In analyzing the growth of an economy or industry, it is important to understand the nature of the growth by tracing its sources. Sources may be such things as an increase in the use of inputs or improved productivity. Study of this sort falls in the analysis of total factor productivity (TFP). Traditionally, TFP is defined as the portion of output growth that can not be attributed to the growth of inputs (labor, capital, etc.). For example, if a manufacturing factory increases its use of inputs by 2% and its output grows by 3.5%, the extra 1.5% increase is defined as TFP. TFP may be the result of improvement in the quality of inputs (more skilled workers, better machines, or streamlined management), or may be due to economies of scale, i.e. better performance of the production system when the size of the operation becomes larger even though the quality in inputs remain the same. In the past, these two sources of TFP were not distinguishable when measured with conventional methods, such as the neoclassical production function approach.

The purpose of this paper is to introduce a new method, the flexible production function approach, that allows a separation of the effect of economies of scale and that of the input quality improvement in measuring increase in productivity. It will compare the new method with the conventional neoclassical production function approach in estimating regional TFP of U.S. manufacturing. The following section discusses conceptual issues pertaining to the conventional and flexible function approaches. The third section compares estimation results from the two different approaches. Section four offers a summary and conclusions.
 

2.1 TOTAL FACTOR PRODUCTIVITY

Early measurement of TFP was conducted by using various index numbers such as the Divisia index, or the Törnqvist index for discrete data. (6) Since Solow's seminal work, (11) researchers have found a close relationship between the index number approach and the neoclassical theory of production. The fundamental notion in the Solow neoclassical approach is to decompose the change in total output into separate components, i.e., the changes associated with various factors of production, and the change associated with new technology. Specifically, for a production function Y=f(L,K,R), where Y is output, L and K are labor and capital, and R is time, total change in output with R can be decomposed into various components as follows:

Define

Substituting [2] into [1] and multiplying both sides by 1/Y, we obtain

Equation [3] can be expressed as

where _i's (i=L, K) are partial output elasticities of the factors of production. In the right-hand side of [4], the first two terms express the effects of changes in capital and labor quantity. The last term represents the effect of quality change in capital and labor with time R. In other words, this is the effect associated with technological change. (11) Denny and Fuss (4) generalize Solow's theory so that R can be defined as either a temporal or a spatial factor. Therefore, the last term of [4] is the change in total output due to variations in technology between two points in time, or between two spatial units. This effect is called intertemporal or interspatial change (4) and is defined as TFP. It can be seen as a residual component in total output change, i.e.
 

2.2 CONVENTIONAL MEASUREMENT OF TFP

For a unit change in R, equation [5] can be expressed as
 

This formula is sufficiently close to Fisher's Ideal Index (6) and Törnqvist's Translog Index: (13)
 

where a's are factor shares of different factors of production in total output value. In the conventional neoclassical production function, linear homogeneity is usually assumed. In other words, there exist no economies (or diseconomies) of scale. As a result, the output elasticities of the factors of production in [6] are equal to the factor shares in [7]. Thus, Solow's work is considered to establish a close relationship between the index number study and the neoclassical production function. Measuring TFP by using the index in [6] or [7] is convenient in that information on factor shares is readily available and the information on factor quantities can be found through use of various proxies. Thus, variations in efficiency, as evidenced by total factor productivity, can be measured without statistical estimation.

2.3 MEASUREMENT OF TFP WITH THE FLEXIBLE FUNCTION

Although the linear homogeneity assumption in the conventional production function provides convenience in mathematical manipulations, production process in the real world may not obey such regularity. When there exist economies of scale in production, their effect on output growth will be left in the residual component R in a growth function, together with the effect of input quality improvement on output growth. Using a flexible production function provides a solution to this problem.

A flexible production function can be defined as the second-order Taylor expansion of a input/output relationship. (3) There are two general flexible functional forms: the general quadratic form and the translog form. (5) For consistency and comparison with the conventional approach, this research only discusses the translog form. Assume a production function

                LnY=f(LnL, LnK, R)                                                                                    [8]

where Ln represents the natural log. The translog function is then
 

where , , and are parameters and Rm is a remainder. The above production function is a second-order Taylor expansion at point LnL=LnK=1.

Suppose two points in R, R₁ and R₂. First, expand [8] at point 1 and evaluate the function at point 2; then expand [8] at point 2 and evaluate the function at point 1. Finding the differential between the two values, we obtain:
 

where f' and f" are the first- and second-order partial derivatives, Rm.1 and Rm.2 are remainders, and is a composite value which contains all differential terms that contain R. Specifically:
 

Here, the index is a measure of total factor productivity. According to Denny and Fuss, (4) if R is a temporal variable, is an intertemporal index number; if R is a spatial variable, is then an interspatial index number.

From [10], the index can be expressed as a residual component, i.e.,
 

In [12], assume zero for the remainder term. If the production function exhibits linear homogeneity, conditions f^"_2.LK-f^"_1.LK=0 and f^"_2.XX-f^"_1.XX=0 (X=L and K) must be met. Equation [12] then collapses into the Törnqvist translog index in [7]. Therefore, the index number approach and the neoclassical approach can be seen as special cases of the generalized flexible production function approach where the second-order derivative differential is zero and the first-order derivatives are equal to factor shares. In situations where linear homogeneity is not observed, the measurement of TFP with the conventional approach is apparently biased due to the missing second-order terms and the simplified first-order terms. Therefore, a more accurate measurement of TFP should rely on the first- and second-order partial derivatives that are statistically estimated, using the general form of the production function given in [9] or the differential form given in [10]. The estimates of parameters are then substituted into [12] to obtain . More importantly, in the flexible function method, the effect of economies of scale can be measured by the magnitude of the second-order terms, while the residual component will contain only the effect of input quality improvement. In addition, the second-order terms for labor and capital, and the interaction term between labor and capital will help point out the source of economies of scale.

The data utilized in estimation are from the Census of Manufactures for 1972 and 1992. The information on labor is proxied by the number of working hours of manufacturing employees while manufacturing capital is proxied by subtracting total manufacturing salaries from total manufacturing value added. The manufacturing capital used here is not a perfect proxy due to the fact that the difference between the total value added and the labor cost also includes payment to other factors of production such as profits and land-use expenses. Nonetheless, researchers have adopted this approach because of difficulty in obtaining direct data on capital. (1, 2, 9, 10) "Capital" in this context actually includes all non-labor inputs. The basic observational unit is the metropolitan area as defined in 1993, which is the basis for the 1992 census. Due to frequent changes in metropolitan definitions, the 1972 data are corrected to conform to the 1993 metropolitan definition. The New England metropolitan areas are not included in the analysis due to inconsistent metropolitan definitions compared to the rest of the country. In total, there are 293 metropolitan areas. They are grouped in three census regions, the North, South, and West. The manufacturing value added and manufacturing wage payments in 1972 and 1992 are corrected to a comparable basis where the 1982-1984 dollars are set to be 100.

3.1 DESCRIPTIVE ANALYSIS OF THE DATA

The period between 1972 and 1992 corresponds to slow growth in the U.S. economy and productivity in general, (7, 8) and manufacturing decline in particular. (12) The presumed high efficiency associated with information technologies had not been able to show appreciable effect in the economy or in manufacturing as a whole. (7, 8) The task now is to analytically decompose the total change in manufacturing value added into various components that positively and adversely affect manufacturing growth.

Data in Table 1 describe general changing patterns of manufacturing in the U.S. census regions between 1972 and 1992. Conforming to the general notion, between 1972 and 1992, the share of manufacturing value added in the North declined from nearly 60% to only 47%, while it increased from 23.5% to 31.1% for the South, and from 16.5% to 21.6% for the West. A noticeable fact is that changes in the inputs deviated from those in value added. For example, although value added in the North declined by 9% during this 20 year period, its labor experienced a much more significant decline of 33.3%. However, the non-labor inputs saw a 5.6% gain. Apparently, some output decline caused by a negative labor input increase was offset by the non-labor input increase. In the South, during these 20 years, labor inputs declined while the non-labor inputs increased, which was largely responsible for the overall increase in the value added. In the West, both labor and non-labor input increase seemed responsible for the increase in value added. Since labor inputs experienced negative increase or smaller increase than non-labor inputs, in all three regions, manufacturing in 1992 was less labor intensive than in 1972.

3.2 TOTAL FACTOR PRODUCTIVITY ANALYSIS

Table 2 lists parameter estimates for the flexible functions and parameters calculated for the neoclassical functions. The parameter estimates for the flexible form are obtained by applying the OLS on the differential function in [10]. These parameter estimates are then substituted into the right-hand side in [12], together with sample mean values, to obtain manufacturing growth that is associated with different components: changes in the use of labor and capital (non-labor inputs), economies of scale, and the residual component (TFP), as listed in Table 3. The parameters in the last two rows for the neoclassical form of Table 2 are calculated by using the factor shares. These parameters are then substituted into [7] to obtain the effect of each of the factors on output and residual component (TFP) in Table 3.

A comparison of the results shows both consistencies and differences between the two approaches. Both approaches show that while there was a negative output growth associated with a decrease in the use of labor input for the national and North and South metropolitan systems, the West metropolitan system saw a positive output as a result of the growth in the use of labor inputs. In addition, both models indicate that there was a positive output growth due to an increase in the use of capital input in all metropolitan systems.

However, results from two methods differ significantly in many respects. Firstly, values in columns 4 and 5 in Table 3 indicate that the neoclassical approach exaggerates the magnitude of output changes associated with changes in labor, and understates the magnitude of output growth attributable to changes in capital.
 

Secondly, the effect of TFP on output growth is significantly larger in the neoclassical model than in the flexible model. This is partly because the neoclassical model is unable to dissect the effect of economies of scale and the effect of input quality improvement on output growth. Therefore, TFP estimated from the neoclassical model is actually the combination of the two. However, this cannot explain the fact that the neoclassical models generate a much larger TFP than the combined effects of economies of scale and TFP from flexible models (e.g. 0.29 vs. 0.09 for the Nation). The answer lies in the previous observation that the neoclassical model understates the effect of capital change and overstates the effect of labor change. Since the effect of capital is positive in all cases and the effect of labor is negative in all models except in the West, the neoclassical models leave an excess amount of effects in TFP. As for the West, while the effect of labor increase on output growth is positive in both types of models, the effect of labor change is significantly larger in the flexible model than in the neoclassical model. Consequently, the flexible models results in a negative value in TFP.

Finally, the effect of economies of scale is missing in the neoclassical model. It is interesting to observe sources of economies of scale estimated from flexible functions. As Table 4 shows, the effect of economies of scale may be due to a sheer accumulation of labor and capital, or due to the interactive effect between labor and capital. The parameter for the interaction term in a flexible production function captures the possible effect, that the effect of one input on output depends on the level of another input used. The model used here estimates a differential parameter on the interaction term. In three out of four models (Nation, North and South), the parameter estimates are statistically significant but negative. This means that the degree of input interdependence in output growth become less significant from 1972 to 1992. That is, a higher level of output can be achieved by a given amount of one input even though there is a lower amount of other inputs used. That is why even though there was a lower level use of labor inputs in 1992 than in 1972, a positive growth in capital use still brought a positive output growth in all three cases. In the West, a statistically insignificant differential interaction term parameter means that the level of input interdependence in output growth remained the same from 1972 to 1992. Thus, there is no output growth associated with the differential input use during the 20 year period.

3.3 REGIONAL SOURCES OF MANUFACTURING GROWTH

The above analyses based on the flexible function show that the period from 1972 to 1992 saw an extremely slight annual manufacturing growth attributable to input quality improvement. Most changes in manufacturing seem to be attributable to the sheer effect of input changes, especially the accumulation of capital. For an average metropolitan area in the nation, the effect of declining labor use was more than compensated by the output growth due to increasing capital use. Furthermore, the effect of increasing economies of scale on output growth between 1972 and 1992 came from declining use of labor, increasing use of capital, and declining interactive effect between labor and capital. This may indicate that there was diseconomies of scale in the use of labor in 1972. Manufacturing growth was largely a result of a leaner workforce, higher capital intensity, and a resultant economies of scale in using labor and capital. The overall effect of improvement in labor and capital on manufacturing growth is slim.

For an average metropolitan area in the North, total output growth declined between 1972 and 1992. This was largely caused by the declining labor use. However, the output would have declined much more if not for the positive effect on output growth from capital accumulation, and to a lesser degree, due to the positive effect on output growth from economies of scale and slight input quality improvement. In addition, increasing economies of scale in the North are largely a result of reducing the use of labor (i.e. reducing diseconomies of scale in labor use), and to a lesser degree, a result of changes in interactive effect between labor and capital. Interestingly, the significantly declining labor use in the North (largest magnitude of all regions) seems associated with the largest increase in the labor-induced economies of scale of all regions. The North also experienced the largest growth associated with input quality improvement of all regions.

In the South, an average metropolitan area grew largely due to an accumulation of capital. The declining labor use did not cause significant amount of output decline. As a matter of fact, the declining output due to the declining labor use was entirely compensated by input quality improvement. The slight increase in economies of scale was largely due to a changing interactive effect between labor and capital.

The West differs from the North and South in a significant way. Both labor and capital accumulated to cause a positive increase in total output. However, increase in labor use may have caused slight diseconomies of scale in labor use. In addition, the West saw no effect of input quality improvement on output growth.
 

The conventional neoclassical production function is based on important assumptions concerning production function behavior. These assumptions provide convenience in assessing economic performance. However, possible misspecification may cause bias in measurement and thus cause possible misleading interpretation. In comparison, the flexible function contains no assumption on functional behavior and thus, provides a more general framework for investigating total factor productivity. The results by using the flexible approach seem to better conform to the general trend of manufacturing development during the period between 1972 and 1992.

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