Tag Archives: Mathematical Economics

A Brief Defense of Economic Models

Economic ModelingIn this write up my goal is to convince some Austrians that mathematical models aren’t as fallacious as they presume. Before I get into my arguments I must first make this point, and it is something that needs to be kept in mind as the readers works through this brief write up. Mathematics, insofar as it is utilized in economic theory, does not predict human action in a deterministic fashion, rather, it captures the laws of human action such that predictions can be made given the constraints of relevant conditions under examination within the theory.

To sum up the Austrian view of models, one could argue that they believe models use false assumptions and lead to fallacious results. On the Austrian view, models cannot capture human action. I am not bold enough to claim that mathematical models can perfectly capture human action, but I do think models are highly useful. Consider the standard model of supply and demand. It gives us proper results, is extremely intuitive, and it is a wonderful teaching tool. Austrians, on the other hand, believe that the assumed continuity of the supply and demand curves are false. As far as I can tell, this is their primary objection to using this model. According to them, because supply and demand are not continuous, it is much more unlikely to reach equilibrium. I would like to point out that adherents of this model do not assume that the supply and demand ever actually reach equilibrium, but rather, prices are always adjusting towards equilibrium. This is a part of this particular model’s usefulness.

But I digress.

Consider this quote out of one of the leading graduate school micro theory books:

“Imagine that you are trying to explain a particular phenomenon with one of two competing theories.  Neither fits the data perfectly, but the first does a somewhat better job according to the standard statistical measures. At the same time, the theory is built on some hypotheses about behavior by individuals that are entirely ad hoc, whereas the second is based on a model of behavior that appeals to your intuition about how people act in this sort of situation. I assert trying to decide which model does a better job of “explaining” is not simply a matter of looking at which fits better statistically. The second model should gain credence because of its greater validity, which brings to bear, in an informal sense, other data” (Kreps, A Course in Microeconomic Theory, pg 8)

Does this quote seem to fit the Austrian argument that Chicago school economists are stat chasers and do not care about the validity of assumptions? Quite the contrary, actually. It seems to me that Chicago School economists are concerned about human behavior/theory first, and stats second. I do not see why any economist should have a problem with this method of economics. As long as the economist will admit some of the draw backs his assumptions might have, why should we completely throw models out the window? Note that when models are introduced through scholarly journals the author will almost always point out some of its downfalls, but explain why it is still useful.

Another reason why I think models are important was touched on in my second paragraph. Models are helpful learning tools because they possess intuitive insights into real world phenomena. Consider supply and demand again. If I were to try and explain to someone why rent controls create housing shortages, it is to my explanatory advantage to draw supply and demand curves, and then show the different implications of keeping rent lower than the market clearing price. It is much easier for the layman to see the consequences as a picture, rather than sorting through all the material in their head. The same goes for the economic analysis of tariffs and the minimum wage. I have a hard time explaining why tariffs are bad even to a relatively knowledgeable audience, but as soon as I draw it out for them it clicks.

The last reason why I believe models are important is because even models that are falsified still help us gain knowledge. This is because learning what assumptions lead to the false conclusions now gives us insight into why these assumptions won’t work for future theories. As long as the economist is trying to encompass human behavior as much as possible, and it is fairly explanatory and predictive, why should anyone reject it? Models can have good explanatory power as to why things happen and to what will happen.

Finally, I would just like to point out that this is barely scratching the surface of how much thought goes into what makes a good economic model.  To simply reject these ideas based off of the use of models alone, is in my view, preposterous

 

– JW

 

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Austrians, Mathematics, and Economics

MathematicsThe purpose of this post is to address many of the statements/arguments made by Austrians (in the tradition of Mises) about math, or rather, their position on mathematical use in economics.

One statement I have been hearing lately is along these lines, “Austrians are not against all math, just SOME math” or, “Austrians are not against math, just mathematical predictions” etc etc. I find these claims to be rather telling of the person arguing on behalf of the Austrian position. These statements tell me right away they do not even know or understand the methodological position of their own school of thought. On their view, these 5 quotes need to be interpreted as only being against some math and/or mathematical predictions. Here are the relevant quotations:

“The only economic problems that matter, defy any mathematical approach” – Ludwig Von Mises

“Now, the mathematical economist does not contribute anything to the elucidation of the market process”  – Ludwig Von Mises

“The equations formulated by mathematical economics remain a useless piece of mental gymnastics and would remain so even if they were to express much more than they really do” – Ludwig Von Mises

(These next two are my favorite)

“The mathematical method must be rejected not only on the account of its bareness. It is an entirely vicious method, starting from false assumptions and leading to fallacious inferences. Its syllogisms are not only sterile: they divert the mind from the study of real problems and distort the relations between various phenomena” – Ludwig Von Mises

“Mathematics cannot and does not enter into measuring ideas or values that determine human action.  There are no constants in these. There is no equality in market transactions. Therefore, mathematics does not apply. The use of mathematics requires constants. Mathematics cannot be used in economic theory” – Percy L. Greaves.

All of these quotes can be found in various articles on mises.org.

I am truly baffled as to how someone claiming to be an adherent of the Austrian School could read these, or any Austrian literature, and conclude that Austrians are only against the use of some math. I have read a lot of Austrian literature, and I personally have never read anything that would support that claim. Of course, quotations cannot be “proof” of anything, but I do think they provide rather strong evidence in favor of my argument. Moreover, the Percy Greaves quotation is in response to the question, “is economics completely divorced from mathematics?” Clearly, from his response he thinks it is.

Another statement I hear from Austrians is that Neo-Classicals do not give them any mathematical propositions they should accept. This seems to be a rather silly statement, and in many ways, entirely meaningless. Austrians should accept all mathematical propositions that are true, from 1 + 1 = 2 to the propositions in set theory or algebraic topology etc.

John NashHowever, to get specific I would like to point out two mathematical fields that have vast applications in economics. First is game theory. Game theory is a branch of mathematics first developed by Emile Borel, and then popularized by the works of Von Neumann, Morgenstern, Nash etc. There is a plethora of economic questions game theory answers. One example of such a question is – how do oligopolies decide on how much to produce given the production of the other firms? Game theory provides the answer to this question.

Second, is functional analysis. In general, functional analysis is the study of infinite dimensional vector spaces. This field answers the question – how can a copper mining company extract Q tons of copper from a mine over T years and maximize its profit? To find this function is one thing, and to prove it is the maximum of all functions is another. I would like to ask an Austrian how to solve this problem without the use of mathematics? In my view, it simply cannot be done.

Mathematics is vitally important to the study of economics, and to denounce it the way influential Austrian scholars have is exactly why I am not an Austrian economist.

 

– JW

 

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Discovery and Mathematics

MathematicsWhen most people think of mathematics, they usually think of algebra, geometry, calculus, differential equations, and that is pretty much it. While there is nothing wrong with this analysis, the truth of the matter is, those subjects are on the bottom of the totem pole of mathematics. There is so much more to mathematics. Topology, for example, is the study of topological spaces. Topological spaces could be anything from the real number line to the 11 dimensional shape of our universe that some theoretical physicists claim it to be. In general, mathematics is the study of patterns. This allows us to do remarkable things, and one example is to “see the unseen”.

Consider the history of black holes. Einstein did not believe they existed because they were too “mathematical” and couldn’t arrive at their existence intuitively. There were other physicists who disagreed with him. Clearly Einstein was wrong here, but how was the debate settled? Black holes have such a large gravitational pull that we cannot see them. They do not even let light escape from their grasp. So without being able to physically observe them, how can we conclude that Einstein was wrong? Well, that is where math comes in. The laws of conservation can tell us many things, and one of them is if we put 10 gallons of water through a hose and only get 9 gallons on the other side we know there must be a hole somewhere in the hose. This is how physicists and mathematicians can “see” black holes. If we observe 10 particles going through a selected area and only 3 emerge on the other side we know there must be a hole somewhere in that area. I am a bit of a nerd so things like this are amazing to me. We can look at a piece of paper with symbols and numbers and literally see a black hole in those symbols and numbers. (it turns out the equations of general relativity hold true under the extreme conditions of black holes and hence, they are the strongest evidence that Einstein’s theory is true)

The powerful tool of mathematics possesses a limitless ability to discover patterns in the world, and this would also include patterns of economics. It is because of the powerful nature of mathematics to discover patterns in the world that I find it bothersome when certain economists act as if it is “silly” to use mathematics in economics. Some economists even argue that those who use mathematics in economics are not doing “real economics.” Mathematical economics is relatively young, and advancements to the methods are being improved constantly. There have been 6 (correct me if I am wrong) mathematicians to win the Nobel Prize in economics for game theory. These methods are being applied to various areas of economics, especially oligopolies, with very good results.

This isn’t to say all we need is mathematics, far from it. Good theory is always the most important part of an economic paper. All I am arguing is that mathematics can be a useful tool to complement the theoretical portion of economics.

It took mathematics to prove one of the greatest physicists was wrong about his disbelief in black holes. Will those who denounce mathematical economics come around if there is a truly significant advancement in economics achieved mathematically?

 

– JW

 

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