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Progress through Regression: The Life Story of the Empirical Cobb-Douglas Production Function

Author(s):Biddle, Jeff
Reviewer(s):Field, Alexander J.

Published by EH.Net (May 2022).

Jeff Biddle. Progress through Regression: The Life Story of the Empirical Cobb-Douglas Production Function. Cambridge: Cambridge University Press, 2020. xii + 334 pp. $110 (hardback), ISBN 978-1108492263.

Reviewed for EH.Net by Alexander J. Field, Professor of Economics, Santa Clara University.


Jeff Biddle has written an intellectual history of the Cobb-Douglas production function. Actually, he argues, it’s a history not of the function itself, which predates Cobb and Douglas, but a method for estimating its parameters using ordinary least squares, to wit, regressing the log of an output measure on a constant and logs of labor and physical capital inputs. To be fair, the study offers some of both.

The concept of a production function is best understood in the context of an agricultural experiment station. The yield of a plant (output) can be thought of as dependent on the quantities of a variety of inputs: water, fertilizer, sunlight, for example. The incremental (marginal) products of these different inputs can be measured through controlled experiment, varying the amount of one input while everything else is held constant. From such data one can hope to identify and describe mathematically a function linking the flow of output to the amounts of various inputs, and the rapidity with which the incremental product declines with additional doses, holding other inputs constant. The production function for a particular plant species would be understood to have a real, objective basis in biochemistry, and could be assumed to be similar for all specimens of the same species.

Substitute a particular physical product for an agricultural crop, and the concept, it would seem, can be extended to manufacturing. The production function would now reflect an underlying engineering reality and could be assumed similar in different establishments or firms producing the same good. Given observational data on different combinations of inputs, the underlying function could in principle be identified.

But that’s not what mathematician Charles Cobb and economist Paul Douglas actually attempted. Initially they ran time series regressions where the unit of observation was the entire U.S. manufacturing sector, followed by a study on Massachusetts and then estimates using data from the state of Victoria in Australia. There were and are multiple problems in extending the biological analogy to these larger aggregates and including on the right-hand side just two aggregated inputs, labor and capital. Both are heterogeneous, capital much more so than labor. One can perhaps argue that labor has a natural metric, the person-hour or person-year, but there is no such metric for the wide variety of physical capital goods. Although one would like measures of capital service flow, one will almost invariably be stuck with gross or net stock data serving as proxies. And even if depreciation in market value of different vintages of different types can be measured accurately, service flow deterioration will in almost all cases run more slowly than the decline in market value (depreciation). The extreme case is Oliver Wendell Holmes’ one hoss shay, which fell apart all at once after 100 years. The service flow (and presumably the rental rate) remained the same for a century, but a 98-year-old shay would still command a much lower market value than one which was new or almost new.

Robert Solow dismissed the prospect of calculating capital service flows as “utopian” (1957, p. 314), but today the Bureau of Labor Statistics and other OECD statistical agencies do make a run at it, distinguishing between productive stocks, constructed to grow pari passu with service flows, and wealth stocks (what the Bureau of Economic Analysis calculates). Economics researchers seem largely unaware of these procedures, or, arguing that deterioration in service flow runs geometrically at the same rate as the declines in market value, maintain that a distinction between productive and wealth stocks is unnecessary.

Assuming one does have measures of productive stocks, should there be a utilization adjustment? Douglas felt as did Solow and others after him, that one was needed. My own view is that its desirability is questionable, given that deterioration of fixed capital service flow is largely unaffected by how intensively buildings or equipment are used. The rate at which a building’s roof wears out, or a machine becomes obsolete, are illustrative of forces other than utilization that can govern both deterioration and depreciation. Douglas regretted he was not able to make such an adjustment; Solow made one.

There remains the problem of how one aggregates a sectoral output consisting of many different types of physical products. Is it acceptable to use value added? And the challenges continue. Assuming one can develop plausible measures of the service flows from aggregated labor and capital, is it reasonable to assume that the production functions used in making different products are all the same? Really? Elasticities of substitution are the same, as are the marginal products of ‘capital’ and ‘labor’? Doesn’t that present serious aggregation problems? And, in a cross-section regression, if the production functions are indeed the same across different products, and both market and input markets are all perfectly competitive, wouldn’t all firms and all sectors exhibit the same proportions of capital to labor (at least in long term equilibrium), in which case the regression would suffer from extreme multicollinearity, making it nearly impossible to estimate parameters of an aggregate production function with any degree of precision?

The challenges seem daunting. And yet I would hazard that a sizable majority of economists and economic historians (present company included) refer in their research and teaching to Cobb-Douglas functions, often acutely aware that they are engaged in some hand waving. In describing the initial and continuing reaction to the Cobb-Douglas enterprise, Biddle’s book discusses almost all of these concerns, and gives economic researchers an opportunity to reflect on the nature of their own handwaving and whether or not it is justified given the uses to which their inquiries are put. It is particularly useful to revisit the language used by scholars such as Solow (pp. 248-49) or Zvi Griliches (pp. 294-95) as they finesse these issues and compare their rationales with one’s own.

The book is primarily focused on developments from the 1920s through the 1970s and is divided into two main parts, each with three chapters, followed by a concluding chapter (part III). The materials studied for the most part are published journal articles, and the method for each is to provide an explication of the key arguments along with varying degrees of commentary and evaluation. Following a brief introduction, chapter 1 covers the initial time series studies as described in Cobb and Douglas’s 1928 American Economic Review article and Douglas’s 1934 book, The Theory of Wages, along with initial reactions and criticisms, and subsequent time series studies by Douglas and coauthors.

Chapter 2 is principally focused on debates with economist Horst Mendershausen, who came at Cobb and Douglas from several directions. The most damaging argument was that one could not simply assume “the” production function remained unchanged over multiple decades, and use variation in capital and labor inputs to identify a function that was in fact a moving target. Mendershausen would not be the last to raise this objection. It is perhaps not accidental that around this time Douglas switched from time series to cross-section studies. This avoided some of the hoary problems of adjusting for price changes of capital goods in accounting for depreciation and net additions in building up an inflation adjusted time series of a physical capital wealth stock. But, in moving to cross sectional data with different industries serving as the unit of analysis, one could still wonder whether all industries faced a similar production function. Cleverly, Douglas used the possibility that a few did not in explaining large residuals (differences between the predicted and actual value of production based on his regression estimates) (p. 100). Other complaints voiced by Mendershausen revolved around the fact that all three of the key series moved upward fairly systematically over time – as well as questions about whether it was reasonable to maintain the hypothesis that output was the dependent variable and labor and capital were independent right-hand variables. A number of authors argued that the question of which was dependent and which independent should be decided by “objective” statistical inquiry, whereas Douglas argued, I think convincingly, that one could use knowledge about how the world works to justify the assumption that labor and capital service flows produced output, rather than vice versa.

Douglas’s academic career was interrupted in 1942 when he enlisted in the Marines and then, following a short postwar coda, ended when in 1948 he ran successfully for the U.S. Senate. Chapter 3, the final chapter in part I, covers Douglas’s presidential address to the American Economic Association (AEA), in which he summed up his contributions, as well as additional commentary and criticism from the 1940s. The latter included Jan Tinbergen’s observation that in the cross-sectional studies, variations across industries in capital-labor ratios could come about only if different industries had different production functions, or if different regions faced different labor or capital supply conditions, and thus different factor prices. But the latter could not be the case if input markets were truly competitive. The chapter continued with discussion of papers by Melvin Reder, Martin Bronfenbrenner (an earlier coauthor with Douglas) and Jacob Marschak and William J. Andrews. In his valedictory address to the AEA, Douglas mentioned none of these (p. 130).

Part II of the book is concerned with “diffusion” of Douglas’s research program, spotlighting research in two areas that grew out of his initiative: agricultural economics (chapter 5) and growth accounting (chapter 6). Prior to developing these two “case studies”, in chapter 4 Biddle covers three somewhat unrelated developments during the 1950s: the treatment of the Cobb-Douglas research program in the first econometrics textbooks, the 1957 critique by E.H. Phelps-Brown, and the development of the CES (constant elasticity of substitution) production function of which the Cobb-Douglas function was a special case. Chapter 5 is organized around the work of Earl Heady and a group at Iowa State University. The appeal of the research program within that subdiscipline has already been mentioned. Even where the studies were observational rather than experimental, the greater prevalence of single product firms and the practical questions farmers were concerned with helps explain why the research program was attractive.

Chapter 6 covers growth accounting. With the burgeoning postwar interest in economic growth came empirical attempts to partition advance into the portion attributable on the one hand to input growth conventionally measured and on the other hand to scientific, technological, and organizational progress. The fundamental growth accounting equation is obtained by differentiating both sides of the Cobb-Douglas function with respect to time and can be estimated by running the change in the log of output against a constant and changes in the logs of capital and labor inputs. In that sense the efforts are an offshoot of the Cobb-Douglas program. Biddle acknowledges however, that few growth accounting studies used this method, instead preferring a pure accounting exercise, with the residual calculated as the difference between real output growth and a weighted average of the two key inputs.

One can still argue that growth accounting has a lineage stretching back to Cobb-Douglas. These weights are usually based on factor shares, and Cobb and Douglas maintained that the coefficients on labor and capital they were estimating should equal marginal productivities, which in turn would be reflected in factor shares. But one can question whether that linkage really matters for growth accounting. One of the attractions of such work is that it can be less demanding of commitment to some of the standard production function baggage (p. 280). Even if one remains agnostic about marginal productivity theory, one can argue that weighting by factor shares remains as good a practice as any, and proceed accordingly. Biddle makes an interesting point in crediting Solow with explicitly tying the growth accounting program to the production function framework, insisting that one was separating shifts of a production function from movements along it (p. 251). Indeed, Biddle sees that as Solow’s most important contribution, since little else in the 1957 article can be said to be truly original.

This book can be read profitably by those with interests in the twentieth century history of economics and econometrics, and, more specifically, in production functions and attempts to estimate them. I acknowledge and respect the efforts of the author to be fair to all participants, but at times I wished for a more consistent balance of exposition and evaluation. A work such as this can add value if the author can capture with more clarity and in shorter compass what the original author(s) argued. Simply recapitulating the main arguments, however, can invite readers to ask whether it would not be better simply to read the original texts, a query I frequently posed and acted upon (almost all of the articles are available on JSTOR). Now and again in the book Biddle offers his own judgments. I would have appreciated more articulation of his point of view.

A minor issue: the author repeatedly uses the words homogenous and homogeneous interchangeably. Only the latter is appropriate in charactering a mathematical function.


Solow, Robert M. 1957. “Technical Change and the Aggregate Production Function.” Review of Economics and Statistics 39 (August): 312–20.


Alexander J. Field is the Michel and Mary Orradre Professor of Economics at Santa Clara University. He is the author of A Great Leap Forward: 1930s Depression and U.S. Economic Growth (Yale University Press, 2011) and The Economic Consequences of U.S. Mobilization for the Second World War (Yale University Press, October 2022).

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Subject(s):Economic Development, Growth, and Aggregate Productivity
History of Economic Thought; Methodology
Geographic Area(s):General, International, or Comparative
North America
Time Period(s):20th Century: Pre WWII
20th Century: WWII and post-WWII