I have drawn a picture of the supply and demand for bubbles. I think this picture might make it easier to understand what I was saying in my previous post on the need for a bubble.
If the natural rate of interest, in an economy without a bubble, is below the growth rate of the economy, then the economy "needs" a bubble. There will be both a demand for bubbles, and a supply of bubbles, at any interest rate between that natural rate and the growth rate.
Economists normally define desired savings as a flow demand for assets. Here I want to think of it as a stock demand for assets. Think of an overlapping generations model in which people desire to accumulate a certain value of a stock of assets for their retirement. And for emergencies and other lean years. I have drawn the savings curve as upward-sloping, so people's desired value of their stock of assets is an increasing function of the rate of interest. But that is not essential to my argument.
Economists also normally define desired investment as a flow. Here I want to think of it as a stock supply of real assets. It's the fundamental value of the stock of capital plus land, where "fundamental value" means the present value of the flow of earnings. The investment curve slopes down for two reasons: because a lower interest rate means a higher present value of a given flow of earnings; and because more investments become profitable at a lower rate of interest.
If all assets were priced at their fundamental values, so there are no bubbles, Ponzi schemes, chain letters, etc., then the equilibrium rate of interest would be the "no bubble" natural rate at the intersection of the savings and investment curves.
The supply curve of bubbles (Ponzi schemes, chain letters, etc.) is perfectly elastic at the growth rate of the economy. Here's why.
A bubble, Ponzi scheme, chain letter, etc., that grows at rate b will also pay an interest rate of b (ignoring transactions costs etc.). If it grows faster than the growth rate of the economy, it is unsustainable. Eventually the total value of the bubble will be bigger than the income of the next generation to buy into the bubble, so they won't be able to keep the bubble growing, and it will burst. But if the bubble grows more slowly than the economy, it can, in principle, continue indefinitely. It is profitable to start a bubble, Ponzi scheme, or chain letter. And if the rate of interest offered by the bubble is greater than the market rate of interest, people will willingly participate. So there will be an infinite supply of bubbles at any rate of interest below the growth rate of the economy.
The demand for bubbles, understood as the value of the stock of bubble assets that people will willingly hold, at the market rate of interest, is equal to the gap between the savings curve and the investment curve. For a given interest rate, a fall in desired investment, or an increase in desired savings, will increase the demand for bubbles.
Suppose initially there are no bubbles, and that all assets are priced at their fundamental values. The economy is at the intersection of the savings and investment curves. The equilibrium market rate of interest equals the no-bubble natural rate.
If the growth rate is below the no-bubble natural rate of interest, then it stays there. The market rate of interest is the no-bubble natural rate. The equilibrium quantity of bubbles is zero.
Now suppose the growth rate is above the no-bubble natural rate, as drawn above. A bubble asset that offers a rate of interest b, just slightly above the market rate, but below the growth rate, will be both supplied and demanded. As the bubble spreads and satisfies the demand for assets, the market rate of interest increases. As the market interest rate rises, the fundamental value of the stock of non-bubble assets falls. The value of household assets, including both fundamental plus bubble assets, may rise (if the Savings curve slopes up, as shown). But the flow of savings under a national accounts definition, which ignores capital gains on bubble assets, will fall and may become negative.
Competition between bubbles drives the market rate of interest up to the growth rate. At this point, all bubbles are on the margin of sustainability.
If something very small goes wrong, the bubbles will become unsustainable, and will burst. The interest rate immediately drops to the no-bubble natural rate. Then when confidence recovers it all starts again.
There's no money in the above model. No liquidity preference, only a loanable funds (plus bubble) theory of the rate of interest. With no medium of exchange, there's no distinction between aggregate demand and aggregate supply of newly-produced goods. So there can't be a recession caused by a deficiency of aggregate demand. That's the next step. But I think you can see that with money in the model it would be very hard for the rate of interest to fall quickly enough to prevent a recession, barring a very aggressive monetary policy.