Conditions for bubble formation Michael Joffe Imperial College London
The importance of bubbles • increasing study of bubbles in recent years – not least because of the perception that the financial crash of 2007 -2009 involved one or more of them • several bubbles in recent decades; some observers believe that their frequency and/or severity may be increasing • they tend to occur in stock markets (e. g. dotcom), or in real estate; in the latter case a major role is played by the financial sector (Shiller “Irrational exuberance”, 2 nd edition)
Bubbles: symmetric or asymmetric? • typical definition: “trade in high volume at prices that are considerably at variance from intrinsic value” [King RR et al. 1993] – a question of how prices are set: how do the seller and the potential buyer come to form a compatible perception of “the going price”? • the metaphor of a “bubble” suggests something that grows steadily and then bursts suddenly • the first is symmetric – can be up or down – but the second is asymmetric: only up; which is correct? – an empirical question • in addition, one needs to ask, is there always an “intrinsic” value?
Aim of the paper • to present a model of bubble formation that predicts asymmetry – my reading of the literature is that the focus has predominantly been on symmetric behaviour, in line with the definition rather than the metaphor • the assumed context: – the market is of a type where the price is not set in direct relation to costs, as is the case e. g. for an established product => a need for information on what is “the going price” – market participants may perceive a trend in prices, upwards or downwards, and extrapolate from that to predict that the trend will continue into the future: “trend extrapolation”
The model I starting from a standard market equilibrium model: D(P) = a – b. P S(P) = c + d. P (1) (2) where P is the price; D(P) and S(P) respectively represent the willingness to buy and to sell the asset now at the existing price; assuming for simplicity that the demand the supply curves are both linear and given by the parameters a, b, c, d, with b, d > 0
The standard market equilibrium model S price D P 1 Q 1 quantity
The model I starting from a standard market equilibrium model: D(P) = a – b. P S(P) = c + d. P (1) (2) where P is the price; D(P) and S(P) respectively represent the willingness to buy and to sell the asset now at the existing price; assuming for simplicity that the demand the supply curves are both linear and given by the parameters a, b, c, d, with b, d > 0
The model II trend extrapolation condition: perception of a price trend expected to continue into the future this brings about a price increment ΔP “now” to take into account the cost or benefit of waiting: the price modified by trend extrapolation is (P + ΔP); ΔP is negative with a falling trend ΔP can be represented by: ΔP = θP (3) where θ is the proportional expected future price increment
The model III for simplicity the perceived trend P’ is regarded as linear θ is given by θ = f(P’) (4) with f(. ) an increasing function: θ<0 for P’<0, and θ>0 for P’>0 for a representation of how θ may relate to P’ in terms of ζ, “investor sentiment” or tendency to buy, see Caginalp and Ermentrout 1990
The model IV The original equations now become: D(P) = a – b. P + mθP S(P) = c + d. P – nθP (1’) (2’) where m, n are parameters (>0) that determine the extent to which the price is affected by trend extrapolation
The model IV The original equations now become: D(P) = a – b. P + mθP S(P) = c + d. P – nθP (1’) (2’) where m, n are parameters (>0) that determine the extent to which the price is affected by trend extrapolation m, n > 0 because if the trend is rising (θ>0), then mθP>0 and nθP>0, and vice versa for a falling trend (θ<0)
The model IV The original equations now become: D(P) = a – b. P + mθP S(P) = c + d. P – nθP (1’) (2’) where m, n are parameters (>0) that determine the extent to which the price is affected by trend extrapolation
The model IV The original equations now become: D(P) = a – b. P + mθP = a – P(b – mθ) S(P) = c + d. P – nθP = c + P(d – nθ) (1’) (2’) where m, n are parameters (>0) that determine the extent to which the price is affected by trend extrapolation
The model IV The original equations now become: D(P) = a – b. P + mθP = a – P(b – mθ) S(P) = c + d. P – nθP = c + P(d – nθ) (1’) (2’) where m, n are parameters (>0) that determine the extent to which the price is affected by trend extrapolation Compare (1’) and (2’) with (1) and (2): D(P) = a – b. P (1) S(P) = c + d. P (2)
The model IV The original equations now become: D(P) = a – b. P + mθP = a – P(b – mθ) S(P) = c + d. P – nθP = c + P(d – nθ) (1’) (2’) where m, n are parameters (>0) that determine the extent to which the price is affected by trend extrapolation Compare (1’) and (2’) with (1) and (2): D(P) = a – b. P = a – P(b) (1) S(P) = c + d. P = c + P(d) (2)
The model V bubble occurrence depends on reversal of the sign of the coefficient of P we now have two conditions for this: if mθ > b, a rise in P will lead to a rise in D(P) if nθ > d, a rise in P will lead to a fall in S(P) (5) (6) under these conditions, therefore, instead of the usual decreasing function in P for D(P) and increasing function for S(P), as represented by equations (1) and (2), the situation is reversed (5) and (6) can be written as θ>b/m and θ>d/n
The model VI recall that b, d, m, n > 0 therefore θ>b/m => θ>0 and θ>d/n => θ>0 the property of reversing the overall direction of equations (1) and (2) only occurs when θ>0, and thus also P’>0 this model therefore predicts asymmetry
The model VII D(P) = a – b. P + mθP (1’) S(P) = c + d. P – nθP (2’) in the case where θ<0, the terms mθP and nθP would have the same signs as b. P and d. P respectively they would merely accentuate the normal decreasing and increasing functions represented respectively by equations (1) and (2)
The model VIII To summarize: • the model predicts that if prices start at or near their intrinsic level, a self-fulfilling and thus selfperpetuating tendency will tend to occur in markets with an upward – but not a downward – price trend • the conditions specified by the model bring about a bubble equilibrium: a rising price reinforces the perception that prices are destined to rise, and in turn this perpetuates the rising trend – for as long as the trend extrapolation perception lasts • bubble equilibria are inherently unstable
The model IX • the bursting of a bubble is not directly covered by the model – the only prediction is that the price deviation cannot continue indefinitely; not the timing and subsequent time course – depends on functional form, and other factors • the return towards the intrinsic value then proceeds according to the standard equations (1) and (2) – not necessarily sudden “bursting” • where there is no clear intrinsic value, the equivalent role would be played by a realisation that prices are no longer affordable
Situational rationality I • a rational calculation in the context of imperfect information • can be seen as a form of bounded rationality, but limited calculating ability is not a feature here • the extrapolated trend is external to each market participant => they have to join in, even with misgivings • it is difficult to distinguish between “behavioral theories built on investor irrationality and rational structural uncertainty theories built on incomplete information about the structure of the economic environment” (Brav & Heaton)
Situational rationality II • Caginalp & Ermentrout: “emotional”; also “groupthink”, “optimism” or “panic” • e. g. Akerlof & Shiller: “Animal spirits”
Situational rationality II • Caginalp & Ermentrout: “emotional”; also “groupthink”, “optimism” or “panic” • e. g. Akerlof & Shiller: “Animal spirits”
Situational rationality II • Caginalp & Ermentrout: “emotional”; also “groupthink”, “optimism” or “panic” • e. g. Akerlof & Shiller: “Animal spirits” • these emotions may occur – but are they causal? – hard to answer • are they invariably present? consider someone buying a property to live in, when property is seen as likely to increase in price – it’s a calculation • emotion is likely to be added to this, with a small group of operators + face-to-face contact: e. g. financial market traders; ? the real estate context
Conclusion • the model predicts asymmetry: the process of progressive deviation from intrinsic value (or from affordability) will occur in a rising but not a falling market, and actual prices will be higher than intrinsic value (or affordability) in such circumstances; until the bubble bursts • there are four conditions for this to occur: 1. price setting is not based on established cost, so that additional information is required 2. “the going price” is based on trend extrapolation 3. (θ > b/m) and (θ > d/n) 4. a fortiori, P’ > 0, i. e. bubbles form when prices are rising
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