If demand sniffs by a constant percentage, i.e. if the quantity associated with an increase in city size is proportional to the relative size difference, we will have Qt = [1 -l'oe(Na/No)]fo(Qo).
46
A. Abouchar, Efficiency in the urban economy
In like manner, the generalized supply function may be written AC, = A(Q,).
,,~s the supply and demand functions shift to the right, the equilibrium intersection P~ will shift. The series of intersections will be termed a Dynamic Product Equilibrium (DPE) curve. Recalling, moreover, that there will be a difl~rent family of demand curves for each income level, we will require a DPE curve for each income. Fig. 3 shows
Q. E .Q "5 D"
OPEl
DPE"
i
- -
0
NO
N1
i
ii
!
,
N2
m
N3
City Size N i
Fig. 3. Dynamic prodt~ct eqt~ilibrimn curve (equilibriurn price vs. city size).
two such curves, denoted DPE' and DPE". (DPE' corresponds to the s~tuation of fig. 2 and the equ:~ibrium prices/~l have, accordingly, been relabeled/F on fig. 3. DPE" relates to a higher income level since the equilibrium price for higher income is lower, with a higher quantiq, than that of the lower income level, as indicated earlier.) The DPE curve is a Cescriptive rather than an optimizing device. It merely traces out the supply-demand equifiorium prices for different size cities. Optimization requires that this product ma~'lcet equilibrium curve be evaluated with respect to the benefit~, which are associated with different size cities, to which we now turn.
47
A. Abouchar, Efficiency in the urban economy
2.2.2. Demandfor city size (amenities) Since cities perform different functiors (seaports, manufacturing cities, agricultural centres, and so en) it is first necessary to de:ermine the function or nature of the city bein~ analyzed. We will consider cities oriented to commerce and fight industry. Their amer:it~ies will vary with their size- parks, newspapers, .,,ocial life, etc. In general it is to be expected that for any given size, the willingness to spend on municipal services is positively associated with income. Larger cities may be expected to have greater environmental diversity and advantages than smaller cities; individuals would be willing to pay more for these advantages out of any given income. The extra expenditure on the various municipal services that a consumer is willing to undertake to live in a larger city may be regarded as a price - the price ofriving in the larger city. A group of expenditure-income curves is shown in fig. 4. Annual expenditure
J
EE4
j:
3
j j
EE EE
~mb
m
,,m
Q. al
0 Q. (u 0 12 v
e. :1
'0 o x w
.
.
.
.
.
o
i
i
mill
_
7
Income ( Dollars Per Capita)
Fig. 4. Per capita expenditure-income curves foz varying city size.
48
A. Abouchar, Efficiency in ,he urban economy
is shown on the vertical axis, and income on the horizontal.. The curves are labeled EEl, the subscript indk'.ating that the curve pertains to city size N~. The expenditure-income function can then be related to city size by drawing a perpendicular at any income level, say ~, and plotting in fig. 5 the points of its intersection with the f ~ i l y of exj~enditure curves. This diagram shows per capita e~:penditu:e on the vertical :~xis and city size on the horizontal. The curve derived in this way is labeled EY. This curve will be upward sloping, under our assumptions, ff people were unwilling to pay more to live in larger cities, the curves in fig. 4 would be displaced downward with higher subscript i, and the El? curve would slope dow:award. We would expect, normally, that the EEl would move upwards, though at a decreasing rate, and eventually either fall or remain constar~. In the last named situation the El? curve will flatten out. If the E£~curves were to start to fall, the El?curve would start to decline as well.
S~
t11 .,.,.
O "D
,,m,
°,m
Q.
O eL
"O ¢= O Q.
DASPE
I(
UJ
Ni City Size N i
Fig. 5. Product-equilibrium, expenditure-income equilibrium, and determination of efficient city size.
,4. Abouchar, Efficiency in the urban economy
49
We have considered the expenditare city-size curve for income of Y. Such a curve can be constracted in like roamer for any income. The curve deri,~ed for > ~'wou~d lie above the EYcurve in fig. 5. We can now compare the amenities or city size dems~td, in the form of the expenditure-income function, with ~ cost curve. We will assume that the demand for each service, as analyzed in sec~lon 2.2.1 is independent of the demand for omers, as seems reasonable. The I ~ E curves are given in terms of prices per unit of service and must be converted into expenditure terms. To do this, we multiply P~ by the equilibrium quantity and divide by city size to determine the per capita equilibrium expenditure on this service. The same is done for all other services. The resulting curves are then summed vertically to give a Per Capita Dynamic All Services Product Equilibrium Curve for a given income level. This curve is labeled D A~PE in fig. 5. Following cons,derations of the relationship between the DPE curves for different income levels, as shown in fig. 3 and explained earlier, the DASPE curve for a higher income would lie to the left ofthe DASPE curve in fig. 5. Also shown in fig. 5 is the expenditure-income function tbr the same income level, T. The optimal city size is shown by the intersection of the two curves at N~. Thus, ~he optimal city size is determined by four factors: (i) a faz~y of demand functions for individual services for varying city sizes (one family for each income level); (ii) a long-run average cost function for each cky size; (iii) the expenditure-income function, one for each city s~ze; (iv) per capital personal income. There are some obvious similarities between the graphical analysis just concluded and the Hicks-Hansen elaboration of the Keynesian system. Thus, the dynamic all services product equilibrium curve (DC) is the counterpart of the investment-savings function which defines a series of equilibria in the real economy. DASPE tells us the equilibrium annual expenditure which equates supply and demand for urban services in any given city size. The E Y curve is the founterpart of the Hicks-Hansen liquidity preference function and may also be interr zeted as a kind of financial equilibrium curve. It defines a series of equilibria in people's apportionment of their budgets between expenditure on urban services and other items. We assume that people have some idea of what is available in different size cities which serve the respective functions (commerce and fight industry, in this case) and how much they would want to have left after urban service expenditures to enjoy being in that city. If per capita income is ~, the population in this region will prefer cities of size ,¥1 for their commercial centers. Sizes of other kinds of cities would be determined in like manner- people will
50
A. Abouchar, Efficiency in the urban economy
prefer agricultural supply cities of some given size, seaport another, heavy industrial cev.ters another, and ~o on. ~ To carry Hicks-Hansen analogy further, just as an increase in the money supply will shift the L M curve ~to the right), an increase in per capita income will shift the EY curve (to the le£~, however). Since the income level is different, the DASPE curve will also move to the left. z Both movements lead to a smaller optimal city size. If E Y m o v e s more than DASPE, the final equilibrium expenditure under the higher income in the smaller city will be greater than that under the lower income in the bigger city. This means that with great wealth, people will avail themselves of the luxury of smaller cities (where the 'physical' market supply-demand equilibrium occurs at a higher price). People will spend more per capita on urban services and also pay more per unit of urban service; but at the same time they will spend relatively less of their income on urban services (see the slope ofthe EE curves in fig. 4). They will thus have a larger share income unoommitted to the provision of urban services with which they can enjoy more benefits offered in the smaller city, which has fewer benefits to offer, than they c.ouild enjoy in the larger city, which has more to offer. A reduction in per capita income in the region will shift to the right, defining an equilibrium at greater city size. The logic is similar to that just described. 2.3. /~alysis for new towns policy
Where new towns policy is concerned ~ e do wish to abstract from the input price, variation which has come to characterize relative factor costs in actually observed cities and to concentrate instead only on the scale economies of the underlying technological processes, l'he changes that must be introduced into the analysis of ti=e preceding section relate first to the individual product supply curves and thence to the DPE and DASPE curves. We assume that land can be bought at constant unit price in the region in which the new city is to be founded, ,~o that the differential effects of city growth over time have not come to be reflected in land values. In this case, doubling the ~izeof a city dump will 0 .a[ ~, l~and cost. Similarly, labour costs can be estimate6 at a given consta-~ ",,~ce,, the assumption that the workers who o:me to this ~ity come from the ' aargin ,n many different cities and, within the city size rar~ge being contemplated, ~ ay number can be attracted at that price. The physical inputs underlying the cost curves ol fig. 2 are now valued at these unit prices to calculate a single LILAC curve for each service. The family of 'That cities of any given function in any given reckon are not, in fact, all the same size is not to deny this theory. The cities existing at any point in time have started at different times, have different special characteristics (even within the stone broadly defined type), and so on. Ovr analysis is l,~tended as a logical abstraction of the forces conditioning optimal city size. 2The dynamic product equilibrium curve for any product will be displaced downwards the less, the lower is the income elasticity of demand for the service in question (the less does the family of demand curves ir fig. 3 shift to the right with rising income) and the greater is the elasticity of the individual cost ~ r v e s in fig. 3.
A. Abouchar, Efficiency in the urban economy
51
demand curves remains as before. The series of intersections with the LP,A C and the demand curve family traces out a static product equilibrium curve, one for each service, relating equilibrium prices to city size. The observation for each city size is then multiplied by the quantity provided and divided by the city size to get a per capita annual expenditure for that service. These are summed over all services to yield a per capita static all services product equilibrium curve (SASPE). Confrontation of SASPE with the expenditure-income function, as before, then defines the optimal city size.
2.4. Ar,alysis for consolidation policy Under what conditions should contiguous municipali~:ies unite ? The decision should take into account the cost reduction that may be achieved through consolida~ion. This ~ usually a managerial economy. Another main source of scale e c o n o m y - price concessions - is not so straightforward; the quantity discounts offered by a supplier working with forty local communities, say, cannot be considered as observations on a statistical supply function which would then be simply extrapolated to a larger community since the existing pricing pattern may reflect a currently existing discriminating sales pattern in which the higher average prices to some will offset the lower prices to others, overhead being apportioned, in part, inversely to the strength of the buyer. For the same reason we could not even assume that the purchases after consolidation could all be made at the price now granted the biggest user. The main source of scale economy, then, would be management. Pecuniary economies are less certain. Other sources of cost variation such as the changing input prices or factor cost variation considered earlier would not arise in the consolidation analysis-land and labour costs in the various municipalities within the same urban region would be much the same before and after consolidation. Consolidation of forty municipalities into a metropolitan district of one million would not witness the same stresses and price changes that would accomF~ny the growth of a ~ity from 25 thousand to one million. The diagrammatic soluticn for consolidation is illustrated in fig. 6. The product-equilibrium curve ~'ould be developed in the following way. A longrun average cost curve for a single municipality is given by $So in fig. 6. A demand curve for this municir ~lity is shown as DDo. Consolidation of two such units would spread the managerial bu~dep, somewhat and would permit a d~wnward shift in the cost curve to SSo. For the two municipalities taken together a new demand curve must be introduced. It will be a simple horizontal summation of the existing curves. This will intersect the SS'o curve at a lower unit cost and higher output. This process continues, with the equilibrium price falling as long as there are economies causing the SS curve to shift downwards. Eventually, management may start to yield diseconomies, causing the SS curves to start moving upwards and raising the intersection with the successively rightward shifting demand curves.
52
A. Abouchar. Efficiency in th~ urban economy
,
0
SS 0 maw
m
e e n m e
m
m
m e
abe
a
m
~
I I I I | I
.
.
.
.
.
.
.
0o
i
o
. . . . . . . . . . . . . .
8
Quantity per year D
p
m
m
m
aam
.,m
~,. '
-
I
I|
0 Quar,tity per year
i
I
I
I
III
Gill
1
II
I
I
I I
I_]11
I
28
Fig. 6. lllustration of effects of consolidation of two municipalities.
III
A. Abouchar, E.Ofciency in the urban economy
53
3. Empirical mmlysis of efficiency The question of what approach should be followed to determine empirically the relations described in sect. 2 is rather straightforward although any actual attempt naturally entails difficult data problems. Many of the studies which have been conducted in the past constitute appropriate formulations of some of these relations, and in this section we will evaluate them in this light. A few preliminary words are in order, however. For some services, such as police or fire protection, recourse is frequently made to expenditure to measure output. Criticism is encountered that this approach confuses demand and supply factors, as noted earlier. From the point of view of our analysis, this is unobjectionable, provided the expenditure by the public authority is in some direct way related to the final consumer through e.g. an annual property tax, or an earmarked city sales tax, for in this case any expenditure observation can be construed as an intersection of the demand and supply curves relevant to a given city size. A downward sloping demand curve can be imagined, and it woul,~ be interpreted as showing the amount that each additional person would be willing to pay for another unit of service. The intersection is the efficient product eqt, ilibrium point for a city of that size, and the series would then constitute an efficient product equilibrium curve just as would one derived from natural physicai indicators. It could then be summed directly for the derivation of an all services ~roduct equilibrium curve into which services measured in physical until would also enter, but only after a transformation to an annual per capita expenditure basis. How may the past studies be grouped within our classificatory scheme ? Most of them shed fight 0n the various modifications of a product equilibrium curve, divided about evenly between new towns and growth policy. While consolidation is the ostensible framework for many analyses, we feel that the studies done under this rubric really apply to new towns policy. Little has been done which could shed light on the expenditure-income relationship. Exan~ples of studies which illuminate various aspects of the dynamic p,'oduct equilibrium curve (in which input prices change with changing city size) for individual services include the recent sewage disposal study by Downing (1969) or the fire protection study by Will (1965) in which the fire protection expenditures per capita may be interpreted as city-size-related, rightward-shifting, supplydemand intersections. The gas study by Lomax (1951) also seems to be relevant to determination of the dynamic product equilibrium curve; it acknowledges the regionally related input price variability and attempts to avoid its complications by studying the cost relationships for three 0ifferent regions. Studies which relate to the question of whetuer and what size new cities should be include police, fire and refuse collection studies by Hirsch (1959, 1965, 1968)St. Louis municipalities- and the Schmandt-Stephens (1960) police protection
54
A. Abouchar, Efficiency in the urban economy
study- Milwaukee county municipalities- althc .~gh many of them ostensibly relate to consolidation policy. The observations in these studies are made on municipal units within fairly uniform urban regions with boundary lines of individual municipalities being, on the whole, rather artificial. Therefore, it would seem reasonable to view determinants of land values or wages as being fairly uniform. That is to say, we would not expect relative land prices (ceteris paribus) in two adjacent communities of size 5,000 and 50,000 respectively to show the same differences that two isolated ml.nic~~alities of these sizes would show. The two adjacent cities both reflect the sa~ne l,Lctors and, but for historical accident, would probably constitute a single city of 55,000. In this case, differences in land prices would relate to different use pa~terns rather than to differences in demands for homogeneous l~nd in two isolated ci:ies of different size. If this view is accepted, investigations of thi~ kind really tell us, up to a constant, what the average cost of functions would be starting from scratch. That is, they yield information on the static production equilibrium curve. They do not :ell us exactly, since, in a new situation, different relative prices might prevail. When starting a city from caw acreage, land prices would be lower than •those in the heavily urban areas of these studies, and the labour/land price ratio might differ. Finally, not~'~g apvears to have been done from which inferences could be made regarding tee expenditure-income function. Moreover, it is not certain just how one should proceed if he wanted to study this relationship. We could not regard the total expenditure at ~liiferent city sizes ~ a series of productequilibrium points without making some heroic assumptions. We could have to assume that everything that has developed in the past is as it should be. Even a small knowledge of the realities of the budgetary process precludes this view. 4. Benefit measurement under size change
The change in observed consumption in moving from a small to a large city is the result of two effects - a price effect and a demand sh~t effect which depends on city size. The point may be most easily pictured diagrammatically by reference to the analysis for new towns policy. Fig. 7 represents a supply curve for the output of some service. Three demand curves are Shown: DDo, DD[ and DDI. The left-most curve represents the demand for each of two cities of size No. The curve DDo is the sum ofthese two demand curves and represents the amount that both cities taken separately would demand. This curve cuts the supply curve at a lower price than the equilibrium price at which each city would supply the goods. However, for a city of size ~/~ --- 2No the demand curve would not be a simple doubling of the individual demand curves, which would be DD'o; rather it shifts further to the right to DDI t'or reasons already noted in the anal,,,sis of sect. 2.2.1. Thus, to measure tl: ~ increase in we]Lfareof the larger city, which attaches directly to the increased level of service, we must think in terms of DDo
tl. Abouchar, Efficiency in the urban economy
55
Sp
Fig. 7. Price effect and vity size, effect on change in consumption of an urban service.
and DD'o and relate a consumer's surplus to these curves. This then ~s given by the shaded area ABEC. The further consumption resulting from the higher level of demand necessitated by the larger city does not represent any benefit to the consumer- it is a cost of living in the larger city which presumably does have other benefits which could be measured. This last result is applicable for new towns or growth policy rather than consolidation. If an urban development authority is considering the construction oI~'either two cities of size No each, or ~ single city of size Art = 2No, this is the relevant analysis; similarl5', if consideri,lg growth from size No to N1. In these two cases the demand of the larger city will be greater than the sum of the d~emands of the two cities each half its size since different population densities are expected. However, when consolidatic,n is at issue, the demand after consolidation should be equal to the sum o:" the demands before consolidation since nothing but organization is changed. References Bird, R., 1970, The determinants of state and local expenditures, A review of U.S. studies, Appendix B, in: The growth of government spending in Canada (Canadian Tax Foundation). Breton, A., 1965, Scale effects in local and metropolitan government expenditures, Land Economics, 371. Downing, P.B., 1969, The economics of urban sewage disposal (Frederick Praeger, New York). Hirsch, W.Z., 1959, Expenditure implications of L~e~xopolitan growth and consolidation, Review of Economics and Statistics 41.
56
,4. Abouchar, Efficiency in the urban' economy
Hirsch, W.Z., 1965, Cost functions of an urban government service: Refuse collection, Review of Economics and Statistics 47. Ifirsch, W.Z., 1968, The supply of urban public services, in: H.S. Perloff ~md L. Wingo, Jr., Issues in uxban economics, 477-527. Lomax, K.S., 1951, Cost curves for gas supply, Bulletin of the Oxford Institute of Statistics 13, 243-246. Musgrave, R.A., 1968, The urban public economy, Discussion of Part Ill, in: H.S, Perloi~"and L. Wingo, Jr., Issues in urban economics, 567-574. Schmandt, H.J. and G.R. Stephens, 1960, Measuring municipal output, National Tax Journal. Will, R.E., 1965, Scale economics and urba, service requirements, Yale ..EconomicEssays 5.