Brad DeLong has a (simple, according to him) model of the financial crisis. He does away with rational investors, has stochastic dividends, and has agents who compare rates of return (at random). I think we can probably get rid of these three assumptions and (maybe) write a more successful model. Here's what I would do.
Investors have the option to invest in a risky asset or a safe asset with return r. There is a "true" value of the risky asset, v, based on the (expected) fundamentals of the asset. This could be the discounted value of expected future dividends for a stock or the discounted value of the expected rent of a house.
Investors can calculate the true value of the asset, although it may be subject to shocks so that the current price of the asset may be above or below the long-run value at any given moment. Investors maximize per period (expected) profit. The key is that investors know that asset bubbles are possible. That is, the price of the risky asset can be above its long-run value for multiple periods?
Why is this? It's because investors are willing to invest in a Ponzi-like scheme. When prices start going up, investors assign some probability, a, that the price of the risky asset will continue on the same growth rate as the previous period, and a probability, (1-a), that the price will fall to its long run value (once the bubble bursts). As long as the probability they assign to prices going up is known and shared by all investors, it may be rational to continue to invest in the risky asset even when above its fundamental value.
The main question in the model, I believe, is how to model a, the expected probability that prices will continue to go up. I think it will work if a is a function of the past growth rate in the price of the risky asset and the distance between the current price of the risky asset and its fundamental value (although we may not need both). That is, a is decreasing in the distance between the current price and the fundamental value and is increasing in the past growth rates.
The model's dynamics come from heterogeneous risk preferences in investors. That is, investors have different cut-off points a* at which they are willing to invest in the risky asset. That is, more risk loving investors are willing to buy the asset with a lower a then are more risk averse agents. More risk averse agents will only be willing to buy the risky asset when a is high.
So what happens? A shock hits the asset's long-run value so that its current price is below the fundamental value (as in DeLong's model). This means that a is positive (but small) and less risk averse investors purchase the asset. As they purchase the asset the price of the risky asset goes up and a increases. This draws more investors into the asset and the price continues to go up. However, as the price gets farther and farther above the fundamental value, a starts first to increase more slowly and eventually to go down.
As soon as the growth rate of the risky asset price is negative, everybody assigns a value of a = 0 and sells the asset. Since everybody wants to sell at once, this may push the price of the asset below v. I think this should also push up the price of the riskless asset and so push down the interest rate, r. Once the asset falls to (or below) the fundamental value, the asset either stabilizes or even starts to rise.
The key is that everybody knows its an asset bubble. They are willing to invest in the risky asset because as long as a is high enough, the expected value of the risky asset is higher than the riskless asset. Investors maximize profit by buying the asset above its fundamental value simply because there's a high enough value that the price will keep going up. They're not at the last layer of the pyramid scheme.
It's not exactly true, of course, that investors are always aware they're in a bubble. But what is true is that once the bubble bursts everybody realizes that it was a bubble and tries to get out. This is easy for stocks but less easy for houses. Thus stock bubbles can burst in a week while housing bubbles may take longer to fully deflate.
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