Posts Tagged ‘production functions’

thoughts on economic models

Sunday, July 10th, 2016

A typical, very simple economic model is a demand curve, Dx = f(Px). It states that, other things equal, an increase in the price of x will result in a decrease in demand for x. I other things are not equal, the assumptions of the model do not hold. Suppose, writes blogger Arnold Kling, that x stands for tuition charged by a school, and we observe that demand went up rather than down when tuition increased. What may have happened is that scholarships became more generous, so the full tuition – net of student aid – fell. Other things did not remain equal.

Most economic models contain more variables. Consider the aggregate production function X = f(K,L). The quantity of output is a function of the quantity of physical capital plus the quantity of labour.

[This model] is used to predict that differences in output per worker will be proportionate to differences in capital per worker. When this fails, there are many possible reasons: workers may differ in their human capital; physical capital may not be measured or aggregated correctly; output may not be measured or aggregated correctly; institutional differences may matter. etc.

In fact, the primary use of the aggregate production function model is to examine its failure, which is called “the residual.” Economists place an interpretation on this residual, calling it “total factor productivity.” They interpret the rate of change in this residual over time as “productivity growth.” They interpret the change in the rate of change in this residual as “change in the trend rate of productivity growth.”

Arnold Kling, “Thoughts on the use of Models in Economics“, askblog, 9 July 2016.

Arnold’s explanation of how economists use models is useful, but I think he is too easy on model-building. It is very difficult – probably impossible – to measure output, labour and capital without using prices. This is necessary, because if prices are used, the model becomes close to a tautology. The value of the quantity demanded, for example, is always equal to the value of the quantity supplied (plus or minus abnormal profits or losses). Similarly, the value of output is equal to the cost of the rental of machines plus the wages of labour (adjusted for any abnormal profit or loss).

Think how difficult it is to measure (without prices!) the heterogeneous output of a factory, or the hours of equivalently productive workers. Physical capital is even more difficult to measure. Without prices or interest rates, what is the unit of measure. Kilos of capital? Number of shovels, hammers or machines?

Economic models are helpful, but only as toy models, useful for understanding, but not predicting, how a real economy functions.

See here for past posts on production functions.

does human (and physical) capital exist?

Wednesday, February 25th, 2015

In the heated bloggers’ debate over human capital, I was surprised by the absence of discussion of how to measure human capital, a point that was crucial in the earlier “Cambridge controversy” over physical capital. Typically, human capital is measured as years of schooling. All this is added up (sometimes with adjustment for levels of schooling – counting a year in High School or University as more than a year in primary school). The sum total is then taken to be a nation’s stock of ‘human capital’, which becomes an input into an aggregate production function for the economy. This seems wrong, to me. Some schools are terrible, whereas others are excellent. How can we take account in differences in the quality of schooling?

Nick Rowe has a clever solution to the problem. “Capital” – whether physical or human – is not really a thing, so it is not necessary (impossible?) to measure it.

We invest in increasing our skills, our strength, our knowledge, etc. It just easier to use the words “human capital” to describe the things we have invested in in the past. And we can talk about how our human capital depreciates too, as we forget stuff, or technology changes so our skills become obsolete.

Now the word “capital” is deeply problematic, because time has many future periods, and so trying to convert a vector into a single scalar number is….tricky. That is true for both human and non-human capital. I prefer to say that “capital” isn’t really a thing; it’s just a name we give to a production process where time matters, because inputs and outputs come at different times. And once you start to think of “capital” as just the time-structure of production, then brain surgeons are just as much capital as blast furnaces. Costs come first, and benefits come later.

Nick Rowe, commenting on Max Speak, “For Noah and Nick”, 24 February 2015.

I find it difficult to understand Nick’s argument. If capital is not a ‘thing’, how can it be an input producing output “in a production process where time matters”? If it is an input, it is certainly not a measurable output.

The implication is that we should do away with the aggregate production function, and much of econometrics. I am sympathetic to this idea, and will give it more thought.

For much, much more, see

Nick Rowe, “Does capital income exist?“, WCI, 24 February 2015, and

Nick Rowe, “Ours the task eternal — investing in human capital“, WCI, 24 February 2015.

the human capital controversy

Sunday, February 22nd, 2015

Back in the 1960s there was a famous debate between economists (led by Joan Robinson and Piero Sraffa) of the University of Cambridge in England and economists (led by Paul Samuelson and Robert Solow) of MIT in Cambridge, Massachusetts. The debate, known as “the Cambridge capital controversy“, was over measurement and aggregation of physical capital. The Cambridge (England) economists argued that aggregate physical capital could not be measured without reference to the rate of return on capital. Cambridge (Massachusetts) generally agreed that the Cambridge (England) side won, though many professors of economics continue to teach aggregate production functions and economic growth theory as though the debate never took place.

A similar debate is now taking place, over human rather than physical capital. Noah Smith (HT Mark Thoma) provides a nice overview for those are interested.

Is “human capital” really capital? This is the topic of the latest econ blog debate. Here is Branko Milanovic, who says no, it isn’t. Here is Nick Rowe, who says yes, it is. Here is Paul Krugman, who says no, it isn’t. Here is Tim Worstall, who says yes, it is. Here is Elizabeth Bruenig, who says that people who say it is are bad.

Noah Smith, “Is human capital really capital?“, Noapinion, 21 February 2015.

Noah Smith offers an alternative view: human capital requires owners to work (give up leisure time) to obtain a return from it, so the more leisure is valued relative to other things, the less valuable human capital is. This will be different for each person. In consequence, you are “entitled to your own modeling conventions and definition of terms. So whether human capital is capital is up to you.”

This is an interesting, complex debate. I am still thinking about it but, as TdJ readers might predict, I am most persuaded by the arguments of Carleton University economist Nick Rowe. Before turning to Branko Milanovic and Nick Rowe, however, I would like to emphasize two points that are not always appreciated by participants in this debate. First, financial capital is not capital in an economic sense. Nick makes this point clearly, but others confuse financial capital with physical capital. Financial capital – stocks, bonds and the like – are just pieces of paper, IOUs. They are claims of lenders, and the loans may even have been made for the purpose of consumption rather than investment.

Second, even if human capital is a useful category of income-producing assets (and I think it is), it is as difficult to measure as physical capital is. In fact, it is probably even more difficult to measure. This does not really matter though, as it is impossible to measure aggregate assets of either asset apart from (only in theory!) the present value of the future income the assets produce. The problems of measurement of human capital are  very similar to the problems of measurement of physical capital. For example, if I purchase an automobile which I use for pleasure, and also – as an Uber driver – to generate income, part of the purchase represents investment (addition to physical capital) and part is consumption. Similarly, part of the expense of schooling represents investment (for the purpose of earning more income than I would without skills) and part is consumption (the satisfaction of obtaining knowledge and the ability to better understand the world in which I live).

There is much, much more at the links above. Bloggers will no doubt continue to debate this issue for weeks and months (years?) to come. Here, to get you started, are brief quotes from Branko Milanovic (on the ”No’ side of the debate, and from Nick Rowe (on the ‘Yes’ side, the one that I support):

If “human capital” and “real” capital are the same thing, how can there be a conflict between labor and capital?  If profits and wages are the same thing, why should we fight about distribution? You have your form of capital (which just happens to look like labor), and I have mine, which just happens to look like T-bills and stocks.

Branko Milanovic, “On ‘human capital’ one more time“, Global Inequality, 19 February 2015.


What we call “labour” is as much capital as labour. The wages on “labour” are as much a return to capital as they are a return to raw labour.

Some labour needs very little investment to make it productive; other labour requires a lot. Some labour gives a high return on investment; other labour gives a low return. It’s all different.

“Human capital” is not a synonym for “labour”. It tells us something important about the investment needed to make labour productive.

Nick Rowe, “Human Capital” and “Land Capital“, Worthwhile Canadian Initiative, 14 February 2015.

Once again, I encourage you to click on the links above, to get a feel for the full debate.

education and growth

Friday, September 5th, 2014

[G]overnments seem convinced that the best way to [stimulate employment and growth] … is to increase the number of students pursuing degrees in the so-called “STEM” subjects (science, technology, engineering, and mathematics). Are they right?

The short answer is no. ….

[T]he case for STEM education is … fundamentally flawed, because it treats an economy as an equation. According to this logic, job creation is a matter of slotting humans into identifiable opportunities, and economic growth is a matter of increasing the stock of human or physical capital, while exploiting scientific advances. This is a dark view of modern economies, and a depressing blueprint for the future. ….

What economies need instead is a boost in dynamism. …. [E]conomies today lack the spirit of innovation. Labor markets do not need only more technical expertise; they require an increasing number of soft skills, like the ability to think imaginatively, develop creative solutions to complex challenges, and adapt to changing circumstances and new constraints. ….

A necessary first step is to restore the humanities in high school and university curricula. Exposure to literature, philosophy, and history will inspire young people to seek a life of richness – one that includes making creative, innovative contributions to society. ….

The humanities describe the ascent of the modern world. Countries worldwide can use the humanities to develop or revive the economies that drove this ascent, while helping individuals to lead more productive and fulfilling lives.

Edmund S. Phelps, “Teaching Economic Dynamism“, Project Syndicate, 2 September 2014.

Columbia University economist Edmund Phelps is a Nobel laureate and author of many books, one of which is a plea for wage subsidies: Rewarding Work: How to Restore Participation and Self-Support to Free Enterprise (Harvard University Press, 1997; second edition 2007).

Professor Phelps avoids the term “aggregate production functions”, but clearly intends to criticize them when he describes the depressing way that the economy is treated as an equation, limited to “a matter of increasing the stock of human or physical capital, while exploiting scientific advances”.

the Cobb-Douglas production function

Thursday, May 10th, 2012

The current issue of the Journal of Economic Perspectives (open access) has a 14-page essay on the Cobb-Douglas regression, a popular form of aggregate production function. About time, I thought, that someone writing in a popular journal exposed this work-horse of econometrics for the fraud that it is. I accessed the essay with great anticipation, only to find it full of praise, with very light – almost non-existent – criticism. Here are the essay’s two concluding sentences:

There remain open questions about the scientific value of this procedure in each of the contexts in which it is applied, some of which are variations of the friendly and unfriendly questions raised by Douglas’s initial critics. However, measured by the extent to which it has been embraced, applied, and elaborated upon by subsequent economists, Douglas’s innovative 1927 idea that one could use statistical analysis to uncover meaningful empirical relationships between inputs and outputs, as well as his specific implementation of that idea using the Cobb–Douglas functional form and least squares regression, was an overwhelming success.

Jeff Biddle, “The Introduction of the Cobb–Douglas Regression“, Journal of Economic Perspectives 26:2 (Spring 2012), pp. 223-236.

Michigan State University economist Jeff Biddle took to heart this advice of MIT economist Franklin Fisher:

[A]ttempts to explain the impossibility of using aggregate production functions in practice are often met with great hostility, even outright anger. To that I say … that the moral is: “Don’t interfere with fairytales if you want to live happily ever after.”

Franklin M. Fisher, “Aggregate Production Functions – A Pervasive, but Unpersuasive, Fairytale“, Eastern Economic Journal 31:3 (Winter 2005), pp. 489-491.

Nowhere does Professor Biddle mention the most damning criticism of aggregate production functions (including the Cobb-Douglas variant): their good fit to empirical data is a statistical artifact – a result of the fact that the functions reflect  the accounting identity between the values of inputs and outputs. In other words, aggregate production functions are almost tautologies – true by definition! This was pointed out independently by two Nobel laureates – Paul Samuelson and Herbert Simon – in articles that were published in 1979, and subsequently ignored by virtually everyone. Here are short quotes from each article:

It is a late hour to raise these doubts about the Emperor’s clothes, but ….

Why use the words “production function” for such an accounting-tautology … ?

Paul A. Samuelson, “Paul Douglas’s Measurement of Production Functions and Marginal Productivities“, Journal of Political Economy 87:5, Part 1 (October 1979), pp. 923-939.


Empirical data on the Cobb-Douglas and ACMS [Arrow, Chenery, Minhas and Solow] production functions have been alleged to provide substantial support for the classical theory of the firm–so substantial that further testing of that theory, as distinguished from elaboration of its detail, was no longer necessary. An examination of the evidence suggests instead that the observed good fit of these functions to data … are very likely all statistical artifacts. The data say no more than that the value of the product is approximately equal to the wage bill plus the cost of capital services. This interpretation of the statistical findings is plausible for both interindustry cross-sectional studies and time-series studies, the latter for either a single industry or a whole economy. (p. 469)

Herbert A. Simon, “On parsimonious explanations of production relations“, Scandinavian Journal of Economics 81:4 (1979), pp. 459-474.

Professor Biddle cites Samuelson’s article, but fails to mention Samuelson’s criticism of the Cobb-Douglas function. Biddle does not even cite Herbert Simon in his essay.

For more posts on this subject, click on the production functions tag.

Solow on education

Thursday, June 16th, 2011

MIT economist Robert Solow participated in a recent IMF conference on “Macro and Growth Policies in the Wake of the Crisis”. Camilla Andersen interviewed him for the IMF publication Finance & Development, asking “What is needed to put people back to work? The role of education in the economic growth of middle-income and low-income countries is an important issue.”

Here is Professor Solow’s response:

We economists tend to measure education by input, not output. We count how many years people have been in school. Instead of worrying so much about quantities of education, we ought to be thinking about the content of the education. What is it that primary school or secondary school kids in poor and middle-income countries need to know? This is not necessarily what they are being taught.

And by the way, the same holds for advanced countries and the United States. We measure our success in generating an educated population in terms of the fraction of the age group that is in college. I would be very interested in other kinds of postsecondary education that are skills-based and would equip people for the jobs that are likely to be available.

That is going to require that employers be involved in the planning of that sort of education. For the United States, and perhaps for much of the world, that is a wholly new idea.

Camilla Andersen, “Rethinking Economics in a Changed World“, Finance & Development (June 2011).

Anderson interviewed two other Nobel laureates – NYU economist Michael Spence and Columbia economist Joseph Stiglitz – and reports their comments as well.

Schooling is often included as an explanatory variable in models of economic growth, because it is believed to be an important determinant of technical progress.

Robert Solow (born 1924) is famous for the “Solow residual”, known also as “total factor productivity”, which is assumed to be a measure of technical change. More accurately, it is what is left ‘unexplained’ after regressing GDP on inputs, i.e. the residual of an aggregate production function.

TdJ has insisted, in numerous posts, that aggregate production functions – and measures derived from them – can only be understood as faith, not science. These posts are titled “economics as faith”; one of them focuses on attempts to measure technical change.

natural resources and economic theory

Tuesday, July 13th, 2010

Martin Wolf questions whether it makes sense for theorists to merge natural resources with manufactured capital. This has been the norm ever since neo-classical economics triumphed over classical economics, about a century ago.

In moving from classical to neo-classical economics — the dominant academic school today — economists expunged land — or natural resources [incorporating them into capital]. ….

Yet it would seem to me that this way of thinking by economists is no longer sensible, if it ever was. Land must again be treated as separate from labour and capital.

First, resource scarcity is an increasingly pressing issue. It shows up in concerns over pollution (including global warming), in the discussion of “peak oil” and so forth. The idea that diminishing returns will become a more significant factor in the next century than in the past two seems to me to be compelling, now that modern economic growth has spread across the globe. So we need to return to economic models that incorporate resources, as a matter of course.

Second, in a globalised economy, taxing labour and capital will become increasingly difficult. That leaves land. The Australian government is right to want to extract the full rental value of its mineral resources for the benefit of the Australian people. Similarly, the people of the UK should wish to extract the rental value of London for their own use. The benefits of infrastructure investments that make London more productive would automatically be recouped if land rents were heavily taxed. Meanwhile, the taxation of capital and land [labour!] could be reduced.

Martin Wolf, “Why were resources expunged from neo-classical economics?”, Martin Wolf’s Exchange, 12 July 2010.

Despite the unfortunate typo (“land” instead of “labour”), Martin provides a splendid introduction to an important topic. (The extract above is only a small part of this introduction.)

Martin Wolf’s Exchange is open to all readers, but you must open a free account with to post a comment.

labour productivity

Saturday, March 6th, 2010

An op-ed published in today’s New York Times takes the US Labor Department to task for grossly overstating labour productivity, “especially in the nation’s manufacturing sector”.

Productivity measures how many worker hours are needed for a given unit of output during a given time period; when hours fall relative to output, labor productivity increases. In 2009, the data show, Americans needed 40 percent fewer hours to produce the same unit of output as in 1980.

But there’s a problem: labor productivity figures, which are calculated by the Labor Department, count only worker hours in America, even though American-owned factories and labs have been steadily transplanted overseas, and foreign workers have contributed significantly to the final products counted in productivity measures.

The result is an apparent drop in the number of worker hours required to produce goods — and thus increased productivity. But actually, the total number of worker hours does not necessarily change.

Alan Tonelson and Kevin L. Kearns, “Trading Away Productivity”, New York Times, 6 March 2010.

This column has value only as a discussion starter for a basic economics course. Instructors might ask students to look for elementary errors in reasoning, of which there is no shortage.

The basic flaw is glaringly simple. Productivity, it is true, is measured as units of output (in practice, value of output) divided by worker hours. But economists measure output as value-added by capital and labour, not as the value of final products. The output of an automobile assembly plant, for example, is measured as the value of the final output less the cost of all purchased parts and paint that go into a fully assembled vehicle, leaving only the value added by workers and the capital equipment they operate. Whether the intermediate parts are purchased locally or imported from afar does not matter for calculation of output, hence productivity.

If the Labor Department actually measured productivity the mistaken way claimed in this column, that would indeed be news!

Kevin L. Kearns has a J.D. from Brooklyn Law School and is the president of the United States Business and Industry Council, an association of small manufacturers. Alan Tonelson, a fellow at the council, holds “a B.A. with highest honors in history from Princeton University” and is author of The Race to the Bottom: Why a Worldwide Worker Surplus and Uncontrolled Free Trade are Sinking American Living Standards (Basic Books, 2000). Neither of the two is a trained economist – fortunately. Otherwise their column would be very embarrassing for the economics profession.

Thanks to Jan Sendzimir for the pointer.

innovation is not R&D

Wednesday, December 16th, 2009

But is R&D innovation? John Kay explains.

When we talk about innovation, we visualise men and women in white coats with test tubes and microscopes. Outside many university cities around the world there are biotechnology estates established by governments that believe high technology is the key to a competitive future. The funds that governments provide to support innovation are all too often appropriated by large companies that are better at forming committees to pontificate about what the global village will want in the future than they are at assessing what their customers want today. ….

Last month the [UK’s] National Endowment for Science, Technology and the Arts picked up this point. For years research and development scorecards have dutifully recorded how much pharmaceuticals companies spend on the search for new drugs and the expenditure of governments on defence electronics. But a Nesta report, presenting plans for a new innovation index has now recognised that most of the spending that promotes innovation does not take place in science departments. The financial services industry may have been Britain’s most innovative industry in the past two decades – perhaps too innovative – but practically none of the expenditure behind that innovation comes under “R&D”. And the same is true in retailing, media and a host of other innovative industries.

John Kay, “Innovation is not about wearing a white coat”, Financial Times, 16 December 2009.

An ungated version of this column will soon be posted here.

A pilot version of pilot version of NESTA’s Innovation Index can be downloaded here.

Data will be added over the next 12 months, and the Index will be extended to incorporate public sector innovation. NESTA promises then to update the Index each year with new data.

I was excited and eager to learn more, but became very disillusioned by the time I reached page 14 of the pilot report:

The second component is what macroeconomists describe as Total Factor Productivity (TFP). This is the measure of productivity growth that is not accounted for by the growth in factor inputs, such as physical capital or labour quality, and is generally attributed to better ways of doing things, including the broader benefits of technological advances and improved processes. In the approach used in the Innovation Index, in which the private benefits of investments such as R&D are captured separately, TFP includes the spillover benefits of innovation investment.

This methodology shows that between 2000 and 2007, labour productivity grew at an annual average of 2.7 per cent per year. Innovation contributed 1.8 per cent, or approximately two-thirds of the growth experienced.

NESTA, “The Innovation Index: Measuring the UK’s investment in innovation and its effects”, November 2009.

TFP is the residual that is left after regressing output on inputs, i.e. the residual of an aggregate production function. I have discussed all the problems with this methodology in a series of seven thoughts titled “economics as faith”. To locate these posts, type “economics as faith” into the search bar, or click on the “production functions” tag.

The first component of the Index is somewhat better. It consists of adding up all expenditure on R&D (really!), plus all expenditure on design, training, market research, etc. etc.. “However, R&D represents only 11 per cent of the investment in innovation ….” In other words, 89% of all innovation expenditure is excluded from the R&D budgets of private firms. So far, so good. But NESTA then uses “a growth accounting approach … to understand the effect of these [expenditures] on productivity growth.” This requires measures of aggregate productivity, reliance again on “economics as faith”.

I did not look at the third and last component of the Index, “a set of metrics that can be tracked to assess how favourable a climate the UK is for innovation”, so will not comment on that. I also do not know how – or whether – the three components will be combined to form a single index.

economics as faith (7)

Monday, October 12th, 2009

The data of most economies are filled with apparently inconsistent series. By choosing among them, one can produce almost any estimate of productivity growth imaginable.

Alwyn Young “Gold into Base Metals: Productivity Growth in the People’s Republic of China during the Reform Period”, Journal of Political Economy 111:6 (2003), p. 22.

As Professor Young acknowledges, all growth accounting exercises should be taken with more than a grain of salt. Nonetheless growth accounting is a growth industry for academics. A recent study of China and India reaches new lows, however, in presenting questionable findings without providing the reader with caveats of any kind. The American Economic Association published it last year in their prestigious Journal of Economic Perspectives. Somehow it made it past the editors. I report only on the authors’ adjustment of labour input for skill levels – a glaring example of the general low quality of the piece.

Growth accounting provides a framework for allocating changes in a country’s observed output into the contributions from changes in its factor inputs—capital and labor—and a residual, typically called total factor productivity. ….

This approach is based on a production function in which output is a function of capital, labor, and a term for total factor productivity. …. [L]abor … is adjusted for … skills; we use average years of schooling as a proxy for skill levels and assume a constant annual return of 7 percent for each additional year of education.

Young (2003) provides a useful overview of Chinese statistics on educational attainment …. [H]is analysis of the relationship between earnings and years of schooling finds surprisingly low returns. ….

As noted earlier, our human capital index assumes that each additional year of schooling raises labor force productivity [in China and India] by 7 percent [even though Young (2003, p. 1246) found effects a fraction of that size for China.] This [7%] figure is based on a large number of empirical studies relating wages and years of schooling.

Barry Bosworth and Susan M. Collins, “Accounting for Growth: Comparing China and India”, Journal of Economic Perspectives 22:1 (Winter 2008), pp. 45–66.

The empirical studies alluded to are ‘Mincer equations’ – the regression of wage rates on levels of schooling and (sometimes) experience – and there are literally thousands of these studies. They almost always show that more schooling is associated with higher wages, but this does not prove that schooling increases productivity.

Suppose that schools exist only to screen students for ability, and that school attendance has no effect at all on productivity. In such a world, schooling is privately profitable but socially wasteful. Workers who complete more years of schooling are more productive and enjoy higher wages than workers who drop out of school. But increasing everyone’s consumption of schooling by a year has no effect on productivity or wages! Once there is a large pool of ‘schooled’ workers, employers will find that they have to demand a university degree for jobs that used to require only a high school diploma, because completion of high school no longer signals sufficient intelligence to handle the job. All this is well-known but is disregarded by the authors, who literally pull a 7% figure out of the air and inappropriately apply cross-section regression results to macro models of growth.

Barry Bosworth is Senior Fellow of Brookings Institution in Washington, DC. Susan M. Collins is Dean of Public Policy, Gerald R. Ford School of Public Policy, University of Michigan.