Mathematical Model of a Bubble

[Alan]

I'm playing with some mathematical models to explore what is needed for a bubble. This is a first version and very simple. I will post more as I keep playing with the numbers.

v1. Assume price increases linearly with demand and demand increases linearly with a change in price.
Demand(t) = 50+PriceChange(t-1)/K
Price(t) = 1000*Demand(t)
PriceChange(t) = Price(t) - Price(t-1)
PriceChange(0) = 1000

Results for K=2000
demand price price change
50.00 50000.00 1000.00
50.50 50500.00 500.00
50.25 50250.00 -250.00
49.88 49875.00 -375.00
49.81 49812.50 -62.50
49.97 49968.75 156.25
50.08 50078.13 109.38
50.05 50054.69 -23.44
49.99 49988.28 -66.41
49.97 49966.80 -21.48
49.99 49989.26 22.46
50.01 50011.23 21.97
50.01 50010.99 -0.24
50.00 49999.88 -11.11
49.99 49994.45 -5.43
50.00 49997.28 2.84
50.00 50001.42 4.14
50.00 50002.07 0.65
50.00 50000.32 -1.74
50.00 49999.13 -1.20
50.00 49999.40 0.27
50.00 50000.14 0.73

The price oscillates up and down, but eventually settles to the base price I created in the function. This is NOT evidence that real property prices will do the same -- this model is way too simple.

As K increases the price settles more quickly since demand responds less to change in price. If K<1000, the oscillations grow to infinity. If K=1000 then the system oscillates at a constant amplitude. The price jumps the entire amplitude each time step. There is something wrong with the model there. Demand probably should not be based only on the last observation. An average of the past few observations might work better. I may also add a delay between price and demand to account for observation delay.

The more interesting change will be to make price increase non-linearly with demand.

[Alan]

After playing with a spreadsheet for a while I am convinced that demand is not a direct function of change in value. I will try to explain why.

Suppose we have some product with demand X based on utility. Each X sells for 100. Some even occurs in the market the either increases demand or reduces supply and results in X selling for 110. The increase in sales prices attracts investors. The higher the appreciation, the more investor demand. An increase of 10% results in an investor increase of Y. In order for a bubble to form, the new demand X+Y must increase the market price of the asset by another 10%. I cannot get this to work out using what I feel are reasonable assumputions.

An example:
Let base demand be 50. Price is 1000 times demand. Investor demand is 100 times the appreciation rate. Assume that some event has just occured causing a 0.01 increase in price. Investor demand is 1 so the next total demand is 51. We have to set the supply/demand curve so that 51 causes a price of at least 50500. That sounds reasonable, but since appreciation is still 1%, the investor demand does not change and we get the same price in the next time step. That causes appreciation to be 0% and investor demand disappears. We go back and adjust the supply/demand curve so that the price at 51 is 51000. That gives us a 2% return rate. Now investor demand is 2. But our price growth has to be greater than 2% to keep driving up prices. We set the price at 52 to 52530 for a 3% growth rate. We have to increasing the price growth rate each step or the 'bubble' pops.

This model does not match at all what I have observed and so I think it is the wrong model.

I think there were some probems with the way I constructed the model.

I think that the function describing demand based on rate of return is much closer to a binary stepwise function than it is to a linear function. If the asset appreciates faster than other investments then there will be close to constant high demand with respect to how far it exceeds other investments. Conversely, if said asset performs worse than other accessible investment vehicles then no one is going to use it for an investment (no one buys a new car as an investment).

But demand does change in a bubble. I think this is a result of confidence in the investment being sound. If you watch an investment for a long time and it consistently has a good return rate then you will feel comfortable investing in it. One can model this mathematically with statistics. Your confidence in the mean of a set of observations is equal to 1/sqrt(N) where N is the number of observations. Lower is better in this case since it means your estimate of the mean is more accurate.

Different investors are comfortable with different levels of risk. As time passes and a bubble continues, it attracts more investors who had not entered previously because they did not have sufficient observations to overcome their own personal risk bias. We could model the risk bias of a population with a gaussian. A graph of the number of people above a certain percieved risk would look something like this:



As perceived risk approaches the mean, the bubble will grow the fastest. But then it will begin to slow down. Eventually, demand becomes flat, appreciation stops, and perceived growth begins to drop. Those who have a low-risk bias have been utterly convinced that this is a safe investment and will likely see price drops as an anomoly. They will be left holding the bag.

I'm going to see if I can get this model working well in a spreadsheet.