Friday, May 15, 2015

Mathiness and the Solow production function

Mark Thoma sends us to Paul Romer's blog post about his paper that defines "mathiness" in economics research -- in particular in growth economics:
The style that I am calling mathiness lets academic politics masquerade as science. Like mathematical theory, mathiness uses a mixture of words and symbols, but instead of making tight links, it leaves ample room for slippage between statements in natural versus formal language and between statements with theoretical as opposed to empirical content.
I think there is no better place to see this than the Solow growth model's Cobb-Douglas production function. A really pure dose of the mathiness can be seen here at MR University. In it, they motivate the equation

Y = A \; K^{\alpha} \; L^{1 - \alpha}

where $Y$ and $K$ are real quantities. First, why are they real quantities? I've never quite found a good answer, but in general most economists think that the price of bread being twice as much as it was decades ago is something that should be adjusted for (taken out of the values), not a fundamental property of economics. Of course, solving that problem (by adjusting for inflation) creates a new problem -- money illusion (why do humans seem to think in nominal terms). Usually when incorporating an idea creates new problems, it's a sign that maybe the idea isn't right. But adjusting for inflation is well-established, so let's not rock the boat too much. Besides, there's other examples we can use.

Second, $A$ (total factor productivity, TFP) is introduced on the MRU videos as representing something that exists in the real world: "ideas" or productivity. It's not just a normalization factor. It also implies that it can change over time or across countries. We are immediately assuming that instead of an unimportant normalization that is constant over time, we have a critical component of growth economics.

Finally, while there are potentially reasons for the constant returns to scale assumption where the exponents of $K$ and $L$ add to 1 -- Alex Tabarrok states it in terms of doubling $K$ and $L$ should lead to a doubling of output -- it represents an additional assumption. In most cases, however, the reason for the assumption is a theoretical reason, not empirical.

Actually, we've made three major assumptions before comparing to empirical data: real quantities $Y$ and $K$, $A$ is a factor of production and constant returns to scale for $L$ and $K$. These assumptions lead to $A$ (the normalization factor in your original model) being the most important thing in growth economics -- it represents most of economic growth. Additionally, the model using these assumptions are used to make the assertion (explicitly in the MRU video, but generally the conclusion in much of growth economics) that Mexico doesn't use its labor or capital as effectively as the US.

Note that the theoretical assumptions create these new mysteries: 1) why is most growth TFP and 2) why do some countries use capital and/or labor better than others.

Now in the much more general information equilibrium approach, we end up with almost the same Solow production function

Y = A \; K^{\alpha} \; L^{\beta}

but without the three assumptions. $Y$ and $K$ could be nominal or real quantities, $A$ is a normalization factor that mostly represents the units of $K$ and $L$, and $\alpha$ and $\beta$ are completely arbitrary. What do we get if we compare this model to empirical data? Almost perfect agreement:

That's for the US and Mexico, using nominal quantities for $Y$ and $K$. $A$ is just a normalization factor and not responsible for any economic growth. In Mexico, $\alpha + \beta \simeq 1.42$ with $\alpha \simeq 0.90$ and $\beta \simeq 0.51$. In the US, $\alpha + \beta \simeq 1.25$ with $\alpha \simeq 0.85$ and $\beta \simeq 0.40$. We draw the exact opposite conclusion as MRU: Mexico grows much faster than the US for a given (relative) input of capital or labor!

This may even resolve another conundrum created by the Solow growth model -- why would the US or other advanced countries invest in Mexico if productivity was higher in the US? Sure, maybe the stock of capital is low and so you get more growth relative to depreciation -- but again that is entirely because of the way the Solow production function is built! In the information equilibrium framework, we see that increases in capital grow the economy a bit more than in the US (the exponent is 0.90 vs 0.85). Why move your labor intensive manufacturing to Mexico? Fractionally increasing labor in Mexico produces fractionally more output than in the US (the exponent is 0.51 vs 0.40). I am thinking that "productivity" may just be a way to preserve the idea that the US economy is the best in the world ... a way steeped in mathiness. And maybe even racism.

Note that making the overall normalization $A$ an important factor and using "real" quantities basically transforms the scale of the unit of account an important property of economic growth. The fact that it's ~ 100 Yen or ~ 10 pesos that is roughly equivalent to 1 US dollar or 1 Euro (completely arbitrary designations) is turned into an important economic fact about productivity -- at least if the information equilibrium view is correct.

If you look at the Solow production function empirically and scientifically, you end up with a pretty simple model that works remarkably well! If you try to add assumptions, you end up having to live with added mysteries and a much more complex model -- that doesn't even work that well. Call it Occam's razor, or call it rooting out mathiness. Just be curious -- why do we make certain assumptions?

Update +15 min: The rather large NGDP growth of Mexico in the 1980s is the result of hyperinflation. The nuevo peso was introduced in 1993, shaving off three zeros from prices. Also added the (relative) in the paragraph after the graphs.


  1. This is an interesting post and illustrates a key difference between physics and economics.

    Physicists study natural phenomena. They give these phenomena names and try to understand them. However, the phenomena themselves exist whether physicists study them or not. Indeed, the phenomena exist whether humans exist or not.

    Economics (and business and engineering) are studies of man-made systems. All concepts in man-made systems are man-made concepts. That means that if we want to study man-made systems, we need to be clear precisely what we do and don’t mean by each and every term we use. From my own experience, failure to do this causes problems in managing systems even at the level of a single business or a single machine.

    Money isn’t a natural phenomenon. It is whatever we decide it is. For example, are government bonds money? If so, why? If not, why not?

    What precisely is GDP? I haven’t read ‘GDP: A Brief but Affectionate History’ by Diane Coyle but it’s an entire book which tries to define what GDP is and is not. Here are a few sentences from one of the reviews of the book on Amazon UK.

    “Gross Domestic Product is a concept so massive, so convoluted, so compromised by rules, exceptions, patches, and qualifications, that only a handful of people in the world fully understand it, and that does not include the commentators and politicians who bandy it about, daily. “There is no such entity out there as GDP in the real world, waiting to be measured by economists. It is an abstract idea.” That sets the tone of Diane Coyle’s excellent and sympathetic examination of GDP. Then it’s on into the impenetrable forest”

    Economists appear to analyse these concepts as though they are natural concepts but they are not. GDP is often used as though it is a synonym of ‘the economy’ but it isn’t. GDP relates to new assets, goods and services. It doesn’t include existing assets such as existing houses. However, very significant amounts of money are created when people take out mortgages to buy existing houses. Why, therefore, is it important to analyse GDP versus money rather than ‘the economy’ versus money?

    If you read economists closely, you will see that they often seem unclear or inconsistent even about the terms in their basic accounting identities e.g. saving, investment, consumption. Is saving just money or does it include bonds (assuming that bonds are not money) or shares or physical assets? I asked this question in a recent blog discussion on whether accounting identities were useful. I was told that different economists have different definitions of ‘saving’. Nevertheless, apparently we’re supposed to believe that the identities add up correctly irrespective of what definition you use!!

    (cont’d below)

  2. You mention productivity. I have tried to work out exactly what productivity means in macroeconomics but I am having problems. At a microeconomic level, productivity is something like ‘factory throughput per hour’ or ‘factory throughput per man hour’. However, at a macroeconomic level that would not make sense so economists appear to use productivity to mean something like ‘financial value created per hour’ or ‘financial value created per man hour’.

    For example, suppose an oil company can extract 100 units of oil per 100 man hours from under the sea. Microeconomic productivity is 100 units / 100 man hours = 1 unit per man hour. Suppose that it can sell the 100 units of oil for £100. Macroeconomic productivity is £100 value / 100 man hours = £1 value per man hour. Now suppose the price of oil falls by 50%. Microeconomic productivity does not change. It is still 1 unit per man hour. However, macroeconomic productivity is now £50 value / 100 man hours = £0.5 value per man hour. Macroeconomic productivity has fallen by half. Nevertheless, when macroeconomists talk about productivity they often appear to blame a fall in productivity on poor management as though they are measuring microeconomic productivity. It is not clear to me exactly what macroeconomic productivity is supposed to measure or whether it is useful. Other examples make the concept even less clear e.g. what is macroeconomic productivity in the public sector where the services supplied do not have a price?

    My conclusion from all of this is that a ‘scientific’ study of the economy would have to be very precise on the definitions of all of these terms. Adding precise mathematics on top on poorly and ambiguously defined concepts doesn’t seem very useful to me. This goes back to a discussion we had some time ago about science being about observables. It’s also why I think that accounting (including the concept of conservation as used in physics and chemistry) is central to any ‘scientific’ study. In a business, it is usually the accountants (and the systems analysts in the IT department) who understand the precise definition of terms. The role of these people is to fit the real world into a model which can cope with all of the anomalies found in the real world. These people think more like Charles Darwin (what is the full diversity of reality and how does it evolve) rather than Isaac Newton (what are the fundamental and unchanging rules of reality).

    1. Your description of physics is a deliberate process of mathematical archeology ... slowly scraping away at reality with experiments and uncovering the fundamental properties of nature. You add that such a process wouldn't be amenable to economics because it lacks those fundamental properties of nature -- they are all human constructs subject to flux.

      I disagree with both of those assessments. Physics is mostly a process of feedback between people looking at empirical regularities and people looking for empirical regularities, using math as the medium. And all things, even human behaviors, can have empirical regularities ... look at the sizes of cities (following a power law), or all of the results about random networks. Randomness and the law of large numbers are incredibly powerful forces. You have to have something really powerful to push back against the law of large numbers while it's pushing you into a normal distribution (usually) with small (relative) fluctuations -- even against your will.

      When you bring up the quote about GDP not being well-defined, I think the law of large numbers comes to save us. There is a funny statistical regularity on business shows in the US that whenever the DOW reaches a new record while the S&P500 has not, they run a story about how the DOW is a terrible measure of stocks and the S&P500 is, like, the greatest.

      In my previous blogging incarnation, I wrote a post how silly this is. The key point was that you actually don't need a properly defined stock index to get the properties you need in a properly defined stock index.

      Similarly any shoddy measure of GDP is going to be highly correlated with the perfect measure of GDP. And current data is not really good enough to tell the difference, anyway!

      Regarding productivity, I'd agree. I personally think productivity is a way to encapsulate the idea of entropy I describe here. Somehow the result of economic production is more than the sum of its parts ... well, one way that works in thermodynamics is entropy.

    2. I’ve been a way for a few days and have only just caught up with your reply here.

      I am sure that you are right in your points about randomness and laws of large numbers. I am not disputing that and I am not saying that maths is not useful. Far from it.

      My main point is that analysing GDP versus money is a very narrow way of looking at the economy. It may tell you something useful. However, it’s fairly arbitrary and may also lead you astray.

      Even at the level of a single business or organisation, different people have very different mental models of a system. An accountant might think of the system in terms of its financial transactions. On the other hand, someone in HR might thing of the system in terms of its people and hierarchical structure. A factor manager has a very detailed mental model of the factory. On the other hand, a planner at head office may think of the factory as a couple of equations in a mathematical model.

      Each of these people has a different perspective. They each see things that the others don’t. They each are oblivious to anything outside their own mental model. There is no ‘correct’ model as might be the case in physics. This is why almost all successful organisations have decision making structures which take account of multiple perspectives. This is also why a common symptom of a dysfunctional organisation (or profession e.g. macroeconomics) is the dominance of one specific perspective. This can come about because of dominant personalities or an insular culture or perverse incentives or many other reasons.

      I am sure that your models are useful if only in showing that they are as accurate as the more convoluted models of professional economists. However, their value is limited by whatever parameters you include in the model. The problem with using maths as the primary tool in economics is that it constrains models to an artificial and limited number of parameters. (As you have said before, the limited amount of macro data is also a constraint). That means that, as with the perspectives of the people in my example above, all current mathematical macroeconomic models are blind to the many missing parameters.

      It’s not clear why ever increasing GDP is necessarily a good thing but that is the assumption underpinning most macroeconomic models. Perhaps by pursuing economic growth we are burning too much carbon and polluting the environment to a destructive level. Alternatively, perhaps we would prefer a more egalitarian society even if it meant giving up on some level of growth. Alternatively, maybe we are nearing the limits of the useful technologies arising from physics and chemistry. Alternatively, maybe growth depends on education levels but only relating to certain specific subjects. Alternatively, maybe wars and natural disasters encourage economic growth through the need to replace whatever is destroyed. Alternatively, maybe globalisation is the dominant force in separating demand (from currently rich countries) from supply (from currently poor countries). Etc etc.

      Diverse perspectives are essential for the study of any complex system.

    3. Hi Jamie,

      It's true that GDP may not be important! I'd say my purpose here is to go through the following research program:

      0) Does GDP obey mathematical principles?
      If no, then we shouldn't care about GDP. Make policy based on something else.
      If yes, then:
      1a) What are those principles?
      1b) What do those principles mean?
      1c) What policy should be decided from the meaning of those principles?

      Since we as a society haven't fully answered 1a or 1b, I'd agree that policy should incorporate a diversity of views, possibly informed by various parital answers to 1a and 1b. It is possible that the principles that answer 1a may not be good enough to make policy. That would answer 1b with "very little" and 1c with "use some other prinicple besides GDP".

      We might not have answers to question 0 yet either, but we shouldn't assume there is no answer.

      Actually the information equilibrium picture seems to answer these questions as:

      0) Yes
      1a) Information equilibrium
      1b) The economy follows a paricular path of GDP regardless of policy (i.e. it is mostly useless) but the path and deviations from that path can have three very specific uses: deviations indicate higher likelihood of recessions, the path indicates the effectiveness of monetary policy, and the path indicates how big a monetary or fiscal stimulus package should be. Over time monetary policy becomes less useful, so fiscal policy should be used to restore lost GDP.
      1c) Don't pay attention to GDP as a part of public policy except as an indicator of recession likelihood, size and most effective remedy


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