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Progress through Regression: The Life Story of the Empirical Cobb-Douglas Production Function
The Cobb-Douglas regression, a statistical technique developed to estimate what economists called a 'production function', was introduced in the late 1920s. For several years, only economist Paul Douglas and a few collaborators used the technique, while vigorously defending it against numerous critics. By the 1950s, however, several economists beyond Douglas's circle were using the technique, and by the 1970s, Douglas's regression, and more sophisticated procedures inspired by it, had become standard parts of the empirical economist's toolkit. This volume is the story of the Cobb-Douglas regression from its introduction to its acceptance as general-purpose research tool. The story intersects with the histories of several important empirical research programs in twentieth century economics, and vividly portrays the challenges of empirical economic research during that era. Fundamentally, this work represents a case study of how a controversial, innovative research tool comes to be widely accepted by a community of scholars.
350 pages, Hardcover
Published November 12, 2020
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December 29, 2022
This is the most unexpected book I encountered in 2022. Economic history is never a hot area of study, but dedicating an entire book to this single function is a feat of its own. Jeff Biddle succeeded marvelously. I will provide a summary of what's going on with the following essay that was meant to publish as part of my work but never made its way to press.
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Prior to the 1870s, the subject of economics was dominated by the Classical School pioneered by David Hume and Adam Smith. There are many important elements in Classical theory, but the most relevant here is their theory about value. According to the Classical School, value is primarily determined by the cost of production. A house is more expensive than a piece of bread because it costs more to produce a house.
Since a Classical economist’s eye value is determined by the cost of production, they are naturally interested in the laws of production and laws of distribution. Land, labor, and capital are three factors of production; economists are interested in how these factors of inputs produce outputs. This is the backdrop of the Cobb-Douglas production function.
There are a couple of issues with the Classical School’s framework of value - the most famous one being the Paradox of Value. We all agree that water is more useful than diamond, but why is water close to being free and diamond so expensive? In order to explain the paradox, which in essence is the paradox of the divergence between utility value and exchange value, the Classical economists think it is because the exchange value is determined by the cost of production whereas utility value is not.
But, the theory has its apparent shortcomings (e.g. cases like the exchange value of high end alcohol, and autographed photos of celebrities). Also, Classical economists cannot explain why producers refuse to raise prices even when the cost of production goes up. Shouldn’t the cost of production determine the exchange value of a product?
Hence the emergence of the Neoclassical School and the “Marginal Revolution”. The Neoclassical School was still being incubated in the 1870s and did not gain mainstream popularity until the mid 20th century. They believed that the Classical School failed to pay attention to consumers when thinking about value. Sure, the cost of production influences value, but the key to address the Paradox of Value is consumers. Neoclassical economists started to develop a consumer theory to complement the Classical production theory. Values are subjective, determined by individual choice.
The Neoclassical School initiated “the Marginal Revolution.” While a Classical economist says that price is determined by supply and demand, a Marginalist says that price is determined by marginal forces. Marginal means relevant; hence, in the previous example of water and diamonds, water is less expensive because the marginal utility of water is lower than that of diamond.
Aside from a consumer theory, the Marginalists developed a new theory of distribution—marginal productivity—that explains price received by sellers of any specific factor of production. The Cobb-Douglas function has interesting properties that fit in this “marginal paradigm”: positive but diminishing marginal utility, a potential measurement of substitution effects, among others.
The theory of marginal productivity was contested by the mainstream economists then, since the theory “required instantaneous and complete adjustment of factor prices to change in output prices and vice versa, something that clearly was not the case.” (Biddle 2021, 30) This is a fair criticism. It became one of the key criticisms against the Cobb-Douglas production function.
The Neoclassical School also pioneered the usage of math in economics. Although only some of the Neoclassical economists believed in math, the true believers held that the first language of economics should be math. This was quite controversial (and still is today), but eventually the Neoclassical School took over academia and the first language of economics indeed became math. As we can see later on, had the Neoclassical School not become mainstream, Cobb-Douglas would have likely fallen into oblivion (after all, it is math!).
It is important to keep in mind that during the entirety of Paul Douglas’ career, the Neoclassical School was not a mainstream school. During the interwar period (1920s - 1940s), there was a time when multiple schools of thought flourished (the pluralist era). Paul Douglas himself actually would not have agreed to the premises of the newer schools and the creative (and prevalent) application of his now famous function. He wanted to address an aggregate production problem, but the function got borrowed into modeling firm and industry behaviors (which is the use case in tokenomics today).
Paul Douglas spent most of his career in Illinois, both in the University of Chicago and as a Senator of Illinois (for 18 years). Douglas was interested in measuring input-output relationships in aggregate production, especially through the usage of statistical data. Douglas himself was not a skilled mathematician, so he enlisted help from Charles Cobb, a mathematician that spent most of his career in Amherst College.
While the collection of data is a norm and institutionalized today, it was not the case back in Cobb and Douglas’ time. Actually, Douglas’ key contribution was not the function itself (the function was first conceived by Knut Wicksell, of the Wicksellian-natural-rate-of-interest fame) (Malinvaud 2003), but using empirical data to support his conclusions. The function is properly called Cobb-Douglas regression because it is one of the first cases of using least squares regression and correlation analysis to understand economic relationships.
Using this panel data, they tried to analyze whether they could find a stable relationship among capital, labor, and total output. Unfortunately, Douglas’ project relies on truly heroic assumptions about the quality of data and the nature of the aggregate economy. The primitive state of data makes the regression analysis a garbage-in-garbage-out analysis. For example, back then, the reported capital employed was based on its original cost. Douglas used a cost index to adjust the data. The method somewhat alleviated the garbage-in-garbage-out problem but did not truly resolve it. Also, Douglas insisted on using only fixed assets (e.g. machinery and land) and ignored working capital (e.g. finished goods inventories, and raw materials). This method likely understated the use of capital.
Importantly, many prominent economists raised concerns about the usefulness of any conclusion at all. Even if Douglas reliably estimated a k (the Cobb-Douglas coefficient in the exponent), k might be incidental to the dataset and will not persist in time. The aggregate economy is an extraordinarily complex collection of industries and activities. If there is an aggregate k, how can we react to it? Can we use it to manage the economy? Also, the marginalist assumptions were being contested by economists then. The function implies a marginalist’s worldview (assuming instantaneous adjustments to prices as marginal costs change). Naturally it was incongruent with the zeitgeist.
It is worth pointing out that despite these criticisms, maybe out of pure statistical coincidence, the duo’s original estimate of k somehow persisted into later studies. In their paper A Theory of Production, Cobb-Douglas estimated that k (the share of Labor in aggregate output) for the period of study was roughly 0.75. Some later studies (like the one done by NBER) concluded a k of 0.741, a number remarkably close to Douglas’ original estimate.
It is noteworthy that Cobb himself published a short article Production in Massachusetts Manufacturing, 1890-1928 in the 1930s to revisit the value of k. He made two changes to the original Cobb-Douglas analysis:
1) estimating the coefficients for capital and labor separately, rather than assuming the two coefficients are linear (sum to one, which means constant elasticity);
2) using diagonal mean regression instead of linear regression.
However, Cobb ended up with estimates of k that are all over the place, including negative values and values greater than one. Since then, Cobb essentially abandoned the project. His paper was in essence a criticism of the Cobb-Douglas regression. However, people often forget about this nuance. Also curiously, Douglas never formally responded to Cobb’s criticisms.
As we can see from the above, the original Cobb-Douglas regression (I am using the word regression instead of function to highlight the statistical nature of the project) had many issues and one of the co-authors (Cobb) abandoned it. But how did the function itself obtain mainstream recognition?
There are many forces in politics and academia that contributed to the function’s ascension. One of the earlier ones was the Cowles Commission of Econometrics Research (currently at Yale). After moving to the University of Chicago in 1939 (recall that Douglas worked for the University of Chicago), the Commission began to pursue mathematical formalism. This is exemplified by the adoption of the Walrasian model of general equilibrium.
The Commission pushed for a “probabilistic” worldview instead of a deterministic worldview. What does that mean? In a deterministic worldview, the relationships between economic variables are deterministic: deviations are caused by measurement error. In a probabilistic worldview (sometimes called stochastic models), the actual relationships between the variables are subject to random fluctuations, but they fluctuate about a central tendency. The difference from measurement to measurement is caused by these random fluctuations. Hence we can tolerate the transient nature of estimates (which includes the Cobb-Douglas estimate of k).
Cowles Commission’s philosophy vastly expanded the use cases of statistical analysis which includes the Cobb-Douglas regression. But what truly propelled the function into unfamiliar territories was agricultural economists between 1944 and 1965. A group of economists out of Iowa State College were enthusiastic about the use of Cobb-Douglas in understanding optimizing input-output and input substitute relationships in agriculture. Douglas invented the regression analysis in aggregate economy; the agricultural economists were using the function at a farm level. Critically, Douglas was using regression for descriptive/positive analysis. He wanted to understand the world as-is. But the agricultural economists started to use the function as a normative tool (which by the way is the way it is used today in tokenomics).
It is worth pointing out that the Cobb-Douglas function incidentally got rid of its original criticisms through its adoption in agriculture economics. The quality of data is much better at a farm level. Also farm products are indeed homogenous; it is conceivable that there is a fixed ratio both among the ingredients themselves and the input-output. The function partly ditched its empirical baggage as it came out of agricultural economics.
However, the agricultural economists achieved limited success in using the Cobb-Douglas production function. Earl Heady (one of the leading agricultural economists that belongs to a normative Neoclassical School) concluded that “for the analysis on a multi-enterprise firm, linear programming was superior to the production function approach.” (Biddle 2021, 209)
Nonetheless, the Cobb-Douglas function continued to gain popularity. As we laid out in the Marginal Revolution above, the Neoclassical School of economics did not gain true popularity until perhaps the 1950s. The Cobb-Douglas function has a few mathematical properties that are quite useful for the Neoclassical assumptions:
1) It is formal math, expressed succinctly in a single formula,
2) The function assumes positive, but diminishing marginal returns,
3) It has a marginal rate of substitution between input ingredients (k represents the elasticity of an ingredient),
4) It models an input-output relationship,
5) It can be analyzed through least squares regression through empirical data.
Concurrently, general equilibrium analysis became quite popular in all sorts of realms, including industry analysis. The Cobb-Douglas function fits beautifully into the general equilibrium analysis, and is transformed into a utility function to model the relationships of consumer budget constraints.
Lastly, as mentioned above, Douglas had a long and successful career as a Senator of Illinois (1949-1967). It was during his senatorship that the Cobb-Douglas production function became a widely accepted tool for empirical research in economics.
Some further thoughts:
1) There is a process of product-market fit for tools. Many times the tools get used in vastly different use cases from its original one (sometimes decades later!);
2) Tools have their limitations and trade-offs. When we use something from the standard library, it is useful to think through the trade-offs in design. Cobb-Douglas’ strength is its mathematical form. The regression methodology is innovative but the function would not have gained popularity had it not been its mathematical properties. The limitation is the lack of strong empirical support for this function for what it intends to measure, whether on an aggregate level or industry level;
3) It takes an ecosystem to truly advance a tool. Had it not been the ascension of the Neoclassical School and its philosophical underpinnings, Cobb-Douglas would have been forgotten, most likely.
--------------
Prior to the 1870s, the subject of economics was dominated by the Classical School pioneered by David Hume and Adam Smith. There are many important elements in Classical theory, but the most relevant here is their theory about value. According to the Classical School, value is primarily determined by the cost of production. A house is more expensive than a piece of bread because it costs more to produce a house.
Since a Classical economist’s eye value is determined by the cost of production, they are naturally interested in the laws of production and laws of distribution. Land, labor, and capital are three factors of production; economists are interested in how these factors of inputs produce outputs. This is the backdrop of the Cobb-Douglas production function.
There are a couple of issues with the Classical School’s framework of value - the most famous one being the Paradox of Value. We all agree that water is more useful than diamond, but why is water close to being free and diamond so expensive? In order to explain the paradox, which in essence is the paradox of the divergence between utility value and exchange value, the Classical economists think it is because the exchange value is determined by the cost of production whereas utility value is not.
But, the theory has its apparent shortcomings (e.g. cases like the exchange value of high end alcohol, and autographed photos of celebrities). Also, Classical economists cannot explain why producers refuse to raise prices even when the cost of production goes up. Shouldn’t the cost of production determine the exchange value of a product?
Hence the emergence of the Neoclassical School and the “Marginal Revolution”. The Neoclassical School was still being incubated in the 1870s and did not gain mainstream popularity until the mid 20th century. They believed that the Classical School failed to pay attention to consumers when thinking about value. Sure, the cost of production influences value, but the key to address the Paradox of Value is consumers. Neoclassical economists started to develop a consumer theory to complement the Classical production theory. Values are subjective, determined by individual choice.
The Neoclassical School initiated “the Marginal Revolution.” While a Classical economist says that price is determined by supply and demand, a Marginalist says that price is determined by marginal forces. Marginal means relevant; hence, in the previous example of water and diamonds, water is less expensive because the marginal utility of water is lower than that of diamond.
Aside from a consumer theory, the Marginalists developed a new theory of distribution—marginal productivity—that explains price received by sellers of any specific factor of production. The Cobb-Douglas function has interesting properties that fit in this “marginal paradigm”: positive but diminishing marginal utility, a potential measurement of substitution effects, among others.
The theory of marginal productivity was contested by the mainstream economists then, since the theory “required instantaneous and complete adjustment of factor prices to change in output prices and vice versa, something that clearly was not the case.” (Biddle 2021, 30) This is a fair criticism. It became one of the key criticisms against the Cobb-Douglas production function.
The Neoclassical School also pioneered the usage of math in economics. Although only some of the Neoclassical economists believed in math, the true believers held that the first language of economics should be math. This was quite controversial (and still is today), but eventually the Neoclassical School took over academia and the first language of economics indeed became math. As we can see later on, had the Neoclassical School not become mainstream, Cobb-Douglas would have likely fallen into oblivion (after all, it is math!).
It is important to keep in mind that during the entirety of Paul Douglas’ career, the Neoclassical School was not a mainstream school. During the interwar period (1920s - 1940s), there was a time when multiple schools of thought flourished (the pluralist era). Paul Douglas himself actually would not have agreed to the premises of the newer schools and the creative (and prevalent) application of his now famous function. He wanted to address an aggregate production problem, but the function got borrowed into modeling firm and industry behaviors (which is the use case in tokenomics today).
Paul Douglas spent most of his career in Illinois, both in the University of Chicago and as a Senator of Illinois (for 18 years). Douglas was interested in measuring input-output relationships in aggregate production, especially through the usage of statistical data. Douglas himself was not a skilled mathematician, so he enlisted help from Charles Cobb, a mathematician that spent most of his career in Amherst College.
While the collection of data is a norm and institutionalized today, it was not the case back in Cobb and Douglas’ time. Actually, Douglas’ key contribution was not the function itself (the function was first conceived by Knut Wicksell, of the Wicksellian-natural-rate-of-interest fame) (Malinvaud 2003), but using empirical data to support his conclusions. The function is properly called Cobb-Douglas regression because it is one of the first cases of using least squares regression and correlation analysis to understand economic relationships.
Using this panel data, they tried to analyze whether they could find a stable relationship among capital, labor, and total output. Unfortunately, Douglas’ project relies on truly heroic assumptions about the quality of data and the nature of the aggregate economy. The primitive state of data makes the regression analysis a garbage-in-garbage-out analysis. For example, back then, the reported capital employed was based on its original cost. Douglas used a cost index to adjust the data. The method somewhat alleviated the garbage-in-garbage-out problem but did not truly resolve it. Also, Douglas insisted on using only fixed assets (e.g. machinery and land) and ignored working capital (e.g. finished goods inventories, and raw materials). This method likely understated the use of capital.
Importantly, many prominent economists raised concerns about the usefulness of any conclusion at all. Even if Douglas reliably estimated a k (the Cobb-Douglas coefficient in the exponent), k might be incidental to the dataset and will not persist in time. The aggregate economy is an extraordinarily complex collection of industries and activities. If there is an aggregate k, how can we react to it? Can we use it to manage the economy? Also, the marginalist assumptions were being contested by economists then. The function implies a marginalist’s worldview (assuming instantaneous adjustments to prices as marginal costs change). Naturally it was incongruent with the zeitgeist.
It is worth pointing out that despite these criticisms, maybe out of pure statistical coincidence, the duo’s original estimate of k somehow persisted into later studies. In their paper A Theory of Production, Cobb-Douglas estimated that k (the share of Labor in aggregate output) for the period of study was roughly 0.75. Some later studies (like the one done by NBER) concluded a k of 0.741, a number remarkably close to Douglas’ original estimate.
It is noteworthy that Cobb himself published a short article Production in Massachusetts Manufacturing, 1890-1928 in the 1930s to revisit the value of k. He made two changes to the original Cobb-Douglas analysis:
1) estimating the coefficients for capital and labor separately, rather than assuming the two coefficients are linear (sum to one, which means constant elasticity);
2) using diagonal mean regression instead of linear regression.
However, Cobb ended up with estimates of k that are all over the place, including negative values and values greater than one. Since then, Cobb essentially abandoned the project. His paper was in essence a criticism of the Cobb-Douglas regression. However, people often forget about this nuance. Also curiously, Douglas never formally responded to Cobb’s criticisms.
As we can see from the above, the original Cobb-Douglas regression (I am using the word regression instead of function to highlight the statistical nature of the project) had many issues and one of the co-authors (Cobb) abandoned it. But how did the function itself obtain mainstream recognition?
There are many forces in politics and academia that contributed to the function’s ascension. One of the earlier ones was the Cowles Commission of Econometrics Research (currently at Yale). After moving to the University of Chicago in 1939 (recall that Douglas worked for the University of Chicago), the Commission began to pursue mathematical formalism. This is exemplified by the adoption of the Walrasian model of general equilibrium.
The Commission pushed for a “probabilistic” worldview instead of a deterministic worldview. What does that mean? In a deterministic worldview, the relationships between economic variables are deterministic: deviations are caused by measurement error. In a probabilistic worldview (sometimes called stochastic models), the actual relationships between the variables are subject to random fluctuations, but they fluctuate about a central tendency. The difference from measurement to measurement is caused by these random fluctuations. Hence we can tolerate the transient nature of estimates (which includes the Cobb-Douglas estimate of k).
Cowles Commission’s philosophy vastly expanded the use cases of statistical analysis which includes the Cobb-Douglas regression. But what truly propelled the function into unfamiliar territories was agricultural economists between 1944 and 1965. A group of economists out of Iowa State College were enthusiastic about the use of Cobb-Douglas in understanding optimizing input-output and input substitute relationships in agriculture. Douglas invented the regression analysis in aggregate economy; the agricultural economists were using the function at a farm level. Critically, Douglas was using regression for descriptive/positive analysis. He wanted to understand the world as-is. But the agricultural economists started to use the function as a normative tool (which by the way is the way it is used today in tokenomics).
It is worth pointing out that the Cobb-Douglas function incidentally got rid of its original criticisms through its adoption in agriculture economics. The quality of data is much better at a farm level. Also farm products are indeed homogenous; it is conceivable that there is a fixed ratio both among the ingredients themselves and the input-output. The function partly ditched its empirical baggage as it came out of agricultural economics.
However, the agricultural economists achieved limited success in using the Cobb-Douglas production function. Earl Heady (one of the leading agricultural economists that belongs to a normative Neoclassical School) concluded that “for the analysis on a multi-enterprise firm, linear programming was superior to the production function approach.” (Biddle 2021, 209)
Nonetheless, the Cobb-Douglas function continued to gain popularity. As we laid out in the Marginal Revolution above, the Neoclassical School of economics did not gain true popularity until perhaps the 1950s. The Cobb-Douglas function has a few mathematical properties that are quite useful for the Neoclassical assumptions:
1) It is formal math, expressed succinctly in a single formula,
2) The function assumes positive, but diminishing marginal returns,
3) It has a marginal rate of substitution between input ingredients (k represents the elasticity of an ingredient),
4) It models an input-output relationship,
5) It can be analyzed through least squares regression through empirical data.
Concurrently, general equilibrium analysis became quite popular in all sorts of realms, including industry analysis. The Cobb-Douglas function fits beautifully into the general equilibrium analysis, and is transformed into a utility function to model the relationships of consumer budget constraints.
Lastly, as mentioned above, Douglas had a long and successful career as a Senator of Illinois (1949-1967). It was during his senatorship that the Cobb-Douglas production function became a widely accepted tool for empirical research in economics.
Some further thoughts:
1) There is a process of product-market fit for tools. Many times the tools get used in vastly different use cases from its original one (sometimes decades later!);
2) Tools have their limitations and trade-offs. When we use something from the standard library, it is useful to think through the trade-offs in design. Cobb-Douglas’ strength is its mathematical form. The regression methodology is innovative but the function would not have gained popularity had it not been its mathematical properties. The limitation is the lack of strong empirical support for this function for what it intends to measure, whether on an aggregate level or industry level;
3) It takes an ecosystem to truly advance a tool. Had it not been the ascension of the Neoclassical School and its philosophical underpinnings, Cobb-Douglas would have been forgotten, most likely.
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