Thursday, March 30, 2017

Explaining versus defining (models vs model definitions)


Nick Rowe tweeted his old post on his "minimalist model of recessions" and comes to the conclusion that: 
This minimalist model of recessions gives us a very simple message: recessions are a reduction in the volume of monetary exchange caused by an excess demand for the medium of exchange. Recessions reduce utility because some mutually advantageous exchanges do not take place.
Emphasis in the original. However, this conclusion is based on the following procedure
  1. Assume three markets: firms producing A, B and a "money" market M
  2. Assume utility functions for A and
  3. Maximize utility subject to constraint
  4. Solve for Nash equilibrium to obtain A = B = 100/P
The question is: Does this explain anything or rather just define recessions as an excess demand for money? Steps 2 through 4 are just mechanical mathematical procedures that effectively transform the assumptions of step 1 into the result of step 4. In fact, you really need nothing more than Walras' law with an aggregate goods market and a money market. An excess demand for money is then equal to a deficit of demand for aggregate goods. The "embroidery" (Rowe's term) the minimalist model adds is just to say that if there are two goods markets, both will suffer from a deficit of demand (Walras' law only tells us at least one must).

As it stands, this model just defines recessions to be an excess demand for money. I think this is part of a more general problem in macroeconomics: instead of developing frameworks to study what a recession is, macroeconomic frameworks just define what a recession is.

On its own, simply positing assumptions that lead to a conclusion via mechanical procedures is really no different than positing the inevitable conclusion. There are two cases where it becomes interesting. The first is where you don't know where the mechanical procedures lead ‒ deriving a completely new result. For the second, I will take you through a a different mechanical procedure that leads to a well-known result.

Let's start by assuming there is a constant acceleration due to gravity that has units of distance/time². Integrating this with respect to time (mechanical procedure) we obtain:

v(t) =  -g t + v₀

Integrating again, we obtain

s(t) = -½ g t² + v₀ t + s₀

This minimalist model of ballistic trajectories gives us a very simple message: trajectories are parabolic functions of time. (I'm intentionally paraphrasing Rowe above.) But does the explanatory power of this procedure derive from the assumptions or the procedure itself? No. It comes from the assumptions and the procedure plus empirical data:


Without the data, I'm just defining the function s(t).

In a sense Arrow-Debreu general equilibrium and Nash equilibrium are, on their own, equally devoid of explanatory power. Both are essentially applications of the Bouwer fixed point theorem (not detracting from these examples as mathematical results, but rather as economic ones). The question is whether the system set up (Rowe's three good economy, Arrow & Debreu's markets in time and space, Nash's N-player games, constant acceleration due to gravity) explain empirical data. They don't have to explain it perfectly (even the gravity model above neglects air resistance), but they do have to match data to some level of precision before they can be considered "explanations" rather than just "definitions" (or if you prefer "model definitions").

In writing that, I think that might be a good phrase to introduce to a wider audience. A model is something that explains data. A model definition is just a collection of assumptions and mathematical procedures that relate variables. A model starts off as a model definition, and becomes a model after it is compared to data.

Nick Rowe's minimalist system above is a model definition. The projectile motion equations I wrote down are a model. The IS-LM model as usually presented is a model definition, as are a great deal of DSGE models out there. In fact most of macroeconomics deals not with models but model definitions. Model definitions are only wrong inasmuch as they contain math errors. Models are wrong if they are rejected by empirical data. Conclusions reached via a model definition do not "explain" anything about the real world any more than defining a new term explains anything.

20 comments:

  1. "Model definitions are only wrong inasmuch as they contain math errors."
    They are also logically wrong when they are completely inappropriate to grasp a phenomenon, being just a playing and distracting toy even though mathematically correct.
    M

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    1. I was making a distinction between what I am calling "models" and "model definitions"; in your case you are talking about a "model" (since it is being compared to a phenomenon, i.e. data). "Models" are wrong if they are rejected by empirical data.

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    2. I have not been talking about a model neither comparing it to a phenomenon- and that should be evident: I guess model definitions are supposed to understand and describe phenomena (regardless of their turning in wrong or right models). However, it is possible to assume that pigs fly and elaborate a mathematically coherent model definition about their flights and play with it. That it is what neoclassic economics does.
      BTW recessions should be expected to occur regularly in capitalism, for its nature and way of working.
      M

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  2. Actually, there is data that Rowe's model is intended to explain: what was called, in the early 19th century, a general glut, what we now call a recession. It was a puzzle, according to economic theory of that time. A glut, an excess supply, of one thing was supposed to indicate an excess demand for something else. A general glut was thought to be impossible, yet it apparently occurred. It was John Stuart Mill, in 1829, who came up with an explanation. There was an excess demand for something, namely "money". Brad DeLong gives an overview here: http://delong.typepad.com/sdj/2010/06/is-macroeconomics-hard.html I read about Mill's explanation earlier on DeLong's site, but this post goes into much more detail than the one I read.

    Anyway, if Rowe's model did not produce an excess supply in both A and B, there would not be a general glut. It would not fit the kind of phenomenon it is supposed to fit.

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    1. I believe this is the same story I am telling in paragraph after the numbered list. But Nick's model by definition says that output is where beta firms buy apples (A) and alpha firms buy bananas (B) with the proceeds from each other.

      Another way to put this is that A and B production are coupled *by definition* so a fall in one leads to a fall in the other. This is really no different (mathematically) from a single aggregate good -- we could create an effective theory (and Nick does mention it) where we look at only A+B and it would have exactly the same dynamics and the only difference would be that the equilibrium (instead of a Nash EQ, it is a AS = AD EQ) is A+B = 200/P. In that case:

      (A+B)^d = 0.5(200/P + (A+B))
      (A+B)^d = 0.5(200/P + 200/P)
      (A+B)^d = 200/P

      But as I mention below in my reply to Nick, what would need to be measured independently would have to be this demand for mangoes ("money").

      This model posits a particular mechanism (excess demand for money) to explain a particular outcome (general glut). In order to show this is a model and not just a model definition (i.e. defining that general glut = excess demand for money), one needs to measure the excess demand for money without basing it on measuring the general glut.

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    2. Well, if we get a general glut without money, then Malthus was right. ;)

      As for and independent measure of the excess demand for money, I think that you are asking too much. You have a system with N degrees of freedom and you are asking for one with N+1 degrees for freedom. Another way of putting Mill's solution is to say that according to the theory of the time the system has N-1 degrees of freedom when it actually has N degrees of freedom.

      Mill's solution allows for a general glut among everything except money. Rowe's model aims to produce (at least for a system with only two commodities besides money) a general glut from an excess demand for money. That's a stronger result.

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    3. I'm not sure I made my point clear. The theory Nick puts forward is effectively a 1-good economy -- you can think of that single good as exchanges of fruit (= A + B) for money. Putting a label on different kinds of fruit doesn't really add anything (B could have apples and we've just called them B-apples and A-apples). It is the coupling of A to B (which is just a definition) that lets you rewrite the entire system in terms of A+B (per above).

      If we have a single good C and then redefine it as two goods A and B based on some arbitrary criterion, we haven't really built a two-good economy. But because we can take A and B in Nick's model and show that it really is just a model of C = A+B, we've shown that in defining A and B, we've only taken C and decomposed it arbitrarily into A and B.

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    4. OK, so Nick got a stronger result by making a stronger assumption, right? :)

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    5. I'm sorry, I have not studied Nick Rowe's model, and he can, and does, speak for himself.

      But if C = A + B in his model, simply affecting C does not necessarily affect A and B the same way. I suppose, in economic terms, you have to at least assume that A and B are in some form of equilibrium which will not be perturbed by (small) changes in the amount of money. Have I got that right?

      But in practice changes in the amount of money have differential effects. Like in '08 Wall Street got bailed out, but not Main Street. The creditors got money, the debtors didn't. I know of the claim that it does not matter who gets the money. Really?

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    6. "... affecting C does not necessarily affect A and B the same way."

      Not in the model as constructed. As it is constructed, A and B are tightly coupled. A shock to A would result in an equivalent shock to B (because A is the buyer of B's output).

      I think at a bare minimum you need a third firm such that if there is a shock to one, the other two could still increase their trade with each other to compensate. In that toy model, there is at least a possibility of not having a general glut. If it occurs in that case, then you have something non-tautological (because firms have a choice of whether or not to join in the general glut).

      I still think it would come down to definitions as to whether or not a general glut happened in the three-firm system, though.

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    7. Ah! Thanks. :)

      Yeah, my sense was that with a multitude of economic agents, you don't necessarily get a general glut from an excess demand for money, but a general glut implies an excess demand for money. Still, I found it interesting that you could get a general glut from an excess demand for money with only two agents.

      BTW, a good example of a long term excess demand for money is the situation with the British colonies in America, before the 18th century. That was pretty much on purpose, to make the colonies economically dependent on Britain. As a result, Spanish dollars circulated so widely in the colonies that both Canada and the US adopted the "dollar" as the name for their currencies. (I don't know about Australia, though.)

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  3. Bill's comment is good. I would add that we also seem to see an increase in barter and home production in those events we normally call "recessions". That fact (though unfortunately the data is not as good as I would like) fits the monetary theory, but is hard to explain with other theories

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    1. Hi Nick,

      Studies like this?

      http://faculty.chicagobooth.edu/erik.hurst/research/time.pdf

      In that paper, 50% of the ~ 2 lost work hours go into leisure and 30% go into home production/non-market work during recessions -- the rest going into various other activities. However that requires a bit of massaging of the data since in the raw data non-market work actually goes down. Let's take that study at face value.

      I do not understand how this is difficult to explain with other theories. For example, people who are idle get bored. I have been home sick with a cold the past two days and increased my normal home production (e.g. cooking, doing dishes) by a few hours. I'm sure if you had a reduction in work hours, you'd probably spend some time working on your cars.

      And even if it were the case that home production increase is a tell-tale sign of an increased demand for money, this isn't in your model above. In your model, people at the alpha firms and beta firms simply sit idle, twiddling their thumbs. This is to say that the U Chicago study linked above provides evidence for a different, more complex, model.

      What would turn your model from a definition into a true model would be some measurement of the increased demand for money that is not based on measuring the fall in output.

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    1. Jason,

      In metalogic or formal systems, two terminologies(proof theory and model theory) are corresponding to your concepts of model definitions and models respectively.

      A proof theory describes purely syntactic requirements and no meaning. A model theory describes interpretation of syntactic true statements.

      We often uses notations "P |- Q" for syntactic inference and "P |= Q" for semantic interpretation.

      A formal system without proof and model theories is not a formal system at all. We cannot verify formal system properties such as semantic completeness, validity, consistency, soundness, etc.

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    2. Peiya,

      Thanks for the information.

      I figured there was probably some formal version of what I was saying here (there always is).

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  5. I agree with Jason here but I think that there is an even more basic problem. What does Nick mean by “demand for money”?

    If I have demand for a bicycle, it means that I want a bicycle which I do not currently possess. Once I have purchased the bicycle, I no longer have demand for a bicycle as that demand has been met.

    What happens if we try to apply the same principle to money i.e. demand for money means that I want money that I do not currently have. In one sense, it is trivially true. We would all like to have more money so the demand for money is effectively infinite. Hardly anyone would turn down a pay rise or a gift of money even if they had no immediate use for that money. More realistically, we might see demand for money as demand for money NOW which I will commit to repay in the future. That is demand for a loan. However, excess demand for loans occurs during booms rather than recessions, so that’s not what Nick means by demand for money.

    So what does Nick mean? My best guess is that he means the absence of demand for new goods and services. If I have €100 and demand for a bicycle priced at €100, it means that I am willing to exchange the €100 for a bicycle. If I have €100 but DON’T have demand for a bicycle (or an alternative new good or service), Nick would say that I have demand for the €100 that I already have. However, that is confusing two different concepts: demand and possession.

    Take Nick’s conclusion:

    “recessions are a reduction in the volume of monetary exchange caused by an excess demand for the medium of exchange”

    Now replace “demand for the medium of exchange” with my interpretation:

    “recessions are a reduction in the volume of monetary exchange caused by an (excess) absence of demand for new goods and services”.

    This is true but not interesting. Even if my interpretation is wrong, Nick’s use of the phrase “demand for money” is confusing and inconsistent with the way the concept of demand is used elsewhere. Nick would need to define precisely what he DOES mean by the phrase “demand for money” before we could engage in further constructive debate on the subject.

    It is stating the obvious to say that we need to agree unambiguous definitions of terminology before we can claim to be doing science. Individual terms are themselves model definitions upon which we can build more complex model definitions about the relationships between terms. We cannot measure something, or calculate something, or forecast something in a mathematical model, if we cannot first agree a definition of that something. Modern economics does not meet this standard in many respects. This is just one example.

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    1. Yes, I think your definition "absence of demand for goods and services" really boils down the fact that "demand for M" in this model is just defined as an absence of demand for A and B. It's tautological.

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    2. Yep, of course it is just a ridiculous tautological claim, one out of many, of a non sense pseudo causal model and insane pseudo theory. A waste of time.
      Actually a lot of people would like to buy and spend money but they don't have enough money in their pocket though apparently are crazily demanding it, lolz.
      M

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  6. Another problem with economist measures of money they are incomplete data series often missing the euro dollar market these models out in garbage and pull garbage out

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Also, try to avoid the use of dollar signs as they interfere with my setup of mathjax. I left it set up that way because I think this is funny for an economics blog. You can use € or £ instead.