Interesting argument. Robert Waldmann would like to develop this idea more formally, so I'm sure he would appreciate helpful comments:
Possible desirable effects of irrational exuberance, by Robert Waldmann: Gross 2007 argues that bubbles may be a good thing. Many people have read Gross 2007. I am not one of them, so maybe I will repeat his arguments or maybe say something new.
There are clear cases of sectors where there was a huge amount of entry and investment and many firms went bankrupt. The classic case is railroads in the 19th century and current interest is from the excess of fiber optic cable laid towards the end of the 20th. Gross probably argued that, although a lot of investors lost their shirts, the country got a railroad system and an information superhighway. I am thinking about translating the argument into math and considering irrational exuberance in the context of growth models.
This is a natural argument because endogenous growth models do not have the features which imply an irrational boom will lead to a crash. The argument against exuberance is that it leads to investment in unsound projects which is largely lost when they are scrapped. This is based on the idea that there are a limited number of socially efficient investments to be made at any given time. While models rule out silly investments for convenience, endogenous growth models do not have a natural limit on socially useful investment pretty much by definition.
In the oldest endogenous growth model (Rome 86, Arrow 62 some other guy in the 50s) capital has decreasing private returns to the investing firm but constant social returns, because of Marshallian spillovers of knowledge obtained by learning by doing. In this model, a period of irrational overinvestment due to over-estimates of the private return to capital will cause a growth path of capital and output which permanently stays above that of the rational expectations equilibrium. This causes increased welfare, since, due to the spillover, socially optimal investment is higher than privately optimal investment.
A similar simple result occurs in models of endogenous growth due to research and development. In this case innovation provides social benefits above private benefits because innovators are monopolists but can't price discriminate. Over-optimistic inventors cause an increase in growth and welfare.
This much is obvious. I think there are more interesting arguments.
The first concerns the phase after an invention when imitators jump into the market. This phase is preceded by a shakeout when all but a few firms go bankrupt. Such a pattern can occur even if everyone is rational as a position as a surviving firm in a growing market is valuable enough to balance the high risk of bankruptcy. Irrational entry of firms into a sector which is a natural monopoly can bring the price temporarily closer to marginal cost which contributes to efficiency. I think it is more important that after the shakeout, the monopolist (or oligopolists) will not find it optimal to scrap capacity which they would not have found it optimal to build. Thus there can be a long lasting effect on the price of the good or service lowering it towards the socially efficient competitive level. If the monopolist does not allow the capacity to depreciate (because a railroad with one rail is worthless say or one with a gap in the tracks) the effect can be permanent. This effect is different from and less obvious than the ones discussed above.
It is also possible that irrationally excessive entry can have permanent benefits if the temporary period of very low prices during the shakeout is useful to other innovators (say web content innovators benefiting from desperate need of hosts for content any content free please) . This can also occur, but to a lesser extent, if everyone is rational. The benefits can include increased welfare as well as increased growth as argued above.
If entrepreneurs underestimate the difficulty of an engineering problem, they may invest in R&D which proves fruitless to them. This may have a beneficial spillover due to increases in general non rival non excludable knowledge. It is generally agreed that pure R&D should be subsidised (that is what the NSF is for not to mention what DARPA used to do before the Bush administration). It is good to have useful pure research financed by irrational managers who think it is applied research.
Finally totally unsuccessful efforts at innovation (there are many like the superconducting supercomputer and the atomic airplane) show what not to do to rational innovators. This can be useful as a subsidy to other innovation.
One quick thought that comes to mind - I'll leave it to all of you to ask the good, penetrating questions - I'd like to hear more about the conditions under which welfare is improved. For example, along one dimension of this, does the higher output growth path also have a higher variance of output and, if so, is it always the case that the higher growth more than compensates for the increased risk? E.g., workers gain and lose jobs over these cycles of irrational exuberance, so the higher growth would, it seems, need to more than compensate for the higher risk.