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

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?

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.