Technological progress, capital deepening, and manufacturing productivity growth in the USA: a regional analysis by E Casetti Environment and Planning A, 1982, Vol. 14, Issue 12, pp. 1577-1585. Environment and Planning A is published by Pion Ltd, London, website: www. pion.co.uk. The permission by
[email protected] is gratefully acknowledged.
Environment and Planning A, 1982, volume 14, pages 1577-1585
Technological progress, capital deepening, and manufacturing productivity growth in the USA: a regional analysis+ E Casetti Department of Geography, Ohio State University, Columbus, OH 43210, USA Received 16 October 1981
Abstract. According to the Verdoorn law productivity rises faster in expanding economic sectors. The validity of the Verdoorn law has been investigated primarily for industries. Does the law also hold with respect to areal aggregates? Specifically, in the USA, are productivity gains larger in regions in which jobs and population expand faster? In this paper a technique for separating productivity gains associated with capital deepening, economies of scale, and neutral technological progress is discussed and is then applied to investigate, in a US setting, the 'spatial' validity of the Verdoorn law. Introduction Shifts of population and jobs out of the snow belt into the sun belt are currently in progress in the United States of America. The empirical regularities sometimes referred to as the Verdoorn law suggest that productivity tends to rise faster in economic sectors that are in the process of expanding. Does this law apply to the current regional shifts in the USA? Specifically, is manufacturing productivity rising faster in those regions which have a comparatively larger expansion of population and manufacturing production? Are the productivity changes associated with capital deepening, neutral technological progress, and economies of scale significantly different between those regions which are expanding and the rest of the country? The investigations presented here are intended as a contribution to the empirical study of these questions. Essentially, state-level data on the percentage rate of change of manufacturing labor productivity are decomposed into 'slices' related to neutral technological progress, capital deepening, and scale effects. Then the original rates and their components are related to population growth and to growth in manufacturing production. Regionally disaggregated data on capital inputs are hard to come by and are less reliable than data on value added, salaries, and number of employees. A technique that does not require capital data is discussed in the first portion of this paper. An analysis of empirical data follows. The model Denote regional manufacturing product by Y, and let L and K indicate, respectively, the labor and capital inputs that generated it. Assume a conventional Cobb-Douglas production function in logarithmic form: InY = H+alnL + blnK ,
(1)
where H is a technological 'level' and in which the parameters a and b are not restricted to add up to 1. Let Y, H, L, and K all be smooth functions of time t. t This research was supported by the National Science Foundation under grant SES 8006 563; and the paper was presented at the annual meeting of the Association of American Geographers to the Commission on Industrial Systems, Los Angeles, CA, USA, 1981.
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The logarithmic derivative of the superscripted variable with respect to time will be indicated by a double prime, so that, for instance,
(2)
r =^ Manufacturing labor productivity, y, is defined as the amount of the regional manufacturing product per unit of labor input: y = Y/L.
(3)
By the substitution of equation (1) into equation (3), and the taking of the logarithmic derivative with respect to time of the resulting equation, it can be shown that the percentage instantaneous rate of change of labor productivity over time, y", is given by the following equation: y"
= h + (a--l)L"+bK"
,
(4)
where dH
(5)
Let (6)
v=a + b.
In other words, i; is a 'scale-economy' parameter since, for instance, doubling the labor and capital inputs in equation (1) will respectively more than double, exactly double, or less than double the manufacturing output Y, depending on whether i; is greater than, equal to, or smaller than one. Indicate capital intensity by k, so that k, K k
= L>
™
is capital per unit of labor. Equations (4), (6), and (7) combine to yield equation (8): y»
=h + (v-l)L"
+ bk" ,
(8)
that splits the percentage rate of growth of manufacturing labor productivity y" into the three components h, (p— 1)L", and bk", associated respectively with neutral technological progress, scale economies, and capital deepening. When appropriate, these components will be denoted by P"1 for neutral technological progress, 5 60 for scale economies, and Kdp for capital deepening: namely, P n t = h, S*0 = (v— l)L", and Kdp = bk". One objective of this paper is to calculate the contributions of P n t , S*0, and Kdp to the change of manufacturing labor productivity for individual regions. A straightforward estimation of the parameters of equation (8) is not possible if capital data on a regional basis are not available. The techniques employed by Johansen (1961) and by Dixon and Thirlwall (1975, page 131 et seq) for estimating the parameters of Cobb-Douglas production functions in the absence of capital data require the assumption of constant returns to scale. A technique that does not require i; = 1 is discussed in the sections that follow. The assumptions on which the technique is based are: (a) that one is dealing with a cost minimizing economy, (b) that all product is allocated to the factors, and (c) that over a short to medium time horizon the average capital to output ratio (ACOR) is approximately time invariant. Assumption (c) implies that a region tends to retain the same capital to output ratio over a short to medium time horizon, but not that different regions possess similar ACORs.
Technological progress, capital deepening, and productivity growth
1579
The cost minimization assumption Whenever factors are combined so that costs are minimized the earnings of the factors are proportional to their marginal product. The assumption that economic units minimize costs constitutes the basis of the rich economic literature that uses factor shares to estimate the contribution of the change in amount and quality of inputs to the change in outputs. In his classical Why Growth Rates Differ Denison (1967, page 34) notes that "... the tendency toward proportionality of factor prices and marginal products under conditions of reasonably high employment is sufficiently strong in the United States and, though perhaps weaker, in Western Europe for distributive shares to provide an adequate basis for analysis of the relative contributions of the various factors to growth". It seems that in an analysis of the productivity growth in US manufacturing, such as is carried out here, it is even more reasonable to assume that economic units combine factors so as to minimize costs, since a drive toward efficiency can be presumed to operate in manufacturing perhaps more than in other economic sectors. Next is demonstrated the relation between labor and capital shares and the parameters of the Cobb-Douglas production function produced by the assumptions that the factors are combined so as to minimize costs, and that the product is allocated to the factors. Let the capital share and labor share of the manufacturing product be indicated respectively by Ksh and Lsh, where Ksh + Lsh = 1 .
(9)
By definition, rK Ksh=^F, Y
(10)
Lsh=~y,
(11)
where r and w are respectively the remuneration to a unit of capital and to a unit of labor. The values of K and L that minimize costs and produce an output of Y can be obtained by solving the following optimum problem: minimize wL + rK,
subject to
Y(L, K) = Y.
(12)
To this effect, first the Lagrangian function F is constructed F = wL + rK-p[Y(L,K)-
Y] ,
(13)
where p is a Lagrangian multiplier. Then the first-order partial derivatives of F with respect to L and K are obtained, set to zero, and solved for L and K to obtain: L = ^ ,
(14)
pbY K = -— . (15) r It can be shown that the L and K specified by equations (14) and (15) do indeed yield a constrained minimum to problem (12). Consequently, if it is assumed that costs are minimized and that the product Y is allocated to the factors, so that wL + rK=Y, then
(16)
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1580
and the labor and capital shares are Lsh=pa, sh
K =pb.
(18) (19)
The ACOR constancy assumption The differentiation of the variables in a dynamic system into 'fast' and 'slow' (Dendrinos and Mullally, 1981) constitutes a most important methodological tool, one recently placed into focus by catastrophe theory (Thorn, 1975; Zeeman, 1977, page 65; Poston and Stewart, 1978). Slow variables are variables that change little over a specified time horizon, as compared with the fast variables, so that as a first approximation they can be regarded as constants. Within catastrophe theory, this distinction allows an investigation of the qualitative properties of the fast dynamic of a system for alternative constant values of the slow variables. However, the fast/slow dichotomy justifies analysing the change of the fast variables under the assumption that the slow variables are constant in much wider methodological contexts than the one encompassing catastrophe theory. In this paper the average capital to output ratio will be assumed to be a slow variable. This implies that ACOR may be quite different in different regions, but tends to change over time, in each region, much less than the fast variables of the system. Arguments in support of this assumption will be reviewed in the paragraphs that follow. Empirical analyses based on country-level data show that ACO R changes very slowly over time (Hood and Scott, 1957; Allais, 1962, page 714 et seq; Klein, 1962, page 195; Kuznets, 1966, pages 76 et seq; Brems, 1968, page 459). However, there are also theoretical reasons suggesting that the change over time of ACOR can be expected to be small when compared to that of Y9 K, and y. ACOR will be indicated by X: K X=y.
(20)
The logarithmic derivative of equation (20) with respect to time gives X" = K"-Y" .
(21)
A high rate of economic growth would be associated with high rates of growth of capital, K"9 and of product, Y", that would tend to offset each other in equation (21) thus producing a much smaller rate of change of X. A low rate of economic growth would be associated with low values of K" and Y", that again would cancel each other in their effects on X". Consequently, X" would tend to be small both at high and at low rates of growth of the economy. In a Cobb-Douglas environment, equation (21) translates into the following: X" =ak"-h + (l-v)K" .
(22)
Equation (22) indicates that capital deepening, k"', and neutral technological progress, h, would tend to cancel each other out in their effects on the rate of change of X over time. Since capital deepening and neutral technological progress would tend to be either both high or both low, respectively, during high/low economic growth situations, they would consistently tend to balance each other off in their effects on X". The term (1 —v)K" in equation (22) identifies a scale-economy effect that adds to the capital deepening effect when diseconomies of scale prevail, and to the technological progress effect if economies of scale exist.
Technological progress, capital deepening, and productivity growth
1581
Dixon and Thirlwall (1975, page 141), in order to estimate the structural parameters of a Cobb-Douglas production function in the absence of capital data, assumed the time invariance of the rental per unit of capital. With the use of the symbols adopted here, their assumption translates to r" = 0. Since from equation (15) r = (pbY)/K, taking the logarithmic derivative of both sides with respect to time gives /•" = K" — Y". Therefore, assuming that r" = 0, as Dixon and Thirlwall do, is equivalent to assuming that X" = K" — Y" = 0, since both assumptions imply that K" = Y". Dixon and Thirl wall's reasoning is very similar to the one articulated here. They note regarding their assumption that r" = 0: "This is not to say that rentals are fixed; rather that the increase in rentals due to technical progress is offset by the decrease in rentals due to capital deepening" (page 141). Calculating the components of productivity change From the assumption that X" = 0, and after a few manipulations of equations (4), (19), and (21), one obtains the equation: ph
» y
~ Lsh + p - \
L " [LsbK\-p)]-l
+
'
(23)
that relates the percentage rate of change of labor productivity, y", to the percentage rate of change of labor inputs, L", and to labor shares, Lsh. Equation (23) in conjunction with equations (6), (17), and (18) provides a basis for obtaining numerical estimates of the production function parameters a, b, v, and h, and of the contributions of capital deepening, scale effects, and neutral technological progress to the dynamic of manufacturing labor productivity. The procedure required for such estimation can be described as follows. Suppose that numerical values of y", L'\ and Lsh are given, and that, based on the y" and L" data, numerical values of M and TV of the equation: y" = M + NL" ,
(24)
have been obtained. By 'matching' the variables and parameters of the 'theoretical' equation (23) to the variables and parameters of the 'empirical' equation (24), and by assigning numerical values to Z,sh, numerical values for a, b, v, p, and h can be calculated. Different implementations of this procedure are possible, depending upon the manner in which numerical values of N and M are arrived at, and also depending upon whether the values of y", L", and L sh used are parameter estimates for a population of regions or measures referring to individual regions. The operationalization of the procedure that is used in this paper combines regression estimates of N with measures of y'\ L" and Lsh by region. Specifically, the parameters m and n of the regression equation y" = m + nL" + e ,
(25)
were estimated by ordinary least squares. Then equation (23) was matched to equation (25) in the manner specified by equations (26) and (27): n =
{T=j-1)
y"-nL"
=
(26)
'
L*+H-i
•
(27)
Equation (26) was solved with the use of Lsh values by region to generate p values also by region. Then, by means of equations (27), (17), and (6) and values of y", L'\ and Lsh for individual regions, regional values for a, b, v, and h are arrived at. It should
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1582
be noted that these parameter values will produce numerical values of the capital deepening, scale, and technological progress components of productivity change that account exactly for the value of y" (plus or minus a small rounding error) for each region. This technique was employed to split growth rates of manufacturing labor productivity by states into capital deepening, scale, and technological progress components that are then analysed in subsequent sections of this paper. The source data used are aggregate number of employees, aggregate payroll, and aggregate value added in manufacturing, for the years 1972 and 1976, for the fifty-one states of the USA, published in the Statistical Abstract of the United States (US Bureau of the Census, 1979, page 804). The symbols used to denote these variables are TI I c$2 ? «ie97?> a?9?2> affi6, a\f12i and a\%A16. The apay and a vad variables were converted into constant 1967 dollars. The derived variables Y", y", and Lsh were generated from the source variables by use of the following equations:
r=ilngr),
(28)
' v a d / M emp \ ^1976/^1976 \ vad / w emp / > a 1972In 1972/
rsh
=
r ? Q . Ay
\ )
(a]\
f -1976 "1972 V ^ad~^vad") 1976 flW2/
•
(30)
The variables Y", y", and Lsh were used to calculate P n t , Kdp, and S*0 components of productivity growth by state. Subsequently, a regression analysis of y" and of its P n t , Kdp, and S** components was carried out with percentage rates of change of population and of manufacturing product used as independent variables. The rationale of the analysis and the analysis itself are presented in the sections that follow. Regional shifts and productivity growth The objective of the regression analyses referred to in the preceding paragraph is to investigate the relations between the shifts of population and jobs out of the snow belt and into the sun belt and the dynamic of manufacturing productivity and of its components. Shifts of population and jobs out of the snow belt into the sun belt, and also away from some larger metropolitan agglomerations into urban centers of intermediate size and into nonurbanized areas are currently in progress in the USA (Beale, 1977; Berry and Dahman, 1977; Chinitz, 1978; Sternlieb and Hughes, 1975; 1978). These shifts have been placed into focus only recently. However, they quickly came to be regarded as an especially interesting and challenging research theme (a) because they constitute an unexpected reversal of long-run trends that shaped the spatial structure of the American economic system (Vining and Strauss, 1977), and (b) since they cannot be explained in terms of cumulative causation and growth-pole theories (Richardson, 1980) that, until recently, came close to represent the conventional wisdom in matters pertaining to regional growth. The growth of the sun belt and the decline or comparative stagnation of the snow belt raises interesting questions regarding the relation of these spatial temporal patterns to those of manufacturing productivity. The Verdoorn law states that productivity grows faster in expanding economic sectors (Verdoorn, 1949; Kaldor, 1967; 1970). The testing of this law has been based primarily on industrial data. It is an interesting question if and to what extent this law is confirmed by areal data, and specifically, whether it translates into higher productivity growth in the expanding
Technological progress, capital deepening, and productivity growth
1583
sun belt regions. If this is the case, it is interesting to ascertain which components of productivity growth are related, and by how much, to the productivity growth in the sun belt regions. These questions will be discussed in the remainder of this section. Alternative definitions of sun belt and snow belt appear in the literature (Rice, 1981). Since regional shifts of population and jobs rather than any one definition of sun belt and snow belt are of concern here, it seemed appropriate to use percentage rates of change of population, P", and of industrial value added, Y", as convenient indicators of these shifts. P" stands for percentage rate of change of population per year between 1972 and 1976, calculated from the population data by states published in the Statistical Abstracts of the United States (US Bureau of the Census,' 1975, page 12; 1979, page 14). As is well known, the old industrial states in the North East and Mid West are characterized by rates of population growth either negative or much smaller than the ones prevailing in the South and the West. The results of regression analyses with manufacturing productivity growth and its components as dependent variables and P" and Y" as independent variables are shown in table 1. In the first two regressions, growth rates of productivity, y", are related respectively to growth rates of value added, Y", and of population,• P". The r-values of both independent variables are significant, and both regression coefficients are positive. This indicates that expansion of manufacturing production and of population tends to be associated with above average productivity growth, and consequently, that the Verdoorn empirical regularity applies here. The regression analyses relating the three components of manufacturing productivity growth to P" and Y" are shown in the third to eighth lines of table 1. As was the case for the regressions with.y" as dependent variable, Y" is consistently a better predictor of Pnt, Kdp, and Sec than P". The regressions in which Y" is the independent variable explain a considerably larger proportion of the variance of the dependent variables, and are characterized by regression coefficients associated with much larger ^-values. However, the coefficients of P" as well as of Y" are significant at the 5% level or better in all the analyses. The main result of the analyses is that there is a strong relation between growth of population and product and growth of manufacturing productivity due to neutral technological progress and to capital deepening. States with larger population growth Table 1. Regression analyses (with ^-values given in brackets). Dependent variable tr
y
Intercept
P"
Y"
R
R2
0-9288
1-2534
0-240
(3-93)
-
0-490
(1-77)
0-725
0-526
-
0-549
0-302
0-3086
0-848
0-719
0-520
0-271
0-748
0-559
0-338
0-114
0-512
0-262
y
0-1748
pnt
0-6301
0-5793
(3-04)
(4-60)
pnt
0-2396 0-3267 (1-09)
Kd»
-0-1006 (-0-42)
sec
sec
(7-37)
-
(1-78) Kdp
0-6399
-
(0-41)
(11-2)
0-7725 ,
(4-26)
-
-0-0276
-0-0993
(-0-42)
(-2-51)
0-0363
-
(0-60)
0-3830 (7-88)
-0-0520 (-4-18)
1584
E Casetti
and manufacturing growth tend to have larger manufacturing productivity growth due to neutral technological progress and to capital deepening. Actually, P" and Y" appear to have a stronger effect on Kdp than on Pnt. For instance, the fourth to sixth lines of table 1 show that a 1% increase in value added is associated on the average with a 0-31% productivity growth due to Pnt, and with a 0-38% productivity growth due to Kdp. This result is consistent with the widely held presupposition that capital shifts toward regions where the marginal productivity of capital is higher constitute a major correlate and determinant of the economic and demographic shifts currently in progress in the USA. The effect of P" and Y" on productivity changes associated with S**, is much smaller than their effect on Pnt and Kdp. Nevertheless, the regression coefficients of P" and Y" in the seventh and eighth lines of table 1 are both statistically significant and, interestingly enough, negative. This means that states in which population and industrial production are expanding faster tend to have either a smaller positive or a larger negative S^ component of productivity growth. According to this analysis, higher productivity growth in the states in which manufacturing is expanding is due to higher rates of capital deepening and neutral technological progress, and takes place irrespective of more unfavourable scale economies/diseconomies effects. Perhaps, regulations and laws such as those for the protection of the environment and of the consumers, that are so widely credited with negative impacts on productivity, are implemented more heavy-handedly on expanding economic activities that are in a better position to withstand their cost. Possibly, these regulations are imposed to a lesser degree upon the stagnating economic sectors that instead resent and oppose them more, and more vocally. Summary The first part of this paper discusses a technique for splitting regional growth rates of manufacturing labor productivity into components associated with neutral technological progress, capital deepening, and scale economies, respectively. The technique does not require capital data, and is based on the assumptions that the economy minimizes factor costs, that all product is apportioned among the factors, and that regional capital to output ratios are approximately time invariant over short to medium time horizons. The technique is then applied to 1972 and 1976 state-level manufacturing data for the USA to generate numerical estimates of neutral technological progress, capital deepening, and scale components of manufacturing labor productivity change, by states. Finally, the rates of change of manufacturing labor productivity and their components are analysed to determine whether the states that are experiencing larger rates of growth of population and of manufacturing production are also characterized by higher growth of productivity and of its components. These analyses indicated that the states where population and manufacturing production is growing faster do indeed experience larger productivity growth, and especially larger productivity growth due to capital deepending and, to a lesser degree, to neutral technological progress. References Allais M, 1962 "The influence of the capital output ratio on real national income" Econometrica 30 700-728 Beale C L, 1977 "The recent shifts of United States population to nonmetropolitan areas, 1970-75" International Regional Science Review 2 113-122 Berry B J L, Dahman D, 1977 "Population redistribution in the United States in the 1970's" Population and Development Review 3 443-471 Brems H, 1968 Quantitative Economic Theory—A Synthetic Approach (John Wiley, New York)
Technological progress, capital deepening, and productivity growth
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Chinitz B, 1978 The Declining Northeast: Demographic and Economic Analysis (Praeger, New York) Dendrinos D S, Mullally H, 1981 "Fast and slow equations: the development pattern of urban settings" Environment and Planning A 13 819-827 Denison E E, 1967 Why Growth Rates Differ: Postwar Experience in Nine Western Countries (Brookings Institution, Washington DC) Dixon R J, Thirlwall A P, 1975 Regional Growth and Unemployment in the United Kingdom (Mac mil Ian, New York) Hood W M C, Scott A, 1957 Output, Labour and Capital in the Canadian Economy Royal Commission on Canada's Economic Prospects (Queen's Printer and Controller of Stationary, Hull, Quebec) Johansen L, 1961 "A method for separating the effects of capital accumulation and shifts in production functions upon growth in labour productivity" Economic Journal (December) 71 775-782 Kaldor N, 1967 Strategic Factors in Economic Development New York State School of Industrial and Labor Relations, Cornell University, Ithaca, New-York Kaldor N, 1970 "The case for regional policies" Scottish Journal of Political Economy 17 337-348 Klein L R, 1962 An Introduction to Econometrics (Prentice-Hall, Englewood Cliffs, NJ) Kuznets S, 1966 Modern Economic Growth: Rate, Structure, and Spread (Yale University Press, New Haven, CT) Poston T, Stewart I, 1978 Catastrophe Theory and Its Applications (Fearon and Pitman, Belmont, CA) Rice B R, 1981 "Searching for the Sunbelt" American Demographics 3 22-23 Richardson H W, 1980 "Polarization reversal in developing countries" Papers of the Regional Science Association 45 67-85 Sternlieb G, Hughes J W (Eds), 1975 Post-Industrial America: Metropolitan Decline and InterRegional Job Shifts (Center for Urban Policy Research New Brunswich, NJ) Sternlieb G, Hughes J W, 1978 "The new economic geography of America" in Revitalizing the Northeast: Prelude to an Agenda Eds G Sternlieb, J W Hughes (Center for Urban Policy Research, New Brunswich, NJ) pp 75-127 Thorn R, 1975 Structural Stability and Morphogenesis (Benjamin, New York) US Bureau of the Census, 1975 Statistical Abstract of the United States: 1975 (US Bureau of the Census, Washington DC) US Bureau of the Census, 1979 Statistical Abstract of the United States: 1979 (US Bureau of the Census, Washington DC) Verdoorn P J, 1949 "Fattori che regolano lo sviluppo economico della produttivita del lavoro" L'Industria 1 45-53 Vining D R, Strauss A, 1977 "A demonstration that the current deconcentration of population in the United States is a clean break with the past" Environment and Planning A 9 751-758 Zeeman E C, 1977 Catastrophe Theory (Addison-Wesley, Reading, MA)
