Math Arguments

The purpose of this post is to address many of the statements/arguments made by Austrians about math or their position on math.

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 the 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 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.  For Austrians should accept all mathematical propositions that are true, from 1 + 1 = 2 to the propositions in set theory or algebraic topology etc.

However, 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?  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.

Cubic Spline Interpolation and Econometrics

This paper was written for my Numerical Analysis class.  It was a small project in which I had to write a code and also answer a real world problem.  One of the problems I face as an aspiring economist is finding appropriate data to run regressions on.  If the data does not have the same time frequency, the model is meaningless.  Hence, I wrote a code for Cubic Spline Interpolation and showed it accurately interpolates data.

The paper requires a bit of mathematical maturity, but the idea can still be clearly understood.

Note: I did not include any of the appendices, and the format of the paper was altered when I turned it into a PDF for some reason.

Click Here for PDF

A Brief Defense of Economic Models

In 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 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 just 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.

But I digress.

If what Austrians said about mathematical models were true, I would not need to write this because I would probably agree with them.  However that is not the case.  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 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 are 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 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.  Same goes for tariffs and 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.

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

Neo-Classical Economics

As of late I have been getting into many debates with adherents of Austrian Economics about how economists should do economics.  One thing I have found to be in common with all these debates is that Austrians don't fully understand how neo-classicals actually do economics.  They tend to think the neo-classical sits at a computer with data and tries to empirically find economic principles from this data.  This could not be further from the truth.  I have written a few economics papers and there is a standard way to go about writing them.

Step one of writing a paper is deciding what you are interested in.  I personally am interested in education, environmental economics, game theory, agriculture and others.  Step two is to come up with a theory involving your area of interest.  This is no easy step.  The theoretical portion of the paper is THE most important part of the paper.  If the theory is wrong then the rest of the paper means nothing.  Up to here, neo-classicals and Austrians agree.  Theory is the most important part of economics.  That being said, I can understand the Austrians grief with mathematical economics.  A lot of mathematics is replacing theory.  This is not good for economics.  This does not mean that mathematics does not belong in economics, or have uses in economics, however.

After the theoretical part is done, step three is to do literature review.  Often times in this part of your paper you will be able to find whether your theory is on target or needs to be adjusted.  Furthermore, during the literature review you see how other economists have addressed this problem.  This part of writing a paper is important as well because here is where you decide whether past economists have made mistakes in their theory or their model and can find areas to improve upon.

Next is to gather data.  This is where neo-classicals and Austrians disagree.  Austrians think that gathering and testing data is fruitless.  They have various arguments why they think it is fruitless, but these arguments are lacking.  I will address why I think they are lacking in a different post.  Gathering data can make or break your paper.  If you cannot find proper data then your paper is meaningless.  The data obviously has to be relevant, have many observations, be recent, and it must be "clean".

After you have found and cleaned your data, it is time to make your empirical model.  Many times this is a regression.  If you have ever written a proper econometrics paper you will understand a lot of effort goes into this.  This part of your paper takes around 20-30 pages of tests to make sure your estimates are not over or under estimated.  After creating your model you can now run the tests and see whether your theory has merit or not.  Then finally, the last step of writing your paper is to point out the flaws in your own paper and model.  No model is ever perfect and can always be improved upon.  This is one of the most important parts that will lead further researchers in the right direction.

Here is an example from a paper I wrote as an undergraduate.  In the movie Food Inc. (which I highly recommend) they talk about how the price of beef goes up with the price of corn and fuel.  This is bad because cows are supposed to be grass fed and it is healthier to eat local where the cows are not grown as fast and as unhealthily as possible.  I decided to test this.  What I found is that the price of corn and fuel, while it did slightly effect the price of beef it was not statistically significant.  I.E., the price of corn and fuel do not significantly affect price changes in beef.

In conclusion, I hope this sheds some light on how neo-classicals do economics.  It is not this radical data worship that Austrians can make it out to be.  Neo-classicals believe theory always comes first.  I personally think running a regression to see if your theory holds is not very radical.  As Rothbard points out, Austrians play mind games holding things constant in their head to try and extract economic principles is the same thing as a regression except regression can test how significant the economic principles are after you have played this mind game.

Discovery and Mathematics

When 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.  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".

If we 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.  So without being able to actually 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 but one 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)

What does any of this have to do with a blog dedicated to economics?  I am working to demonstrate the powerful tool of mathematics and its limitless ability to discover patterns in the world.  It is because of the powerful nature of mathematics to discover patterns in the world that it really irks me 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?

Mises Institute Article

Today I read an article on the Mises Institute entitled "Monopoly Through Austrian Lenses" that left me truly dumbfounded.

In the beginning of the article, Newman feels the need to "bash" a tool that neo-classical economists use.  He argues against the introductory monopoly model used to teach students why monopolies are inefficient, and then he acts as if this is as rich as monopoly theory gets to the neo-classicals.  If one wants to argue against a certain school of thought, should they not be arguing against the most advanced and highly celebrated theory?  Certainly they should not be arguing against the learning tool for college freshman.  Also, the differences aren't just compared to the perfect competition model either, they are compared to the oligopolistic model, and the monopolistic competition model.  But that is neither here nor there, just one more thing he forgets to mention, as if the only models are monopolistic and perfect.

In essence what he is doing is using a teaching tool to argue that the methodology is wrong.  This is similar to me bashing the "broken window fallacy" to say that Austrian Economics is the wrong way to study.  The broken window fallacy does not take into account the desire for the shopkeeper to want a new suit, or hat or whatever his desires are.  Since all individuals are different with different degrees of desires, how can we conclude that the shopkeeper won't use a credit card to buy what it is he desires, or take out a loan?  Now he has to fix the window, which leads to increased spending and he buys the suit he so desperately desires.  Hence the multiplier is real and the broken window fallacy is a farce.  Austrian economics must be the wrong way to do economics right?  But of course this argumentation is not right.

The next thing he does left me as perplexed as the first.  He compares it to individuals selling their labor.  This surprised me because the monopoly model used in the article holds zero bearing when considering individuals selling their labor.  There is an entire different field of study for that.  It is called Labor Economics and recognizes that each person has a monopoly on their own labor.  This again goes back to what I mentioned in the blog I wrote a few weeks ago.  If Austrians want to start getting taken more seriously they need to start making coherent arguments.  One must have taken two economics courses, labor and micro, to realize this argument does not mean anything.  So what do you think someone with a PhD in economics thinks of it?

Just a few more things to consider.  First he talks about calling things "unjust" or "unfair".  I hope he is not bashing neo-classicals with this argument but rather the interviewer.  No where will you find a good economist in any school of thought talk about things being unjust or unfair when they are doing economics.  However, this might not be the case if they are talking about political philosophy.

One last thing I would like to consider is the following statement, because Austrians seem to make this claim all the time.

"Finally these valuations are entirely subjective in two ways:  Jones's utility or satisfaction from wearing  a pair of Oakley's cannot be compared quantifiably to his satisfaction from wearing a pair of Ray-Bans, even by his own inspection.  Similarly, Jones's satisfaction from wearing a pair of Oakley's cannot be compared to Smith's satisfaction from wearing an identical pair of Oakley's.  Interpersonal utility comparisons are impossible, and even intrapersonal preferences are only ordinally ranked."

No economist would disagree with the Austrians that we cannot put a number on our "happiness" and then compare them.  There have been economists that have given us tools to look at this in a different way, such as Von Neumann and Morgenstern in TGEB.  Austrians have never addressed these arguments as far as I have seen (I plan on writing about this in a different post).

Here is one way how Jones can do it.  Say for example the Ray-Bans cost $150 while the Oakley's cost $130.  If he buys the Ray-Bans, clearly he prefers them by at least $20.  And in his head it is completely plausible for him to think "I'd pay up to $180 for these!"  I do this in my head all the time.  Almost every time I purchase something I do this.  Think about the next time you go to McDonald's you are either going to buy off the value menu or not.  If I do, that means I don't value a Big Mac at its original price.  However, if it is 2 for $3.33 I might buy them.  I can then compare the utilities as prices.

Finally Jones's satisfaction of wearing Oakley's to Smith's can be done the same way.  They both know what they would be willing to pay and compare the prices.  It's not rocket science.

This of course is exactly what airlines do when setting their prices.  Clearly someone buying tickets 3 months out from a flight is not going to be willing to pay as high of a price as someone whose parent gets sick and needs to fly out immediately.  Hence the airline charges more to the more needy person.  Their "satisfaction" from getting the ticket is higher than the person who has 3 months to make other arrangements. 

I do not understand why there has to be a separate way to look at monopolies for the Austrians.  Other schools of thought have already absorbed the proper Austrian Theory into their theories, why can't Austrians do the same?

Here is the link to the article I am referencing:
http://mises.org/daily/6468/Monopoly-Through-Austrian-Lenses

Litigation Fear

I asked our twitter followers if there was anything in specific they would like me or my brother to write about and someone asked us to contemplate the question "Does fear of litigation force doctors to overcompensate?"

The immediate response in my head was "it has to force them to overcompensate".  Here is why:

If we consider this through a decision theoretic approach it is easy to see why doctors are likely to overcompensate.  Decision theory is similar to game theory except instead of considering multiple individuals making decisions against each other, we consider a single individual making decisions against "states of the world".  The possible states of the world we must consider are: x = not overcompensating and correct diagnosis, y = overcompensating and correct diagnosis, z = not overcompensating and wrong diagnosis, w = overcompensating and wrong diagnosis.

Now consider what "states of the world" are most favorable.  Obviously it is either x and y, and to the doctor the difference between them is negligible even if to the patient they are not.  This leaves z and w as the two least favorable situations, of them, w is more favorable to the doctor and z is more favorable to the patient ( z >w for the patient because lower medical bills, w>z for the doctor because "they covered all their bases" and is less likely to lead to litigation).  From this we can see the rational decision for the doctor is to overcompensate because it maximizes his expected "payoff".  (payoff being keeping his job, not getting sued etc.)  The payoff is greater because the payoff of x=the payoff of y, but the payoff of w > payoff of z.

What effect does this have on society is the next question.  Well for one, it increases costs to the patient.  In the state of Illinois women had to buy psychiatric coverage, unlimited overnight stays in the hospital, OBGYN coverage, unlimited mammograms and a plethora of other things.  The reason why they had to buy these is because doctors have to "cover their bases".  This obviously increases health insurance costs.  It seems absurd that a woman who is sane, not trying to have a baby shouldn't be allowed to buy basic coverage.  This is why, well part of the reason why, so many people could not afford health insurance.  This of course leads to all sorts of other costs to society, such as medicare/medicaid, for example.

This is how I think of this problem and hope it helps everyone understand the situation more clearly!

Austrians and Game Theory

My favorite area of economics and mathematics is Game Theory.  It is a fascinating subject with a wide variety of applications.  As someone who used to consider myself an adherent of the Austrian school of economics I know it deals with some of the issues Austrian Economists have with normal General Equilibrium methods of economics. Also, it just so happened to be "co-founded" by an Austrian economist, Oskar Morgenstern.  I knew Mises was not a fan of game theory, however, I was interested to see what current Austrian economists think of it and that is what motivated this post.  Needless to say there is a lack of understanding.

I went searching on mises.org to find as many articles I could, this first led me to the mises.org blog.  People there had varying opinions of game theory but the overwhelming understanding of game theory was completely mistaken.  One person mentioned how game theory is largely based off the prisoners dilemma and therefore is a useless thing to study.  This is very frustrating because game theory has roots all the way back to Emile Borel and other mathematicians, it's most influential work was published in 1944, and the prisoner's dilemma as we know it is credited to Luce and Raiffa in 1957!  This, amongst other fallacious arguments, were very common in these blog discussions.

Okay, so people writing on the blogs misunderstand game theory, but the trained economists couldn't be so drastically mistaken could they?  In an article entitled "The Games Economists Play" by Robert Murphy he attacks the conclusions of game theory in a clearly misinformed fashion.  He analyzes the aforementioned prisoner's dilemma and uses it to claim that because of this we should not accept game theory in general.

In the prisoners dilemma, two players are accused of committing crimes, one minor crime in which their guilt can be proven with out a confession, and major crime for which they cannot be convicted unless at least one confesses.  The confessor will go free but the other will go to jail for 6 years.  If neither confess, they will go to jail for only 1 year.  If they both confess they will go to jail for 5 years.  In this game without communication, the Nash Equilibrium is for each player to confess and hence go to jail for 5 years.  This, to Robert Murphy, is the downfall of game theory because each person could increase their non-jail time time by not confessing.

What Dr. Murphy does not understand is that Nash Equilibrium does not tell you the outcome will be optimal, only the strategies that rational players will make.  The reason the prisoner's dilemma is so famous is because it was the first example of an inefficient Nash Equilibrium (at least that I have found in my research)  Furthermore, he says

"Even here, the game theorists orthodox analysis is not entirely appealing: real world players often do cooperate even in a one-shot prisoner's dilemma"

This made me wonder if he is completely unaware of the study of games with communication?  Or even the study of cooperative game theory.  In the situation described above it is assumed that the prisoner's cannot communicate and have zero way of knowing what the other will do.  Hence, it certainly becomes much more plausible to confess (I have watched enough First 48 on A&E to see that people often do confess).  Also, if we analyze the game properly the outcome makes complete sense.  Since the criminals are not cooperating or communicating in any way, as soon as criminal A thinks criminal B will not confess, criminal A has all the incentive to confess.  His choice becomes either go to prison for one year or zero years.  Likewise for the other criminal.  Now one could argue that many criminals would rather go to jail than be a "rat".  This is where I would like to point out that the focal point effect already deals with this objection.  

Now, if we look at this game through a cooperative game theory lens, the outcome changes entirely.  Through this lens, we can consider any way in which the criminals will cooperate.  Consider the possibility that there is a contract signed before hand in which the criminals agree to not confess otherwise face a punishment worse than prison.  In this game, the person does not have any incentive to cheat because the time he spends free will be worse than time in prison due to the punishment.  Hence, the equilibrium now becomes both criminals not confessing.  

Two more points to consider: first, he mentions people using the prisoner's dilemma to argue for government intervention.  Again, these people do not understand the fact that the criminals have no way to communicate or cooperate.  Hence, their argument is invalid.  Further, the fact that Robert Murphy would actually use this argument to argue against game theory is intellectually dishonest.  He is misrepresenting something he should have studied while getting his PhD.  Any game theorist knows the prisoner dilemma can actually be used to argue for LESS government.

Finally, he gives a formidable representation of "backward-substitution".  Again, I am wondering if he is unaware of the vast literature on the subject of repeated games.  In 1982 Kreps, Milgrom, Roberts and Wilson constructed a way to show that "non-confession" strategy will be employed given an initial certain doubt and actions during the game.  The explanation gets very technical with a lot of game theoretical terms so I will not go into it here, but it is discussed in full detail in "Game Theory: Analysis of Conflict" by Roger B. Myerson in section 7.6.  Also, there are game theorists who study forward induction as well.

In conclusion, it is clear to see Robert Murphy builds up a straw man representation of game theory in order to tear it down.  No where does he address any of the advancements of game theory in the last 20-30 years.  He does not address perfect equilibrium, proper equilibrium, sequential equilibrium, subgame perfect equilibrium, trembling hand perfect equilibrium, the focal point effect, repeated games and the list could go on.  He does however, address Nash equilibrium and the prisoners dilemma, two aspects of game theory developed in the 1950's.  Both of which have been greatly advanced.