Tuesday, March 3, 2009

A tipping point model of crises

I recently reviewed the book Animal Spirits by George Akerlof and Robert J. Shiller. Despite the mediocre score I gave it overall, it contains a number of interesting ideas on incorporating psychology into macroeconomics. I think every macroeconomist should read it.

I also recently blogged about my own idea to contributing a psychological element to economics: tipping point expectations to explain economic crises. It just so happens that the first chapter of Animal Spirits discusses just such a concept in the confidence multiplier.

Akerlof and Shiller argue that confidence is about trust, and "the very meaning of trust is that we go beyond the rational". More interesting is how they see trust disintegrating within an economy:
As long as people remain trusting, their impulsiveness will not be evident. But then, when the confidence disappears, the tide goes out. The nakedness of their decisions stands revealed.
To me, this sounds like a tipping point problem, and so I present here a simple tipping point model of confidence, where optimism can collapse into financial ruin quickly.

Let there be a group of N investors who are simple information updaters. Each starts with a belief in the state of the economy x0, with probability y0 that the view is optimistic and 1-y0 that its pessimistic.

At time 1, each is given a random piece of information about the economy Y1 that is positive about the economy with probability y1 and negative about the economy with probability 1-y1. They each then update their beliefs to x1=x0*Y1.

In period 2, they are given a new piece of information Y2 that is positive about the economy with probability y2 and negative about the economy with probability 1-y2 and each then updates their beliefs to x2=x1*Y2.

This continues until at a certain period, call it period t, a greater than a zt percentage of individuals have a pessimistic view of the economy. At this time t, each individual changes their belief, x(t-1), of the condition of the economy to xt = x(t-1)*(1-zt). That is, they ignore whatever information yt they are getting and focus only on others beliefs.

In period t+1, there are then z(t+1)>zt percentage of individuals that have a pessimistic view of the economy. Everyone then updates their belief of the condition of the economy to x(t+1) = xt * (1-z(t+1)). This continues as a downward spiral until everyone is pessimistic about the economy.

This model can be easily programed in Excel, which you can find my attempt here. I assume information x0 and Yt is obtained randomly from the 0 to 1 interval, with yt=0.80 and zt=0.50. Pressing F9 refreshes the probabilities and so reruns the model. An intesting outcome of the specific model I use is that crises of confidence happen about 14% of the time. Here is one run that looks quite nice:

In this example, the average level of optimism about the economy is high in the beginning. Then, when poeple see enough others are pessimistic about the economy, they begin changing their updating by focusing on the negative. Eventually, optimism, and the economy, tanks.

I like this model because it explicitly captures how people pay attention to their own information until they see enough other people disagreeing with them, then they radically change their updating. Like in the real world, it takes a lot of others saying we're wrong before we believe it, and then if the information is bad enough, we'll drop everything to get on the bandwagon.

Does this model imply that people are irrational? Not necessarily. If I have limited information about the state of the economy, it may be rational for me to use the data I get until enough other people disagree that I am forced to believe my information is wrong. In a world of incomplete information, it is difficult to know for certain what is going on, and so we must rely on the information we have, and a few heuristics to get us by.

I'm not sure how original this model is. I've looked around and can't find anything that looks quite like it, but I've been known to be behind the curve sometimes. If anyone knows of a similar approach, or a better one, please let me know.

Or, given that I'm not sure what to do with this from here on out, if anyone is interested in formalizing this for a wider audience, or knows someone that might be, please contact me.

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