It’s done. I’m in. I mean, I am in discussing the Cobb-Douglas production function, which seems to be some sort of initiation path for any economist who wants to be considered as a real scientist. Yesterday, in my update in French, I started tackling the issue commonly designated in economics as ‘the enigma of decreasing productivity’. Long story in short words: capital and labour work together and generate the final output of the economy, they contribute to said final output with their respective rates of productivity, and those factor contributions can be summed up in order to calculated the so-called Total Factor Productivity, or TFP for close acquaintances. With all those wonders of technology we have, like solar panels, Hyperloop, as well as Budweiser beer supplying rigorously the same taste across millions of its bottles and cans, we should see that TFP rocketing up at an exponential rate. The tiny little problem is that it is actually the opposite. The database I am quoting and using in my own research so frequently, namely Penn Tables (Feenstra et al. 2015), is very much an attempt at measuring productivity in a complex way. I made a pivot out of that database, focused exclusively on TFP. You can find it here https://drive.google.com/file/d/0B1QaBZlwGxxAZ3MyZ00xcV9zZ1U/view?usp=sharing and you can see by yourself: since 1979, total productivity of production factors has been consistently falling.
There are a few interesting things about that tendency in the TFP to fall: it seems to be correlated with decreasing a velocity of money, and with increasing a speed of depreciation in fixed assets, but it also seems to be structurally stable. What? How can I say it is structurally stable? Right, it deserves some explanation. Good, so I explain my point. In that Excel file I have just given the link to, I provide the mean value of TFP for each year, across the whole sample of countries, as well as the variance of TFP among these countries. Now, when you take the square root of variance (painful, but be brave), and divide it by the mean value, you obtain a coefficient called ‘variability of distribution’. In a set of data, variability is a proportion, or, if you want, a morphology, like arm length to leg length in the body of Michael Angelo’s David. In statistics, it is very much the same as in sculpture: as long as the basic proportions remain the same, we have more or less the same body on the pedestal, give or take some extra thingies (clothes on-clothes off, leaf on or off, some head of a mythical monster cut off and held in one hand etc.). If the coefficient of variability in a statistical distribution remains more or less the same over time, we can venture to hypothesise that the whole thing is structurally stable.
If you care to analyse that pivot of TFP that I placed on my Google disc (link provided earlier in this update), you will see that the variability of cross-sectional distribution in TFP remains more or less constant over time, and very docile by the way, between 0,3 and 0,6. Nothing that the government should be informed about, really. So we have a diminishing trend in a structurally stable spatial distribution. Structure determines function: it is plausible to hypothesise that it is the geography of production, understood as a structure, which produces this diminishing outcome. This is an extremely interesting path to follow, which, by the way, I already made a few shy steps into (see http://researchsocialsci.blogspot.com/2017/08/life-idea-research-some-facts-bloody.html ).
Whatever the interest in studying empirical data about TFP as such, I decided to track back the way we approach the whole issue, in economic sciences. I decided to track it back in the good old biblical way, back to its ancestry. The whole concept of Total Factor Productivity seems to originate from that of production function, which, in turn, we owe to Prof Charles W. Cobb and Prof Paul H. Douglas, in their common work from 1928. As their paper is presently owned by the JSTOR library, I cannot link to it on my disk. Still, I can make available for you the documents, which seem to have been the prime sources of information for prof. Cobb and prof. Douglas, and this can be even more fun, as it shows the context of their research, in all its depth and texture. The piece of empirical data, which seems to have really inspired their whole work seems to be a report issued in 1922 by the US Department of Commerce, Bureau of the Census, and entitled ‘Wealth, Public Debt, and Taxation: 1922. Estimated National Wealth’. You can find it at my Google Disc, here: https://drive.google.com/file/d/0B1QaBZlwGxxAdWFhUE83eEdSbHM/view?usp=sharing . Besides, the two scientists seem to have worked a lot (this is my interpretation of their paper) with annual reports issued by the Federal Trade Commission. I found the report for the fiscal year ended on June the 30th, 1928, so basically published when that paper by prof Cobb and prof Douglas had already been released. You can find it there: https://drive.google.com/file/d/0B1QaBZlwGxxAbXRlS1JBZk51YUE/view?usp=sharing .
Provisionally, that Census report from 1922 seems to be The Mother of All Production Functions, as it made prof Cobb and prof Douglas work for six years (well, five, they must have turned the manuscript in at least half a calendar year before publication) on their concept of production function. So I open that report and try to understand what did the poet mean. The foreword starts with the following statement: ‘When the statistician attempts to measure the wealth of a nation, he encounters two distinct difficulties: First, it is hard to define the term “wealth”; second, it is by no means easy to secure the needed data’. Right, so the headache, then, back in the day, consisted in defining the substance of wealth. Interesting. Let’s stroll further.
The same foreword formulates an interesting chain of ideas, fundamental to our present understanding of economics. Firstly, ‘wealth’ points at two distinct ideas. Firstly, private individuals have some private wealth, and, secondly, the society as a whole has some social wealth. Private wealth is, using the phrasing of the report, practically coextensive and nearly equal in value with private property. The value of private property is the market value of the corresponding legal deeds (money, bonds, stocks etc.) minus the debt burdening the holder of those titles. On the other hand, we have social wealth, and here, it becomes really interesting. The report states: ‘Social wealth includes all objects having utility, that is, all things which people believe will minister to their wants either immediately or in the not too distant future. In this category are included not only those goods which are scarce or which cost money, but also those which are free, as, for example, water, air, the sun, beautiful scenery, and all those gifts of nature which gratify our desires. This is the kind of wealth to which we generally refer when we say that a nation is wealthy or opulent. It is the criterion that should be used if we wish to ascertain whether a nation is becoming richer or poorer. No other concept of wealth is more definite or more real, yet, from the standpoint of the statistician, this definition of wealth has one very serious drawback – no one has yet devised a satisfactory unit which can be applied practically in measuring the quantity of social wealth’.
Gotcha, Marsupilami! We are cornered with the concept of social wealth, or utility in objects we have and make. My intuition is that it was the general point of departure for prof Cobb and prof Douglas. Why? Well, let’s quickly read the introductory part of their 1928 paper. The two authors state their research interest in the form of five questions. Firstly, can it be estimated, within limits, whether the observable increase in production was purely fortuitous, whether it was primarily caused by technique, and the degree, if any, to which it responded to changes in the quantity of labour and capital. Secondly, may it be possible to determine, again within limits, the relative influence of upon production of labour as compared to capital? May it be possible to deduce the relative amount added to the total physical product by each unit of labour and capital and what is more important still by the final units of labour and capital in these respective years? Is there historical validity in the theories of decreased imputed productivity? Can we measure the probable slopes of the curves of incremental product, which are imputed to labour and to capital and thus give greater definiteness to what is at present purely a hypothesis with no quantitative values attached? Are the processes of distribution modelled at all closely upon those of the production of values?
In order to have a reliable picture of the context, in which prof Cobb and prof Douglas formed their theory, it is useful to shed light upon one little phrase, namely the timeline of data they started with. The 1922 report from the Census bureau, which seems to have caused all the trouble, gives data for: 1912, 1904, 1900, 1890, 1880, 1870, 1860, and 1850. Just to give an idea to those mildly initiated in economic history. The time we have covered here is the time when American railroads flourished, then virtually collapsed fault of sufficient financing, and almost miraculously rose from ashes. What rose with them, kind of in accompaniment, was the US dollar considered, finally, as a serious currency, together with the Federal Reserve Bank, one of the most convoluted financial structures in the history of mankind. This is the time, when the first capitalistic crisis, based on overinvestment, broke, and brought a wave of mergers and acquisitions, and on the crest of that wave, Mr J.P. Morgan came to the scene. Europe said ‘No!’ to the stability of the Vienna Treaty, and things were getting serious about waging war at each other.
Shortly, the period, statistically referred to in that 1922 Census report, had been the hell of a ride. Studying the events of that timeline must have been a bit like inventorying the outcomes of first contact with an alien civilisation. In that historical hurricane, prof Cobb and prof Douglas tried to keep their bearings and to single the statistics of productive assets out of the total capital account of the nation, so after accounting for other types of fixed property. What came out was a really succinct piece of empirics, namely three periods: 1889, 1899, and 1904. It is all important, in my opinion, because it shows a fundamental trait of the Cobb-Douglas production function: it had been designed, originally, to find underlying, robust logic as for how social wealth is being generated, against a background made of extremely turbulent economic and political changes, and that logic was being searched with very sparse data, covering long leaps in time, and necessitating a lot of largely arbitrary, intellectual shortcuts.
The original theory by Charles W. Cobb, Paul H. Douglas was like a machete that one uses to cut their way out of a bloody mess of entangled vegetation. It wasn’t, at least originally, a tool for fine measurements that we are so used to nowadays. Does it matter? My intuition tells me that yes, it does. It is the well-known principle of influence that the observer has on the observed object. When we study big objects, like big historical jumps and leaps, the methodology of measurement we use is likely to have little influence on the measurement itself, as compared to studying small incremental changes. When you measure a building, your error is likely to be much smaller, in relative terms, than the measurement of cogwheels inside a Swiss watch.
Thus, my point is that the original theory of production function, as formulated by Charles W. Cobb, Paul H. Douglas, was an attempt to make sense out of a turbulent historical change. I think this was precisely the reason for its subsequent success in economic sciences: it gave a lot of people a handy tool for understanding what the hell has just happened. It is interesting to see, how those two authors perceived their own theory. At the end of their paper, they formulate directions for further research. I am repeating the whole content of the two paragraphs I judge the most interesting: ‘Thus we may hope for: (1) An improved index of labour supply which will approximate more closely the relative actual number of hours worked not only by manual workers but also by clerical workers as well; (2) a better index of capital growth; (3) an improved index of production which will be based upon the admirable work of Dr. Thomas; (4) a more accurate index of the relative exchange value of a unit of manufactured goods. In analysing this data, we should (1) be prepared to devise formulas which will not necessarily be based upon constant relative “contributions” of each factor to the total product but which will allow for variations from year to year, and (2) will eliminate so far as possible the time element from the process’.
The last sentence is probably the most intriguing. Charles W. Cobb, Paul H. Douglas clearly signal that they had a problem with time. I mean, most people have, but in this precise case the two scientists clearly suggest that the model they provide is a simplification, and most likely an oversimplification, of a phenomenon not-quite-clearly-elucidated as for its unfolding in time. The funny part is that today, we use the Cobb-Douglas production function for assessing exactly the class of phenomena those two distinguished, scientific minds had the most doubts about: changes over time. They clearly suggest that the greatest weakness of their approach is robustness over time, and this is exactly what we do with their model today: we use it to assess temporal sequences. Kind of odd, I must say. Mind you, this is what happens when you figure out something interesting and then you die. Take Adam Smith. Nowhere in his writings, literally nowhere, did he use the expression ‘invisible hand of the market’. You can check by yourself. Still, this stupid metaphor (how many hands does a market have?) has become the staple of Smithsonian approach. There are two ways out of that dire predicament: you don’t die, or you don’t invent anything interesting. The latter seems relatively easier.
Right, time to go back forward in time. I mean, back to the present, or, rather, to a more recent past. Time is bloody complicated. My point is that I take that Total Factor Productivity from Penn Tables 9.0, and I regress it linearly, by Ordinary Least Squares, on a bunch of things I think are important. As I study any social phenomenon that I can measure, I assume that three kinds of things are important: the functional factors, the scale effects, and the residual value. The functional factors are phenomena that I suspect being really at work and shaping the value of my outcome value, the TFP in this precise occurrence. I have four prime suspects. Firstly, it is the relative speed of technology ageing, measured as the share of aggregate amortization in the GDP. ‘AmortGDP’ for friends. Secondly, it is the Keynesian intuition that governments have an economic role to play, and so I take the share of government spending in the available stock of fixed capital. I label it ‘GovInCK’. Now, my third suspect are ideas we have, and I measure ideas as the number of resident patent applications per million people, on average. Spell ‘PatAppPop’. Finally, productivity in making things could have something to do with efficiency in using energy. So I causally drop the energy use per capita, in kilograms of oil equivalent, into my model, under the heading of ‘EnergyUse’.
Now, I assume that size matters. Yes, I assume so. The size of what I am measuring is given by three variables: GDP (no need to abridge), population (Pop), and the available capital stock (CK). When size matters, some of that size may remain idle, out of reach of the functional factors. It just may, mind you, it doesn’t have to. This is why at the beginning, I assume the existence of a residual value in TFP, independent from functional factors and from scale effects. Events tend to get out of hand, in life as a whole, so just for safety, I take the natural logarithm of everything. Logarithms are much more docile than real things, and the natural logarithm is kind of knighted, as it has that Euler’s constant as base. Anyway, the model I am coming up with, is the following:
ln(TFP) = a1*ln(GDP) + a2*ln(Pop) + a3*ln(CK) + a4*ln(AmortGDP) + a5*ln(GovInCK) + a6*ln(PatAppPop) + a7*ln(EnergyUse) + a8*Residual_value
Good, I am checking. Sample size: n = 2 323 country/year observations. Not bad. Accuracy of explaining the variance in TFP: R2 = 0,643. Quite respectable. Now, the model:
ln(TFP) = 1,094*ln(GDP), standard error (0,035), significance p < 0,001
-0,471*ln(Pop), standard error (0,013), significance p < 0,001
-0,644*ln(CK), standard error (0,028), significance p < 0,001
-0,033*ln(AmortGDP), standard error (0,03), significance p = 0,276
-0,127*ln(GovInCK), standard error (0,013), significance p < 0,001
-0,04*ln(PatAppPop), standard error (0,004), significance p < 0,001
-0,03*ln(EnergyUse), standard error (0,013), significance p = 0,018
Residual_value = -3,967, standard error (0,126), significance p < 0,001
Well, ‘moderate success’ seems to be the best term to describe those results. The negative residual value is just stupid. It does not make sense to have negative productivity. Probably too much co-integration between the explanatory variables. Something to straighten up in the future. The scale effect of GDP appears as the only sensible factor in the equation. Lots of work ahead.
 Feenstra, Robert C., Robert Inklaar and Marcel P. Timmer (2015), “The Next Generation of the Penn World Table” American Economic Review, 105(10), 3150-3182, available for download at http://www.ggdc.net/pwt
 Charles W. Cobb, Paul H. Douglas, 1928, A Theory of Production, The American Economic Review, Volume 18, Issue 1, Supplement, Papers and Proceedings of the Fortieth Annual Meeting of the American Economic Association (March 1928), pp. 139 – 165