This post is a sketch of a model of secular stagnation; and of bubbles that burst and get replaced by different bubbles. I don't formalise the model mathematically, because I don't have a comparative (or absolute) advantage at that sort of thing. But I think it could be formalised fairly easily.
Start with a model where the equilibrium interest rate is below the growth rate of the economy. For example, an overlapping generations model, where the young produce consumption goods, and want to save for their old age, but real investment opportunities either don't exist, or else yield a very low rate of return.
Now suppose there are some intrinsically useless shells that the young can pick up from the beach and store. The shells come in lots of different colours. There is a fixed stock of shells of each colour on the beach.
Every period there is a sunspot that is observed by all. The sunspots come in lots of different colours too, just like the shells. Every period there is a small probability that the sunspot will change colour. The new colour is determined randomly. The sunspots are intrinsically irrelevant events, just like the shells are intrinsically useless assets.
The equilibrium (or one of many possible equilibria) looks like this:
If the sunspot suddenly changes colour to red, the red shells become valuable and all other shells become worthless. Young agents spend part of their time collecting red shells from the beach and storing them. Next period, when that cohort of young agents become old, and if the sunspot is again red, they sell those red shells to the next cohort of young agents, who again sell them to the next cohort of young the following period, and so on.
If the sunspot suddenly changes colour to green, the red shells become worthless, and are thrown away on the beach. The old agents who held those red shells are impoverished. The young agents collect green shells from the beach and store them, and the cycle continues.
We need to assume there are strictly positive costs of storing shells, otherwise the red shells would not become worthless when the sunspot turns green, because agents would buy them and hold them, waiting for the next red sunspot.
We need to assume that the probability of the sunspot changing colour is small, and the cost of storing shells is small, otherwise it would not be profitable for the young to collect and store red shells when they see a red sunspot.
In the limit, as the probability of the sunspot changing colour approaches zero, and as the cost of storing shells approaches zero, the rate of return on holding shells approaches the growth rate of the economy.
We can have an equilibrium in which every individual is behaving rationally, and where intrinsically worthless assets are valuable, but where bubbles burst and get replaced by a new bubble, and all agents know that bubbles eventually burst.
If you are an old agent when the sunspot changes colour, you are worse off than you would be without a bubble. Because you bought red shells when you were young, but they are now worthless. Otherwise, all agents are better off with a bubble. Because they earn a higher rate of return on their saving.
If it is costly to collect shells from the beach, that activity will be measured as "investment", and so investment will be inefficiently high whenever the sunspot changes colour (thanks to Steve Williamson for that point).
One solution might be for the government to issue one trill perpetuity (promising to pay an annual dividend of one trillionth of NGDP forever), maybe broken into a million bits to help liquidity.
But I wonder why intrinsically useful land does not dominate intrinsically useless shells as an infinitely-lived asset. Perhaps land is less liquid?
[I wrote this post because Steve Williamson asked what Paul Krugman might be saying. So I tried to model it. It might be something like this. Or it might not.]