# Wobegone Finance

Robert Waldmann

The average person thinks he has higher than average intelligence. This is an empirical fact [citation needed]. It is not incorporated into standard finance theory, but it matters a lot. It has been argued that the high volume of transactions on financial markets can be understood easily. If two people with equal quality information each think they have better information than the other, each will think that he or she benefits when they take bets against each other [citation needed plus why didn’t I think of this].

This psychological fact can also explain bubbles. A bubble can grow and burst if people don’t recognise that it is a bubble. A bubble can also grow and burst even if people recognise there is a bubble, but each thinks that he or she will be the first to detect the peak. This theory of bubbles fits the sort of things traders say during bubbles. The standard phrase for this sort of reasoning is the greater fool hypothesis.

Human overconfidence makes us very tempted by the greater fool hypothesis, even if we are the greatest fool. Given the risks, overconfident people self select into finance. Given personel management at financial firms, people who have been lucky will have power over lots of money — there is no way to tell luck from skill. They will also tend to be overconfident — there is no way to tell luck from skill.

It is trivially easy to write a model in which it is very important that everyone thinks they will detect something sooner than everyone else. It doesn’t even matter much what that something is.

This model is unusually pointless. The point should be obvious given the discussion above. In fact, I think the point was obvious to many people long before I wrote this post. The model clarifies nothing. If you understand the point it is supposed to illustrate, you might be able to follow it and gain nothing.

A very simple model of bubbles. Time is discrete. Agents are risk neutral. There are two assets. One is a risk free asset which pays interest rate r. The risk free asset is a storage technology, so the amount of risk free asset is not in fixed supply. There is one share which pays dividend r each period. Population is a continuum normalized to one (each agent is infinitesimal compared to the market). Agents are not allowed to hold short positions (borrowing is shorting the storage technology). The last assumption is just needed to keep each agent’s demand bounded given risk neutrality.

There is a random variable X_t which is equal to 0 with proability a(t) and equal to zero with probability 1-a_t. I am going to be very vague about a(t) but just note that it grows so there is some big T such that a(T)=1. This will amount to saying that everyone knows the bubble must burst by T+2.

Agents do not see this variable instantly. If the variable is 0 at period t, then agents perceive that it is zero in period t+2. Agents know this is true of all other agents and was true of him in the past, but each thinks that he has extra alertness and perceives X_t in period t+1. For the moment agents don’t understand that other agents are over confident. They think the other agents know that the other agents detect X_t in period t+2.

What can happen to the price P_t of the share in this model ? It is easier to ask what can’t happen, since many things can happen.

One possibility is that P_1 = 1 all the time. Both assets are, in practice, riskless paying return r with certainty.

It is alos possible that there will be an unsustainable speculative bubble. P_t can be greater than one when X_t-2 = 1 and fall to one when X_t-2 = 0.

First the standard assumption. All agents are rational. No agent is over confident. Each knows he sees X_t after the same 2 period lag as everyone else.

There can be a sunspot equilibrium only if P can go to infinity.

X_t-3 = 0

X_t = 1

1)x_(t-3) = 0, P_t-1=0

2) X_(t-2)= 1 P_t= P_t

3) if X_(t-1) = 0, P_(t+1) = 1

if X_(t-1)= 0 P_(t+1) = (P_t(1+r)-a(t-1)P_(t-1) + r)/(1-a(t-1))

Formula 3 also gives P_(t+1) conditional on P_(t+2) and X_t=1 and etc.

This would work fine except for the assumption that there is a T such that a(T) = 1. That assumption amounts to the assumption that everyone knows the bubble must burst by period T+2 at the latest. There will be a zero in the denominator of the formula for X_(T+1).

So by backwards induction, there can be no bubble.

Ignoring that, imagine an overcondent agent. An overconfident agent is sure that in period t+1 the price will be greater than one so long as X_(t-2) = 1. An overconfident agent thinks he is seeing the signal one period before he really sees it, so he thinks this means X_(t-1)=1. This means an overconfident agent will put any amount of his money (up to all of it) in the risky asset in period t so long as equation 4 holds

4) X_(t-2)= 1 P_(t+1) = (P_t(1+r) – r)

Uh oh. There is no a in that equation. An overconfindent agent thinks the probability of the bubble bursting has nothing to do with his choice in period t, since he is sure that the bubble won’t burst in period t+1.

Actually even if agents know all other agents are overconfident, it doesn’t matter. It is enough that I think that the other guys definitely won’t see the sunspot that says “bubble bursting now” for 2 periods.

Finally, it is not necessary for agents to over estimate the spead at which they detect the signal. If agents actually see the signal after one period, but each thinks that all other agents see it after two periods, then everything works the same except that the subscripts on X are a little less irritating.

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