Mathematical Modelling

Mathematics and modelling in particular is very powerful, and hence can be very risky if you get it wrong, as in mainstream economics. But is modelling inappropriate – as has been claimed – or is it just that it has not been done well enough?

As a mathematician who has dabbled in modelling and economics I thought I’d try my hand at modelling economies. What harm could there be?

My first notion is that actors activity is habitual.

My second is that habits persist until there is a ‘bad’ experience, in which case they are revised. What is taken account of, what counts as ‘bad’ and how habits are replaced or revised are all subject to meta-habits (habits about habits).

In particular, mainstream economists suppose that actors seek to maximise their utilities, and they never revise this approach. But this may be too restrictive.

Myself, I would add that most actors mostly seek to copy others and also tend to discount experiences and lessons identified by previous generations.

With some such ‘axioms’ (suitably formalised) such as those above, one can predict booms and busts leading to new ‘epochs’ characterised by dominant theories and habits. For example, suppose that some actors habitually borrow as much as they can to invest in an asset (such as a house for rent) and the asset class performs well. Then they will continue in their habit, and others who have done less well will increasingly copy them, fuelling an asset price boom. But no asset class is worth an infinite amount, so the boom must end, resulting in disappointment and changes in habit, which may again be copied by those who are losing out on the asset class., giving a bust.  Thus one has an ’emergent behaviour’ that contradicts some of the implicit mainstream assumptions about rationality  (such as ‘ergodicity’), and hence the possibility of meaningful ‘expectations’ and utility functions to be maximized. This is not to say that such things cannot exist, only that if they do exist it must be due to some economic law as yet unidentified, and we need an alternative explanation for booms and busts.

What I take from this is that mathematical models seem possible and may even provide insights.I do not assume that a model that is adequate in the short-run will necessarily continue to be adequate, and my model shows how economic epochs can be self-destructing. To me, the problem in economics is not so much that it uses mathematics and in particular mathematical modelling but that it does so badly. My ‘axioms’ mimic the approach that Einstein took to physics: it replaces an absolutist model by a relativistic one, and shows that it makes a difference. In my model there are no magical ‘expectations’, rather actors may have realistic habits and expectations, based on their experience and interpretation of the media and other sources, which may be ‘correct’ (or at least not falsified) in the short-run, but which cannot provide adequate predictions for the longer run. To survive a change of epochs our actors would need to be at least following some actors who were monitoring and thinking about the overall situation more broadly and deeply than those who focus on short run utility. (Something that currently seems lacking.)

David Marsay

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About Dave Marsay
Mathematician with an interest in 'good' reasoning.

3 Responses to Mathematical Modelling

  1. David Chester says:

    Mathematical modeling is not an end in itself but simply a more concise way of expressing a complicated situation in a short-hand way. What is behind it is the need to be logical and then it is natural for the mathematics to follow. Without logic we cannot formulate a theory of economics or anything else. It is so important to be logical that we cannot accept an axiom as a starting point in an argument without said axiom having been checked and double checked by both common sense and deep philosophical understanding. This make axiom building both an honorable and a dangerous profession.

    The special axioms applicable to macroeconomics are self contradictory yet meet this need. “Man seeks to satisfy his needs with the least exertion” ,and, “Man’s desires are unlimited”–both by Henry George in 1879 book “Progress and Poverty” which sold 3 million copies and is still in print (he should of course included women too). These 2 axioms define macroeconomics as a combination of two opposing effects and it is only by taking this attitude that we can make progress in understanding how this particular subject works. Past confusion in it were by people who were unable to see it in this way.

    My recent book “Consequential Macroeconomics” takes the logical process into the need for building a model as if our subject were an engineering one, having a logical core. (write to me at chesterdh@hotmail.com for a free e-copy. See also my working paper SSRN 2865571 “Einstein’s Criterion Applied to Logical Macroeconomics Modeling” which explains the way this most simple yet complete (another combination of opposites) model can be built using only 6 entities and 19 mutual flows of money verses goods, services, rights, taxes, etc.

    “If I am not for myself, who is for me? And being for myself alone, what am I? And if not now, when” Hillel the Elder, cira about 0 CE.

    • Dave Marsay says:

      Thanks for your thoughtful comments. In any empirical discipline logic can be used to try to rationalise current understanding, or more directly to understanding the domain better. Thus in physics there is a clear distinction between theoretical and experimental arms, with real progress requiring both.

      The proposition ‘Man seeks to satisfy his needs with the least exertion could be taken to mean that man seeks an optimal cost/benefit trade-off, but following Herb Simon’s concept of ‘bounded rationality’ it seems consistent with the notion that Man is a creature of habit, so long as he is not aware of any undue down-side. At least, it is important to note that under some circumstances Man can fail to satisfy his needs, or be hugely wasteful. I agree that our current system is largely driven as if Man’s desires were unlimited, but this seems to me a problem to be overcome rather than necessarily an ‘axiom’. If you want to make it an axiom then surely we need to pay close attention to the dimensions in which Man’s desires are unlimited. Not, I think, always just money-wealth.

  2. David Chester says:

    Our way of modeling is necessarily logical and to try to include the irregularities about wasteful use of opportunities is almost out of bounds when model building. All we can do is to include a factor which is less than 100% to allow for loss. With this approach, the two axioms still apply but with some added aspects in the model such as use of by-products and the cost and effort of cleaning-up.

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