I think studying the Solow-Swan model from student economists is important and necessary. The Solow–Swan model is an economic model of long-run economic growth by looking at capital accumulation, labor or population growth, and increases in productivity or technological progress.

In my view, it is important to give the undergraduate student a rough orientation what he can expect in graduate macroeconomic studies. This helps to select students who are interested in macroeconomics from these who want to earn money in business and other occupations. Of course, growth theory is a crucial subject for graduate studies, but the Solow model is a milestone, and it should be at least mentioned in an undergraduate curse which gives an overview on growth theory in general. I figures out that I agree with the two former partiipant in this discussion.

The Solow model is essential reading at an entry level to the theory of economic growth. His theory is a necessary antidote to theories which suggest the introduction of new intellectual property (innovation) provides additional capital services to the existing production function. Solow promotes the role of IP as changing the inputs to give a new function, with no material inputs from the IP. What the IP may do is introduce new capital assets which act as access devices to the new idea.

To teach ONLY Solow-Swan would be somewhat misleading, because it overemphasizes the substitutability among factors of production. In my opinion the question should be: " Why does growth theory matter for undergraduate students?", and the answer in this case is far from easy.

I support Andrea. It is necessary to teach Solow-Swan growth model, only because it is the most famous growth model. But if you only teach it and do not add comments on the problems that the model contains, we must say that you are intellectually dishonest.

It depends much how many hours your can spend on growth theory. If you have three hours., you can teach in the first hour the model, the claimed meaning (total factor productivity = technological development, etc.) and the arithmetic exercise to calculate total factor productivity.

In the second hour, you can tell your students that there are many objections on Solow's claim. One of most useful objections must be that of Hebert A. Simon 's paper (1979) "On Parsimonious Explanations of Production Relations" The Scandinavian Journal of Economics 81(4): 459-474. Simon thought his objection so important that he published this paper at the moment of his Nobel Prize reception. He claimed that the Cob-Douglas function only reflects accounting identity at the base of phenomena. For curious advanced students you can refer to a book by J. Felipe and J. S. L. McCombie (2013) The Aggregate Production Function and the Measurement of Technical Change, Edward Elgar.

If you have a third hour, Solow's Growth model is a good example you can teach that some econometric procedure can give apparently good but erroneous results if observers do not study the in-depth causal structure beneath the apparent phenomenon. This might be the most important lesson that you can teach your students who want to be data analysts or business analysts.

Good morning.

As an economist-statistician writing from the side of the "economists-as-plumbers", to use Duflo's definition of practical and action oriented economics, I am happy to read that Yoshinori Shiozawa recalls that the main objective of teaching under-graduate economics is providing adequate practical analytical tools, with an emphasis on quantitative methods, in order to find a job.

It is not as obvious as many thinks: at least in my country (France), the teaching of no-nonsense economics has almost disappeared from the curriculum and you may end with young economists ignorant of national accounts or unable to read and understand a company's balance sheet or a country's balance of payments.

The nice feature of Solow-Swan is that it derives naturally from national accounts: GDP growth is just an increase in total value-added, which is itself the remuneration of capital and labour services. A tautology, perhaps, and not free of endogeneity issues, but it led to lots of interesting derivation about productivity.

I also agree with Andrea Salanti that Solow-Swan should not be the only model. Harrod-Domar preceeds Solow-Swan, and the comparison between a long-term Keynesian demand-oriented approach and a long-term Neoclassic Supply-Side approach is an important teaching device, in my opinion.

Solow-Swan parameters may not be econometrically water-tight, but the main idea behind the model is key for understanding also heterodox growth models (disequilibrium or constrained growth paths). For example, the heterodox Two-Gaps model reflects the scarcity of savings to finance K in developing countries, leading to constrained growth and/or balance of payments crisis. So, even if you don't like Solow-the-Neo-Classic, as an heterodox you need to know and understand him.

The teaching of convergence and convergence clubs needs also to refer to Solow-Swan. Endogenous growth would not exist without a critic to the limitation of this model. And so on and so forth: I am sure there are many other examples.

In brief, long live to the teaching of Harrod-Domar, Solow-Swan and other old timers...

The notion of "economist as plumber" was new to me. (Thanks to Hubert Escaith ) I learned that it was a term used by Esther Duflo in her Ely Lecture:

Esther Duflo (2017) The Economist as Plumber, a Richard T. Ely lecture

https://economics.mit.edu/files/12552

Although I do not believe that "randomized control trial approach" is the unique way to make economics evidence-based science, the work as plumber must enrich economics. Armchair economists like me must learn much and sincerely from that. This does not imply, I hope, that pure theoretical work in economics has lost all relevance. By no means, economic theories must be reorganized from their very basis and that requires pure theory work.

See also Davio Ruccio's post in Real-World Economics Review Blog:

https://rwer.wordpress.com/2017/02/15/economists-as-plumbers/

Dear Yoshinori Shiozawa : Nothing new in economists as plumbers: Physiocrats invented Hydraulic macroeconomics, basically the study of the economy that treats product and money as liquids that circulates through the economic plumbing. Modern days hydraulic macro would be called “stock-flow” analysis, I suppose.

More recently, William Phillips, a Keynesian economist famous for his curves, simulated the economy through an analog kind of computer using hydrolics. An analogy devoid of theory, but the main driving force behind much of the economic policy during the 1960s and 1970s. Until Friedman came with a new narrative, similarly devoid of strong scientific background.

This said, I fully agree that theory is important. But it happens that policy often follows fashions rather than science.

> But it happens that policy often follows fashions rather than science.

Indeed!.

It is a pity, that economic science seems - rather than policy - to follow fashions, but most economic fashions are rather old-fashioned. I see the Solow model as a serious error. What policy advice can be drawn from it? It proposes an increase of saving, i.e. a reduction of consumption in order to boost capital stock. It recommends to increase pressure on the population (by reducing unemployment remunerations and increase pension age) such that more people are obliged to stay employed or enter employment. And, it proposes measures to increase working time. If this causes low wages, then - in this model - this will lead to higher growth, because this will result in higher profits, higher savings and investment (classical twins, although they will often react in different directions to policy). This is really nonsense (which does not avoid that one gets a Nobel Price for that). Nevertheless, I am not against teaching the Solow model, but it is important to show that it has nothing to do with (any) reality.

By the way, economics in general remains to be (mainly) focused on growth, although most other sensible people think that high growth causes much more problems than it seems to solve.

With regards to Solow model itself, Anton is right. It is quite natural that many people infer the policy implications like those Anton depicted. However, there are two problems in Anton's argument. First, it is dangerous to estimate a theory (including models) by their policy implications alone. The second point is the significance of Solow's research. Main lesson from Solow's research was that the contribution of capital K and labor L is rather small and major part of economic growth seemed to be caused by technological change. Let me explain each point more in detail.

(1) Plausibility of Solow(-Swan) model

There are several methods by which to estimate the accuracy of Solow model. In the simplest form, Solow model assumes Cobb-Douglas production function:

Y = A Lα Kβ where α ＋β = 1.

When we have a time series Y(t), L(t), K(t) for various t's, we can estimate α and β as independt variables by using A as adjusting parameter. By this estimation, α ＋β is very close to 1, for example in 10.2 or 0.99. As Herbert Simon pointed it, this is rather rare to get this accuracy. As for α itself, it gives labor share of total income according to marginal theory of productivity (or marginal producivity theory of distribution). The index α is near to 0.7 and this fits to the real labor share we get from a system of national accounting. These are main (or at least two substantial) reasons that economists estimate that Solow model with Cobb-Douglas production function is good.

However, Anwar Shaikk showed (before Simon) that any income distribution that has average income share gives similar results in his paper:

Laws of Production and Laws of Algebra: The Humbug Production Function, The Review of Economics and Statistics 56(1): 115-120.

There are other reasons that we can claim as Simon (in his Scandinavian Journal of Economics) that Cobb-Douglas production function reflects the accounting identity

Y = W + R = w L + r K

where W is the total of wages, R is the business surplus considered to be the total profit produced by capital; w is the wage rate and r the rate of profit. Index α simply reflects the mean of labor shares W/Y. This is the main contents that can be taught in my first answer on 3 days ago (February 5?) if Jacob Smith has the second hour to teach.

(2) Total factor productivity and technological change

If regression of time series is made assuming A to be constant, fitness of the regression is not good. This was the reason why it was necessary to assume A to be time dependent. This part was very big as it explained 60 to 70 percents of the variance of the total production Y(t) . Solow named this part total factor productivity and interpreted it that it is caused by technological change. But he could not show why it represents technological change. If there were no technological change (including capital goods improvement and amelioration of production processes), the constant A must be constant. The fact that A is not constant does not imply that total factor productivity is a result of technological progress. It can be caused by other factors. We can also argue that Solow model with Cobb-Douglas production function itself lacks its theoretical basis.

In 1960's, there was a big debate called capital measurement controversy (or Cambridge capital controversy). Neoclassical production function (including Cobb-Douglas production function and CES [constant elastic substitution] production function) assumes that capital can be measured independent of the state of distribution. In other words, it assumes that K can be measured independent of profit rate r. This is strange because value of capital goods changes (renewal or book value) changes when r changes. This seriously questions the validity of neoclassical marginal productivity theory of distribution.

The custom to identify total factor productivity and technological change is adopted even among many heterodox economists. Development economists who define themselves structuralists often use the concept of total productivity to indicate the importance of technological change. On this point, I believe two theoretical developments are necessary.

One is to construct a theory of income distribution that does not rely on marginal productivity theory of distribution. I believe our new results explained in Chapter 2 of the book

Book Microfoundations of Evolutionary Economics

gives a new foundation for the renewal of the theory of distribution. Simpler version is also given in

Conference Paper The Revival of Classical Theory of Values

(This is a downloadable working paper.)

The second necessity is to develop a theory based on national accounting system a similar concept like total productivity. This may not be difficult, because, as Herbert A. Simon pointed it, Cobb-Douglas production function is but a multiplicative surrogate of additive accounting identity.

N.B. This post relies on my memory. If there are any inexact points in my argument, please point us.

I think it is legitimate and sufficient, to critizise a theory or model, because the policy advice drawn from it leads to the contrary of which it pretends. But it is easy to see that usual production functions are far from reality. One just needs a simple thinking experiment. Assume that a firm invests in a machine, which enables to produce a certain quantity of goods with a certain number of employed. Let us assume that this machine is imported from a country, the exchange rate of which has fallen by 10% against the year before. Does that mean that its productivity is lower than the year before. According to depreciation rules (for a single enterprise: IAS or tax rules, for National Accounts: some percentage of the book value) the capital stock incorporated in this machine falls year by year, but the productivity will not change until the end of its lifetime, meaning an increase of capital and/or total productivity over time in a statistical estimation. Maybe that there is a slump of demand for the product, what will happen? If the slump is assumed only short run, production might be continued, but going on stocks, which - in general - means lower valuation (only production cost, whereas for saler there is a profit mark-up). In statistics, this would show up as a fall in productivity. If the reduced demand is assumed for a longer term, the firm may have different opportunities to adjust (most of which could take more or less time according to the employment laws of a country and the ecpectations for the future. Most of these adjustments are likely to show up as a fall observed capital productivty, although this productivity has not changed. I think, one can find numerous examples why real developments are far from production function theories. An aggregate production function (even with a 99% R²-fit) can and will never be a base of decisions of individual firms.

Good morning. I have difficulties in following Anton Rainer ‘s argument here. On the one hand (the quantity one) he is right. But most if not all the examples in the post come from the monetary dimension: the tax rule allows to create a reserve for depreciation not because the existing machine is less productive but because its expected life is decreasing and the firm will have at some point in time to buy a new one or to repare the old one. Firms do not take their investment decision on the basis of quantity but on the basis of money (the internal rate of return calculated on the expected productive life of the machine). When the cost of a machine increases, the profitability of buying it may drop tothe point where the firm decides to use another production technology (e.g, more labour intensive) or to do something else.But it remains a financial issue, not a productivity one (the physical productivity of a machine tool is not affected if nobody buys it; but the gross benefit of the firm producing it goes down).

This said, most publicly available data are in the monetary dimension and the economists dedicated to measuring aggregated production functions (e.g, for producing the Penn tables) must do this type of hypothesis (e.g, The stock of capital is measurable, The value of the service provided by the existing stock of capital decreases according to its age, ...) I am sure they would use another accounting approach if it was better rooted in theory and easy to implement.

I believe Anton and Hubert are bringing up a difficult issue:

What is productivity?

I my previous post, I used the term productivity, but it is troublesome and difficult term to define. If we admit that productivity means always value-added productivity per something (e.g. person or unit of work hours), it is a clear idea and easy to measure it. But, productivity means also efficiency of a plant, a firm, or workforce. It sometimes stands for the goodness of a production technique. If I call them physical productivity, we often think by using this physical productivity concept.

Let us take an example. I often read comments like this: As the productivity of the service sector is low, the wage of service sector is low (in comparison to manufacturing sector). Is this a good reasoning?

Suppose two persons: Anna works as a nurse and Bob works as a metal worker. Suppose they work the same hours by the same shift. Anna gains 200 thousand dollars per month and produces 300 thousand dollars of added value. Bob earns 300 thousand dollars per month and produces 450 thousand dollars of added value. Is it correct to argue that Anna's wage is low because Anna's (labor) productivity is low? I do not think so.

Suppose Anna works in a city-governed hospital and the city hall decided to raise her wage to 300 thousand per month, Anna probably produces at least 400 thousand dollars of added value per month and it is possible that she produces 450 thousand dollars of added value per month. But the way Anna works did not changed at all. She is working very busy as before.

In my opinion, it is not reasonable to use value-added productivity to judge whether a wage rate in an industry is sufficient or deficient in comparison to the wage rate of other industry. But we often use this kind of reasoning when we use the concept productivity.

Dears. At some point here, we need to be practical and accept we live in a second-best world.

Anna's value-added in Yoshinori Shiozawa post is set by National Accounts convention to be equal to her salary, because it is the way we measure the value-added of public employees. In a country with a tiny public sector, her true contribution to the social well-being is probably higher (e.g., in Switzerland or USA), while in a country with many civil servants (e.g., France, Egypt), her contribution is probably lower because in these countries, civil service is a way to deal with unemployment.

It is just a convention, but we must accept it because nobody can actually measure the social utility of a civil servant, unless we look at their time-sheet and measure their output in terms of actual social utility. Thus, being practical, we must accept that if Anna's salary is increased, GDP will automatically rise.

Bob, being employed by the private sector, is expected to create more commercial value (id est, mode value-added at basic or market prices) than his full salary, because the convention is that her employer is not a masochist and she will not employ people just for the pleasure of losing her money. Again, this is a convention. And if Bob's salary increases while the value of output does not increase in proportion (no mark-up pricing), then GDP remains constant and someone else must lose in the process (id est, the firm's owners or the tax-people).

We may not like these conventions, but National Accounts and Growth Accounting are based on such conventions.

Again, it is not enough to say this is not perfect (everybody agrees, in particular the colleagues at the Groningen University in Holland, who are contributing to the Penn tables), but if we are unhappy with this situation, we have to provideGrowth Accountants with a better option, which needs to be both theoretically strong and empirically implementable when dealing on cross-country comparison (the Penn table cover 182 countries between 1950 and 2017, see: https://www.rug.nl/ggdc/productivity/pwt/ ).

For the time being, I do not know of any plausible alternative.

Dear Hubert Escaith ,

economic statistics are full of conventions. As you put it in the answer three posts above, it must be easy to implement. I am not complaining that a System of National Accounting is not theoretically ideal. I am complaining the often observed arguments that explain low wage rate of an industry because it has low productivity. If we know that value-added is defined to be equal to the wage, it is just a tautology to say that the wage is low because it has low productivity.

Hubert, have you never read that sort of arguments? In Japan, hélas, we read and hear it very often. Such a statement is even uttered by high-ranked business persons of some leading companies.

I agree, at least at firm level (not sure on average). Wages do not depend only on individual worker's productivity.

Given a social state of class struggle (the Post Keynesian way of understanding the distribution of value-added between wages and profit) it is much better to be a worker in a firm that enjoys some degree of market monopoly (the Kalecki approach to mark-up pricing). Or, to use a more positive view (I do not remember if it is in Pasinetti, Sraffa or Morishima --my memory is getting worse with time), you are much better as a worker in a branch which enjoys a growing demand, because mark-up price is a function of cost of production plus a premium for activity growth.

By the way, this approach is also fully consistent with the New Keynesian understanding of reservation and efficiency wages (if as an employer, you are expecting an increasing volume of production, you will pay a higher wage to lower the turn-over of your employees and make sure they don't cheat you).

If two antagonist schools of thought arrive to the same conclusions by different way, there is some chance that it may be a plausible option. An hypothesis to be tested and falsified, as all good Popperians should do.

The problem with falsificationism is that, because the main aim of macroeconomics is to support the effectiveness of certain (macroeconomic) policy interventions, it would be needed to have sound evidence of the causal mechanisms that are supposed to be at work. But, for a number of reasons, this is by no means the case. The empirical evidence usually provided ends up by being inconclusive because unable to select specific hypotheses from pools of competing and empirically equivalent causal hypotheses.

It may not be a question of hypothesis testing. I am questioning the validity of physical productivity concept. This problem remains even if we admit that value-added productivity is defined in statistics by convention that it is added-value per person or per hour.

In my feeling, all problems depend on the one-good economy assumption. This assumption is widely accepted not only in neoclassical or new classical economics (e.g. marginal productivity theory of distribution), but also in Kaleckian and most often in Keynesian economics. This is one of the reasons why I asked my question:

https://www.researchgate.net/post/Does_Post_Keynesian_Economics_need_no_theoretical_foundations

Hubert is, of course, right that for any accounting – be it on the level of a single firm, of a group of firms or on a wider level (National Accounts) - conventions are needed. In NA, all products are combined into one indicator (GDP) by adding-up the value-added of all producers which itself is (in principle) calculated by deducting input cost (px) from sales (py), with vectors p for prices, x for quantities of inputs and y output quantities. GDP is treated as one single good with a deflator as its price. If p is held constant for some time (I do not want to go into indicator problems deeply here), then one arrives at real GDP, which is clearly a quantity measure for total production. I think that the conventions for calculating GDP are reasonable, and I do not question the convention for the calculation of the capital stock: K=(1-d)*K(-1)+I, where I is real investment.

But one cannot use K for explaining value-added in a CD- or CES-production function. This is very clear for a single firm even with a very simple production process (one good, one input, one machine). Whereas K falls over time, production capacity would be rather constant until the lifetime of the machine. But even if one assumes far reaching substitution, i.e. that it is possible to change the output of the machine by variations of labour input (which is a rather restrictive assumption), one can never arrive at a mainstream production function. Therefore, usual production functions cannot fit on the micro level, and I think one cannot find a function, which will lead to such a function for an aggregate. For my thesis, finished in 1975, I estimated several production functions (trend, CD, CES, VES with and without “technical progress”) for the Austrian manufacturing industry and 12 of its branches on the base of data from 1954-1968. Mostly, the time trend “explained” more than 90% of the production data and the coefficients for K and L were insignificant or far from acceptable values.

What makes production functions so attractive is, that they allow to derive any productivity measures from it. But from nonsense one can derive nothing but nonsense!

In contrast to supply-oriented models, the aggregation is much less problematic in Keynesian demand models, because these models assume that supply=production adjusts to demand.

What concerns the relation between productivity and wages, I am with Yoshinori: Production and productivity rather depend on wages than the other way round. The concept of equality of wage and marginal productivity is Humbug. I think, that mostly nobody can know the contribution of an additional employee on production. Hubert brought the expression “social utility”. It is interesting to see, that often income and with it the contribution to GDP is inverse to this social utility. A successful financial adviser will likely earn a lot by contributing to a more uneven income distribution (which is a social disadvantage), whereas a nursery-school teacher may earn only a fraction of his income, although her job is socially very useful. One should also see, that productive activities outside markets are more or less undervalued in National Accounts.

Let me finally repeat that economic mainstream theories are too much focused on growth as if growth had no disadvantages and as if economies could and should grow without limits.

Aleš, you present a rather strange biological-physical-economic(?) theory.

1. Men are not only biologically driven and fitness is not our only aim. I, e.g. like to go to a cinema, a theatre or a consert. Or having a good meal with 1, 2 or 3 glasses of wine etc. What has that to do with my fitness. Of course, I try to stay fit (makes life easier and hopefully longer), but I will never reach the fitness level of 40 years ago. Individually and socially, fitness and growth are quite different things.

2. Venus has a similar mass like earth and it gets more solar energy. According to your theory, there you could have almost infinite high growth. Nevertheless, I would not like to live thare - apart from I would not live very long.

Never heard of global warming, climate change, waste problems etc.?

Obviously, you mean by growth what is often called "qualitative growth" (not only non-fossil, but also resource-saving and waste-reducing or waste-recycling). I would rather call it "development", because it may not show up as increased GDP in National Accounts.

Let us now return to the original question:

• Why does the Solow Model matter for undergraduate students?

What became clear from above arguments is that the concept/formulation of production function (either in the form of Cobb-Douglas, CES [constant elasticity of substitution ], or in a more general form as f(L, K) ) is highly questionable. We should not base our analyses on this sort of production functions.

This concept was questioned and was concluded invalid in the capital controversy in 1960's and yet it resurrected in the macroeconomics textbook as if no such controversy ever took place. Students should be taught this fact before (or at least after) Solow model is mentioned.

I wonder how Jacob Smith and Hubert Gabrisch think on this point.

Sure, why not. Although I do not advocate the Solow growth model in my own research and thinking, I am convinced that it plays an imminent role in the history of modern economic thinking. What lacks in undergraduate teaching is simply an overview on this history. A reader or professor should present his own views in graduate courses. That is simply all!

Yoshinori Shiozawa Pedagogically I think the solow model is a first take at teaching the lesson of how baseline conditions lead to specific welfare outcomes and evolves.

Its the simplest sort of model it introduce to someone that has zero background in thinking about these questions. Its the same reason why we discuss the Malthusian model with undergraduate students.Its a lesson with putting pieces of the economic story together and what we can say about the future of an economy that runs strictly to how we defined it to be.

In terms of my original question with specific career applications (which would be relevant for undergraduate students not intending to go to graduate school) I have my own answer. I don't think its the best answer but its the answer I'm currently going with it (it may have been stated above).

Id say the Solow model is relevant for these workforce oriented students as it teaches the lesson that doing good things (productive things) the same way over and over will only be beneficial up to a point. It teaches a lesson about real economic structures (whether that a small business, communal organizations or cities) that progress isnt made by doing the same thing again and again, this is demonstrated by the limits of capital accumulation in the model.

It teaches that you got to be much more complex (by coming up with ideas) to improve the material conditions of your organization.

I dont think this is the best answer, but its the best one I know of so far!

I think Jacob has got it (for me).

The debate becomes quite interesting. Jacob Smith , have you read my first comment on February 6? I have mentioned there Herbert A. Simon's paper in 1979 and Felipe and McCombie's book in 2013. Solow's model is not a case of simplest models, but a model constructed on a wrong theory.

To teach such a model with no mention on how Solow's model contains conceptual or theoretical problems is to teach it as truth and deprives those "workforce oriented students" the chance to know that Solow's model and marginal productivity theory which underlies the model are a wrong theory.

If a economic theory is wrong or good is an academic problem that must be decided among economists as questions of science. However, the trouble with economic theory is that it has some persuasive powers over people. In the case of Solow model, it is easy to derive that wage rate w is determined by the formula

w = ∂f(L, K)/∂L .

If this is interpreted as inevitable truth, it can be used persuasively in the argument whether the nurses' and care workers' wages are too low or not.

There are no natural or objective law which determines the relative level of wage rate with respect to wage rates in other industries or trades. Marginal productivity theory (of income distribution) is often used wrongly to persuade people to unjustified low wages.

If Jacob is really thinking to empower "workforce oriented students" , he should think of giving them knowledge that will be helpful for them. In that case, Jacob must not simply teach Solow model (and how to estimate the model from given data) without mentioning how it contains flaws.

Dear Robin Lynch

please read Simon's paper that he prepared for Scandinavian Journal of Economics at the time he received the Nobel Prize in Economic Sciences:

https://pdfs.semanticscholar.org/3774/a0c32f4011ca9a08e90efcc5d526b1fa3006.pdf

In his saying, the question was so important that he preferred to publish a full paper instead of referring briefly in his prize receiving speech.

We can argue all within national accounting framework without appealing to Solow model.

The Solow model and especially the macroeconomic production function (MPF) has spoiked many economist's brains. In fact, the MPF is a rather stupid concept. One should not teach it without presenting its defects.

Typing error: It should be "spoiled" instead of "spoiked" (which, by the way, does not sound so bad)

Yoshinori Shiozawa you have given me some good resources to look at and I would need to read them and then reply.

I have found a freely downloadable paper that treated arguments exchanged between Robert Solow and Herbert Simon:

Article "On the Cobb-Douglas and all that..": The Solow-Simon corres...

The correspondence between the two was mentioned in Smion's Scandinavian Journal of Economics paper that I have cited on April 10.

@ Yoshinori Shiozawa very very interesting article

I do not know whether Hubert Gabrisch , Aleš Kralj , and Robin Lynch had the time to read Simon's paper in The Scandinavian Journal of Economics. It explains beautifully how the underlying accounting identity is forged into Cob-Douglass production function. As it is hinted in the paper and as you can know more in detail by reading Scott Carter's paper, Solow responded in an unfaithful way.

The fact that Solow model depicted the importance of technological change is important but is not exposed in a correct way (or at least in a defensible way). We can explain Solow's finding without using aggregate production function.

Thank you Yoshinori Shiozawa, I understand better now, having read the Simon article. My main focus was that innovation changes inputs (intermediate or

capital assets) in the production process. The capital assets can be access devices which carry into practice the innovation. I will continue my research!

I think that the Solow model is an important one in macroeconomics for sure. It can give an overall picture of the macroeconomy and allows for a nice way to compare countries.

However one of the main issues with the Solow model and the aggregate production function that underlies it is the fact that it is NOT a corroboration of the marginal productivity theory of distribution. The notion of "aggregate capital" is tenuous which means its marginal productivity does not strictly speaking exist.

The Solow model is a nice (because it is simple and one-sided), but false way to compare countries.

I don't think that "false" is the right way to frame the matter because a lot of economics is "false", including in my view convexity of either preferences or technologies, and even the notion of the margin itself inasmuch as that is purported to be an economically determinant category. Certainly the Solow Model and Aggregate Production Functions generally have been used precisely for cross country comparisons as well as analysis in a single country. For cross country comparisons a lot depends on how the real data is measured and whether it is consistent across countries, e.g. by using PPP, adjusting for exchange rates, etc.

Whether any of this tells the "true" story both within and across countries is a different matter altogether. All it does is give a simple view in my estimation. But it is much more accurate to eschew aggregate production functions altogether and instead use sectoral and more disaggregated data if the "true" story is to be told.

I agree with Scott Carter . To obtain the "total factor productivity" seems not difficult to obtain without using aggregate production function. (Solow residual may be acceptable but the notion of "total factor productivity" must be a wrong concept.)

Solow's calculation starts from the formula

ΔQ/Q = ΔA/A + w ΔL/L + r ΔK/K,

where

w = (∂f/∂L)*(L/Q) and r = (∂f/∂K)*(K/Q).

Notations are slightly modified. Let us do a small calculation:

ΔQ = Q(t+1) - Q(t)

= {w(t+1) L(t+1) + r(t+1) K(t+1)} - {w(t) L(t) + r(t) K(t)}

= {w(t+1) L(t+1)- w(t) L(t) } + {r(t+1) K(t+1) - r(t) K(t)}

= (w(t+1)- w(t)) L(t+1) + w(t)(L(t+1)- L(t))

+ (r(t+1)- r(t)) K(t+1) + r(t)(K(t+1)- K(t))

= {(w(t+1)- w(t)) L(t+1) + (r(t+1)- r(t)) K(t+1)}

+ {w(t)(L(t+1)- L(t)) + r(t) (K(t+1)-K(t))}.

Hence,

ΔQ/Q = {((w(t+1)- w(t)) L(t+1) + (r(t+1)-r(t)) K(t+1)}/Q(t)

+ {w(t)(L(t+1)- L(t)) + r(t) (K(t+1)-K(t))}/Q(t)

When there is no change of technology and if wage and profit rates remain constant, we can calculate the contribution to the increase of Q(t) as follows:

ΔQ~/Q = [{w(t) L(t+1)+ r(t) K(t+1)}-{w(t) L(t)+ r(t) K(t+1)}]/ Q(t)

= {w(t)(L(t+1)-L(t)) + r(t) (K(t+1)-K(t))}/Q(t).

The Solow residual can be defined as the difference between ΔQ/Q - ΔQ~/Q.

Hence,

(SR) = {w(t+1)- w(t)) L(t+1) + (r(t+1)- r(t)) K(t+1)}/Q(t).

We can simply interpret this as the effect of factor price changes due to technological change. If I am right, the Solow residual can be defined and understood without using any notion of aggregate production function.

I wonder if the new definition gives very different results than Solow and his followers's calculations. Jacob Smith , would you like to try the both methods and let us know the results?

Dear Yoshinori,

My first question concerning your calculation: What is the function f? Obviously, it is for a not clearly defined variable V such that V=f(L,K), because otherwise the derivatives for L and K would be meaningless. If V=Q, we have the usual neoclassical production function, if not, we need your explanation. For Q=f(L,K), the calculation (or definition?) of w (and r) is rather strange. An increase of L would increase w (which I think is the wage per hour or employee), a higher Q would lead to a lower w. This is similar for r, where it is unclear, what this r is in practice. The variable K is questionable, too: It is a (nominal) variable from accounting which should show, how much money an enterprise has spent for (productive?) assets. To take into account the assets' lifetime, one usually writes off a certain percentage p.a. But, if you purchase a machine with, say, a lifetime of 10 years, its "productivity" will, of course, not fall according to its diminishing book-value.

I think, the Solow residual is necessarily connected with a (aggregate) production function. If we speak of Total Factor Productivity (TFP), we should ask ourselves, what this means on a micro level. If you e.g. asked the director of a factory to tell you the factory's TFP, I assume, s/he would not understand. The same would be true for questions about the marginal productivity of labour and capital or even the productivity of her/his secretary, one of the workers (if they are working in groups) or of a bookkeeper etc.

It is more realistic and much simpler to assume that technological progress is embedded in the production factors and simply measure Q/L as an indicator (instead of TFP) for productivity, but always be aware that this measure is highly dependent on demand (maybe more than on technology). I am sure, that in many countries this figure, but also TFP will be lower for 2020 than for 2018 and 2019. That is, of course, not due to a reduced productivity (or potential production), but to the restrictions and reduced demand in the course of the Corona pandemy.

Dear Anton Rainer ,

> It is more realistic and much simpler to assume that technological progress is embedded in the production factors and simply measure Q/L as an indicator (instead of TFP) for productivity.

I agree with you. As long as labor share remains constant (which is the condition to get Cobb-Douglas production function), Q/L can replace TFP. But how can you persuade that there is no contribution of capital for the increase of total income per labor? If your argument is effective, there is no need to argue more than a half century on Total Factor Productivity (TFP).

One of my motives that I came to be interested in Solow's fomulation is that

two of my colleagues (one a specialist in the history and philosophy of industrial technology and another a specialist in development economics) use TFP in their arguments. They are not people much influenced by neoclassical economics. They are rather heterodox in their economic thinking. Even though, they feel they have to use TFP as evidence that technological progress is much influential than the increase of capital inputs (or increase of labor / capital ratio). Heterodox economics (Post Keynesians, marxists, French regulationists and Austrians)

have no framework to analyze technological progress. In order to end neoclassical dominance, it is imperative to present an alternative theory and measurement.

In the above post, I have explained that TFP or Solow residual is measurable without appealing to the notion of aggregate production function.

In the next post, I will explain the background theory that supports the above measurement or formula.

Anton Rainer

> What is the function f? Obviously, it is for a not clearly defined variable V such that V=f(L,K), because otherwise the derivatives for L and K would be meaningless

The function f that appears in the two formulae after "where" is that of Solow. I do not use any function of the type V = f(L, K). I have cited them only to compare with my calculation.

In my calculation, Q(t), L(t), K(t), w(t) and r(t) are respectively the gross domestic product, labor, capital, wage rate and profit rate which are all time series assumed to be measurable. (There are long argument on the measurability of K).

I do not assume any production function, but I assume an economy where there are many firms, many products, many method of production, and many households. Q(t) is the product or income that is the sum of all product net of intermediate inputs at the accounting period t. L(t) and K(t) are also sums of labor hours and capital used in each firm. The idea of my calculation is that wage and profit rates change by technological progress. (w must be measured as the wage rate at constant prices).

The pure effects of technological change is

(w(t+1)- w(t)) L(t+1) + (r(t+1)- r(t)) K(t+1).

If we divide by Q(t), we get (SR).

The reason why and how the (real) wage rate changes as effects of technological change is given in my paper

A new framework for analyzing technological change.

This paper is not yet published. Readers who want to read it are requested to ask me (If you open EXPERIENCE page of my profile in RG, you can find a button on your right down.)

Addendum:

If the labor and profit shares are constant, there is no need to measure r(t) and K(t). Indeed, if the labor share is α (and profit share is 1-α), then

r(t) K(t) = {(1－α)/α} w(t) L(t). (1)

Then, we get

(SR) = {w(t+1)- w(t)) L(t+1) + (r(t+1)- r(t)) K(t+1)}/Q(t)

　　 ≒ (1/α){w(t+1)- w(t)) L(t+1)}/Q(t).

Note also that (1) is the fundamental assumption when Solow and others use Cobb-Douglas production function, because α is the power index of the Cobb-Douglas function.

I hope Jacob Smith , Scott Carter , Hubert Gabrisch will check whether my proposal is right enough.

The problem with SR = {w(t+1)- w(t)) L(t+1) + (r(t+1)- r(t)) K(t+1)}/Q(t) is that it is only a complicated calculation of a questionable indicator. The reason for that is that w and r are of different character. Whereas w is mainly fixed by contracts in advance, r is calculated by dividing profits (=Q-wL) by K (in general, the "book-value" of the beginning of the period, mostly year), i.e. r=(Q-wL)/K. If you substitute r in SR, you get

SR={[(w(t+1)-w(t))L(t+1)]+[(Q(t+1)-(w(t+1)L(t+1))/K(t+1)-(Q(t)-w(t)L(t))/K(t)]K(t+1)]/Q(t)=

={w(t+1)L(t+1)-w(t)L(t+1)+Q(t+1)-w(t+1)L(t+1)-[Q(t)-w(t)L(t)](1+%K)}/Q(t) with 1+%K=K(t+1)/K(t), i.e. 1 plus the percentage change of K.

Therefore, SR={Q(t+1)-Q(t)-(Q(t)-w(t)L(t))*%K}/Q(t)=%Q-Profit(t)/Q(t)*%K. I think and hope my calculation is correct, but I cannot interpret this indicator.

I think one cannot and should not split GDP development between labour, capital and technical progress. Therefore, the simple Q/L is the best measure for productivity and one should forget production functions, TFP or the Solow residual. I know that Q/L is far from a perfect productivity measure, too, because Q is mainly dependent on demand and L be cannot adjusted to what is demanded without problems (dependent on employment laws, habits, expectations etc.).

But to use Q/L has also the advantage to widen the view: that higher productivity need not necessarily be used for growth, but to reduce working time.

Looking mainly on productivity leads to overstressing potential production and to proposals that, in fact, are against growth and - what is more important - against employment (if you analyse them in a Keynesian framework).

Anton Rainer

>The reason for that is that w and r are of different character. Whereas w is mainly fixed by contracts in advance, r is calculated by dividing profits (=Q-wL) by K.

I do not object to r and K part. I added Addendum mainly on the same reason. As for w, you are mistaken. In this kind of calculation, one normally uses "constant price". Even if the wage rate at money term is fixed by contract, when the prices go down (or more precisely if the price index goes down), the wage rate w goes up. At a fixed wage rate, w(t+1) and w(t) can be different.

> I think one cannot and should not split GDP development between labour, capital and technical progress. Therefore, the simple Q/L is the best measure for productivity and one should forget production functions, TFP or the Solow residual. I know that Q/L is far from a perfect productivity measure, too, because Q is mainly dependent on demand and L be cannot adjusted to what is demanded without problems (dependent on employment laws, habits, expectations etc.).

It is your opinion. I am also thinking that production function is a sheer fiction that is better avoided. But, a question remains. Why TFP or Solow residual attracted such great interest and continues to do so? Something different from Q/L is wanted, isn't it?

As you know well, to claim that higher the productivity higher the income is almost tautology. We should have a theory that can distinguish the productivity growth and the capital increase contribution and a method to measure them.

Please rad my paper:

A new framework for analyzing technological change.

You need to understand the background idea.

Dear Yoshinori,

Before having finished reading and studying your "new framework"-article, I want to add some remarks on your last answer.

You are correct that real wage w is not only determined by contracts and laws, but from price development, too. But may I, nevertheless, refer to my derivation of the SR indicator from your proposed procedure: SR=%Q-Profit(t)/Q(t)*%K. Profit here is "real", i.e. real production Q minus real wage sum wL, but if one assumes the same deflator for Q and w, the profit share will be the same in nominal and real terms. %K is the percentage change of the capital stock, i.e. (I-D)/K(-1) with I gross investment and D depreciation. All variables are in real terms, valued at prices of year t (or t-1?). Apart from these calculations, I see severe problems when interpreting SR. If I

Thanks to Emiliano Alvarez,

https://www.researchgate.net/publication/348578756_Economic_Growth_and_Population_Dynamics_a_Discrete_Time_Analysis

I found three papers that report an astonishing fact. Here are exerpts from abstract of each of them:

Jess Benhabib and Mark M. Spiegel 1994 The role of human capital in economic development: Evidence from aggregate cross-country data. Journal of Monetary Economics 34: 143-17

"Our results indicate that human capital enters insignificantly in explaining per capita growth rates."

Lant Pritchett 2001 Where has all the education gone? The World Bank Economic Review 15(3): 367-391.

"Cross-national data show no association between increases in human capital attributable to the rising educational attainment of the labor force and the rate of growth of output per worker. This implies that the association of educational capital growth with conventional measures of total factor production is large, strongly statistically significant, and negative."

Chandra Shekhar Kumar 2006 Human capital and growth empirics. The Journal of Developing Areas 40(1): 153-179.

"In all the above estimation methodologies, the positive and significant effect of human capital is not observed."

There must be various reasons to explain the fact. Pritchett examines three possible reasons. But, before arguing the reasons, one inquiry seems to be missing. It is the question of validity of Solow augmented production function and by consequence the validity of endogenous growth theory (at least that based on the Lucas-Uzawa model). Three papers above cited imply that there is no significant positive effect of human capital accumulation on the increase of per capita product.