*****WHOOPS Made some mistakes in analysis. Update HERE*****

**Check out PDF for Equations**

**And Code to run yourself**

**Introduction:**

“This Time is different” by Reinhart and Rogoff is an
empirical history of financial crises and panics. It describes many economic events such as;
inflation, bubbles, and defaults.

The theme that ties these events together is the idea that
people have an unrealistic expectation of the future because they believe “This
time is different”. With this new belief, they act in a way as to cause the
next crisis.

While the book dealt with the empirical investigation of the
matter, it didn’t describe the theme itself in intricate detail. This post will
attempt to fill that void by giving a mathematical interpretation of the theme.

There are 3 aspects of the model I’d like to capture,

1.
prior beliefs act guide and tell us how to make
decisions

2.
decisions have an effect on the future

3.
people update our beliefs when new information
becomes available

**Economic Maximization Problem:**

I came up with a simple driving example that describes these
properties. Suppose we’d like to
maximize the speed of a car and minimize the probability of a crash (or
maximize probability of not crashing). However, the probability of a car
crashing is dependent on the speed. This can be described as a utility
maximization problem described below, where we maximize subject to a
constraint.

Maximize:

Utility(Speed,Probability of Not Crashing) = (Speed^a)*(Probability of not crashing ^(1-a))

Utility(Speed,Probability of Not Crashing) = (Speed^a)*(Probability of not crashing ^(1-a))

Subject to:

Probability of Not Crashing = B*log(1/speed-1) + C

Where B and C are the constant and slope parameters of an
inverse logistic function and a is tradeoff for utility function between the
two goods (Probability of not crashing and Speed).

(Look in PDF for optimal solution)

**Bayesian Updating Problem:**

Described so far the problem is a fairly straightforward
maximization problem. However, in this model, the parameters B and C are unknown to the individual. While there is
a true function for the relationship between Speed and probability of not crashing (that is a true B and C)
the individual doesn’t know it.

Bayesian statistics seems like a natural to model the individual’s
beliefs over these parameters. With this prior belief over parameters, he
maximizes his utility subject to it and makes a decision in each time period.
When new information becomes he updates his prior information and uses this to
make a new decision. In this sense, the system is dynamic and a time series can
be produced.

Unfortunately, I don’t believe there is a closed form
solution to updating Bayesian logistic regression. Therefore, I used RSTAN to
simulate from the posterior distributions (Code shown at end).

As you can see, the individual is learning the true
parameters as he goes through time. When a crash occurs, he dramatically
reduces his decision and if a crash doesn’t occur, he slightly increased his
decision. This can be seen as a smoothed over “this time is different” where
the actor adjusts his decision more optimistically until a negative event
happen, in which case, he changes his decision more abruptly.

The optimum decision is shown in the blue horizontal line,
and while the actor doesn’t achieve it in this example, he will eventually (… I
think) as this is the equilibrium.

**Conclusion:**

The purpose of this post was to describe a mathematical
interpretation of “This Time is Different”. I did this by solving a
maximization problem, with unknown parameters, and then updating those
parameters using the Bayesian Framework.

While the example I used was only relation to speed and car
crashes, I don’t think its much of a stretch to use this model to describe
economic events. An example of might be that a bank wants to have an optimal
amount of amount of capital requirements vs event of bankruptcy for a bank. (The
smaller capital requirements, the higher the expected profits, and higher
probability of bankruptcy.)

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