Estimation and recent behavior of urban population and employment density gradients

JOURNAL

OF URBAN

ECONOMICS

l&251-260

(1985)

Estimation and Recent Behavior of Urban Population and Employment Density Gradients’ MOLLY K. MACAULEY Resources for the Future, I61 6 P Street, N. W., Washington. D. C. 20036 Received August 16,1983; revised December 28,1983 The monoccntric model of urban structure predicts that urban population density declines with distance from the central business district. Using the negative exponential function to approximate the decline, Mills (E. S. Mills, “Studies in the Structure of the Urban Economy,” Johns Hopkins Press, Baltimore, Md. (1972)) estimated population and employment density gradients from 1948 to 1963 for a sample of 18 SMSAs. This paper updates Mills’ estimates and examines recent patterns in population and employment suburbanization. The updated series estimated here is obtained using a “corrected” version of Mills’ method. The original procedure incorporated a bias which Mills noted and later corrected (E. Mills and K. Ohta, in “Asia’s New Giant” (Patrick and Rosovsky, Eds.), The Brookings Institution, Washington, D.C. (1976)). The comparability of the series begun by h4ills and extended here is not interrupted, however, because, as Mills suspected, the bias is indeed small prmided SMSA data are used. On the other hand, Urbanized Area definitions of metropolitan areas cause the original and corrected versions to yield significantly different results. This finding has implications for the appropriate choice of data for urban studies. o 1985 Academic PWS. 1~.

INTRODUCTION The monocentric model of urban structure predicts that urban population density declines with distance from the central business district. Using the negative exponential function to approximate the decline, Mills [7] estimated population and employment density gradients from 1948 to 1963 for a sample of 18 SMSAs. The gradients Mills obtained, and gradients calculated by the method he proposed, have provided empirical evidence for the discussion of topics such as the suburbanization over time of people and jobs (71, the location of manufacturing employment [5], and wasteful commuting [4]. This paper updates Mills’ estimates and examines recent patterns in population and employment suburbanization. Mills’ “two-point” technique for calculating the gradients has also contributed to debate on the theoretical appropriateness and empirical performance of the negative exponential function, in general, and two-point estimates of it, in particular (see, e.g., [2-4, 6-8, lo]). The crux of the “pro” argument for the use of two-point measures is summarized by Mills and ISpecial thanks to Bruce Hamilton, innovative insights.

Jim Stokes, and the referee for careful critiques and 251 00961190/85

$3.00

Copyright 0 1985 by Academic Rcss, Inc. All rigbu of reproduction in any form rcxrvcd

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Ohta [8] and Hamilton [4], who remark that two-point estimates are theoretically appealing in constraining the integrated gradients to predict population and employment correctly.* McDonald and Bowman [6], however, obtain reasonable estimates by imposing this restriction in the form of a loss function in multiple regression, although Frankena [2] observes(1) an inherent upward bias in their OLS estimates, a bias imparted by a nonrandom pattern in census tract data, and (2) that under certain functional specifications, estimates do not pass the integration test. Meanwhile, White [lo] performs a Monte Carlo experiment to compare two-point, nonlinear, and least-squares estimates given hypothesized error structures and concludes that two-point values are fairly good. Ceteris paribus, then, and as economists, perhaps a vote should be cast for the two-point schemeby yet another criterion-true to its nickname, the method is parsimonious in its data requirements. In fact, the technique is a convenient, if not sole resort, for estimating employment densities, becausein this case the census tract or alternative data required for regression are generally unavailable. Of course, that the negative exponential function aptly illustrates employment densities is moot; for opposing arguments, see[4] and [5]. The updated series estimated in this paper is obtained using a “corrected” version of Mills’ method. The original procedure incorporated a bias which Mills noted, many discussed (e.g., [S] and [lo]), and which was later corrected in [8]. The comparability of the series begun by Mills and extended here is not interrupted, however, because, as Mills suspected, the bias is indeed small provided SMSA data are used. On the other hand, Urbanized Area definitions of metropolitan areas cause the original and corrected versions to yield significantly different results. This finding has implications for the appropriate choice of data for urban studies. METHODOLOGY The technique suggested by Mills and Ohta [8] is used to estimate the parameters y and D, in (1).3 D( 24)= Doewyu

(1)

*In fact, population density gradients estimated by OLS generally do not ‘I integrate up” to total population over the appropriate geographic area. See, for instance, McDonald and Bowman’s and Hamilton’s calculations regarding this shortcoming of gradients for Muth’s series [9]. 3The only other differences in methodology are (a) Mills interpolated census population data, collected decennially, to make them correspond with employment data, collected every 5 years. I did not interpolate the data, but the relative magnitudes of gradients across population and employment sectors over time, discussed in the text, are inferable if the gradients flatten monotonically-likely to be true in most cases here. (b) Central city and SMSA land areas were obtained from published and unpublished Bureau of Census data for years commensurate with the date of the censuses; Mills scaled all census data to boundaries existing in 1950.

DENSITY GRADIENTS

253

where D,, is population or employment density at the center of the city, D is density at distance u from the center, and y, which is positive, is the rate at which D declines with distance. Mills’ 1972 method applied NewtonRaphson to (2), obtained by integrating (1) over a circular central city of radius K and net of a wedge of 27r - I#Iradians:

N(k) = (PD, y2 (1 -(l + yk)ePyk). Mills set +Do/y2, obtained by integrating (2) to infinity, equal to N, total SMSA population (or sectoral employment), hence requiring that density fall asymptotically to zero as the edge of the SMSA extends to infinity. However, to the extent that both urban outskirts and rural areas have a finite density, Mills noted that a too small value substituted for @Do/y2 biases estimates of y (and therefore Da) upward.4 Mills and Ohta avoid this assumption by using Newton-Raphson to estimate y and D,, from (3), the ratio of central city to SMSA (radius k) population: N(E) _ (1 --(I + y~)ee7') N(k) (1 -(l + yk)ewyk ’ The bias in (2) has been the subject of discussion by Kemper and Schmenner [5] and White [lo], although Mills, in 1972, conjectured that the bias would be small. He was correct, provided SMSA central-city and suburban data are used. If Urbanized Area (UA) Census Bureau definitions are used, however, estimates of gradients indicate a sign&ant and large bias. Table 1 lists parameters obtained in two tests for the magnitude of the bias, first using the Mills and Mills/Ohta methods for SMSA population densities in 1940 and 1980, and then using these methods for UA population densities in 1980. Since the original Mills’ method assumesthat urban population is zero as the edge of the metropolitan area extends to infinity, the 1940 and 1980 data were used to investigate whether the bias increases over time as populations suburbanize, rendering the assumption more critical. SMSA and UA data usually differ significantly in the statistical portrait they describe; as to which measure of a metropolitan area is appropriate, the UA definition better reflects what the urban economist has in mind in depicting models of the city. Unfortunately, however, UA employment data are not generally available, precluding tracing the movement of people and jobs. 4See [5] for an algebraic proof that the direction of the bias is upward.

254

MOLLY

K. MACAULEY TABLE

1

Population Density Parameters 1940 SMSAs Mills/Ohta

Milk

Y Albuquerque Baltimore Boston Canton Columbus Denver Houston Milwaukee Philadelphia Phoenix Pittsburgh Rochester Sacramento San Antonio San Diego Toledo Tulsa Wichita 7

0.90 0.55 0.30 0.74 0.83 0.68 0.54 0.53 0.32 0.69 0.28 0.80 0.96 0.79 0.31 0.64 1.01 1.16 0.67

1980 SMSAs

Do

Y

Do

8984 0.90 8986 58125 0.55 58688 36582 0.28 35518 20595 0.74 20606 42869 0.77 38532 30217 0.68 14034 24161 0.54 24164 67351 0.51 65455 51555 0.32 51579 14041 0.69 14034 26501 0.28 26501 44898 0.80 44941 27640 0.96 27637 33864 0.79 33896 9108 0.31 9113 44859 0.64 44756 31107 1.00 31091 30568 1.16 30546 0.66

Mills

1980 UAs

Mills/Ohta

MilIs/Oht;

Milk

Y

Do

Y

Do

Y

DO

Y

Do

0.47 0.24 0.20 0.37 0.23 0.21 0.14 0.20 0.19 0.17 0.19 0.29 0.18 0.29 0.11 0.29 0.23 0.41 0.25

16003 21849 21858 8690 9043 11338 8845 17337 27249 7246 12722 13038 5560 13938 7285 10734 5698 11068

0.47 0.23 0.19 0.36 0.23 0.21 0.14 0.20 0.19 0.17 0.19 0.29 0.18 0.29 0.11 0.29 0.23 0.41 0.24

16005 21708 21212 8634 8935 11325 8732 17074 27041 7236 12630 13036 5519 13873 7139 10700 5691 11066

0.54 0.28 0.21 0.54 0.31 0.22 0.17 0.23 0.21 0.19 0.22 0.42 0.21 0.35 0.12 0.50 0.40 0.71 0.32

19162 25223 22104 11182 12552 10048 11149 19714 29750 7770 13778 16710 6279 18868 7399 19166 11378 24819

0.36 0.24 0.19 0.35 0.10 0.10 0.07 0.22 0.19 0.04 0.18 0.33 0.003 0.14 0.08 0.34 0.19 0.33 0.19

1179 2245 20941 845 505 670 541: 19191 26771 326: 12511 1405: 293( 667: 562: 12431 47% 873’

The results in Table 1 indicate these conclusions: While the Mills y’s are equal to or larger than Mills/Ohta y’s for commensurate years, the average y’s are insignifkantly different for the SMSA data. Despite well-documented suburbanization of these cities, the 1940 and 1980 estimates also suggest that the bias has not necessarily increased over time. An explanation for the small bias is included in the Appendix; essentially, population density at an infinitely distant edge of the area is empirically small--N is not a bad statistic to estimate y. The results when UA data are used imply quite a different conclusion, however. The original Mills’ procedure predicts an average y 68% larger than the Mills/Ohta method. An SMSA is defined primarily by population and commuting pattern criteria relative to central city limits, and SMSA suburbs include only whole countries. Both definitional constraints render the SMSA generally more geographically extensive than an UA and, in particular, extend the SMSA to relatively low-density outskirts. An UA, however, usually requires contiguous territory of a certain population density and ignores county boundaries. The sample mean 1980 UA land area is 462 square miles and the radius is 13.2 miles; for the sample mean 1980

DENSITY GRADIENTS

255

SMSA, the land area is 3539 square miles and the radius is 33.9 miles. The discrepancy between the parameters estimated by the different data, then, is likely attributable to a large difference in the tails of the respective gradients.5 The Appendix discussesthe discrepancy further. Updated Estimates The estimates of the density gradients for 1970 and 1980 for population and 1972 and 1977 for employment sectors are listed in Table 2, organized to correspond to Mills’ Table 11, pp. 40-41.6 Several patterns which Mills observed among gradients over time and acrosssectors continue: 1. The flattening of the gradients indicates continuing decentralization of urban areas, although there are 10 instances of steeper employment gradients. Seven of these gradients are for Albuquerque and Wichita, which were also among Mills’ incidences of steeper gradients. 2. The tendency for the rate of decentralization to slow, hinted at toward the end of Mills’ series, is more prevalent in the recent data. The unweighted average population gradient, listed in Table 3, flattened by 0.013 points per year during 1963 to 1970. This rate was also the 1954-1958 rate. However, the rate slowed noticeably to 0.005 points per year in the most recent decade. It is interesting to note, from an even longer time perspective, that this rate is the same as that for a six-SMSA sample for which Mills constructed population density gradients back to 1920; these estimates and their updates are presented in Table 4. In this long time series, the 1970-1980 rate is the slowest rate exhibited-it is even slower than that during the heretofore slowest period of decentralization, the 193Os,when the average gradient fell 0.007 points per year. Slower decentralization is consistent to some extent with Mills’ regression estimates of the determinants of density functions. He demonstrated that income and population growth, rather than the passageof time, explained the flattening of the gradients. Patterns among the updated gradients also indicate regional variation, perhaps attributable to the major central-city population loss in the Northeast in the 197Os, and, perhaps, a role for additional explanatory variables such as demographics, the energy crisis, slowing urbanization, and a preference for housing renovation in central cities. 51 thank the referee for noting that recent changesin SMSA definitions could affect the size of the discrepancy calculated for post-1980 SMSA data. The Census Bureau, in June 1983, revised SMSA geographic boundaries, in part specifically to exclude extremely low-density counties (see [l, p. 3421).Excluding low-density counties, particularly if they have large land areas, would tend to increase the estimation bias for SMSA density gradients. %ee footnote 3(a) regarding the noncommensurate years for population and employment estimates.

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MOLLY

K. MACAULEY TABLE

2

Estimates of Density Gradient Parameters Population 1970 Albuquerque Baltimore Boston Canton Columbus Denver Houston Milwaukee Philadelphia Phoenix Pittsburgh Rochester Sacramento San Antonio San Diego Toledo Tulsa Wichita

0.51 13797 0.28 29510 0.20 24930 0.41 10838 ‘D, 0.27 Y 12062 Do 0.29 Y 16331 Do 0.19 10862 LO 0.22 Y Do 21326 0.22 Y Do 36004 0.23 Y 8186 Do 0.21 Y Do 16344 0.33 Y Do 16495 0.20 Y 5576 Do 0.39 Y DO 21194 0.12 Y 6175 Do 0.33 Y Do 13238 0.27 Y 6379 Do 0.48 Y Do 14279

Y Do Y Do Y Do

Manufacturing

Retailing

Wholesaling

!ktViCeS

1980

1972

1977

1972

1977

1972

1977

1972

1977

0.47 16005 0.23 21708 0.19 21212 0.36 8634 0.23 8935 0.21 11325 0.14 8732 0.19 17074 0.19 27041 0.17 7236 0.19 12630 0.29 13036 0.17 5519 0.29 13873 0.11 7139 0.29 10700 0.23 5691 0.41 11066

0.59 656 0.32 3354 0.20 2297 0.60 3336 0.31 1605 0.29 1250 0.19 965 0.23 3282 0.22 3794 0.29 953 0.23 2028 0.63 9006 0.31 371 0.41 919 0.17 589 0.40 2371 0.38 1024 0.39 925

0.59 816 0.28 2294 0.28 4006 0.61 3420 0.25 989 0.21 777 0.20 1280 0.17 2196 0.19 2484 0.26 885 0.22 1781 0.61 8604 0.26 302 0.34 722 0.15 604 0.38 2159 0.34 1080 0.34 912

0.81 0.75 2320 2658 0.28 0.22 1684 1145 0.24 0.22 2270 1923 0.54 0.47 1044 873 0.32 0.28 1057 927 0.32 0.21 1368 754 0.22 0.20 935 1139 0.28 0.18 1628 1126 0.20 0.17 1667 1298 0.23 0.20 559 585 0.25 0.23 1209 1127 0.36 0.28 1189 823 0.23 0.20 417 446 0.41 0.35 1401 1255 0.12 0.12 377 446 0.38 0.34 1067 849 0.38 0.36 711 810 0.66 0.59 1573 1638

0.39 417 0.35 1410 0.38 3084 0.64 413 0.40 894 0.42 1365 0.26 939 0.28 1010 0.28 1618 0.29 532 0.39 1358 0.59 1183 0.31 328 0.48 934 0.18 432 0.53 736 0.53 717 0.73 889

0.43 575 0.30 1067 0.35 2832 0.63 452 0.32 627 0.31 941 0.23 1149 0.25 918 0.21 1133 0.26 537 0.34 1273 0.58 1256 0.29 366 0.42 836 0.17 513 0.46 678 0.53 892 0.72 1081

0.74 651 0.40 1112 0.29 1252 0.58 343 0.36 469 0.50 1429 0.31 984 0.23 497 0.27 1174 0.34 410 0.35 822 0.55 757 0.39 347 0.54 850 0.17 140 0.50 584 0.45 415 0.74 795

0.65 639 0.31 657 0.25 867 0.58 365 0.31 374 0.34 836 0.29 1044 0.19 419 0.21 698 0.30 419 0.32 754 0.47 551 0.36 359 0.45 666 0.17 187 0.41 414 0.48 625 0.59 640

3. From Table 3 can also be calculated the average rate of flattening of the gradients in each employment sector. In the most recent period, 1972-1977, this rate ranges from 30 to 60% of the 1958-1963 rate. Mills noted a rapid convergence of the average manufacturing gradient toward the mean population gradient. In the 197Os,manufacturing con-

257

DENSITY GRADIENTS TABLE 3 Averages of Gradients by Sector and Year” Sector

1948

1954

1958

1963

1910/1972b

Population Manufacturing Retailing services Wholesaling

0.58 0.68 0.88 0.97

0.47 0.55 0.75 0.81 0.86

0.42 0.48 0.59 0.66 0.70

0.38 0.42

0.29 0.34 0.35 0.41 0.43

1.00

0.44 0.53 0.56

1977/1980' 0.24 0.32 0.30 0.38 0.37

“1948-1963 data: [7, Table 12, p. 421. bNoncommensurate years (see text).

tinued to decentralize but at a rate closely matching population suburbanization. These rates slow during the decade and are outpaced by more rapidly decentralizing retaining and wholesaling. 4. Mills also observed that the average gradients converge both absolutely and relatively across population and job sectors. The updated averages imply that the convergence has continued. According to Mills’ absolute measure, which is the difference between the largest and smallest y’s, the columns of Table 3 indicate that the difference across sectors by the end of his series was at a minimum, 0.18; the updated estimates show that by 1977/1980, the difference is 0.14. The difference is likely underestimated and overestimated for 1970/1972 and 1980/1977, respectively, because of the noncontemporaneous population and employment data. In any case,the pattern of convergence is largely due to the fast flattening of y’s which were

TABLE 4 Averages for Six-SMSA Set” Sector Population Manufacturing Retailing SlXViSXS

Wholesaling

1920 1929 1939 1948 1954 1958 1963 1970/1972’

1977/1980’

0.46 0.67 0.73 0.81 0.89

0.23 0.31 0.23 0.35 0.32

0.73

0.61

0.57

0.95 0.82 N/A 1.02 N/A N/A N/A 1.43

0.84

0.77 0.90 1.12 1.24

0.76 0.76 0.88 1.01

0.41

0.36

0.60 0.58 0.70 0.77

0.48 0.41 0.55 0.59

0.28 0.35 0.30

0.41 0.41

Note. N/A = not available. “See footnote to Table 3. ‘1920-1963 data: [7, Table 14, p. 461; also, see his notes, that Table.

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MOLLY K. MACAULEY

steep in 1948. Given the flat y’s of the recent periods, the difference in y’s relative to the largest y for the 1970s equals that for the early 1960s. 5. Mills did not note that a similar convergencecharacterizes gradients across SMSAs, again due to the rapid flattening of steep 1948 y ‘s. From Table 3 the absolute difference betwen high and low y’s across SMSAs in 1980 was about 50% less than in 1948. 6. By 1977, the ranking by degree of suburbanization among employment categories changed from the order which had lasted from 1948 to 1963. Retailing and service sectors were respectively most and least suburbanized in 1977, although the difference between the sectors was half its 1963 value. That this observation is probably least general, however, is suggested by the six-SMSA sample, in which employment sector rankings are inconsistent over time. There are some interesting updated estimates among individual SMSAs. Rapid decentralization-y flattened 46 to 67%-appears to have happened in Columbus, Sacramento, San Diego, and Toledo during the 1960s and 1970s. Suburban population increased in these areas by 32 to 67% and, even though in Columbus and Toledo, the population increase was associated with increases in suburban land area, the net effect most likely caused the two-point fitting of the gradient to rotate it downward. The population density gradient for Boston is also interesting. When the estimated parameters are used to predict the distance at which population density is 100 people per square mile, the predictions are consistent with census tract data in all casesexcept Boston.7 This density, roughly lo-acre lot size, occurs inside each SMSA boundary (although inside by as little as 4 miles in Philadelphia and by as much as 28 miles in Phoenix) except Boston’s where the point is 28.7 miles from the city center, or 7 miles outside the SMSA edge. Inspection of Boston census tract data for 1970 indicates that population density near the edge of the SMSA exceeds 100 per acre, but observers have remarked that lo-acre lots 7 miles outside the SMSA overestimate density-i.e., the estimated gradient is too flat.* One source of bias, if it exists, may be as follows. To the extent that land used at the city center is nonresidential, estimating urban population density gradients from the center to its edge-that is, from 0 to the edge-biases the gradient downward. Particularly in the case of Boston, the tract data indicate that this bias may be large. But sliding the center-city endpoint of Boston’s gradient out as far as lf miles, as suggested by the ‘This test of the reasonablenessof the estimates is used by [4]. sIf Boston’s gradient is biased, its ommission from the calculation of the 1970 and 1980 averages in Table 3 lowers them by 0.001 and 0.003, respectively.

DENSITY

GRADIENTS

259

census tract information, and holding all else constant, yields a y of 0.22 and lo-acre lot size at 24.7 miles, still 3 miles beyond the SMSA limit. That Boston violates the assumption of monocentricity may also account for the (literally) far out density prediction. In fact, Boston’s central-city definition is under review by the Census Bureau because of the number of smaller, “central-type” cities such as Cambridge, Lynn, and Waltham near the city of Boston.9 On the other hand, the transportation convenience of Rt. 128 may explain a flat gradient, as Hamilton notes. APPENDIX The difference between (1) integrated to intinity and integrated to the urban edge indicates the bias of the original Mills version:

Nb)

- N(k) = (1 + y,+,-?k N(a)

(IA)

which is a small number for typical values of y and k. This result suggests that the tail of the negative exponential distribution-that is, density as the city extends to infinity-is small. In fact, the parameters derived by the Mills/Ohta method predict that density at the SMSA edge in 1940 averaged eight people per square mile (27.8 if Boston is included-see text for discussion), or about 160-acre lot size. For 1980, the tail includes about 14 people per square mile (34.5 with Boston), or 92-acre lot size. Over time, y has flattened and k has increased. The former effect works to increase the disparity in (1A) and the latter to decrease it, ceteris paribus, because derivatives with respect to y and to k are negative and, for the sample data, y and k have respectively decreased and increased monotonically (except for Albuquerque, over the period 1948-1963, according to Mills’ estimates). The relative magnitude of the effect of proportional changes in y and k is such that increases in k alter (1A) more than a flatter y. But increases in k have been less than proportional to decreasesin y, on average, and the effects seem to wash. The above discussion applies when the two-point dichotomization is made on the basis of SMSA central-city and suburban area information. As discussed in the text, however, the bias can be large and significant when Urbanized Area census data are used. In this case, the mean y predicted by the Mills procedure is nearly 70% larger than the prediction by the MiIIs/Ohta method. Certain cities, namely Albuquerque, Columbus, Denver, Phoenix, Sacramento, and San Antonio, contribute to the large discrepancy in the mean y. For these cities, the bias in (1A) is pronounced, probably because these areas have an average ratio of central-city to urban %ee [l, pp. 343-3441.

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area land area of 0.5 and an average population ratio of 0.6-that is, the cities have a fairly even population density. In the remainder of the sample, the difference between SMSA and UA estimates is insignificant except to the extent estimates of Do are affected. For this set of cities, the corresponding land area and population ratios are 0.3 and 0.5, respectively. REFERENCES 1. Federal Committee on Standard Metropolitan Statistical Areas, “Documents Relating to the Metropolitan Statistical Area Classification for the 1980’s,” Washington, D.C., (August 1980), mimeo. 2. M. W. Frankena, A bias in estimating urban population density functions, J. Urban Econ., 5, 35-45 (1978). 3. D. L. Greene and J. Bambrock, A note on problems in estimating exponential urban density models, J. Urban Econ., 5, 285-290 (1978). 4. B. Hamilton, Wasteful commuting, J. Pol. Econ., 5, 1035-1053 (1982). 5. P. Kemper and R. Schmenner,The density gradient for manufacturing industry, J. Urban Econ., 1, 410-427 (1974). 6. J. F. McDonald and H. W. Bowman, Some tests of alternative urban population density function, J. Urban Econ., 3, 242-252 (1976). 7. E. S. Mills, “Studies in the Structure of the Urban Economy,” Johns Hopkins Press, Baltimore, Md. (1972). 8. E. Mills and K. Ohta, Urbanization and urban problems, in “Asia’s New Giant” (Patrick and Rosovsky, Eds.), The Brookings Institution, Washington, D.C. (1976). 9. R. Muth, “Cities and Housing,” Univ. of Chicago Press, Chicago (1969). 10. L. J. White, How good are two-point estimates of urban density gradients and central densities? J. Urban Econ., 4, 292-309 (1977).