Abstract
Although many classical political economists such as Mill (1848) recognized that self-fulfilling prophecies could happen in speculative markets, they viewed them as inevitably irrational because they supposedly deviate from long-run equilibria as determined by tastes, technology, and factor endowments. The first to suggest that a speculative bubble might somehow be a rational self-fulfilling prophetic equilibrium in its own right was Keynes (1936, Chap. 12) in his “beauty contest” example where market participants judge the judgments of their fellow judges in the beauty contest.
“Then we, As we beheld her striding there alone, Knew that there never was a world for her Except the one she sang and, singing, made.”
Wallace Stevens, 1947
“The Idea of Order at Key West”
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Marcet and Sargent (1988, 1989a, 1989b) examine the behavior of several adaptive learning mechanisms and their convergence to rational expectations equilibria. A favorite mechanism in this literature is least-squares learning, originally due to Ljung (1977). Evans and Honkapohja (1994) show that least-squares learning can converge on an explosive AR(1) solution.
Backward induction is a controversial problem in rational game theory. Binmore (1987, 1997) argues that rational players will not use the backward induction strategy if they expect a deviation from it to occur. Aumann (1995) counters that prior common knowledge of their rationality will lead players to follow the backward induction strategy. It can be argued here that in some sense rationality itself becomes a self-fulfilling prophecy.
These arguments derive from models by Radner (1967) and Bewley (1972) of general equilibrium with infinite commodities.
R may reflect a stream of consumption utility for something like a rare postage stamp or fiat money, both of which are difficult to observe or measure.
The condition that r0 = r < n can be relaxed and still have an efficient equilibrium with a rational bubble if there is risk with high risk-aversion (Bertocchi, 1991), the speculative behavior in effect replacing insurance markets, or in the case of a suitably specified endogenous growth model where bubbly “excessive optimism” based on naive expectations can lead to a higher growth path than that arising from simple rational expectations (Nyssen, 1994).
Despite the claim of that rational negative bubbles cannot exist, they appear in laboratory experiments about asset market behavior (Noussair, Robin, and Rufuex, 1998). Such negative bubbles often end with “anti-crashes” in which prices suddenly increase sharply. Defenders of rationality can criticize such experiments as not being “for real” with the subjects just treating it as a game for fun and thus engaging in behavior that they would not in real asset markets.
Tirole (1985) distinguishes “financial fundamental,” equaling the present discounted sum of future net income, from “market fundamental,” representing the present discounted sum of future net utility. His ruling out of negative bubbles can be interpreted as saying that a market fundamental must be at least as great as a financial fundamental.
Farmer’s assumption also provides a solution to the “cake eating problem” of how to eat a cake in infinite continuous time (Gale, 1967; Artstein, 1980; Romer, 1986a).
For his virtually textbook presentation of this concept within a broader discussion of bubbles, sunspots, multiple equilibria, and so forth, see Blanchard and Fischer (1989, Chap. 5).
Matsuyama (1991) shows for similar models that non-separability implies a continuum of equilibria, some exhibiting bounded oscillations with some of these chaotic. The downward motions can be deflationary bubbles, with the boundedness allowing fulfillment of the transversality condition. He (1990) also shows the general possibility of sunspot equilibria in money-in-the-utility function models.
Such an upper bound on the utility of money might not exist for money hungry “Scrooges.” A similar limiting condition is an upper bound on world wealth, noted to the author by Rogoff (1988) in a personal communication.
We assume that the bubble crashes if a generation appears for whom current real balances exceed their upper utility bound.
In a study of the pork market, Chavas (1995) estimates that only 23% of market participants show behavior consistent with rational expectations while the remaining 77% exhibit some backward-looking element in their expectations.
The usual claim is that chartists lose money. However, Brock, Lakonishok, and LeBaron (1992) showed that traders could outperform the overall US stock market for 1897–1986 by using any of 26 chartist technical trading rules. Such an finding parallels the argument of DeLong, Shleifer, Summers, and Waldmann (1991) that some noise traders can not only survive but outperform the market. The result of Brock, Lakonishok, and LeBaron has been challenged by Sullivan, Timmermann, and White (1998) who argue that these rules do not perform well for post-1986 data, perhaps a case of the market learning about the rules, or perhaps simply a reflection of broader changes in the nature of the market.
Ironically one of the major critiques of Zeeman’s stock market model by Zahler and Sussman (1977) was its allowing for agents who did not have rational expectations, the chartists. This was near the high water mark of when such a criticism would have been considered fatally devastating, whereas today it merely seems silly.
Caplin and Leahy (1994) present a three stage market dynamics model in which there is asymmetric information but in normal functioning private information remains concealed. Then it begins to leak and then there is a herd response following its revelation. They see this as explaining sudden collapses such as bank runs, international debt crises, and political collapses, but make no direct reference to speculative bubble crashes.
Other applications of the cusp catastrophe to financial models include Ho and Saunders’ (1980) model of bank failures and Gregory-Allen and Henderson’s (1991) model of corporate failures.
Gennotte and Leland (1990) suggest a cusp catastrophe interpretation of their discontinuous dynamics results in their model of the 1987 crash. Their model has three categories of agents and focuses on hedging strategies.
For more extensive discussion of market mediator behavior see Day (1994, Chap. 11) and Gu (1995).
An alternative approach to chaotic dynamics in stock markets is that of Shaffer (1991) who assumes that firms pay a constant share of realized profits as dividends with the remainder being einvested according to a linearly declining marginal efficiency of investment schedule. Chaotic dynamics arise from certain parameter values with small shifts leading to large changes in volatility, possibly explaining the 1987 stock market crash.
One way of attempting to test the robustness of Lyapunov exponent estimates is through bootstrapping, (Efron, 1979; Li and Maddala, 1996) which generates an IID null through resampling of the data in a special way. Blank (1991) found positive Lyapunov exponents for a series on soybean futures which he supported through the use of bootstrapping.
This argument is a kissing cousin of that of Grossman and Stiglitz (1980) that informationally efficient markets are impossible because then there would be no incentive to trade, but without trades there would be no information.
This has also been labeled the “peso problem” after Rogoff s (1979) and Krasker’s (1980) explanation of the consistent underprediction of future values of the Mexican peso by the forward markets in the 1970s. Rational agents perceived a skewed underlying distribution with a tail toward deep devaluations. In small samples when this tail was not observed, the sample mean could exceed the population mean, and the market seemed not to be rational. In 1982 the peso was sharply devalued, as it has been since several times, thus confirming the rationality of expecting such a skewed distribution.
See LeRoy (1989) for a survey of volatility studies.
Cointegration was originally inspired by a test for rational bubbles in the gold market (Diba and Grossman, 1984).
Mandelbrot (1983) claims that this theory was the initial inspiration for his development of fractal geometry.
Classical Hurst tests finding positive persistence have been done for the gold and silver markets (Booth and Kaen, 1979), foreign exchange markets (Booth, Kaen, and Koveos, 1981), and commodities futures markets (Helms, Kaen, and Rosenman, 1984).
It was in this paper that Flood and Garber (1980) articulated the problem of the “misspecified fundamental.” They attempted to deal with it in this case by ignoring the last five months of the hyperinflation data because of Cagan’s (1956) argument that during this period a possible regime switch was under discussion.
It was disgust with this long and apparently inexplicable appreciation of the dollar that led policymakers to abandon the purely floating exchange rate system and impose the “managed floating” system for the major currencies under the Plaza Accord. It could be argued that this was a rejection of Milton Friedman’s position that floating rates would be stable and not subject to speculative bubbles.
The chaotic bubble models of foreign exchange rates developed by De Grauwe and Vansanten (1990) and De Grauwe, Dewachter, and Embrechts (1993) derive their dynamics from an interaction between fundamentalist and chartist traders with sufficiently great “J-curve” lags operating. Jeanne and Masson (1998) show the possibility of chaotic dynamics in the expectations of devaluation in a model of sunspot attacks on currencies.
Kindleberger (1989) argues that most historical bubbles have involved at least two assets whose rise fed off each other. In 1990, after the Japanese stock market had collapsed but real estate had not yet, he accurately predicted to this author in a personal communication that real estate would crash as well.
The author finds this argument more credible for the Mississippi bubble where John Law was carrying out and propagandizing the alleged benefits of his scheme. This was not the case for the much more fraud-ridden South Sea bubble. Barsky and De Long (1990) extend Garber’s argument to the major bull and bear runs of the 20th century US stock market.
De Long and Shleifer (1991) cite this as evidence that the 1929 US stock market was indeed a bubble, although it is only directly evidence of a bubble in the closed-end funds themselves.
This suggests a disjuncture and possible asymmetry between the perceptions of international investors who buy the CECFs and domestic investors who buy the underlying assets generally. Frankel and Shmukler (1996) use this to argue that the 1994 crash of the Mexican peso was triggered by Mexican insiders fleeing the currency rather than outsiders, the evidence being the emergence of premia in the three Mexico funds just before the crash as NAVs dropped while fund values held up.
See Gastineau and Jarrow (1991) and Koppl and Yeager (1996) for the theory of the Big Player in speculation.
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Rosser, J.B. (2000). Speculative Bubbles and Crashes II: Rational and Semi-Rational. In: From Catastrophe to Chaos: A General Theory of Economic Discontinuities. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-1613-0_5
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