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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