Are There Returns to Scale in City Size?

Are There Returns to Scale in City Size?Author(s): David SegalSource: The Review of Economics and Statistics, Vol. 58, No. 3 (Aug., 1976), pp. 339-350Published by: The MIT PressStable URL: http://www.jstor.org/stable/1924956 .

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ARE THERE RETURNS TO SCALE IN CITY SIZE?

David Segal*

I. Introduction

A question central to the issue of optimal city size is whether scale economies exist

in urban production. Over a third of all Ameri- cans who currently live in metropolitan areas live in one of the dozen largest. If it can be shown that these areas have a significant production advantage over the remaining ones, then welfare analysis could be applied to learn whether the extra output produced in these metropolitan areas outweighs the drawbacks of

'their greater disamenities. Economists have long recognized that wages

and output per worker in large cities exceed those in smaller ones (Alonso, 1970; Fuchs, 1967; Hoch, 1972; and Izraeli, 1973). Work- ers have apparently known about this too, and as a result the size distribution of cities has been shifting in favor of the largest cities for much of the past century. But before one can pass welfare judgments about this trend it is necessary to have in hand an empirically-based theory of production and income in urban areas and of the role, if any, played by city size. Our main concern is to develop such a theory -to explain variation in worker incomes across a set of metropolitan areas. We do this for 58 areas using data for 1967. As a part of the study capital stock data were estimated for each of the areas. A marginal productivity theory of factor incomes is assumed and justified empirically. Two-thirds of the variation in gross metropolitan income per worker is explained.

We found that the largest SMSAs-those with a population of two million or more had a return to factors 8% higher than the remaining SMSAs. The reason for this was not increasing returns to scale in production

-we observed constant returns both across the entire sample and within several smaller city/larger city partitionings of the sample. Rather, there is an "agglomeration effect" that seems to obtain for areas of more than two million -a change in the constant term causing a shift in the production function. The reason for this effect, apparently, is that economies exist in transport and communica- tion in the very largest cities with the result that the benefits from agglomeration more than offset congestion costs.

A marginal productivity theory of distribution in cities is developed briefly in the next section. This is followed by an empirical section which describes the data and relates the findings.

II. Theory of Production and Distribution in Cities

Imagine that a city is like a firm or a farm, where factors such as capital, labor and land are transformed into output. The amounts of the factors available to any particular city change over time, depending upon whether their marginal products are higher or lower than in other cities. There exists neither per- fect mobility nor immobility of factors, for example, entrepreneurship, so that at any given point in time the system may not be in equilibrium but simply tending toward equilibrium.

Like firms or farms, cities have differing natural endowments -climate, waterfront or airport facilities, scenic amenities, and so on. Over time, of course, these endowments become joint factors in production with land or capital, and local landlords will emerge as their owners - and pricers.

A model perhaps better suited to the analysis of production differences among cities than that of the individual firm is the aggregate production model. Employing Hicks' "theorem of aggregation," we assume that the returns to scale of a city as a whole are a weighted average of the returns of individual firms, corrected for the positive and negative externalities they confer on one another, where the weights are

Received for publication December 6, 1974. Revision accepted for publication October 31, 1975.

* The research was supported by the Center for Popu- lation Research, National Institute of Child Health and Human Development. Thanks are due to Thomas Stein- meier, Kurt Hausafus, Irving Hoch, Joshua Kay, Terry Murray, and an anonymous referee who provided useful comments on an earlier draft.

[ 339 1



340 THE REVIEW OF ECONOMICS AND STATISTICS

the shares of total city income generated by each firm.1

Using an aggregate production model, how would one interpret the fact that average real income or output per worker in the largest cities is higher than in the remaining cities in, the country?2 There are three possible explana- tions. First, capital/labor ratios may increase with city size. Higher ratios of money wages to profit rates (because money wages must compensate for non-marketed disamenities) in larger cities would imply higher capital/labor ratios. So this is an explanation that has some a priori appeal. Perhaps too the quality of the labor force is higher in the biggest cities. Were either of these conditions to prevail we would want to correct for inter-city differences in the specification of the aggregate production function.

A second possible explanation is that increasing returns to scale exist in city size. A third possibility is that small and large cities have the same production function (same coefficients for labor and capital) but that a different multiplicative constant term applies to the two categories of cities.

Which of these three cases - or what com- bination of them - describes reality? While this might be viewed largely as an empirical question, we can nonetheless make certain observations a priori. If we assume that the initial distribution of cities is at equilibrium, then we pretty well have to rule out increasing returns. By equilibrium we mean that each city is at its own equilibrium size, and that productive factors have no incentive to move to new loca- tions. The system is more- likely to be at equilibrium if either capital or labor (or both) is perfectly mobile and if adjustment by the mobile factor(s) is instantaneous.

The question as to whether there is one production function or several is not so easily resolved. In conceptual terms it is easy to imagine that every city has its own separate production function; and that the constant term for each city reflects its own unique bundle of environmental and site characteristics, age and density of habitation, proximity to other cities, infrastructure, and so on. Aside from this we know that the city system is being reshuffled continuously, through factor migration, in favor of the largest groupings of population. Size in and of itself can be thought of as a site characteristic, imbedded in the constant term of a city's production function. The essential question is whether the impact of this characteristic is positive, negative or neutral across the size distribution of cities, once other factors differentiating cities have been con- trolled for. Because of higher real incomes in the larger cities a reasonable hypothesis is that size exerts a positive influence on city product, ceteris paribus.

If this is the case, then we can apply marginal productivity concepts to analyze city differences in productivity, as shown in figure 1.

FIGURE 1.-WAGE-RATE DETERMINATION BETWEEN

SMALL AND LARGE CITIES

WAGE RATE

(I + Y) w,

WL t

W?s

VM PL (KL)

? LLSLL EMVMPS?(Kl

0 ~~~LISLs LL LLI EMPLOYMENT

This figure contains two value of marginal product of labor (VMP) curves for a typical small and large city. The bracketed terms, KS and KL, indicate the capital stocks of each of these cities - assumed constant in the short

1 Hicks (1946), P. 50. Hicks' theorem requires that the relative prices of goods produced in different cities be constant across cities. This may not everywhere be the case, however. For example, the remoteness of some cities will cause the prices of exported goods net of transportation costs to be lower.

2 Throughout the analysis we shall be considering only real income or output, corrected for inter-city price differences. As noted in the next section average land rent in a city -together with the prices of all land-using activities, including labor, which use residential land-is an increasing function of city size. For similar reasons, capital stock will be considered only in real terms.



ARE THERE RETURNS TO SCALE IN CITY SIZE? 341

run. Wage earners, numbering Ls in the small city, earn a wage w,; while those in the larger city earn WL > WS. Points r and t on the two VMP functions are short-run equilibrium positions.

Are they also long-run equilibrium positions? Hoch (1972) believes that they may well be, that there is an upward-sloping long-run labor supply curve, passing through points such as r and t, which reflects the fact that workers require wage compensation for the non-pecuniary costs of city life. These costs crime, pollution, congestion, and noise - are associated with density, which in turn is an increasing function of city size.

An alternative interpretation depends on the notion that there is some disequilibrium in the national labor market. If the true long-run labor supply curve were more elastic than a function passing through r and t, then the largest cities should grow larger still, at the expense of those of lesser size. In the extreme and improbable case of an infinitely elastic long-run labor supply curve, equilibrium wage and labor force sizes in the small and large cities would be we, Lse and LLe. The careful specification and estimation of the long-run labor supply function across cities of increasing size is beyond the scope of this paper.

Aggregate urban output in city i is determined by the following production function:

= A S'YC-6KiaLVk8kqik (1) where Qi is real output or value added in production; Ki is the city's capital stock; Li is its employment; and qi is a vector of labor quality variables, reflecting the composition by education, sex, race, and age of the city's work force. A, S, and Ci taken together form the constant term of the production function. Ci is a vector of site characteristics in city i, reporting special features such as climate, natural resources, and the importance of the city as a regional center, and 6 is the corresponding vector of elasticities. A is a transformation coefficient and S is a dummy variable for size, with y its elasticity. If we set S equal to e for the largest cities, and 1 for all the rest, we are actually assuming an

identical production function for all cities but for a "size effect" -the presence of net ag-

glomeration economies in the larger cities ex- ceeding those in the smaller ones.

If we accept the marginal productivity theory of income distribution among factors, then the wage rate in city i will equal the first- order partial derivative of Qi with respect to Li. That is,

wi _ Qi ( B ik (9L, A S-C-bKiaL,,Xk>: kqik-1 (2)

The two curves in figure 1 are graphic repre- sentations of equation (2) for values of K ap- proximating those in typical small and large cities. Because of the size effect, if wage earners in a given small city instead belonged to a work force of similar size in a typical large city, they would earn a wage rate y per cent higher. The fact of the higher real wage rate in larger cities has caused employment in them to grow (movement along VMPL(KL) to t) so that in equilibrium the money wage rate in the larger cities, corrected for price differences, is only (WL/WS- 1) < y per cent more than in the smaller cities.3

We turn now to the empirical side of the paper.

III. Empirical Analysis and Findings

The empirical work is presented in three sections. First, we outline a strategy for ascer- taining whether there is a scale effect in city size and, if so, how to determine it. Second, we comment on the data which are used. Finally we present the results.

A. Strategy Before one can even consider estimating a

production function such as equation (1) it is necessary to correct some of the data for inter- city price differences. This is not a problem with the employment figures, which are in real terms. But the output and capital stock data are both in value terms and do require adjusting.

3 This is not to suggest a historical, process of declining wages in the largest cities. In actuality the capital stock has grown simultaneously with the labor force in the largest cities. This means that the VMPL(KL) curve has shifted out at the same time that a growing work force has moved along it. Shifts of the function have dominated -movement along it so that wages in the largest cities have risen over time.




Output price differentials among cities are determined simultaneously with wage differentials. In his work on this subject, Izraeli (1973) tells us that wages vary with city size only because some environmental goods vary with city size. Ideally the impact of environmental goods, both natural and publicly produced, should be measured explicitly. We opt instead for the simpler device of letting city size and geographic region be proxies for Izraeli's environmental goods. Accordingly, a can of cat food will cost less in San Diego where lower supermarket wages and nearby recreational and scenic amenities get passed on as lower distribution costs - than in Los Angeles, where disamenities and higher land rents associated with larger city size push costs up. However, the cat food will be less expensive in California, ceteris paribus, than in one of the New England states for similar reasons.

If we let QR and KR denote raw output and capital, then the price corrected levels of these variables can be found simply by normalizing them:

Q= Q

f (D, S) and

KR 3

g (D, S)(3 where f and g are normalization functions described in the next section; D is a set of regional dummy variables, and S is a city size variable. Both D and S are proxies for environmental characteristics.

Theory tells us that K and L are highly correlated in a sample such as ours, so a pre- ferred estimation technique is to regress output per worker on the capital-labor ratio and other variables; that is, to divide both sides of equation (1) by L before proceeding with the estimation:

(L - -AS i L- L i (4)

where a has the same value as a in equation

(1) and bi- J8kq,7c + a - 1.

The strategy we shall pursue in analysing empirically equations such as (1) and (4) has several component parts. First, we shall be interested in the explanatory power of equation

(4), and whether the coefficient of S is significant and positive as we would expect. We shall also be interested in whether, as we would anticipate, constant returns to scale prevail across the entire sample of cities. That is, we shall be interested to see if the coefficients a

and L73Pkqik sum to unity. Throughout this paper we have shown an

interest in economic differences, from a production viewpoint, between small and large cities (because the smallest metropolitan area in our sample has a quarter-million population, the distinction is really one between the "large" and the "very large"). In the empirical work we use a dummy variable for size to distinguish between the two size categories. When it comes to comparing production relations in the small and large cities, we note three possibilities. Both the large and small cities may have the same production function, with the same constant term. Second, the production function may be identical across the two size categories, but the constant terms may differ. Finally, the production functions themselves may differ. In structuring the estimation problem as we do, in equations (1) and (4) we assume that the second of these possibilities prevails that only the constant term, to which the size dummy belongs, differs. Logic requires that if we choose the second possibility, we must at the same time reject the first and third.

A final aspect of our strategy will be to see if the model supports the marginal productivity hypothesis by explaining wage and salary incomes. We shall deem that it does if earnings in a city are accurately estimated by multi- plying output by the estimates for labor's share. Moreover, as we are employing a two- factor model, we shall assume that (i) if the model confirms the marginal productivity theory of wages and (ii) if constant returns to scale are also confirmed, then the model simultaneously supports- a marginal productivity of capital theory of returns to capital - here the non-wage components of value added.

The marginal productivity hypothesis, using our terminology, states that

wiLi Z/kqikASCiKiaL kqik (5)




If in substituting the parameter estimates for /3, y, 8 and a from equation (1) in the right- hand side of equation (5) we are provided with good estimates of earnings from one city to the next, then we may emerge from our study not only with a statement about the economics of city size but also with some insights into the determination of earnings income within cities.

B. The Data The initial sample size was 73 SMSAs, this

being the number of areas represented in at least four consecutive Censuses of Manufac- tures (a sequence, as we shall see, desirable for the creation of capital stock data for manufacturing). The sample size was reduced to 58 SMSAs because data were lacking on several of the observations, and because in the case of several New England SMSAs too much land area was included for the SMSA/core city relationships to be maintained.4 The sample has a slight bias toward areas with a manufacturing orientation, a bias that is more pro- nounced for the smaller cities.

Listed in appendix B are comments on the sources from which each of the data were drawn, and special assumptions which were required in order to compile them. As noted at the outset, capital stock data were assembled for each of the SMSAs. This was by far the largest part of the task. Whatever their flaws may be, these appear to be the only existing data on the capital stocks by SMSA.

The capital stock data were inferred from investment data using a standard approach.5 The technique recognizes both depreciation and technological change as functions of time. We assume that at time t an investment outlay made in year v, It, is worth

Kt.vz[ (1 d) t-v( 1 + k)vIv]+def (v, t) (6)

where d is a depreciation rate, k an obsolescence or technical innovation rate, and def (v,t) a time series deflator for the base year - t. To get total effective capital stock at time

t we simply sum such investments for a sequence of vintages. We employed values of d -.025 and k - .02, values similar to those employed by Solow.

As suggested in equation (3) it is necessary to correct both the value added and the capital stock data for inter-city price differences. How were the deflators constructed? In the case of the value-added deflator, f (D,S), we regressed the cost-of-living index for 38 cities for a middle class family of four on 1970 SMSA population and a South regional dummy, and obtained the following equation (standard errors in parentheses) :6

Index 98.41 + .677 (SMSA population, (0.89) (.258) in millions)

-7.93 (South (1.3 5) dummy)

R2 0.59 R2- 0.55.

J was then defined for any given SMSA as the estimated index, using the above regression coefficients divided by 100.

A similar approach was available to correct for inter-city differences in the prices of capital goods. As noted in appendix B, F. W. Dodge construction cost indexes were obtained for each of the SMSAs in the sample, with New York City - 100 as the numeraire. If one regressed the index on the same variables that the index for value added was regressed upon, the following results would emerge:

Index 87.90 + .49 (SMSA population, (0.75) (.28) in millions)

-11.56 (South (1. 19) dummy)

R2- .62 R- .60.

Because of the availability of a complete set of the Dodge deflators, it was decided to use them instead of the estimated deflators that might have been obtained using the above equation. The Dodge indexes were applied just to the construction component of capital stock and

4 Providence-Pawtucket--Warwick, R.I.-Mass.; Worcester, Mass.; Bridgeport, Conn.; New Haven, Conn.; and Water- bury, Conn.

5 The approach was introduced by Solow (1962). The treatment of technological progress and obsolescence was refined in Thurow and Taylor (1966).

6 Similar results are obtained by Hoch (1972), p. 310. He includes dummy variables for Northeast and North Central, but these are not significant. Neither were they in our analysis. Izraeli (1973, p. 23) obtained comparable explanatory power in a model which enumerates environmental amenities and disamenities.




not to equipment, furniture, fixtures and in- ventories.

The adjusted capital estimates that emerged seem reasonable. The average capital-output (capital stock/value added) ratio for the SMSAs in the sample is 3.2. This number would be somewhat higher if data on land and airports had been included in the capital stock estimates. The average capital stock per worker in U.S. manufacturing has been reported as $30,000. For employment and capital stock overall in the SMSAs in the study this number is $28,960 -a number which, to repeat, understates true capital (land included) per worker.

Nevertheless, it is a pity that the property insurance sector, which collects and makes available data on insured property values by state, does not break down these data by metropolitan area. Were this the case we would have an enormously valuable asset in constru- ing area capital stocks. The Census of Govern- ments publishes data on taxable property values, but it does not yet separate structures from land, reflecting the fact that many munic- ipalities do not yet do so.

A few specific comments are in order regard- ing the labor quality variables (q), specific city characteristics (C) and the city size variable (S). We tried four measures of labor quality, described in appendix B and reported in table 1. Education and sex were significant; race and age were not.

We looked to variations in the industrial structures of SMSAs for clues concerning differential productivity, as indicated in the previous section. Data on structure must be approached with caution however. From economic base analysis we -know that, ceteris paribus, the larger its size, the greater the share of a city's total output is aimed at ful- filling local needs. For this reason it would be wrong simply to seek out data on the export orientation of an SMSA's economy and to neglect the effects of size. Accordingly, we used industrial structure data for different sectors, reflecting the excess employment in these sectors, over and above minimum local requirements (Ullman, 1969, table V, pp. 72-74). Specifical-ly these-were 1970 data on the excess

(export) industrial employment as a per cent of total employment.

We included as city characteristics orientation (above Ullman's calculations of minimum requirements) toward manufacturing, services, and mining. We also included a variable which reported on the importance of a city as a regional business and administrative center - as an "entrepot" for the regional economy. This variable summed the excess (export) employment in public administration and wholesale and retail trade.

The manufacturing and service variables proved to be somewhat correlated with two of the labor quality variables, FEMPCT (per cent women) and EDUCATION (r - -.51 and +.29, respectively). As a result the labor quality variables were included in the model, and the industry characteristics were excluded, because this seemed more in keeping with the nature of the model.

Excess mining activity (MINING) and regional importance (REGION) were significant as we shall see. The hypotheses were OQI aMINING < 0, and Q/laREGION > 0. That is, it was thought that mining within a metropolitan area would raise real production costs because of the pollution associated with it. The entrepot and distribution sector activities con- ducted in regional centers, on the other hand, would provide positive externalities which would lower the cost curves of firms doing business in such cities.

On the matter of city size, various two- and three-way demarkations between cities of different sizes were tried, using a size dummy with a value of e for the particular size category into which the city fell and a value of 1 otherwise.

C. The Findings Several versions of equations (1) and (4)

were estimated using ordinary least squares on their logarithmic form. The equations differ from one another mainly as to the number of labor quality variables included. The results are presented in table 1.

First, the findings point to constant returns to scale across the entire sample. The coefficients of capital, labor, and the labor/labor-




quality interaction terms7 in column (1) sum to 0.991, which turns out not to be significantly different from unity, at the 5 % level. The notion of constant returns to scale is supported by the fact that the coefficient of labor is not significantly different from zero in columns (2) through (4).

Second, the coefficient for size is positive and significant in all equations in which cities are grouped into two categories, for a range of partitionings of the sample beginning at SMSAs of 1.9 million and continuing to SMSAs of 3 million. The best demarkation in terms of the significance and contribution- to explanatory power of the SIZE variable is at SMSAs of 2 million population. This division, which is the one reported in table 1, puts eleven SMSAs in the large-city category. The coefficient for SIZE is about 0.08 at this partition- ing, and slightly lower for the others. Several three-way groupings were examined, using two

size dummies. When one of the divisions was put at cities of 2 million population or greater, the other size dummy- no matter where it was put -proved not to be significant. De- creasing the number of SMSAs in the largest size category (and hence increasing average SMSA size in this category) failed to render significant any of a set of two-way size distinctions that were attempted for the remaining SMSAs. Some speculation on the effects of size on output throughout the entire range of U.S. SMSAs is offered in the concluding section.

The equations illustrate the impact of labor quality variables on output per worker. FEMPCT is significant and has the expected sign in all equations. The same is true for EDUCATION in columns (1) and (2). WHITEPCT, in columns (3) and (4), has the expected sign, but the coefficient size is no greater than its standard error. AGE, in the last column, has a negative sign and is not significant.8 Other measures of age reflecting the U-shaped effect of age distribution of employment proved no more successful. AGE is slightly correlated with EDUCATION, due probably to relatively high migration of younger groups.

The coefficients of REGION and MINING both had the hypothesized signs and significance, although the impact and significance of REGION were somewhat less than presup- posed. Several variables reflecting climate were included in estimates of equation (4) but had little impact. The underlying hypothesis was that, when climate is good, less brick and mortar and other factors are needed to produce a given level of output, and factor produc- tivities are enhanced. The failure to gain significance on the weather variables might in part be explained by the fact that some of the South/non-South differences in output, whether measured or real, were reduced when the- out-

TABLE 1. - PRODUCTION FUNCTION ESTIMATION RESULTS

(1) (2) (3) (4) Dependent Variable Q QIL QIL Q/L

Independent Vaciable

CONSTANT 7.933 7.933 7.777 7.751 (0.243) (0.243) (0.284) (0.290)

K .116 (.026)

L .891 .0077 .023 .043 (.058) (.047) (.049) (.068)

L* EDUCATION .080 .080 .048 .047 (.032) (.032) (.044) (.044)

L* FEMPCT -.096 -.096 -.096 -.099 (.025) (.025) (.025) (.026)

L* A GEMEDIAN -.013 (.029)

L* WHITEPCT .022 .019 (.020) (.021)

SIZE .078 .078 .082 .081 (.025) (.025) (.025) (.026)

K/L .116 .128 .131 (.026) (.028) (.029)

REGION .0036 .0036 .0046 .0045 (.0022) (.0022) (.0025) (.0025)

MINING -.033 -.033 -.034 -.034 (.008) (.008) (.008) (.008)

R2 .998 .665 .666 .661 S.S.E. .043 .043 .043 .044

Note: Q, Q/L, K, L, K/L, SIZE and the CONSTANT are all estimated in logarithmic form. Numbers in parentheses are estimated standard errors.

7 Each of the individual labor quality variables is normalized around unity.

8 Griliches (1967, pp. 299-300), achieved generally similar results with the same labor quality variables. In the Griliches study -an estimation of production functions in U.S. manufacturing -the age variable was significant only as long as occupational differences were identified and con- trolled for. The loss of the significance of age once these distinctions were dropped is attributed to "differences in the age of establishments or the age of (more finely defined) industries," which become blurred at more aggregate levels of analysis.




put data were normalized. That is, in normalizing the data we might have taken out some of the impact of climate.

We also attempted a variable to reflect average vintage of the capital stock. The underlying notion was that perhaps newer cities might reveal a more productive capital stock. This variable proved not to be significant, perhaps because, when it came to measuring capital stock, we had once again corrected for the influence in question, namely vintage.

We have thus far asserted with some evidence that constant returns to scale obtain across the sample size and that only the constant term differs between small and large cities. We shall try now to strengthen this as- sertion by ruling out alternative inferences from the data.

We began with the question of whether in fact there may be but one production function, with the same constant term, for the entire sample of cities. We were led to reject this because a t-test, applied to the coefficient of the size dummy, showed that the constant term was significantly higher for larger cities.

But if the constant terms differ, might not the production functions differ as well for the smaller and larger cities? We estimated the production functions separately for several sets of two subsamples of cities (excluding SIZE from each), and applied an F-test to the residuals. The purpose of the test was to see whether at standard significance levels we could reject the hypothesis that the production functions for any given pair of subsamples were in fact the same. Table 2 shows the outcome of the application of an F-test to the residuals for a range of sample splits that includes the 47 smaller/l1 larger division reported in table 1.'

As the critical value of F at the 5 % significance level is 2.31 (see footnote 9), we see from table 2 that at this level we cannot reject the hypothesis that the production functions are the same for a substantial range of city-size breaks.

Finally, even if we are prepared to accept that the production functions are the same in the two subsamples and that only the constant terms differ, we still require assurances that the production function (s) exhibits constant returns to scale. As evidence on this point we refer to the labor term in columns (2) through (4) of table 1. The coefficient of this term is /30qiO (i.e., qio = 1) in equation (4). For the coefficient not to be significantly different from zero suggests constant returns to scale.

A simultaneity problem is frequently en- countered in estimating production functions with cross-sections data - across firms or farms. The essence of the problem is that firms or farms have differing managerial endowments and failure to measure and correct for this can easily lead to spurious constant returns to scale (see Hoch (1958) and Massell (1967)). This has been referred to as the ''management bias" problem.

A problem emerges in our study if there is a failure correctly to identify, measure, and in-

TABLE 2. - F-TEST ON THE RESIDUALS AT DIFFERENT

SAMPLE PARTITIONINGS

Number of Cities in Number of Cities in Smaller Size Category Larger Size Category F-Statistic

51 7 a 49 9 2.13 47 1 1 2.20 45 13 1.07 43 15 0.51 41 17 0.54

a Cannot be computed because there are insufficient degrees of free- dom for the regression using the data of the larger cities.

9 If we look at the division between the 47 smaller SMSAs and the 11 larger ones, the F-test requires that we compare the value of

L SSR58 - (SSR11 + SSR47) (n-2k2 - 1

(SSR11 + SSR47) j k. J

with some critical value of F, where SSR58 is the regression sum of squared residuals when the estimation is across all 58 observations, SSR11 is the regression sum of squared residuals when the equation is estimated only on the largest 11 sample observations, and so on; n is the number of observations in the entire sample; k1 is the number of

restrictions or independent variables in the model; and k2 (= k, + 2) augments the number of independent variables by two, recalling that dummy variables for size and region were used in order to normalize the output and capital stock data.

We found SSR8=.09434; SSR1 = .00659; SSR47= .06478; n = 58; k1 = 6; and k2 = 8. The six independent variables or restrictions included in k1 are L, K, L* EDU- CATION, L* FEMPCT, REGION and MINING. The F value that is generated from these data is 2.20, which is lower than the critical value for F at the 5%o significance level, 2.3 1.




clude all of the variables differentiating cities in the production function equation -as part of the constant term. At one extreme it could be argued that all cities differ from one another in terms of natural endowments, climate, and so on, and that there are as many components to the vector C as there are cities. If this were the case then a time series of cross-section data sets might be the appropriate remedy.

Our study is located at the opposite extreme. The model is highly aggregative in nature; that is, we emphasize what SMSAs have in common rather than what differentiates them. We believe that including variables such as SIZE, MIN- ING and REGION helps reduce simultaneous equations bias.

Finally, let us turn to the question of the marginal productivity hypothesis and whether the production function and data of this paper support it. The hypothesis was described in equation (5) and the text following it. The technique that we used was to compare (i) the error sum of squares, when the production function parameter estimates from equation (1) were used to predict earnings, and when the predicted earnings were then compared with actual earnings, with (ii) the residual sum of squares, when actual earnings were regressed on the production function variables, as in equation (5). We found that there was not a significant difference in the sum of squared error using the two approaches. As a result we are unable to reject the hypothesis, at standard significance levels, that the marginal productivity hypothesis, as stated in equation (5), is valid."0

IV. Summary and Conclusions

Are there economies of scale in city size? On a priori grounds we suggested there might not be, and the empirical work presented here seems to bear this out. Rather, constant returns to scale obtain across cities of different size; and an "agglomeration effect," imbedded in the constant term of the production function for the largest cities, makes units of labor and capital 8% more productive in these cities. The presumed reason for this is that there are pro- nounced net benefits of agglomeration in production in large metropolitan areas -areas with over two million population in 1967.

The other principal finding is that we have empirically confirmed to a limited degree a marginal productivity theory of income distribution for metropolitan areas. Our production model appears to be a reasonably good predic- tor of aggregate wages and salaries in different SMSAs. The model itself shows that output per employee in an SMSA depends upon the capital/labor ratio, labor quality, the importance of the area to the regional ecoiLomy, and how much, if any, mining takes place.

The empirical work here, as noted earlier, is concentrated at the large end of the city-size spectrum. We are comparing SMSAs in the size category of 2 X (105-to-106) with those in the 2 X (106-to-107) range. Hoch has argued that there is a linear relationship between the logarithm of city size and the logarithm of money income, corrected for prices. If so, this might suggest that areas in the 2 X (105-to- 106) category are 8% more productive than those in the 2 X (104-to-105) interval, and so on. Without the data we cannot confirm this.

Finally, a comment is in order about whether or not the inter-urban labor market is in equilibrium. This would be the case only if the long-run labor supply curve passes through points such as r and t in figure 1. That is, if workers require an earnings increase of exactly 8% to offset greater commuter costs, pollution and the like, before migrating from an SMSA of 500,000 to one of 5 million population, then the inter-urban labor market is in equilibrium even though the larger SMSAs offer higher earnings.

10 Once again an F-test can be applied. The coefficients of column (1) in table 1 were used in equation (5) to estimate w*Li for each SMSA, i. These estimates were then subtracted from the actual values of w*L* and the differences were squared and summed. This error sum of squares we shall call SSR1. In contrast, we estimated the regression of equation (5), by regressing earnings on the variables of the right-hand side. The regression residuals, squared and summed, we refer to as SSR2. The essential question, then, is whether the sample variance using the first approach, SSR1, is significantly different (larger) from the sample variance using the second, SSR2.

We evaluated [(SSR1 - SSR2)/SSR,] ((n - k -1) K), finding it to equal 1.1375. The critical value of Fn_ k-i k

at the 5%o significance level is 1.60, so we cannot reject the hypothesis that the sample variance is the same.




APPENDIX A 58 SMSAs INCLUDED IN SAMPLE

(listed from smallest population to largest) AND THEIR CAPITAL STOCKS

SMSA 1967 Capital Stock ($1 million)a SMSA 1967 Capital Stock ($1 million)

Erie, Pa. 2,150.0 Phoenix, Ariz. 9,607.0 Reading, Pa. 1,677.8 Portland, Oreg.-Wash. 11,307.8 Chattanooga, Tenn.-Ga. 3,660.1 Tampa-St. Petersburg, Fla. 11,061.7 Lancaster, Pa. 1,955.2 New Orleans, La. 13,430.0 Utica-Rome, N.Y. 1,682.8 San Jose, Calif. 14,476.0 Peoria, Ill. 2,890.7 Indianapolis, Ind. 10,324.2 Wilkes-Barre-Hazleton, Pa. 2,094.9 San Bernardino, Riverside- Davenport-Rock Island-Moline, Ontario, Calif. 13,687.6

Iowa-Ill. 3,335.1 Denver, Colo. 14,360.6 Canton, Ohio 2,515.5 Kansas City, Mo.-Kans. 13,866.7 Wichita, Kans. 3,730.3 Buffalo, N.Y. 10,480.2 Tulsa, Okla. 3,756.5 Cincinnati, Ohio-Ky.-Ind. 13,470.3 Flint, Mich. 6,205.8 Atlanta, Ga. 19,326.7 Richmond, Va. 7,311.8 Milwaukee, Wis. 14,961.6 Springfield-Chicopee-Holyoke, Seattle-Everett, Wash. 21,316.7

Mass.-Conn. 4,466.0 Dallas, Tex. 21,274.1 Youngstown-Warren, Ohio 4,867.1 Minneapolis-St. Paul, Minn. 20,769.6 Grand Rapids, Mich. 4,602.0 Houston, Tex. 28,832.1 Nashville, Davidson, Tenn. 6,716.0 Cleveland, Ohio 23,045.9 Allentown-Bethlehem-Easton, Baltimore, Md. 21,168.0

Pa.-N.J. 4,143.3 St. Louis, Mo.-Ill. 19,677.8 Syracuse, N.Y. 4,911.1 Pittsburgh, Pa. 18,020.1 Akron, Ohio 7,145.6 Boston, Mass. 32,184.6 Toledo, Ohio-Mich. 5,792.1 Washington, D.C.-Md.-Va. 36,506.5 Albany-Schenectady-Troy, N.Y. 4,980.3 San Francisco-Oakland, Calif. 42,883.6 Birmingham, Ala. 7,444.6 Detroit, Mich. 45,217.7 Fort Worth, Tex. 8,326.9 Philadelphia, Pa.-N.J. 38,148.6 Memphis, Tenn.-Ark. 8,815.3 Chicago, Ill. 82,309.7 Louisville, Ky.-Ind. 8,833.0 Los Angeles-Long Beach, Calif. 123,160.6 Dayton, Ohio 8,064.6 New York, N.Y. 130,776.5 Rochester, N.Y. 8,815.3 Mean for 58 SMSAs 17,708.2 Columbus, Ohio 10,532.7

a All capital stock data are deflated for inter-city differences as reported in the text and in appendix B.

APPENDIX B SOURCES OF DATA

Measurement, Name of Variable Units, Year Sources Remarks

Unadjusted Total 1967 personal Survey of Current Interest, dividends, and transfer payments Output (QR) income less transfer Business, May 1968 were excluded because there was little a priori

payments less interest expectation that this kind of income originated and dividends plus con- in the SMSA under observation. An imputed tributions for social in- value for housing services was included using surance for each SMSA procedures and assumptions identical to those

used by the Bureau of Economic Analysis, De- partment of -Commerce, in calculating national income.

Output 1966 base year for Monthly Labor Review, The method used to construct the deflator is Deflator (f) cost-of-living data; April 1969, table 3 described in the text.

1970 data on SMSA (U.S. Department of population Labor, Bureau of La-

bor Statistics) ; 1970 Census of Population




APPENDIX B (continued) Measurement,

Name of Variable Units, Year Sources Remarks

Unadjusted 1967 Various -see below The capital stock data for each SMSA were Capital an amalgam of data on the capital stocks of Stock (KR) five different sectors, listed below. Airport data

were unavailable at time of writing. The stock data were inferred from investment flow data as described in the text. The data for inter- censal years were interpolated, using SMSA percentages of national data for these years; in several instances a similar procedure was used in extrapolating to earlier years before area data were compiled. In all instances land data were excluded.

1. Local 1967 Census of Governments Data were capital outlays for schools, roads, Government public utilities, etc.

2. Housing 1967 Census of Housing In the case of housing, and of the local government and private non-manufacturing sectors as well, the stock data were increased by a percentage ranging from 20% to 35% so as to in- corporate the value of plant and equipment, and furniture and fixtures.

3. Manufacturing 1967 Census of Manufac- tures

4. Private Non- 1967 Ccnstruction Reports Manufacturing structures were excluded from manufacturing construction data because they were gathered,

as noted from the Census of Manufactures. Data from Construction Reports, which date back to World War I, are based on the value of construction permits issued. To compensate for undervaluation in the reported data, residential and non-residential figures were increased by the ratio of value of "construction put in place" to the value of "building permit authorization." These data appear in Construc- tion Reports for the nation as a whole. For residences the ratio has recently averaged 1.25; for non-residences, 1.43.

5. State Roads 1967 State Highway Expen- (portion ditures Within -Coun- within ties Comprising SMSAs) SMSAs: 1960-67 (pub-

lished by the Bureau of Public Roads).

Capital Stock 1967 "United States Con- New York = 100 was used as the numeraire. deflator (g) struction Costs," Dodge The method used to construct the deflator is

Building Cost Calcu- described in the text. lator and Valuation Guide (Oct. 1973)

Employment (L) Total non agricul- Employment and Earn- and tural employment and ings: States and Areas, Earnings (wL) earnings for SMSAs for 1939-71 (U.S. Depart-

1967. ment of Labor, Bureau of Labor Statistics, 1972).

Indicators of Labor 1970 Census of Population, Four different measures of labor quality for Quality (q) 1970 employment in each SMSA were used in the

study: median years of education (ED UCA- TION), percentage of those employed who were women (FEMPCT), percentage of those employed who were white (WHITEPCT), and an age variable (AGE) which was represented, alternately, as median age of the employed population and percentage of total employment which was in the 25-44 age br.acket.




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Are There Returns to Scale in City Size?