**DeLong and the shorts**

Schopenhauer correctly pointed out that “Buying books would be a good thing if one could also buy the time to read them in: but as a rule the purchase of books is mistaken for the appropriation of their contents”. If you’re wondering why the book reviews promised “tomorrow”, several days ago, aren’t here, that’s why. I am working on a solution to this problem, which will be posted soon.

But in the meantime, I started reading my own archives, and was impressed by two things. First, how good my comments section is (no really, I love you guys, sob), and second, how much better this weblog was when I used to stick to the rubric and write more stuff about economics than anything else. The shift wasn’t really a matter of policy; it’s just that the economics stuff takes rather a while to write and I have less spare time than I did a year ago. So I’m going to try and keep this short; if it is too dense and incomprehensible as a result, say so in comments and I’ll have another go. I’ve posted it here rather than Crooked Timber because I think it fits better.

The question I’d like to address is one that Brad DeLong asked a while ago with respect to the dot com bubble. Now, Brad actually wrote one of the seminal papers on how it is that markets can get so wildly out of control without someone coming in a turning a profit by taking the other half of all the suckers’ trades and waiting them out until they run out of money (“Noise Trader Risk”, or something it’s called and it’s a goody). But he confessed on his weblog to still being a bit surprised that the DeLong/Shleifer^{1}/Summers^{1} effect seemed to dominate so completely. The question he asked was “Why are there no rational, well-financed players to come in and bring the market back to its senses?”

My answer to this question is that the reason there are none is that *it is logically impossible for the kind of market participant Brad is thinking of to exist*.

Consider the very simplest model of a stock market bubble there could be. (I have a mathematical version of this first bit on paper, so either it definitely works (10% chance) or I have made a mathematical error and maybe it works (90% chance)). There are two players; “the crowd”, who are noise traders, and one single player (call him “Soros”) who is rational and has access to unlimited investable capital. The crowd have the normal Blanchard-style bubble dynamics (non economists; basically, every day, the market has to either crash, or rise by enough to compensate the crowd for bearing the risk of a crash). On any given day, unless a crash has already happened, Soros can decide to commit his capital to short positions. If he does this, he sucks up all the excess demand for stock and a crash is inevitable on the next day.

So, in this model, when does Soros decide to pop the bubble? Well, think about it this way. Number the days 0, 1, 2, ..n, and call P(t) the profit for Soros from the strategy “Wait until day T and then pop”. Call B(t) the level which the market has reached by day T as long as the bubble is still going (one can read this out of the standard bubbles model) and R(t) the probability of a spontaneous collapse on day t conditional on no collapse before t. I’ll use cum(R(t)) to refer to the cumulated version of R(t), ie the unconditional probability as of T=0 that the market has collapsed before T. Clear as mud? Thought so.

Anyway we can say the following things about P(t) as a function of t:

- P(t) is equal to B(t) conditional on no collapse before t and 0 otherwise. Therefore:
- As B(t) is assumed a monotonically positive function of t, for every day that passes, Soros can commit more capital to the market when he decides to pop, and earns a greater return on every unit of capital committed. This would tend to make P(t) increase with respect to t.
- Since cum(R(t)) is also a monotonically positive function of t, every further day also increases the risk that the eventual profit will be zero.
- Whatever happens, Soros can’t
*lose*money in this deal, because he is assumed to have perfect information that the market is in fact overvalued. Therefore there is an option-like structure here, meaning that there is a positive reward to waiting. Note that there is*no*“noise trader risk” in this model.

I think you can solve this model for the optimal value of t in P(t), and that this value could be quite large. The idea is that even for a rational well-financed arbitrageur, it is actually rational to see if a bubble gets bigger so that you get more bang for your buck when popping it (this is the flipside of the noise trader risk in Brad’s paper; there, the danger was that it would get bigger and bust smaller arbitrageurs).

So anyway, we’ve established that a Soros-type investor – a monopolist in the provision of arbitrage – will not pop bubbles but will allow them to get bigger. But a single large monopolist of capital is not the typical way in which we think about stock markets. Typically, we assume that there is a very large number of investors on an equal footing. And obviously, if there is an infinity of Soroses, things are very different. If we call the optimum bubble-popping day for the monopolist T*, then if there is a second Soros in the market, he knows that he can get a smaller return, but take it all for himself by popping on T*-1. But a third Soros could pop on T*-3 and so on, until we get the standard Steve Ross view of the world in which the Soroses are competing with each other tooth and nail, and are content to scalp the tiny return they can get by jumping on the market the moment it gets even a tiny bit out of line with fundamentals.

But hang on … isn’t there something a bit funny about assuming an infinite number of Soroses? Well yes. What we’re being asked to believe here, is something akin to the “small firm assumption” of perfect competition; that each individual Soros is so small relative to the total community of Soroses that he can’t influence the price (can’t prop up the market on T*-1 so as to ensure he can be the one to pop it on T*). But this assumption is indefensible. *The Soroses are part of the market*. It is not consistent to both assume that any given Soros is large enough to be a price maker with respect to the whole market (a necessary condition for him to be a Soros in this model), but simultaneously small enough to be a price taker with respect to a subset of the market (Soroses). This is the logical inconsistency above; if an arbitrageur is big enough to pop bubbles, he won’t do so until he’s good and ready. And if he doesn’t have the liberty to wait until he’s good and ready, then he may call himself an arbitrageur, but he’s actually part of the crowd.

There is a lot more to this; particularly, I have to deal with the cases where there is neither a monopoly nor a continuum of Soroses, but rather a small number of them interacting strategically. But I am afraid that, Fermat-like, I have to just say that the solution to that problem is truly marvellous, but is slightly to long for the 45 minutes (1321 words, yo JQ) I have allocated to this post.

^{1}Imagine me booing and hissing like the crowd at a pantomime at the mention of these two names ….

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