Mathematics Versus Mathiness
In "Protecting the Norms of Science in Economics", Paul Romer writes:About math: I have studied physics as an undergraduate. I’ve seen clear evidence that math can facilitate scientific progress toward the truth.I'll see his undergraduate degree in physics, and raise him by a Ph.D. in applied mathematics (Control Systems Theory). Undergraduate physicists study highly polished and elegant models which have been chosen because they describe aspects of physical behaviour quite well. Conversely, engineers study complex systems which may or may not appear to follow those simplified laws of nature*, and are forced to make educated guesses on how to simplify the analysis of the systems. Economics is dealing with complex systems that are a lot closer to the reality of engineering than the ivory tower of undergraduate physics.
If you think that math is worthless or dangerous, I’m sure that there are people who will be happy to discuss this with you. I’m not interested. I’m busy.
One needs to push down the expectations of what mathematics can accomplish. Mathematics consists of two things:
- sets; and
- mathematical (formal) logic, which determine the rules on how we can manipulate sets to create new sets. (One can also set new rules for mathematical logic and see what happens, but that is largely the domain of pure mathematicians.)
For example, I took a first year graduate course on Linear Systems (taught by George Zames) when I was an undergraduate, He asked "What is a function?", and nobody in the class knew the correct answer (including me). People were coming with definitions involving rules, mappings, et cetera. The real answer: a function is a set of ordered pairs with some extra properties tacked on. If we write y=f(x), that is really just specifying that f is a set of the form {(x,y)}, with the usual property that if (x,y) is an element of f, we cannot have another element (x,z) in the set, with z<>y. Once we fix the allowed logical rules, everything in mathematics is a set, and all you can do is manipulate said sets.
When you read modern mainstream economics, this seems to be lost. You read about things like "equilibrium", "households", "agents" as if they are concepts within mathematics. The exception that I am aware of is Stock-Flow Consistent (SFC) modelling, as those authors actually write down the system equations in a fashion that is consistent with the established norms of mathematics.
As a very basic example, take two optimisation problem models.
- Model M1: Find x such that the function U(x)= -x^2 is maximised.
- Model M2: Find x such that the function U(x) = -(x+1)^2 is maximised.
It is straightforward to see that x=0 is the solution to model M1, and x=-1 is the solution to model M2. Even though there is a function with the same name (U(x)), and it is a function of a variable with the same name (x), most undergraduates will have no difficulty in understanding that these are actually different objects (sets). To what extent these mathematical models and solutions correspond to real-world optimisation problems, both models can be viewed as "valid models" (but presumably not for the same real-world system).
However, Dynamic Stochastic General Equilibrium (DSGE) macro has a hard time dealing with this distinction. Results from optimisation problems on one set of mathematical systems ("microeconomics") are imported at random into another optimisation problem (the DSGE model). Which random set of results which are imported is based on the political biases of the DSGE modeller. To translate this to my simplified example above, this is equivalent to saying that x=0 is the solution to M2, just because it is the solution to M1, and the variables have the same name. I would be hard-pressed to call this "mathiness"; "Calvinball" is a better description.
As an example, I will refer to the example in my recent discussion of fiscal policy in DSGE models that was published at the Real-World Economic Review.
All we can do with mathematical models of dynamical systems:
- Write them down clearly, specifying all the sets involved in the system.
- Determine the solution(s) to the system.
- See whether these system results correspond to real-world behaviour.
Growth Theory
This outburst by Paul Romer was largely triggered by some academic turf fight within the field of Growth Theory. The theory attempts to characterise the determinants of long-term growth trends. I have spent little time looking at growth theory, since it appears self-evidently useless to me. The main recommendations seem to be: "grow your population", and "be more productive, somehow!".
The long run is made up of short runs. At present, all of the major countries have put in place policy settings that leaves a large portion of their working population in unemployment or underemployment. This obviously constrains growth, yet I have not seen any part of "Growth Theory" that addresses this problem.
The Tyranny Of Consensus
In "Don't get mathy with me, or I'll give you a good shunning", Stephan Williamson responds in what I view as effective fashion. It should be noted that Williamson is a Monetarist, not a school of thought of economics that I normally have a lot of positive things to say about.
Williamson writes:
Think of truth as existing at the top of a mountain. Once we get to the top of the mountain we'll know it, as we'll be able to see a long way, but while we're climbing the mountain we're in a fog, and we can't see the top of the mountain. But we might be able to discern whether we're moving up, down, or just sitting in one place. Paul thinks that we can't just let scientists run loose to take various paths up the mountain with different kinds of gear, and with different companions of their choosing. According to him, we have to organize this enterprise, and it's absolutely necessary that we write down a set of rules that we will abide by, come hell or high water. And when he says "reputational equilibrium" most economists will know what he has in mind - there will be punishments (imposed by the group) for deviating from the rules.
Simon Wren-Lewis objects, defending Romer's views.
But why is it important to have an idea of what that plurality is and acknowledge it? I can think of three reasons. First, it presents an honest picture to those learning the discipline. Second, it is very important that policy makers are told which ideas are widely agreed and which are the views of a small minority. That does not stop policy makers going with the minority, but they should know what they are doing (as should voters). The public’s trust in economics might also increase as a result. Third, it helps the unity of the subject, mutual understanding and progress. It becomes clear why those who do not accept the views of the “plurality” disagree, and what they need to do to convince the plurality that they are wrong.
Wren-Lewis' view is highly appropriate, given his status as a professor of Oxford. I hold a doctorate from the University of Cambridge, but I have no illusions about the intellectual history of the Oxbridge system. Throughout most of their history, Universities were not a centre of independent thought; they trained people to think at a high level within socially accepted limits. Variations of his argument have been intoned by Oxbridge dons for centuries.
To use political terminology that was probably obsolete when I was an undergraduate, Universities are part of "the Establishment". Independent thought was largely pursued by people working outside the university system. If the thinkers were part of a university, they developed their thinking alone in the library, not in a collegial atmosphere. Camille Paglia's book of essays Sex, Art, and American Culture: Essays describes this quite well.
Professor Wren-Lewis cites the use of New Keynesian models by central banks as support for his thesis that they are useful. This is actually an example of my thesis. Central banks are part of "the Establishment", and so whatever is popular in academia will almost by definition be used by central banks. And what did modern mainstream macro offer central bankers?
And:To use political terminology that was probably obsolete when I was an undergraduate, Universities are part of "the Establishment". Independent thought was largely pursued by people working outside the university system. If the thinkers were part of a university, they developed their thinking alone in the library, not in a collegial atmosphere. Camille Paglia's book of essays Sex, Art, and American Culture: Essays describes this quite well.
Professor Wren-Lewis cites the use of New Keynesian models by central banks as support for his thesis that they are useful. This is actually an example of my thesis. Central banks are part of "the Establishment", and so whatever is popular in academia will almost by definition be used by central banks. And what did modern mainstream macro offer central bankers?
- They were freed from the boring job of regulating the financial system.
- DSGE models were set in a such a fashion that central banks allegedly have to be independent of the political process, and are not subject to oversight like any other agency of government.
- Any forecast errors are not the fault of the central bank; they are due to "unforecastable shocks".
- The neoliberal bias within DSGE modelling (fiscal policy is allegedly ineffective) helps justify too-tight fiscal policy, leading to the low inflation rates that central banks can take credit for.
In summary, mainstream macro is structured to make central banks powerful and completely unaccountable -- which is a wonderful combination for any bureaucratic organisation. Is it surprising that they supported DSGE macro?
By contrast, anyone who has a job that depends upon being correct about the economy -- for example, market participants -- ignores DSGE macro.
Groupthink And The Natural Rate Of Interest
Mainstream DSGE macro is built around the concept of the natural rate of interest. I have noted before that this concept is circular, but I have not yet had discuss this in detail. James Montier of GMO has written about this in the article "The Idolatry of Interest Rates Part I: Chasing Will-o’-the-Wisp"
I do not necessarily endorse everything that Montier writes, but the following characterisations appear correct.
One could take the view that so many bright individuals all coalescing around a single framework was evidence of the wisdom of crowds. However, rather than representing the power of consensus, it appears to me to be evidence of extreme groupthink – it is very telling that not one of the aforementioned luminaries has questioned the framework itself.
According to the Wicksellian perspective so beloved by Bernanke et al., the natural rate of interest is simply assumed to exist. This has disturbing parallels with perhaps the oldest joke concerning economists that I know. When I first studied the subject nigh on 30 years ago, one of my teachers told us the old hackneyed tale of the shipwrecked academics: An engineer, a chemist, and an economist were all stranded on a desert island with no implements and a can of food. The engineer rigs up a Heath Robinson-like contraption with stick levers and vine pulleys. The chemist suggests using salt water to erode the can and then heating it once it is weakened. They then turn to the economist who merely says, “Let’s assume we have a can opener!”A consensus has built up around a theory that the economy is driven by an non-measurable natural rate of interest. Meanwhile, this natural rate of interest moves in such a fashion so that it is always consistent with observed data, and its movements cannot be forecast. This is non-falsifiable, and should thus be rejected as "anti-Scientific". However, the powerful figures in academic macro have built their reputations using this non-falsifiable theory. Any realistic assessment of academic politics tells us that innovations will have to come from outside of the University system.
Footnote:
* I had a summer job when I was a senior undergraduate at a research lab for large "electrical machines" (motors or generators). The macroscopic behaviour of those motors sometimes changed, even though the electro-mechanical properties of the system were supposedly fixed.
(c) Brian Romanchuk 2015
Really great post! You're on a roll with the last few commentaries. And on top of that you've got a book on the way! How you do it, I don't know.
ReplyDeleteI totally agree with your statement about math being a neutral tool of analysis (I would add a communicative tool also). As for the other debate about economics being a science or not (which I can't recall if you've written about it before), I think this too is an unfortunate debate. The way I see it (as someone who's spent quite a bit of time studying economics and other social sciences), I believe economics can and should make progress (improve itself as time goes by). To me, the search for progress is what differentiates sciences from the "arts" and other non-scientific endeavors (hobbies, etc). But getting back to your post, do you mean to say something similar when you say that economics is similar to engineering?
Yes, although I realise that I need to be careful about the analogy. I have in mind the analysis of systems, and not the building of them. A mechanical engineer changing a prototype is in a much different position than a government attempting to adjust its economic policies.
DeleteControl engineers typically are not involved in the design of the physical system (unless its control is highly problematic), rather they are taking an existing physical system, which is presumably complex, and then attempting to build a control law (a mathematical rule) to make it operate in a satisfactory fashion. Control systems is therefore built around seeing what are the limits of mathematical models, which may not be how most people visualise engineering.
Returning to your question, I hope that we can gradually get a better handle on how economic systems evolve over time. At the very least, we can rule out techniques that do not work. I believe that you need to work with approximations, and not expect the resulting models to be highly elegant.
The tools used in control engineering may not applicable to economics, but the principles might be. Unfortunately, the latest application of control engineering is cringe-worthy. "Optimal control" was abandoned as a bad idea by control engineers in the 1960s, but it has been revived by economists. I wrote about optimal control earlier on this site.
Optimal control was replaced by robust control (which I studied), which was built around taking into account model error. These techniques are a way of dealing with "uncertainty" (as opposed to "randomness"). I may have mentioned the relationship to uncertainty, but I am unsure whether I explained this concept in an article.
The problem with maths is that it is a communicative tool in the same way that Latin is a communication language.
DeleteThere is a reason that translating the bible into English was considered an heretical act. It eliminated the power of the elite to translate the information via their political filters.
The economic system is an Information System and we never use mathematics to model those because there are about four people in the entire world who can do the necessary mathematics at that complexity without making dozens of mistakes.
And therefore only about four people who can ever understand the description properly.
Within academia, it can be used as a tool for communications. What it allows is the ability to specify exactly what you are writing about, and to judge whether a contribution is novel. With purely verbal descriptions, a field can go in a loop forever. (This is not inevitable, but you need strong commitments to standards, which I am unsure that the modern university system can provide.)
DeleteWhether or not this is working out in practice in economics is debatable. Even in control systems, the average quality of publications could be quite low despite the mathematics.
Neil - I have my doubts that the math alone is the issue. Although one could predict that something bad was going to happen, predicting the precise effects of the meltdown in 2009 would have been extremely hard. There was a lot of crazy financing out there, and it was impossible to tell how deeply involved investors were with it (until the unravelling began). If we are stuck with partial models, those partial models need not be very complex, but they will not explain everything.
PS: And about Montier's article and the ongoing debate about real rate or natural rate being negative, I still don't get what the big deal is all about. Back in the day, secstag was about a shortage of investment opportunities yielding a rate of return acceptable to investors. Today, they say the real rate of interest compatible with full utilization is negative, and not consistently achievable. Seems to me like a difference without much of a distinction.
ReplyDeleteI am less familiar with the history of the idea, but yes, I have a hard time seeing what the kerfuffle is about. I think it is mainly the result of Larry Summers dredging up a cool phrase which is topical. The recent papers seem to blur into one another after reading them.
DeleteTrue enough! The real debate is whether, moving forward, we are going to see persistent secular stagnation caused by low demand or the so-called 'demographic timebomb', which essentially implies too much demand. It's incredible (in the true sense of the world) that both extremes are as widely discussed.
DeleteYes really nice post from my perspective, a physical scientist with an interest in economics.
ReplyDeleteReminded me of another econ joke.
https://pbs.twimg.com/media/A_3UCatCIAETDAA.gif:large
Thanks. To a certain extent, you are part of my ideal reading audience. People who want to learn economics, but have some math training. (This was were I was when I started in finance; to a certain extent, my writing is guided by what I would have found useful then.) I do not use a lot of math here (my reports will have more), but I do put up simulation results that I hope that are not just black boxes for my readers.
DeleteFrom your background in control engineering, I would assume that you focus on feedback loops that can be used to influence future movement.
ReplyDeleteMy most recent hobby is seismic activity detection using a home built seismometer. Economics seems to me to have many seismic characteristics. The most striking similarity is the impulse effect of the initial event. I compare the creation of new money to an earthquake. Both events are an "impulse" with the creation of new money being a little more predictable.
Following the initial impulse, seismic analysis is a tracing of the effects of the released energy distribution. I analyze economics the same way, trying to follow the effects of each money impulse event. Individual spending then becomes an impulse event, with bank loans just another example of impulse economic activity.
Now, bringing this comment back to your post on macroeconomic "mathiness", I agree that math that does not have clearly understood links to observable patterns fails to be persuasive and may be misleading.
Perhaps the seismically similar nature of economic activity is yet another path that can be explored.
Roger,
DeleteI work in the seismic processing industry, so am perhaps qualified to comment here. I think that perhaps the greater similarity between the two is that some of the processing techniques which are used with seismic (time series analysis, filters and so on) are also common to the toolset which can be used to look at economic data.
As you perhaps know, a seismic trace is modelled by an impulse response convolved with a series of delta functions representing stratigraphc layers. Whether or not you could use such a model to simulate the economy I don't know, but it's an interesting idea.
I took a course on Semiconductor Theory in the late 1980s. In a semiconductor material the charge carriers are electron-hole pairs. The population of charge carriers is characterized by specifying the rate of generation versus destruction (recombination) of electron-hole pairs. In the early 1990s I studied law. The population of financial instruments increases when markets net create asset-liability pairs in the attempt to convert money into financial investments. This population decreases rapidly when society attempts to convert financial investments into money. The financial intermediation system is not designed to create money unless it can roll-over liabilities, and it cannot roll over liabilities when markets try to convert investments into money. The electron-hole analogy works best for me, however, in finance the change in population of instruments is driven by psycho-biological changing perceptions of balance sheet positions tied to cash flows that depend on balance sheet expansion.
DeleteBrian,
ReplyDeleteRegarding your comments on growth theory, to my mind one of the biggest omissions of growth theory is its utter lack of any physical basis, as I belive the current models are descendants of the Swann-Solow model, which involves only labour, capital and "technology", whatever that is. As such it is completely unable to comprehend issues like rising resource costs, peak oil and what have you. As such, I feel that this kind of leaves us navigating through dangerous waters, not only without a map, but crewed by people who deny the existence of coral reefs.
I fit within the "peak oil" camp, so I agree with you from a very practical point of view. (What should society be doing?)
DeleteBut if we look at things like debt dynamics, nominal GDP growth matters. What we see is that it is the same as Gross Domestic Income. As long as people are doing something, they are earning income. Since we have a hard time aligning nominal income to real world production, it is entirely possible that it will appear that nominal and real GDP can keep growing.
I wrote about this in an article "Sustained Growth on a Finite Planet."
Yes, I read that article and found myself with broad agreement with your position.
DeleteI suppose the problem comes from the fact that there are two fundamentally different ways of looking at the economy, which are perhaps difficult to reconcile. On the one hand, one can look at it purely in terms of a physical system, meauring energy usage, material throughput, efficiencies and so on. Or one can regonise that it largely exists to cater to human needs, values and preferences, which is much more intangible and is measured in terms of dollars, which are ultimately a nonphysical unit and is instead an abstract symbol used for accounting and signalling within the system. To what extent the two views are able to talk to one another is, I suppose, the fundamental question.
One economic theory is that innovation will remove constraints imposed by scarcity. For example the development of relatively abundant fossil fuels replaced burning scarce whale oil. In the Bible there is a story about the karma of energy and money: Joseph stores up grain (energy source) during the years of plenty and dispenses grain in exchange for farm animals and money during the famine to restore the land. Farm animals are a sink of energy and source of natural pollution when kept in high concentrations driven by the profit motive.
DeleteThis is a great blog for those in the economics profession but with some kind of mathematical background. Keep it up!
ReplyDeleteA few comments.
1) I too am wholly uncomfortable with quantities like the neutral interest rate which are not observable and measurable. To accept this fundamental aspect without some degree of scepticism makes me think that economics has a religious element to it. That is, accepting statements on faith without evidence.
2) So why is this the case? Economics has a strong policy orientation, i.e., the economy is "broke" and people need to fix it without really understanding the dynamics in the first place. Too many of the models are self-justification for the role of policy and don't really explain the "why" or "how" with much depth or insight. Moreover, economists are poorly equipped to deal with the nitty-gritty of problem solving, e.g. data collection, analysis, doing this efficiently with some programming etc. Graduate school tends to be the first time they have to do some proper mathematics which means they probably didn't learn too much in their undergrad, they are stuck with economics as a field and don't have much flexibility to move, and their toolkit is really quite limited for problem solving. It's just really shocking at the lack of rigour despite the major impact economics has on people's daily lives. I think the whole discipline is in desperate need of reform.
3) Economic historians are under appreciated but more data never hurt.
End rant.
Thanks!
DeleteEconomic history is important, particularly in finance. The same cycles repeat, just with added complexity that just distracts from the underlying forces. For the style of analysis that I am doing here, knowing economic history is more useful than the theory.