Estimation of scale economies underlying growth and productivity: the empirical implications of data aggregation.

• Author: Morrison Paul, Catherine J.

1. Introduction

   Using production theory models to represent the behavior of sectors has been recognized as problematic since well before Robert Solow's (1957, p. 312) seminal work on the measurement of technical change, in which he stated that "it takes something more than the usual 'willing suspension of disbelief' to talk seriously of the aggregate production function."

   Very little has changed since Solow's (1957) statement, including the difficulties of trying to justify this type of endeavor using "fancy theorems on aggregation and index numbers" (p. 312). Ultimately, the appeal of this kind of aggregate economics depends on a number of factors, including the issue at hand and the data that are available for the empirical analysis. As Solow indicated, there are benefits and drawbacks to such an approach. Thus, he "belongs to both schools" (p. 312).

   We also belong to the large group of applied economists who sit on this somewhat uncomfortable fence. Aggregation theory still does not provide a solid basis for representing aggregates by production and cost functions. However, this is the basis for most research on productivity and growth, at least in part because questions of interest to economists are largely about overall trends in a sector or country. Also, useful conclusions may be drawn from the results even if they are not explicitly interpretable as stemming from the technology and behavior for a particular firm. Thus, aggregate economics is widely pursued, and generating insights about how data aggregation affects empirical findings is an important goal.

   This paper addresses the implications of aggregation for measures of scale economies deriving from technological and knowledge capital factors. We base our analysis on a cost function framework that allows us to consider whether identifiable patterns in the results emerge for measures based on data at different levels of aggregation. We focus on the measurement of various aspects of scale economies since this has been an important issue in a number of fields in economics, especially in the endogenous growth literature. The emerging importance of such economies in the theoretical and empirical literature highlights the difficulties of measuring their existence and determinants since the interpretation of results for a heterogeneous sector as those for a representative firm is justified by the assumption of constant returns to scale.

   To pursue this, we use four-digit, two-digit, and total SIC level U.S. manufacturing data to compare estimates of short/long-run and internal/external scale economies and their determinants. Even though the four-digit SIC estimates are still based on industry-level data rather than plant- or establishment-level data, one would think that the breakdown at this level results in sufficiently homogeneous groups for the results at this level to be considered disaggregated. Accepting this assertion is not necessary, however, since the exercise we carry out is meant to provide information on the impact of further aggregation, not to compare micro to macro data (which is ultimately infeasible due to the number of units involved).

   Scale effects appear important conceptually, theoretically, and empirically, so the assumption of constant returns is simply unacceptable for most applications. Measures of scale effects provide important information about the technology, behavior and economic performance of sectors. Since it is difficult to generate such measures without relying on econometric implementation of the theory of the firm, our goal is to gain information about the effects of aggregation on these estimates by considering the associated empirical patterns.

   The patterns we find provide provocative evidence about the impacts of data aggregation on estimates of scale relationships and the impacts of knowledge capital factors via elasticity measures. The estimates also emphasize that important scale effects are reflected in the data, even if they are not directly interpretable as scale economies in the sense of the slope of a firm's long-run cost curve.

2. More About the Issue and Our Approach

   The fact that the specification and estimation of aggregate production and cost functions rests on a shaky theoretical foundation poses a number of problems. First, disaggregated data more appropriately represented by the theory of the firm is limited by availability, quality, and confidentiality restrictions. The quality issue is particularly problematic, and it is sometimes asserted that problems with irrelevant data points will average or wash out in more aggregated data. These problems are similar to those involving measurement error. Measurement error plagues any type of econometric exercise but may be particularly likely to emerge at more disaggregated data levels where errors in reporting are likely and where specific or unique economic conditions facing the establishment are not obvious from the raw data.

   Perhaps an even more critical issue involves the questions that are of interest to applied economists and to the users of their empirical results and statistics. Most of these questions involve overall sectoral trends. For example, growth and production/cost models have been used to evaluate the effects of various types of external (knowledge) and internal capital/technological factors, education (human capital), energy prices, trade policies, and other economic conditions on productivity, employment, investment, obsolescence, and other indicators of the technology, behavior, and performance of firms within a sector. These kinds of issues apply to the sector as a whole, where sector in this case might be a particular industry, region, or country.

   Information on various heterogeneous plants might be of interest to identify differences within a sector but is not applicable to questions about overall impacts, that is, they do not necessarily generalize.(1) Thus, addressing these issues often requires the development and use of more aggregate data and implementation of a production and cost structure that allows consideration of technological and behavioral patterns of firms on average within the sector, which is the sense in which sectoral data reflect the representative firm.

   Consideration of aggregate trends within a sector using the rich structure of production and cost theory provides important information on substitution patterns and economic performance. The difficulties arise because the use and interpretation of aggregated data for empirical estimation are not well guided by economic theory.

   Theories regarding the construction of such data are quite well developed, if difficult to accommodate satisfactorily. These theories provide guidance both on what characteristics of the data make it amenable to aggregation (Hicks aggregation theories suggest that, if prices move similarly, the goods may reasonably be aggregated) and how to carry out such aggregation even if these requirements are not met (such as using chained or Divisia indexes in lieu of simpler measures).(2) Aggregation theory indicating how these data might be used to generate interpretable measures of overall behavior in a sector has, however, not been as forthcoming.

   Typically the approach used has been to minimize the conceptual difficulties by imposing assumptions that facilitate interpretation of empirical results as those for a representative firm. One of the most prevalent of these is the assumption of constant returns to scale (CRTS). Imposing this assumption is useful for a number of reasons, including the ease of developing theoretical results and, more importantly for our purposes, the appropriateness of using aggregate data when this restriction holds. Specifically, when CRTS exists for each firm, it is justifiable to imply that it also exists in aggregate; if a technology exhibits a minimum possible level of average costs, all firms will be at this level in long-run equilibrium with CRTS. However, if nonconstant returns to scale (NCRTS) exists, firms will be at different cost levels due to their scale of production, and it is not possible to add them up and thus to theoretically justify the notion of a representative firm.

   The assumption of CRTS is quite restrictive, however, since various types of scale effects exist theoretically, conceptually, and empirically. Take, for example, the assumption of full equilibrium in all input markets (an assumption that Solow [1957, p. 312] says is "an assumption often made" so "the price may not be unreasonably high"). If this assumption is violated due to short-run fixities, the firm will be on a short-run cost curve and (short-run) scale economies will be observed. This has been a major focus in economics since Marshall's work over a century ago but is often ignored in empirical applications.

   Technological conditions may also cause long-run scale economies to exist, perhaps due to the long-term fixity of entrepreneurial inputs or other limited factors. In addition, other (both internal and external) scale economies may stem from the expansion of information technology, research, education, public goods, and other knowledge factors.

   The existence and nature of scale effects has also provided a rich basis for (nonacademic and academic) analysis in the recent past. One needs only to pick up a newspaper to find a reference to firms downsizing or expanding their scale of operations. The effects of deregulation and policies that promote free trade on firms, workers, and consumers have generated considerable debate. The economic motivations for these market and policy responses to economic conditions hinge, to a large extent, on the prevalence, importance, and determinants of economies of scale.

   The notion of external scale effects derived from knowledge factors has also been raised as an important stimulus to economic growth. This has provided the basis for the new endogenous growth literature, which has been...