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Endogenous regime switching in speculative markets
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STRUCTURAL CHANGEAND ECONOMIC DYNAMICS Structural
EUEVlER
Change
and Economic
Dynamics
7 (1996)
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Endogenous regime switching in speculative markets Rajiv Sethi Drpartment
qf Economics.
Barnard Collegr. Columbia Uniwrsity, New York, NY 10027, USA
3009 Broadway,
This paper examines a nonlinear disequilibrium model of price formation in speculative markets with two groups of speculators, fundamentalists and chartists. The behavior of the fundamentalists is stabilizing, whereas that of the chartists is destabilizing; a sufficiently high share of wealth in the hands of the latter group can cause attracting periodic orbits to arise. If information about fundamentals is costly to obtain, then chartists obtain higher pay-offs in a stable regime, while fundamentalists perform better in an unstable regime. This gives rise to endogenous regime switching: the market alternates between periods of stability and instability, with the transition from one regime to another determined endogenously through the evolution of wealth shares. Keywords:
Speculation; Financial instability; Nonlinear dynamics
JEL classijicution: D84; G12
1. Introduction
Do movements in the prices of financial assets broadly reflect changes in their underlying fundamental values, or are there important determinants of such movements that are unrelated to fundamentals? This question has attracted a great deal of empirical and theoretical attention over the past decade. While Shiller’s view that “increasingly, there is statistical evidence that suggests the stock market may have a life of its own to some extent, unrelated to economic fundamentals” (Shiller, 1990, p. 398) remains controversial, it appears to be gaining ground. Even those who saw the international stock market crash of October 1987 as “an adjustment to a change in fundamental values” (Fama, 1989, p. 81) concede that “the long, orderly upside 0954-349X/96/$15.00 ,(:‘ 1996 Elsevier SSDI 0954-349X(95)00040-2
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of the swing preceding October 19th” could well represent a prolonged departure from fundamentals. On the theoretical front it is now well known that persistent divergence from fundamentals can be consistent with self-fulfilling expectations as well as the no-arbitrage condition, both in deterministic and stochastic models (Flood and Garber, 1982; Blanchard and Watson, 1982). In contrast to this work on rational bubbles, there is also a fairly substantial and growing body of literature based on the assumption of heterogeneous agents, some of whom may not have self-fulfilling expectations (Beja and Goldman, 1980; Frankel and Froot, 1986; De Long et al., 1990a,b; Chiarella, 1992). The heterogeneous agent literature is itself diverse. Beja and Goldman (1982) and Chiarella (1992) consider disequilibrium models in which two types of traders, fundamentalists and chartists, interact. The former base their decisions on the deviation of the asset price from fundamentals, and the latter on the trends they discern from past observations of the data. For certain parameter values the equilibrium in which prices are equal to fundamentals is locally unstable. The Beja/Goldman model is linear and therefore this instability is global, but Chiarella’s nonlinear version gives rise to a stable limit cycle, with prices oscillating around but never converging to fundamental values. Frankel and Froot (1986) also consider the behavior of fundamentalists and chartists, but the expectations of the two groups are aggregated through the activities of a third group, the portfolio managers, who give more weight to the forecasts of the group with the best recent forecasting performance. In this model prices can converge to values different from fundamentals, so that the price may be ‘high and stuck’; the authors argue that this was indeed the case with regard to the U.S. dollar in the mid-1980s. De Long et al. (1990a) consider ‘noise traders’ who misjudge returns by a random amount in each period; they are confronted in the market by unboundedly rational but risk-averse traders who correctly forecast true returns, taking full account of the presence of noise traders. Prices can fluctuate even if fundamentals remain constant, and it is also possible for the noise traders to earn greater returns than the unboundedly rational agents if, on average, their miscalculations induce them to carry more risk than they believe themselves to be carrying. De Long et al. (1990b) consider a different model in which unboundedly rational speculators interact with chartists; in this case the rational agents destabilize prices to exploit the adaptive behavior of the latter. The model developed below bears a close relationship to Chiarella’s nonlinear version of the Beja-Goldman model. In Section 2 a generalized version of the Chiarella (1992) model is presented which takes into account the fact that the demands of fundamentalists and chartists depend not only on prices and expectations but on their respective shares of total market wealth. As in Chiarella (1992), prices are assumed to adjust in response to excess demand in the asset market, and chartist expectations adjust adaptively. The model has a unique equilibrium with prices equal to fundamental values. It is shown that if price adjustment is sufficiently sluggish and chartist demand relatively insensitive to changes in expectations, then the price dynamics are stable. If these conditions are not met, then the equilibrium will be stable if and only if the chartist share of market wealth is sufficiently small. The chartist wealth share is therefore identified as a bifurcation parameter that governs
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the stability properties of the dynamics. In Section 3 it is shown that with nonlinear chartist demand, trajectories converge to a limit cycle in the locally unstable case. In Section 4 the model is generalized to allow the share of chartist wealth in the market to evolve endogenously over time, as a result of trades between the two groups. Since explicit trading between groups of agents with heterogeneous beliefs and the corresponding evolution of balance sheets is seldom taken into account in economic models, this represents a significant departure from the existing literature. The result is a four-dimensional dynamical system in prices, expectations, asset holdings and money holdings. It is shown that only two outcomes are possible in the long run: either prices converge to fundamental values as trades between the two groups diminish chartist wealth over time, or fundamentlist wealth declines to zero. The former possibility is the more interesting and plausible one, and may arise even when initial chartist wealth is well above the bifurcation value identified in Section 2. The rationale behind the result is that when the price dynamics are oscillatory, fundamentalists increase their share of wealth over a complete cycle, even though chartists may outperform fundamentalists during some phases of the cycle. In Section 5 the model is generalized further to allow for the possibility that fundamentalist speculation requires greater information acquisition and processing costs than chartist activity. The costs of information acquisition and processing are seldom taken into account in the economics literature, although some recent attempts have been made to model such costs explicitly (Conlisk, 1980, 1988; Evans and Ramey, 1992; Sethi and Franke, 1995). With costly fundamentalism, chartists are favored when the regime is stable, and fundamentalists when the regime is cyclic. Since the extent of chartist activity is itself a crucial factor in determining stability, this gives rise to the intriguing possibility that the economy may switch back and forth between stable and unstable regimes as the chartist share of wealth contracts in the latter and expands in the former. The resulting time series is characterized by periods of excess volatility and mean-reversion, punctuated by periods in which the market is efficient in the standard sense, with the transition from one phase to another being the endogenous outcome of shifting wealth across the two groups of investors.
2. A dynamic model of speculation
There is a single asset in fixed supply, and p denotes the logarithm of its price. We allow for two different types of speculator which, following Zeeman (1974), Beja and Goldman (1980), Frankel and Froot (1986) and others, are referred to as &&amentalists and chautists. Chartists perceive a trend in price movements which is denoted by rc, and their demand DC(n) for the asset is an increasing function of this perceived trend:
DC(n)= KY(n), where p E [0, l] is the share of total market wealth in the hands of chartists. It is
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assumed that the function g( .) satisfies g(0) = 0,
g’ > 0.
In other words, if chartists expect prices to rise (fall) their demand for the asset is positive (negative). The greater the perceived trend in price movements, the greater is chartist market participation, which is motivated by the prospect of short-term capital gains. In addition, demand from chartists is proportional to the share of total market wealth, ,u, that they command; this is assumed exogenous for the moment. Fundamentalist demand D’(p) depends on the difference between the current price p and what fundamentalists believe to be the fundamental value, f, of the asset. Without loss of generality, we use the normalization f = 0, so that p measures prices as deviations from perceived fundamentals.’ For simplicity, this function is assumed to be linear: D’(p) = -(l
- p)Bp.
(2)
In other words, fundamentalists buy (sell) when the asset is priced below (above) its perceived fundamental value. The greater the deviation between current prices and fundamentals, the greater is fundamentalist market participation. This strategy reflects a long-term perspective since demand is independent of perceived short-term expectations of price movements. In addition, fundamentalist demand is proportional to the share of wealth (1 - p) in fundamentalist hands. The parameter 0 measures the strength of fundamentalist reaction to departures of prices from (perceived) fundamentalist values. Aggregate excess demand for the asset, D(p, rc), is simply the sum of fundamentalist and chartist demands: D(P, n) = D’(7c) + of(p) = &J(x) - (1 - p)@p.
(3)
Continuous market-clearing is not assumed. Instead, the asset price adjusts under pressure of excess demand as follows: d = P&4
n).
(4)
Since excess demand need not be zero at all times, not all intended trades will be executed, unless there are other market participants such as market-makers who satisfy excess demand out of inventory (Day and Huang, 1990). If we consider a market consisting only of speculators, then without instantaneous market clearing one (or both) groups of traders will be rationed. The continuous market-clearing case can be approximated to any degree of precision by allowing /I to approach infinity. These issues are taken up in greater detail below when trades are considered explicitly. Chartist expectations are revised in an adaptive manner, on the basis of the difference between current and expected price changes: 7i = ct($ - 7-c) = WmP,
n) - Tc).
(5)
The parameters (x, fi, and H are all assumed to be strictly positive. ’ Note
that
p and ,f refer to logarithms.
so that negative
values
are admissible.
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Eqs. (4) and (5) represent a two-dimensional differential equation system in the state variables p and x, with a unique equilibrium (p*, rc*) = (0,O) provided that p # 1.’ In other words, unless the entire market wealth is in the hands of chartists, the only rest point of the system (4)-(5) occurs when prices equal fundamental values and chartists expect them to stay there. The conditions under which this equilibrium will be dynamically stable are given by the following result, where c is used to denote g’(O), the slope of the function g( .) at the equilibrium value of chartist expectations.3 Proposition 1. Assume that p E [0, 1). Then: (a) If jk I 1, the unique equilibrium of the system (4)-(5) is locally asymptotically stable for all values of TX,9 and p. (b) If/k > 1, thenfor any given values of%, j, t? and c there exists p0 E (0,l) such that the unique equilibrium qf the system (4)-(5) is locally asymptotically stable when p < p0 and repelling when p > pO. (c) The bifurcation value pLo(u,fl, 0, c) satisfies 2p0/dcr < 0, Zp0/8/I < 0, dpO/CYl> 0 and 8p0/Zc < 0. The information regarding the local stability properties of the system (4)-(5) contained in the above result is summarized in Table 1. According to Proposition 1, if the adjustment of prices under pressure of excess demand is sufficiently slow, and the responsiveness of chartist demand to small perceived price trends is weak (PC < l), then the equilibrium will be locally stable irrespective of the share of wealth held by each group. If these conditions are not met, then the stability of the equilibrium depends on the share of wealth held by chartists. Specifically, there is a bifurcation ualue of this wealth share, denoted pO, such that the equilibrium is locally unstable if p exceeds ,u~ and stable if pLoexceeds p. This bifurcation value itself falls as a, p and c rise (enlarging the region of instability in parameter space) and rises as 0 rises. To summarize, local instability is possible if the adjustment of prices is rapid, if chartist demand is highly sensitive to changes in expectations, and the share of wealth in chartist hands is sufficiently large. If the
Table Local
1 stability
of equilibrium
’ To see this, observe that the equilibrium condition 6 = 0 implies, from (4), that D( p*, x*) = 0. This in turn implies from (5) and the equilibrium condition ti = 0 that z* = 0. Hence g(n*) = 0 and thus D’(n*) = 0, so from (3) we have D’(p*) = 0 and p* = 0. If p = 1, then any point (p, 0) is an equilibrium, since D’(p) = 0 for any p. 3 Proofs of all formal propositions are collected in the appendix.
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population consists only of fundamentalists (11= 0), then the system is stable under all circumstances. Local instability need not, of course imply unbounded divergence from fundamentals, since we are dealing with a potentially nonlinear system. The global behavior of the model in the locally unstable case is examined in the next section.
3. Global analysis If the relationship between chartist expectations and demand, as expressed by the function g( .), is linear, then the system (4))(5) as a whole is linear and local instability implies global instability. Prices would eventually exceed any stated bound or become arbitrarily close to zero, even if fundamentals remain constant over time. On the other hand, if g( .) is nonlinear, local instability can coexist with boundedness and non-negativity of prices. We shall assume here that although chartist demand increases as the perceived trend in prices rises, it does not do so without bound. This is to be expected for at least two reasons. First, investor expectations will not be held with subjective certainty, so that if investors are risk-averse, they will not be willing to take infinitely large arbitrage positions. Secondly, even if investors are assumed to be risk-neutral or to hold expectations with subjective certainty, they will be faced with constraints on borrowing and on short sales. Both these factors will ensure the boundedness of speculative demand. This effect may be captured by allowing g( .) to be nonlinear, with the following conditions imposed: lim g(7c) = g” < x8, I- II
lim x*--r
g(n)=
-g’>
-m.
(6)
These conditions are sufficient to ensure that prices can only deviate by some finite (possibly very large) amount from fundamental values. yielding: Proposition 2. Assume that p E [0, l), and g( ‘) satkfies conditions (6). Then, if the unique equilibrium of‘ the system (4)-(5) is locally unstable, all trajectories that do not originate at the equilibrium point converge to a periodic orbit. Recalling Proposition 1, this implies that if the adjustment of prices and chartist expectations is sufficiently rapid, then all trajectories converge to a limit cycle provided that the share of wealth held by chartists is sufficiently high. Propositions 1 and 2 together provide sufficient conditions for the existence of a stable limit cycle. The result can be illustrated graphically by making the properties of the isoclines fl = 0 and ti = 0 explicit. The first of these isoclines is given by
p =
y(n)=
e(7c) &l
-
cl)'
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I\ Fig.
1. Direction
of the vector
field in
n-p
space.
Clearly, y(O) = 0, y’ > 0, lim, _ ~ y(n) = y”, and lim, _ _ r y(x) = -g’. isocline, corresponding to the condition ti = 0, is defined by
The second
Again, it is easily verified that z(0) = 0, lim, ~ 3, Z(X) = - cx: and lim, _ - x z(n) = ccj. It is clear that z’(n) < 0 whenever 7t is sufficiently large in absolute value. It may be shown, furthermore, that if the unique equilibrium of the system (4)-(5) is locally unstable, then z’(0) > Oa4 The isocline fi = 0 thus has a shape that is qualitatively similar to that of g( .); it is upward sloping, passes through the origin, and lies between g1 and g”. The isocline ti = 0 is negatively sloped when n is sufficiently different from zero, and positively sloped in some neighborhood of the origin if the equilibrium of the system is unstable. These properties are depicted for specific numerical values in Fig. 1. The arrows showing the direction of motion may be deduced directly from Eqs. (4) and (5); above the p = 0 isocline prices are falling, and above the 7t = 0 locus the expectations of chartists are falling. From the direction of the vector field, it is clear that the rectangle shown in the figure is an inoariant set, i.e. no trajectory originating within this rectangle can ever leave it. Furthermore, all trajectories originating at points outside the rectangle eventually enter it. If the unique equilibrium at the origin is unstable, then, by the Poincark-Bendixson Theorem, all trajectories not starting at the origin must converge to a limit cycle. A simulated realization of a cyclic trajectory is shown in Fig. 2. The behavior of prices and expectations over the cycle can be described as follows. Suppose initially that prices equal fundamentals but chartists expect prices to rise. This expectation ’ To see this. observe that [I,uc < 1 is sufficient
that z’(O) > 0 if and only if /I/K > I. and check for the stability of the system (4)-(5).
from
the proof
of Proposition
1
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space.
induces a rise in prices since it causes excess demand to be positive. Initially prices rise even faster than chartists expect them to, so that the perceived trend is revised upwards. Eventually chartist perceptions of the trend overtake the actual trend, which itself is slowing as a result of the negative asset demand of fundamentalists and the nonlinearity in the model. Beyond this point prices are still rising but at a slower rate than chartists expect, so that this expectation is being revised downwards. Eventually prices peak and start to decline toward fundamentals, with the trend perceived by chartists chasing the actual trend downwards and reinforcing this decline. At some point the trend perceived by chartists becomes negative, and at some point thereafter the price drops below fundamentals. It continues to fall, since chartists are now expecting a declining trend. Eventually chartist expectations catch up with the actual trend downwards and subsequently chartists start to become more optimistic. Prices reach their trough and then start to rise under pressure of fundamental demand, despite the fact that chartists are still expecting a decline. Chartist expectations now chase the price trend upwards and eventually chartists begin to expect a positive trend. Prices eventually reach fundamentals, at which point chartist expectations are once more buoyant, and the cycle begins again. Hence two regimes are possible: a fundamentalist regime, which is dynamically stable and in which the chartist share of wealth is low, and a limit cycle chartist regime in which the fundamentalist share is low. Which of these prevails depends on the value taken by p, which so far has been taken to be exogenous. Since the two groups differ with regard to their expectations, however, there arises a possibility for trade to occur between them. As a consequence of this trade the share of wealth held by each group will evolve over time, with the more effective strategy bringing about an increase in relative wealth. However, the relative effectiveness of the two strategies will itself depend on the prevailing regime: as we shall see below, a fundamentalist strategy is highly effective in the long run if the regime is cyclical. If the persistence of a given regime gives rise to changes in ,u in such a direction as to cause it to cross
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the threshold value pO, then the regime cannot persist indefinitely and we may speak, from the point of view of the smaller system (4))(5), of an endogenous regime switch. These issues are explored in the following section.
4. Endogenous
wealth
dynamics
If the share of wealth ,Uheld by chartists is to be fully endogenized, taking account of trades between the two groups, then the payoffs obtained by each group must be made explicit. We proceed under the simplifying assumption that speculator wealth is held either in the tradeable asset considered above, or in money. Let EC and MC denote the stocks of the asset and of money held by chartists at any point in time, and let E’ and M’ denote the corresponding stocks for fundamentalists. The aggregate stocks of both assets are considered fixed, and denoted by E and M respectively. Total chartist wealth is IV” = MC + ePEC, and total fundamentalist wealth is W’ = Mf + epEf, where ep is the asset price. 5 Note that the aggregate wealth of market participants is price dependent even though all asset stocks are fixed: E = WC + W’ = M + ePE. The balance sheets of the two groups are then given by Table 2. The share of wealth in the hands of chartists will depend on the asset price as well as on asset and money holdings, which will themselves evolve as trades occur. From the balance sheets depicted in Table 2, we have6 p(p, EC, MC) = !!c?ex
(7)
M t ePE Table 2 Balance sheets Assets
Liabilities
MC epEc
WC
Fundamentalists
M’ @E‘
W’
Total
M + r”E
W
Chartists
5 Recall that p represents the logarithm of the asset price. 6 In the previous two sections p was treated as an exogenous parameter. independent of asset holdings and prices. If asset holdings evolve only gradually over time as a result of trades. then their influence on 11 can be neglected as a first approximation. However, changes in p can have large effects on p, even if there are no trades, if the port@io composition of chartists differs substantially from that of the fundamentalists. If. on the other hand, the portfolio compositions of the two groups are identical. so that MC/E’ = M’IEr, then it can be shown that p is independent of p. The analysis of Sections 2 and 3 may therefore be understood as proceeding on the basis of the simplifying assumption of identical portfolio compositions.
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To take account of the fact that M” and EC are to be stale variables in the following discussion, and that p(p, EC, MC) is no longer an exogenous parameter, aggregate excess demand as defined in relation (3) may be written as D(p, rc, EC, MC) and the dynamic Eqs. (4) and (5) as P = BD(p, MC, MC),
(8)
fi = ci[fiD( p, XL,EC, MC) - n].
(9)
The variables EC and MC will evolve over time as transactions occur. Since the market is not assumed to clear, however, the volume of transactions that are made at any moment need to be specified. We shall assume that the only market participants involved in active trading are the two groups of speculators. It is clear that fundamentalists will wish to buy if the price is below fundamentals and sell if it is above. Chartists. on the other hand, wish to buy when they perceive a rising trend and sell which they consider that prices will fall. Transactions can occur only if the two groups are on opposite sides of the market, otherwise both will want to buy (sell) and there would be no sellers (buyers). When prices are above (below) fundamentals and believed by chartists to be on a rising (falling) trend, chartists will be purchasing assets from fundamentalists, so that EC > 0 (kc < 0). The volume of trading, T(p, rc,E”, MC), interpreted as the number of units of the asset purchased by chartists from fundamentalists, will be determined by the short side of the market as follows:
T(p, x, EC, MC) =
min{D”, -D’),
if Df < 0 < DC, MC > 0, EC < E,
max(D’, - D’j,
if DC < 0 < D’, MC < M, E” > 0,
I 0,
otherwise,
where p = p(p, EC, MC) as defined in (7) above, with DC(n) and D’(p) as defined in (1) and (2) in Section 2. The magnitude T is equal to net realized chartist purchases of the asset, and will be positive (negative) when chartists are buying (selling) the asset. In addition, some boundary conditions have been imposed on trades. Purchases by chartists can only be effected if their money balances are positive and they do not own the entire stock of equities. Similarly, sales by chartists can only occur if their stock of equities is positive and they do not own the entire stock of money. The evolution of chartist asset and money holdings is then simply given by I? = T(p, n, EC, MC).
(10)
With regard to the evolution of the fourth state variable, MC, we may proceed under two alternative assumptions. In one scenario, money is used only to effect purchases of the asset and the activities of the two groups with regard to expectation formation and trading are executed costlessly. In the second, there are differences in the real resource costs of financing fundamentalist and chartist activity. We consider the former case in this section. Under the assumption that speculation does not involve any real resource costs, the dynamics of chartist money holdings, MC, is easily described. Assuming that all
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purchases must be paid for in cash, the following is implied by the traders’ budget constraints: Id’ = -epic
= -ePT(p,
x, EC, MC).
(11)
Eqs. (8)-(11) together constitute a four-dimensional differential equation system in the variables p, n, EC and MC. Although the global properties of four-dimensional nonlinear differential equation systems are not generally easy to characterize, the above system has a very special structure which enables its long-run behavior to be unambiguously deduced. First, there is a continuum of equilibria corresponding to all points (0, 0, EC, MC). In other words, irrespective of the distribution of assets, a state in which prices equal fundamentals and chart&s are expecting no change is an equilibrium.’ It may be conjectured, on the basis of the analysis in the preceding sections, that if EC and MC are sufficiently small, then the corresponding equilibrium will be stable. In fact, a stronger result can be proved: regardless of initial conditions, either prices must converge to fundamentals or the chartist share of total wealth must converge to 1. Proposition 3. Along
any
trajectory:
of the system (8)-(1 l), either lim, _ ~ p = 0, or
lim z+ x AP, EC,MC) = 1. Proposition 3 states that only two outcomes are possible in the long run: either fundamentalist wealth vanishes or prices converge to fundamentals. In other words, as long as fundamentalists survive in the long run, instability cannot persist. This convergence of prices to fundamentals can occur even if the initial value of chartist wealth exceeds the bifurcation value identified in Section 2, so that prices diverge from the fundamentals to begin with. The rationale is the following. If chartist wealth is initially high enough to cause the subsystem (4)-(5) to be unstable, asset prices will fluctuate around fundamentals. These fluctuations themselves, however, will tend to diminish chartist wealth over time, as fundamentalists consistently ‘buy low and sell high’. Eventually chartist wealth declines sufficiently to move the market into a stable region. The chartist share of total wealth in the limiting state will be positive. but sufficiently small to ensure that the corresponding equilibrium is stable. A typical trajectory, starting with a high share of chartist wealth is depicted in Fig. 3. High chartist wealth leads to cyclic trajectories, which in turn erode chartist wealth gradually over time. Eventually, the bifurcation value ,uO is crossed and the price movements start to stabilize. The above result may be summarized as follows: if speculators whose behavior is stabilizing survive in the long run, then prices must converge to fundamentals. This is a weak version of Friedman’s (1953) influential argument: “People who argue that speculation is generally destabilizing seldom realize that this is largely equivalent to saying that speculators lose money, since speculation can be destabilizing in general ’ In addition, are in possession
there is a continuum of the entire stock
of equilibria of wealth.
of the type
(p. 0, E, M)
for arbitrary
p, where
chart&s
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ASSET PRICE (p)
I CHARTIST
SHARE OF WEALTH
(p)
Unstable Bifurcation
0.50
Regime
value
Stable Regime
Fig. 3. Elimination
of instability
through
the diminution
of chartist
wealth.
only if speculators on the average sell when the currency is low in price and buy when it is high” (Friedman, 1953, p. 175). Friedman’s view of a rational speculator is one who buys low and sells high, behavior which is described above as fundamentalism. That this behavior is unambiguously stabilizing, irrespective of the various adjustment speeds and elasticities in the model, has already been demonstrated in Proposition 1. However, it is optimal only under very restrictive conditions, and more generally it is entirely possible for optimal behavior to be destabilizing. In the present model, a speculator who buys on the way up, even when prices exceed fundamentals, but dumps her holdings just before the market turns down, would outperform the fundamentalists.* Although stabilizing speculation (fundamentalism) may not be individually rational in the sense that there are more profitable strategies available, there is a sense in which it is superior to chartism if investor horizons are sufficiently long. This need not be the case if it is costlier to obtain information about fundamentals than to form expectations based on extrapolating trends using historical data on asset prices.
5. Costly
fundamentalism
The long-run stability of the system need not obtain if the real resource costs of fundamentalism are greater than those of chartism. This is indeed likely to be the case, since knowledge of the fundamental value of an asset requires the acquisition of a considerable amount of information regarding future economic conditions,
a De Long et al. (1990b) describe individual speculators who have consciously such destabilizing strategies, and develop a model to explain their success.
and successfully
adopted
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competitor behavior, technological developments, and other uncertain events. In the model as developed so far, fundamentals have been assumed to be time-invariant and known to fundamentalists. In fact they will vary over time rather erratically and keeping abreast of changes in fundamentals is likely to be both time-consuming and expensive. The information costs incurred by chartists are negligible in comparison, since chartism requires the use of widely and freely available data on prices; it is the equipment and expertise required to manipulate and extract information from these data that represent the costs of chartist activity. Without denying that these costs may be significant, they are likely to be well short of the costs associated with gathering detailed asset-specific information. Assume that fundamentalist outlays on the acquisition of information are proportional to their share of total market wealth, with the factor of proportionality denoted by E. Hence, the flow of expenditures on information acquisition and processing at any moment is (1 - ~1)s. To maintain the assumption that the aggregate stock of money held collectively by speculators is constant, it is assumed that this outflow is matched by an inflow into the market of the same amount, (1 - ~)a. This inflow is distributed among fundamentalists and chartists in proportion to their current wealth shares. Hence, the net effect of money flows is that chartists gain ~(1 - ~)a, while fundamentalists lose (1 - F)E - (1 - I*)‘& = ~(1 - P)E. The net gain is zero, so the aggregate money stock in the market is constant, as in Section 4. The only modification to the dynamical system is that Eq. (11) has to be replaced by ltiC = --eqp,
71,EC, M’) + ,U(l - /J)&.
(12)
The revised system, (8))(10) and (12) has an equilibrium at (0, 0, 0, 0), with prices equal to fundamentals and with chartist wealth eliminated. Let us call this the F-equilibrium of the system. In addition, there is a continuum of equilibria of the form (p, 0, E, M), where chartist expectations are stationary at some arbitrary price while fundamentalist wealth is zero. Of the equilibria in this continuum, there is exactly one with prices equal to fundamentals: (0, 0, E, M). Call this the C-equilibrium of the system. We then have: Proposition 4. If‘ /lc > 1 + E/x(M + E), the C-equilibrium of the system (8)-( lo), (12) is locally unstable. Therefore, if price adjustment is sufficiently rapid or the expectations elasticity of chartist demand is sufficiently high, then prices will converge to fundamentals along a trajectory only if they converge to the F-equilibrium of the system where chartist wealth diminishes to zero. Is this a likely outcome? It is less so than in the case of costless fundamentalism, since the introduction of the cost E causes a net outflow of wealth from fundamentalists to chartists whenever both groups have positive wealth. If the initial value of chartist wealth is higher than the bifurcation value identified in Section 2, then the price dynamics will be unstable and oscillatory. Provided that E is sufficiently low, this will tend to raise the wealth held by fundamentalists since they achieve a positive net gain over a complete cycle. This in turn will act to reduce price volatility as in the case of costless speculation. However, as the movement of
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0.20
CHARTIST
n
Bifurcation
SHARE OF WEALTH
(p)
Unstable
Regime
n
value Stable Regime
Fig. 4. Alternating
stable
and unstable
regimes
with
costly
fundamentalism.
prices stabilizes, the relative advantage of fundamentalist strategies declines, and the costs of acquiring information start to dominate the dynamics of wealth shares. If price movements largely reflect changes in the underlying fundamentals, then it is more cost-effective to extract information about fundamentals from prices rather than from the more direct sources used by fundamentalists. Chartism therefore becomes a more effective strategy and the share of chartist wealth will start to climb. Once the bifurcation value, pO, is exceeded, the dynamics of the system again begin to display the excess volatility which favors fundamentalist strategies in the long run. This intuitive argument is illustrated in Fig. 4, which shows simulated time-series output for the asset price and the chartist share of wealth as the system @-(IO), (12) evolves over time. The chartist wealth share shows a decreasing trend during volatile periods, which causes prices to stabilize. However, the persistence of tranquility itself favors the strategy of chartists, and causes their share of wealth to rise. Only when the rise has caused the bifurcation value to be crossed, and the resulting volatility to become manifest, does the behavior of fundamentalists again becoming rewarding. Hence with costly fundamentalism one would expect periods of tranquility to be interrupted by periods of volatility in speculative markets, as the endogenous dynamics of wealth shares lead to the decline of whichever group currently dominates the price dynamics.
6. Conclusions
Whether or not changes in asset prices primarily reflect movements in the underlying fundamental values depends on a number of factors, both institutional and behavioral. In a market consisting of two distinct investor groups, fundamentalists and chartists, the share of wealth held by the latter is a bifurcation parameter which,
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if exceeded, transforms a convergent regime into a persistently cyclical one. In the cyclical regime, fundamentalism is a successful strategy against chartism in the long run, which causes the share of wealth in chartist hands to decline if it is initially high. However, if information about fundamentals is costly to acquire, then chartism is a successful strategy against fundamentalism in the stable regime. Hence behavior that is conducive to stability is profitable under conditions of instability and vice versa, giving rise to a phenomenon that has been described here as endogenous regime switching: the bifurcation parameter of a two-dimensional system itself evolves (as part of a larger dynamical system) in such a way as to cause the smaller system to alternate between stable and unstable regimes. Although the argument above has been presented in terms of a highly stylized model, the economic intuition underlying the results is straightforward. Behavior conducive to stability (fundamentalism) may be most profitable when the market is unstable, and behavior conducive to instability (chartism) may be rather lucrative in a stable market when it is costly to collect information about fundamentals. There is a sense in which the two groups of speculators enjoy a symbiotic relationship with each other: each group benefits from an increase in the numbers of the other. Fundamentalists prey on the adaptive behavior of the chartists when chartists dominate the market, and chartists extract information about fundamentals costlessly when fundamentalists dominate the market. As a result, one would expect heterogeneity of practices to persist in the long run, with the dominance of one set of practices giving way to that of the other. The implication for speculative markets is that periods of tranquility are likely to be punctuated from time to time by periods of excessive volatility.
Acknowledgements I thank Carl Chiarella, Englebert Dockner, Peter Flaschel, Reiner Franke, Peter Skott, seminar participants at the University of Bielefeld and the University of Aarhus, and two anonymous referees for helpful comments on an earlier draft.
Appendix
A.1. Proof of Proposition 1 Local stability may be deduced from the Jacobean matrix J which, evaluated at the equilibrium, is given as follows:
J=
1
-B(l -/de Bw 1 -cQ(l - /QO X(&C - 1)
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Writing A for the determinant
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and Economic
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of this matrix, we have the following:
A = @VI( 1 - /J) > 0. The determinant is strictly positive for all admissible parameter values, so that the equilibrium cannot be a saddle-point. The only generic possibilities are therefore an attractor or a repeller. Which of these obtains depends entirely on the sign of the trace t of J, given by T = r&K - pe(1 - /L) - a. Stability of equilibrium requires that the trace be negative. Since p E [0, I), if fit I 1, then the first term is smaller in magnitude than the third, so the equilibrium is stable for all parameter values, proving part (a). Now assume that PC > 1. Then T > 0 when p = 1 and r < 0 when p = 0. Treating T as a function of the various parameters of the model, we obtain the following:
By the continuity and monotonicity of r in the population composition I*, there will be a critical value plo E (0,l) such that the system is stable when p < p0 and unstable when ,u > pO. This bifurcation value is given by
Differentiating
(Al) with respect to x and simplifying, we have
which must be negative if PC > 1. Differentiating simplifying, we have
Differentiating
(Al)
with respect to /I and
(Al) with respect to 8 and simplifying, we have
which, under the assumption that PC > 1, is positive. Finally, differentiating respect to c and simplifying, we have
which completes the proof of part (c).
0
(Al) with
R. SethilStructural
A.2. Proqf
of Proposition
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and Economic
Dynamics
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115
2
Assume that p # 1. Define the sets S, and S, as follows:
and let S = S’ x S,. The unique equilibrium (0,O) of the system (4), (5) is contained in S. We first establish that S is an invariant set by showing that at all points on the boundary of S the direction of the vector field points toward the interior of S. If p I -pg’/Q( 1 - CL),then Df 2 ,ug’ from (2). Since DC > -pg’ from (l), this means that D(p, X) > 0, regardless of x. Hence, from (4), d < 0. An analogous argument establishes that if p 2 pgh/O(l - p), then fl < 0. Hence p must eventually enter the interior of S,, regardless of initial conditions. Now suppose that p E S,. Then Df E [ -pgh, pg’]. Since DC E (-pg’, pgh), it must be the case that D(P, 70 E (-Ad
+ s”>, Ag’ + sh>).
From (4), therefore P E (-/MY
+ gh)3 BAS + gh>).
If rt I -/Ip(g’ + gh), then n < & so from (5) we have ti > 0. Analogously, if 712 pp(g’ + gh), then ri > 0. Hence, 7c must eventually enter the interior of S,, regardless of initial conditions, and the limit sets of all trajectories must lie in S. Since we are dealing with a planar flow with a unique equilibrium, there are only three possible classes of limit sets: fixed points, closed orbits, and homoclinic orbits (Guckenheimer and Holmes, 1983, p. 45). Homoclinic orbits are trajectories that connect a fixed point to itself, and may be ruled out in the present model since they require that the equilibrium be a saddle-point, which is impossible since lJ\ > 0 for all parameter values. Hence all trajectories converge either to the unique equilibrium or to a closed orbit. If the unique equilibrium is locally unstable, then there exists and open neighborhood U of the equilibrium such that the set S n 0 is itself invariant, where 0 represents the complement of U in R2. The limit sets of all trajectories not originating at the equilibrium must lie in S n 0, which contains no equilibrium point. Hence the limit set of all trajectories not originating at the equilibrium must, by the PoincarC&Bendixson Theorem (Arrowsmith and Place, 1982, p. 110) be closed orbits. 0 A.3. Proqf
qf Proposition
3
Let us define the function I/ = MC + EC, which measures the total wealth of chartists, with equities valued at fundamentals rather than current prices. The trading rule defined in Section 4 ensures that V remains non-negative along any trajectory of the system (8))(11). Furthermore, p = &fc + EC = (1 - eP)gc.
(43
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If p < 0, then Df > 0 so that E I 0. Similarly, if p > 0, then D’ < 0 so that J?’ 2 0. Together with (A2), these conditions imply that 6’ I 0. Since C’ is nonincreasing and bounded from below, it must be the case that as t tends to infinity, 9 goes to zero. This in turn implies that, in the limit, either p = 0 or hi’ = .E? = 0 (or both). The first of these conditions implies the second, so that the latter condition always holds in the limit. Hence trades cease in the limit and MC and EC converge to some magnitudes MC and EC respectively. To prove the proposition we need to establish that unless lim, _ m p = 0, R” = M and Ec = E, respectively (fundamentalist wealth vanishes in the limit). Suppose prices do not converge to fundamentals. We consider two cases separately: Case 1. Prices converge, but not to fundamentals. Let the limiting price be p # 0. Since prices converge, Ij = 0, which in turn implies from (8) and (9) that n = 0 and then DC = 0. If fundamentalist wealth is nonzero, then Df > 0, which implies that D( p, 0) > 0 and hence, from (8), that @# 0. This contradicts the hypothesis that prices converge. Hence, fundamentalist wealth must be zero. Case 2. Prices do not converge. Suppose that fundamentalist wealth is nonzero. Then D’ > 0 whenever p < 0 and D’ < 0 whenever p > 0. In order that trades do not occur, either chartist wealth must be zero, or x < 0 whenever p > 0 and 7c> 0 whenever p < 0. If chartist wealth is zero, then prices converge to fundamentals from Proposition 1, contradicting the hypothesis that they do not converge. If n < 0 whenever p > 0 and 7c> 0 whenever p < 0, then rj < 0 whenever p > 0 and 6 > 0 whenever p < 0. This too implies convergence of prices to fundamentals, contradicting the hypothesis. Hence fundamentalist wealth must be zero. C A.4. Proof qf Proposition 4
The Jacobean of the system (8)-(lo),
(12) is given by the following matrix, where
p = ,a(~, EC, MC) as in (7), D = D(p, 71,EC, MC) as in (3), T = T(p, r, EC, MC) as defined in Section 4, and a subscript i denotes the partial derivative of the function
with respect to its ith argument:
The trace of the above matrix is t = /3Dl + x(/?Dz - 1) + T3 + ePT, + p&l
From (3) we have
- 21.1).
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At the C-equilibrium p = rr = 0 and p = 1, and since g(0) = 0 and g’(0) = c, we have D, = 0 and D, = c. The derivatives & are not defined at all points since T is not differentiable everywhere. However, the derivatives T3 and T, do exist whenever p = 71= 0, since, from (1) and (2): dD’jaE’
= p2g(z) = p2g(0) = 0,
ZD’/aE’ = &3p = 0, dD’/8M’
= p3g(n) = pJg(0) = 0,
dD’/aM’
= /Qp = 0
Hence, T, = T, = 0 at the C-equilibrium. From (7) it is readily verified that p3 = l/(M + E) at the C-equilibrium, since p = 0. A sufficient condition for the local instability of the C-equilibrium is r > 0 which, since D, = 0, D, = c, and ,u = I, occurs whenever CY(/!JC - 1) - E/(M + E) > 0 or /?c > I + E/x(M + E), as required.
0
AS. Simulation details The simulation details are g(z) = 5 tanh(n/5), t, = 1, and E = 0.005.
E = 100, M = 200, c( = 3, /I = 3,
References Arrowsmith. D.K. and C.M. Place, 1982, Ordinary differential equations (Chapman and Hall, London). Beja. A. and M.B. Goldman, 1980, On the dynamic behavior of prices in disequilibrium, Journal of Finance 35,235-247. Blanchard, O.J. and M.W. Watson, 1982, Bubbles. rational expectations, and financial markets, in: P. Wachtel, ed., Crises in the economic and financial structure (Lexington Books. Lexington, MA). Chiarella, C., 1992, The dynamics of speculative behavior, Annals of Operations Research 37, 101-123. Conlisk, J., 1980, Costly optimizers versus cheap imitators, Journal of Economic Behavior and Organization 1, 275-293. Conlisk, J., 1988, Optimization cost, Journal of Economic Behavior and Organization 9, 213.-228. Day, R. and W. Huang, 1990, Bulls, bears and market sheep, Journal of Economic Behavior and Organization 14, 299-329. De Long, J.B., A. Shleifer, L. Summers and R. Waldman. 1990a, Noise trader risk in financial markets, Journal of Political Economy 98, 703-738. De Long, J.B., A. Shleifer, L. Summers and R. Waldman, 1990b, Positive feedback investment strategies and destabilizing rational speculation, Journal of Finance 45, 379-395. Evans. G.W. and G. Ramey, 1992, Expectation calculation and macroeconomic dynamics, American Economic Review 82, 207-224. Fama, E.F., 1989, Perspectives on’october 1987, or, What did we learn from the crash?, in: R.W. Kamphuis, R.C. Kormendi, and J.W.H. Watson, eds., Black Monday and the future of financial markets (Dow Jones-Irwin, Homewood, IL).
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Flood, R.P. and P.M. Garber, 1982, Bubbles, runs, and gold monetization, in: P. Wachtel, ed., Crises the economic and financial structure (Lexington Books, Lexington, MA). Frankel, J.A. and Froot, K.A., 1986, Understanding the US dollar in the eighties: The expectations chartists and fundamentalists. Economic Record, Special Issue on Exchange Rates and the Economy, 24438. Friedman, M.. 1953, The case for flexible exchange rates, in: Essays in positive economics (University Chicago Press, Chicago, IL). Guckenheimer, J. and P. Holmes, 1983, Nonlinear oscillations, dynamical systems, and bifurcations vector fields (Springer, New York). Sethi, R. and R. Franke, 1995, Behavioral heterogeneity under evolutionary pressure: Macroeconomic implications of costly optimization, Economic Journal 105, 5833600. Shiller, R.J., 1990, Market volatility (MIT Press, Cambridge, MA). Zeeman, E.C., 1974, On the unstable behavior of stock exchanges, Journal of Mathematical Economics I. 39949.
in of
of of
.
.
I
.H -= --
B
STRUCTURAL CHANGEAND ECONOMIC DYNAMICS Structural
EUEVlER
Change
and Economic
Dynamics
7 (1996)
99-118
Endogenous regime switching in speculative markets Rajiv Sethi Drpartment
qf Economics.
Barnard Collegr. Columbia Uniwrsity, New York, NY 10027, USA
3009 Broadway,
This paper examines a nonlinear disequilibrium model of price formation in speculative markets with two groups of speculators, fundamentalists and chartists. The behavior of the fundamentalists is stabilizing, whereas that of the chartists is destabilizing; a sufficiently high share of wealth in the hands of the latter group can cause attracting periodic orbits to arise. If information about fundamentals is costly to obtain, then chartists obtain higher pay-offs in a stable regime, while fundamentalists perform better in an unstable regime. This gives rise to endogenous regime switching: the market alternates between periods of stability and instability, with the transition from one regime to another determined endogenously through the evolution of wealth shares. Keywords:
Speculation; Financial instability; Nonlinear dynamics
JEL classijicution: D84; G12
1. Introduction
Do movements in the prices of financial assets broadly reflect changes in their underlying fundamental values, or are there important determinants of such movements that are unrelated to fundamentals? This question has attracted a great deal of empirical and theoretical attention over the past decade. While Shiller’s view that “increasingly, there is statistical evidence that suggests the stock market may have a life of its own to some extent, unrelated to economic fundamentals” (Shiller, 1990, p. 398) remains controversial, it appears to be gaining ground. Even those who saw the international stock market crash of October 1987 as “an adjustment to a change in fundamental values” (Fama, 1989, p. 81) concede that “the long, orderly upside 0954-349X/96/$15.00 ,(:‘ 1996 Elsevier SSDI 0954-349X(95)00040-2
Science
B.V. All rights
reserved
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of the swing preceding October 19th” could well represent a prolonged departure from fundamentals. On the theoretical front it is now well known that persistent divergence from fundamentals can be consistent with self-fulfilling expectations as well as the no-arbitrage condition, both in deterministic and stochastic models (Flood and Garber, 1982; Blanchard and Watson, 1982). In contrast to this work on rational bubbles, there is also a fairly substantial and growing body of literature based on the assumption of heterogeneous agents, some of whom may not have self-fulfilling expectations (Beja and Goldman, 1980; Frankel and Froot, 1986; De Long et al., 1990a,b; Chiarella, 1992). The heterogeneous agent literature is itself diverse. Beja and Goldman (1982) and Chiarella (1992) consider disequilibrium models in which two types of traders, fundamentalists and chartists, interact. The former base their decisions on the deviation of the asset price from fundamentals, and the latter on the trends they discern from past observations of the data. For certain parameter values the equilibrium in which prices are equal to fundamentals is locally unstable. The Beja/Goldman model is linear and therefore this instability is global, but Chiarella’s nonlinear version gives rise to a stable limit cycle, with prices oscillating around but never converging to fundamental values. Frankel and Froot (1986) also consider the behavior of fundamentalists and chartists, but the expectations of the two groups are aggregated through the activities of a third group, the portfolio managers, who give more weight to the forecasts of the group with the best recent forecasting performance. In this model prices can converge to values different from fundamentals, so that the price may be ‘high and stuck’; the authors argue that this was indeed the case with regard to the U.S. dollar in the mid-1980s. De Long et al. (1990a) consider ‘noise traders’ who misjudge returns by a random amount in each period; they are confronted in the market by unboundedly rational but risk-averse traders who correctly forecast true returns, taking full account of the presence of noise traders. Prices can fluctuate even if fundamentals remain constant, and it is also possible for the noise traders to earn greater returns than the unboundedly rational agents if, on average, their miscalculations induce them to carry more risk than they believe themselves to be carrying. De Long et al. (1990b) consider a different model in which unboundedly rational speculators interact with chartists; in this case the rational agents destabilize prices to exploit the adaptive behavior of the latter. The model developed below bears a close relationship to Chiarella’s nonlinear version of the Beja-Goldman model. In Section 2 a generalized version of the Chiarella (1992) model is presented which takes into account the fact that the demands of fundamentalists and chartists depend not only on prices and expectations but on their respective shares of total market wealth. As in Chiarella (1992), prices are assumed to adjust in response to excess demand in the asset market, and chartist expectations adjust adaptively. The model has a unique equilibrium with prices equal to fundamental values. It is shown that if price adjustment is sufficiently sluggish and chartist demand relatively insensitive to changes in expectations, then the price dynamics are stable. If these conditions are not met, then the equilibrium will be stable if and only if the chartist share of market wealth is sufficiently small. The chartist wealth share is therefore identified as a bifurcation parameter that governs
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the stability properties of the dynamics. In Section 3 it is shown that with nonlinear chartist demand, trajectories converge to a limit cycle in the locally unstable case. In Section 4 the model is generalized to allow the share of chartist wealth in the market to evolve endogenously over time, as a result of trades between the two groups. Since explicit trading between groups of agents with heterogeneous beliefs and the corresponding evolution of balance sheets is seldom taken into account in economic models, this represents a significant departure from the existing literature. The result is a four-dimensional dynamical system in prices, expectations, asset holdings and money holdings. It is shown that only two outcomes are possible in the long run: either prices converge to fundamental values as trades between the two groups diminish chartist wealth over time, or fundamentlist wealth declines to zero. The former possibility is the more interesting and plausible one, and may arise even when initial chartist wealth is well above the bifurcation value identified in Section 2. The rationale behind the result is that when the price dynamics are oscillatory, fundamentalists increase their share of wealth over a complete cycle, even though chartists may outperform fundamentalists during some phases of the cycle. In Section 5 the model is generalized further to allow for the possibility that fundamentalist speculation requires greater information acquisition and processing costs than chartist activity. The costs of information acquisition and processing are seldom taken into account in the economics literature, although some recent attempts have been made to model such costs explicitly (Conlisk, 1980, 1988; Evans and Ramey, 1992; Sethi and Franke, 1995). With costly fundamentalism, chartists are favored when the regime is stable, and fundamentalists when the regime is cyclic. Since the extent of chartist activity is itself a crucial factor in determining stability, this gives rise to the intriguing possibility that the economy may switch back and forth between stable and unstable regimes as the chartist share of wealth contracts in the latter and expands in the former. The resulting time series is characterized by periods of excess volatility and mean-reversion, punctuated by periods in which the market is efficient in the standard sense, with the transition from one phase to another being the endogenous outcome of shifting wealth across the two groups of investors.
2. A dynamic model of speculation
There is a single asset in fixed supply, and p denotes the logarithm of its price. We allow for two different types of speculator which, following Zeeman (1974), Beja and Goldman (1980), Frankel and Froot (1986) and others, are referred to as &&amentalists and chautists. Chartists perceive a trend in price movements which is denoted by rc, and their demand DC(n) for the asset is an increasing function of this perceived trend:
DC(n)= KY(n), where p E [0, l] is the share of total market wealth in the hands of chartists. It is
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assumed that the function g( .) satisfies g(0) = 0,
g’ > 0.
In other words, if chartists expect prices to rise (fall) their demand for the asset is positive (negative). The greater the perceived trend in price movements, the greater is chartist market participation, which is motivated by the prospect of short-term capital gains. In addition, demand from chartists is proportional to the share of total market wealth, ,u, that they command; this is assumed exogenous for the moment. Fundamentalist demand D’(p) depends on the difference between the current price p and what fundamentalists believe to be the fundamental value, f, of the asset. Without loss of generality, we use the normalization f = 0, so that p measures prices as deviations from perceived fundamentals.’ For simplicity, this function is assumed to be linear: D’(p) = -(l
- p)Bp.
(2)
In other words, fundamentalists buy (sell) when the asset is priced below (above) its perceived fundamental value. The greater the deviation between current prices and fundamentals, the greater is fundamentalist market participation. This strategy reflects a long-term perspective since demand is independent of perceived short-term expectations of price movements. In addition, fundamentalist demand is proportional to the share of wealth (1 - p) in fundamentalist hands. The parameter 0 measures the strength of fundamentalist reaction to departures of prices from (perceived) fundamentalist values. Aggregate excess demand for the asset, D(p, rc), is simply the sum of fundamentalist and chartist demands: D(P, n) = D’(7c) + of(p) = &J(x) - (1 - p)@p.
(3)
Continuous market-clearing is not assumed. Instead, the asset price adjusts under pressure of excess demand as follows: d = P&4
n).
(4)
Since excess demand need not be zero at all times, not all intended trades will be executed, unless there are other market participants such as market-makers who satisfy excess demand out of inventory (Day and Huang, 1990). If we consider a market consisting only of speculators, then without instantaneous market clearing one (or both) groups of traders will be rationed. The continuous market-clearing case can be approximated to any degree of precision by allowing /I to approach infinity. These issues are taken up in greater detail below when trades are considered explicitly. Chartist expectations are revised in an adaptive manner, on the basis of the difference between current and expected price changes: 7i = ct($ - 7-c) = WmP,
n) - Tc).
(5)
The parameters (x, fi, and H are all assumed to be strictly positive. ’ Note
that
p and ,f refer to logarithms.
so that negative
values
are admissible.
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Eqs. (4) and (5) represent a two-dimensional differential equation system in the state variables p and x, with a unique equilibrium (p*, rc*) = (0,O) provided that p # 1.’ In other words, unless the entire market wealth is in the hands of chartists, the only rest point of the system (4)-(5) occurs when prices equal fundamental values and chartists expect them to stay there. The conditions under which this equilibrium will be dynamically stable are given by the following result, where c is used to denote g’(O), the slope of the function g( .) at the equilibrium value of chartist expectations.3 Proposition 1. Assume that p E [0, 1). Then: (a) If jk I 1, the unique equilibrium of the system (4)-(5) is locally asymptotically stable for all values of TX,9 and p. (b) If/k > 1, thenfor any given values of%, j, t? and c there exists p0 E (0,l) such that the unique equilibrium qf the system (4)-(5) is locally asymptotically stable when p < p0 and repelling when p > pO. (c) The bifurcation value pLo(u,fl, 0, c) satisfies 2p0/dcr < 0, Zp0/8/I < 0, dpO/CYl> 0 and 8p0/Zc < 0. The information regarding the local stability properties of the system (4)-(5) contained in the above result is summarized in Table 1. According to Proposition 1, if the adjustment of prices under pressure of excess demand is sufficiently slow, and the responsiveness of chartist demand to small perceived price trends is weak (PC < l), then the equilibrium will be locally stable irrespective of the share of wealth held by each group. If these conditions are not met, then the stability of the equilibrium depends on the share of wealth held by chartists. Specifically, there is a bifurcation ualue of this wealth share, denoted pO, such that the equilibrium is locally unstable if p exceeds ,u~ and stable if pLoexceeds p. This bifurcation value itself falls as a, p and c rise (enlarging the region of instability in parameter space) and rises as 0 rises. To summarize, local instability is possible if the adjustment of prices is rapid, if chartist demand is highly sensitive to changes in expectations, and the share of wealth in chartist hands is sufficiently large. If the
Table Local
1 stability
of equilibrium
’ To see this, observe that the equilibrium condition 6 = 0 implies, from (4), that D( p*, x*) = 0. This in turn implies from (5) and the equilibrium condition ti = 0 that z* = 0. Hence g(n*) = 0 and thus D’(n*) = 0, so from (3) we have D’(p*) = 0 and p* = 0. If p = 1, then any point (p, 0) is an equilibrium, since D’(p) = 0 for any p. 3 Proofs of all formal propositions are collected in the appendix.
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population consists only of fundamentalists (11= 0), then the system is stable under all circumstances. Local instability need not, of course imply unbounded divergence from fundamentals, since we are dealing with a potentially nonlinear system. The global behavior of the model in the locally unstable case is examined in the next section.
3. Global analysis If the relationship between chartist expectations and demand, as expressed by the function g( .), is linear, then the system (4))(5) as a whole is linear and local instability implies global instability. Prices would eventually exceed any stated bound or become arbitrarily close to zero, even if fundamentals remain constant over time. On the other hand, if g( .) is nonlinear, local instability can coexist with boundedness and non-negativity of prices. We shall assume here that although chartist demand increases as the perceived trend in prices rises, it does not do so without bound. This is to be expected for at least two reasons. First, investor expectations will not be held with subjective certainty, so that if investors are risk-averse, they will not be willing to take infinitely large arbitrage positions. Secondly, even if investors are assumed to be risk-neutral or to hold expectations with subjective certainty, they will be faced with constraints on borrowing and on short sales. Both these factors will ensure the boundedness of speculative demand. This effect may be captured by allowing g( .) to be nonlinear, with the following conditions imposed: lim g(7c) = g” < x8, I- II
lim x*--r
g(n)=
-g’>
-m.
(6)
These conditions are sufficient to ensure that prices can only deviate by some finite (possibly very large) amount from fundamental values. yielding: Proposition 2. Assume that p E [0, l), and g( ‘) satkfies conditions (6). Then, if the unique equilibrium of‘ the system (4)-(5) is locally unstable, all trajectories that do not originate at the equilibrium point converge to a periodic orbit. Recalling Proposition 1, this implies that if the adjustment of prices and chartist expectations is sufficiently rapid, then all trajectories converge to a limit cycle provided that the share of wealth held by chartists is sufficiently high. Propositions 1 and 2 together provide sufficient conditions for the existence of a stable limit cycle. The result can be illustrated graphically by making the properties of the isoclines fl = 0 and ti = 0 explicit. The first of these isoclines is given by
p =
y(n)=
e(7c) &l
-
cl)'
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105
I\ Fig.
1. Direction
of the vector
field in
n-p
space.
Clearly, y(O) = 0, y’ > 0, lim, _ ~ y(n) = y”, and lim, _ _ r y(x) = -g’. isocline, corresponding to the condition ti = 0, is defined by
The second
Again, it is easily verified that z(0) = 0, lim, ~ 3, Z(X) = - cx: and lim, _ - x z(n) = ccj. It is clear that z’(n) < 0 whenever 7t is sufficiently large in absolute value. It may be shown, furthermore, that if the unique equilibrium of the system (4)-(5) is locally unstable, then z’(0) > Oa4 The isocline fi = 0 thus has a shape that is qualitatively similar to that of g( .); it is upward sloping, passes through the origin, and lies between g1 and g”. The isocline ti = 0 is negatively sloped when n is sufficiently different from zero, and positively sloped in some neighborhood of the origin if the equilibrium of the system is unstable. These properties are depicted for specific numerical values in Fig. 1. The arrows showing the direction of motion may be deduced directly from Eqs. (4) and (5); above the p = 0 isocline prices are falling, and above the 7t = 0 locus the expectations of chartists are falling. From the direction of the vector field, it is clear that the rectangle shown in the figure is an inoariant set, i.e. no trajectory originating within this rectangle can ever leave it. Furthermore, all trajectories originating at points outside the rectangle eventually enter it. If the unique equilibrium at the origin is unstable, then, by the Poincark-Bendixson Theorem, all trajectories not starting at the origin must converge to a limit cycle. A simulated realization of a cyclic trajectory is shown in Fig. 2. The behavior of prices and expectations over the cycle can be described as follows. Suppose initially that prices equal fundamentals but chartists expect prices to rise. This expectation ’ To see this. observe that [I,uc < 1 is sufficient
that z’(O) > 0 if and only if /I/K > I. and check for the stability of the system (4)-(5).
from
the proof
of Proposition
1
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induces a rise in prices since it causes excess demand to be positive. Initially prices rise even faster than chartists expect them to, so that the perceived trend is revised upwards. Eventually chartist perceptions of the trend overtake the actual trend, which itself is slowing as a result of the negative asset demand of fundamentalists and the nonlinearity in the model. Beyond this point prices are still rising but at a slower rate than chartists expect, so that this expectation is being revised downwards. Eventually prices peak and start to decline toward fundamentals, with the trend perceived by chartists chasing the actual trend downwards and reinforcing this decline. At some point the trend perceived by chartists becomes negative, and at some point thereafter the price drops below fundamentals. It continues to fall, since chartists are now expecting a declining trend. Eventually chartist expectations catch up with the actual trend downwards and subsequently chartists start to become more optimistic. Prices reach their trough and then start to rise under pressure of fundamental demand, despite the fact that chartists are still expecting a decline. Chartist expectations now chase the price trend upwards and eventually chartists begin to expect a positive trend. Prices eventually reach fundamentals, at which point chartist expectations are once more buoyant, and the cycle begins again. Hence two regimes are possible: a fundamentalist regime, which is dynamically stable and in which the chartist share of wealth is low, and a limit cycle chartist regime in which the fundamentalist share is low. Which of these prevails depends on the value taken by p, which so far has been taken to be exogenous. Since the two groups differ with regard to their expectations, however, there arises a possibility for trade to occur between them. As a consequence of this trade the share of wealth held by each group will evolve over time, with the more effective strategy bringing about an increase in relative wealth. However, the relative effectiveness of the two strategies will itself depend on the prevailing regime: as we shall see below, a fundamentalist strategy is highly effective in the long run if the regime is cyclical. If the persistence of a given regime gives rise to changes in ,u in such a direction as to cause it to cross
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the threshold value pO, then the regime cannot persist indefinitely and we may speak, from the point of view of the smaller system (4))(5), of an endogenous regime switch. These issues are explored in the following section.
4. Endogenous
wealth
dynamics
If the share of wealth ,Uheld by chartists is to be fully endogenized, taking account of trades between the two groups, then the payoffs obtained by each group must be made explicit. We proceed under the simplifying assumption that speculator wealth is held either in the tradeable asset considered above, or in money. Let EC and MC denote the stocks of the asset and of money held by chartists at any point in time, and let E’ and M’ denote the corresponding stocks for fundamentalists. The aggregate stocks of both assets are considered fixed, and denoted by E and M respectively. Total chartist wealth is IV” = MC + ePEC, and total fundamentalist wealth is W’ = Mf + epEf, where ep is the asset price. 5 Note that the aggregate wealth of market participants is price dependent even though all asset stocks are fixed: E = WC + W’ = M + ePE. The balance sheets of the two groups are then given by Table 2. The share of wealth in the hands of chartists will depend on the asset price as well as on asset and money holdings, which will themselves evolve as trades occur. From the balance sheets depicted in Table 2, we have6 p(p, EC, MC) = !!c?ex
(7)
M t ePE Table 2 Balance sheets Assets
Liabilities
MC epEc
WC
Fundamentalists
M’ @E‘
W’
Total
M + r”E
W
Chartists
5 Recall that p represents the logarithm of the asset price. 6 In the previous two sections p was treated as an exogenous parameter. independent of asset holdings and prices. If asset holdings evolve only gradually over time as a result of trades. then their influence on 11 can be neglected as a first approximation. However, changes in p can have large effects on p, even if there are no trades, if the port@io composition of chartists differs substantially from that of the fundamentalists. If. on the other hand, the portfolio compositions of the two groups are identical. so that MC/E’ = M’IEr, then it can be shown that p is independent of p. The analysis of Sections 2 and 3 may therefore be understood as proceeding on the basis of the simplifying assumption of identical portfolio compositions.
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To take account of the fact that M” and EC are to be stale variables in the following discussion, and that p(p, EC, MC) is no longer an exogenous parameter, aggregate excess demand as defined in relation (3) may be written as D(p, rc, EC, MC) and the dynamic Eqs. (4) and (5) as P = BD(p, MC, MC),
(8)
fi = ci[fiD( p, XL,EC, MC) - n].
(9)
The variables EC and MC will evolve over time as transactions occur. Since the market is not assumed to clear, however, the volume of transactions that are made at any moment need to be specified. We shall assume that the only market participants involved in active trading are the two groups of speculators. It is clear that fundamentalists will wish to buy if the price is below fundamentals and sell if it is above. Chartists. on the other hand, wish to buy when they perceive a rising trend and sell which they consider that prices will fall. Transactions can occur only if the two groups are on opposite sides of the market, otherwise both will want to buy (sell) and there would be no sellers (buyers). When prices are above (below) fundamentals and believed by chartists to be on a rising (falling) trend, chartists will be purchasing assets from fundamentalists, so that EC > 0 (kc < 0). The volume of trading, T(p, rc,E”, MC), interpreted as the number of units of the asset purchased by chartists from fundamentalists, will be determined by the short side of the market as follows:
T(p, x, EC, MC) =
min{D”, -D’),
if Df < 0 < DC, MC > 0, EC < E,
max(D’, - D’j,
if DC < 0 < D’, MC < M, E” > 0,
I 0,
otherwise,
where p = p(p, EC, MC) as defined in (7) above, with DC(n) and D’(p) as defined in (1) and (2) in Section 2. The magnitude T is equal to net realized chartist purchases of the asset, and will be positive (negative) when chartists are buying (selling) the asset. In addition, some boundary conditions have been imposed on trades. Purchases by chartists can only be effected if their money balances are positive and they do not own the entire stock of equities. Similarly, sales by chartists can only occur if their stock of equities is positive and they do not own the entire stock of money. The evolution of chartist asset and money holdings is then simply given by I? = T(p, n, EC, MC).
(10)
With regard to the evolution of the fourth state variable, MC, we may proceed under two alternative assumptions. In one scenario, money is used only to effect purchases of the asset and the activities of the two groups with regard to expectation formation and trading are executed costlessly. In the second, there are differences in the real resource costs of financing fundamentalist and chartist activity. We consider the former case in this section. Under the assumption that speculation does not involve any real resource costs, the dynamics of chartist money holdings, MC, is easily described. Assuming that all
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purchases must be paid for in cash, the following is implied by the traders’ budget constraints: Id’ = -epic
= -ePT(p,
x, EC, MC).
(11)
Eqs. (8)-(11) together constitute a four-dimensional differential equation system in the variables p, n, EC and MC. Although the global properties of four-dimensional nonlinear differential equation systems are not generally easy to characterize, the above system has a very special structure which enables its long-run behavior to be unambiguously deduced. First, there is a continuum of equilibria corresponding to all points (0, 0, EC, MC). In other words, irrespective of the distribution of assets, a state in which prices equal fundamentals and chart&s are expecting no change is an equilibrium.’ It may be conjectured, on the basis of the analysis in the preceding sections, that if EC and MC are sufficiently small, then the corresponding equilibrium will be stable. In fact, a stronger result can be proved: regardless of initial conditions, either prices must converge to fundamentals or the chartist share of total wealth must converge to 1. Proposition 3. Along
any
trajectory:
of the system (8)-(1 l), either lim, _ ~ p = 0, or
lim z+ x AP, EC,MC) = 1. Proposition 3 states that only two outcomes are possible in the long run: either fundamentalist wealth vanishes or prices converge to fundamentals. In other words, as long as fundamentalists survive in the long run, instability cannot persist. This convergence of prices to fundamentals can occur even if the initial value of chartist wealth exceeds the bifurcation value identified in Section 2, so that prices diverge from the fundamentals to begin with. The rationale is the following. If chartist wealth is initially high enough to cause the subsystem (4)-(5) to be unstable, asset prices will fluctuate around fundamentals. These fluctuations themselves, however, will tend to diminish chartist wealth over time, as fundamentalists consistently ‘buy low and sell high’. Eventually chartist wealth declines sufficiently to move the market into a stable region. The chartist share of total wealth in the limiting state will be positive. but sufficiently small to ensure that the corresponding equilibrium is stable. A typical trajectory, starting with a high share of chartist wealth is depicted in Fig. 3. High chartist wealth leads to cyclic trajectories, which in turn erode chartist wealth gradually over time. Eventually, the bifurcation value ,uO is crossed and the price movements start to stabilize. The above result may be summarized as follows: if speculators whose behavior is stabilizing survive in the long run, then prices must converge to fundamentals. This is a weak version of Friedman’s (1953) influential argument: “People who argue that speculation is generally destabilizing seldom realize that this is largely equivalent to saying that speculators lose money, since speculation can be destabilizing in general ’ In addition, are in possession
there is a continuum of the entire stock
of equilibria of wealth.
of the type
(p. 0, E, M)
for arbitrary
p, where
chart&s
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ASSET PRICE (p)
I CHARTIST
SHARE OF WEALTH
(p)
Unstable Bifurcation
0.50
Regime
value
Stable Regime
Fig. 3. Elimination
of instability
through
the diminution
of chartist
wealth.
only if speculators on the average sell when the currency is low in price and buy when it is high” (Friedman, 1953, p. 175). Friedman’s view of a rational speculator is one who buys low and sells high, behavior which is described above as fundamentalism. That this behavior is unambiguously stabilizing, irrespective of the various adjustment speeds and elasticities in the model, has already been demonstrated in Proposition 1. However, it is optimal only under very restrictive conditions, and more generally it is entirely possible for optimal behavior to be destabilizing. In the present model, a speculator who buys on the way up, even when prices exceed fundamentals, but dumps her holdings just before the market turns down, would outperform the fundamentalists.* Although stabilizing speculation (fundamentalism) may not be individually rational in the sense that there are more profitable strategies available, there is a sense in which it is superior to chartism if investor horizons are sufficiently long. This need not be the case if it is costlier to obtain information about fundamentals than to form expectations based on extrapolating trends using historical data on asset prices.
5. Costly
fundamentalism
The long-run stability of the system need not obtain if the real resource costs of fundamentalism are greater than those of chartism. This is indeed likely to be the case, since knowledge of the fundamental value of an asset requires the acquisition of a considerable amount of information regarding future economic conditions,
a De Long et al. (1990b) describe individual speculators who have consciously such destabilizing strategies, and develop a model to explain their success.
and successfully
adopted
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competitor behavior, technological developments, and other uncertain events. In the model as developed so far, fundamentals have been assumed to be time-invariant and known to fundamentalists. In fact they will vary over time rather erratically and keeping abreast of changes in fundamentals is likely to be both time-consuming and expensive. The information costs incurred by chartists are negligible in comparison, since chartism requires the use of widely and freely available data on prices; it is the equipment and expertise required to manipulate and extract information from these data that represent the costs of chartist activity. Without denying that these costs may be significant, they are likely to be well short of the costs associated with gathering detailed asset-specific information. Assume that fundamentalist outlays on the acquisition of information are proportional to their share of total market wealth, with the factor of proportionality denoted by E. Hence, the flow of expenditures on information acquisition and processing at any moment is (1 - ~1)s. To maintain the assumption that the aggregate stock of money held collectively by speculators is constant, it is assumed that this outflow is matched by an inflow into the market of the same amount, (1 - ~)a. This inflow is distributed among fundamentalists and chartists in proportion to their current wealth shares. Hence, the net effect of money flows is that chartists gain ~(1 - ~)a, while fundamentalists lose (1 - F)E - (1 - I*)‘& = ~(1 - P)E. The net gain is zero, so the aggregate money stock in the market is constant, as in Section 4. The only modification to the dynamical system is that Eq. (11) has to be replaced by ltiC = --eqp,
71,EC, M’) + ,U(l - /J)&.
(12)
The revised system, (8))(10) and (12) has an equilibrium at (0, 0, 0, 0), with prices equal to fundamentals and with chartist wealth eliminated. Let us call this the F-equilibrium of the system. In addition, there is a continuum of equilibria of the form (p, 0, E, M), where chartist expectations are stationary at some arbitrary price while fundamentalist wealth is zero. Of the equilibria in this continuum, there is exactly one with prices equal to fundamentals: (0, 0, E, M). Call this the C-equilibrium of the system. We then have: Proposition 4. If‘ /lc > 1 + E/x(M + E), the C-equilibrium of the system (8)-( lo), (12) is locally unstable. Therefore, if price adjustment is sufficiently rapid or the expectations elasticity of chartist demand is sufficiently high, then prices will converge to fundamentals along a trajectory only if they converge to the F-equilibrium of the system where chartist wealth diminishes to zero. Is this a likely outcome? It is less so than in the case of costless fundamentalism, since the introduction of the cost E causes a net outflow of wealth from fundamentalists to chartists whenever both groups have positive wealth. If the initial value of chartist wealth is higher than the bifurcation value identified in Section 2, then the price dynamics will be unstable and oscillatory. Provided that E is sufficiently low, this will tend to raise the wealth held by fundamentalists since they achieve a positive net gain over a complete cycle. This in turn will act to reduce price volatility as in the case of costless speculation. However, as the movement of
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0.20
CHARTIST
n
Bifurcation
SHARE OF WEALTH
(p)
Unstable
Regime
n
value Stable Regime
Fig. 4. Alternating
stable
and unstable
regimes
with
costly
fundamentalism.
prices stabilizes, the relative advantage of fundamentalist strategies declines, and the costs of acquiring information start to dominate the dynamics of wealth shares. If price movements largely reflect changes in the underlying fundamentals, then it is more cost-effective to extract information about fundamentals from prices rather than from the more direct sources used by fundamentalists. Chartism therefore becomes a more effective strategy and the share of chartist wealth will start to climb. Once the bifurcation value, pO, is exceeded, the dynamics of the system again begin to display the excess volatility which favors fundamentalist strategies in the long run. This intuitive argument is illustrated in Fig. 4, which shows simulated time-series output for the asset price and the chartist share of wealth as the system @-(IO), (12) evolves over time. The chartist wealth share shows a decreasing trend during volatile periods, which causes prices to stabilize. However, the persistence of tranquility itself favors the strategy of chartists, and causes their share of wealth to rise. Only when the rise has caused the bifurcation value to be crossed, and the resulting volatility to become manifest, does the behavior of fundamentalists again becoming rewarding. Hence with costly fundamentalism one would expect periods of tranquility to be interrupted by periods of volatility in speculative markets, as the endogenous dynamics of wealth shares lead to the decline of whichever group currently dominates the price dynamics.
6. Conclusions
Whether or not changes in asset prices primarily reflect movements in the underlying fundamental values depends on a number of factors, both institutional and behavioral. In a market consisting of two distinct investor groups, fundamentalists and chartists, the share of wealth held by the latter is a bifurcation parameter which,
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if exceeded, transforms a convergent regime into a persistently cyclical one. In the cyclical regime, fundamentalism is a successful strategy against chartism in the long run, which causes the share of wealth in chartist hands to decline if it is initially high. However, if information about fundamentals is costly to acquire, then chartism is a successful strategy against fundamentalism in the stable regime. Hence behavior that is conducive to stability is profitable under conditions of instability and vice versa, giving rise to a phenomenon that has been described here as endogenous regime switching: the bifurcation parameter of a two-dimensional system itself evolves (as part of a larger dynamical system) in such a way as to cause the smaller system to alternate between stable and unstable regimes. Although the argument above has been presented in terms of a highly stylized model, the economic intuition underlying the results is straightforward. Behavior conducive to stability (fundamentalism) may be most profitable when the market is unstable, and behavior conducive to instability (chartism) may be rather lucrative in a stable market when it is costly to collect information about fundamentals. There is a sense in which the two groups of speculators enjoy a symbiotic relationship with each other: each group benefits from an increase in the numbers of the other. Fundamentalists prey on the adaptive behavior of the chartists when chartists dominate the market, and chartists extract information about fundamentals costlessly when fundamentalists dominate the market. As a result, one would expect heterogeneity of practices to persist in the long run, with the dominance of one set of practices giving way to that of the other. The implication for speculative markets is that periods of tranquility are likely to be punctuated from time to time by periods of excessive volatility.
Acknowledgements I thank Carl Chiarella, Englebert Dockner, Peter Flaschel, Reiner Franke, Peter Skott, seminar participants at the University of Bielefeld and the University of Aarhus, and two anonymous referees for helpful comments on an earlier draft.
Appendix
A.1. Proof of Proposition 1 Local stability may be deduced from the Jacobean matrix J which, evaluated at the equilibrium, is given as follows:
J=
1
-B(l -/de Bw 1 -cQ(l - /QO X(&C - 1)
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Writing A for the determinant
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and Economic
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of this matrix, we have the following:
A = @VI( 1 - /J) > 0. The determinant is strictly positive for all admissible parameter values, so that the equilibrium cannot be a saddle-point. The only generic possibilities are therefore an attractor or a repeller. Which of these obtains depends entirely on the sign of the trace t of J, given by T = r&K - pe(1 - /L) - a. Stability of equilibrium requires that the trace be negative. Since p E [0, I), if fit I 1, then the first term is smaller in magnitude than the third, so the equilibrium is stable for all parameter values, proving part (a). Now assume that PC > 1. Then T > 0 when p = 1 and r < 0 when p = 0. Treating T as a function of the various parameters of the model, we obtain the following:
By the continuity and monotonicity of r in the population composition I*, there will be a critical value plo E (0,l) such that the system is stable when p < p0 and unstable when ,u > pO. This bifurcation value is given by
Differentiating
(Al) with respect to x and simplifying, we have
which must be negative if PC > 1. Differentiating simplifying, we have
Differentiating
(Al)
with respect to /I and
(Al) with respect to 8 and simplifying, we have
which, under the assumption that PC > 1, is positive. Finally, differentiating respect to c and simplifying, we have
which completes the proof of part (c).
0
(Al) with
R. SethilStructural
A.2. Proqf
of Proposition
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and Economic
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115
2
Assume that p # 1. Define the sets S, and S, as follows:
and let S = S’ x S,. The unique equilibrium (0,O) of the system (4), (5) is contained in S. We first establish that S is an invariant set by showing that at all points on the boundary of S the direction of the vector field points toward the interior of S. If p I -pg’/Q( 1 - CL),then Df 2 ,ug’ from (2). Since DC > -pg’ from (l), this means that D(p, X) > 0, regardless of x. Hence, from (4), d < 0. An analogous argument establishes that if p 2 pgh/O(l - p), then fl < 0. Hence p must eventually enter the interior of S,, regardless of initial conditions. Now suppose that p E S,. Then Df E [ -pgh, pg’]. Since DC E (-pg’, pgh), it must be the case that D(P, 70 E (-Ad
+ s”>, Ag’ + sh>).
From (4), therefore P E (-/MY
+ gh)3 BAS + gh>).
If rt I -/Ip(g’ + gh), then n < & so from (5) we have ti > 0. Analogously, if 712 pp(g’ + gh), then ri > 0. Hence, 7c must eventually enter the interior of S,, regardless of initial conditions, and the limit sets of all trajectories must lie in S. Since we are dealing with a planar flow with a unique equilibrium, there are only three possible classes of limit sets: fixed points, closed orbits, and homoclinic orbits (Guckenheimer and Holmes, 1983, p. 45). Homoclinic orbits are trajectories that connect a fixed point to itself, and may be ruled out in the present model since they require that the equilibrium be a saddle-point, which is impossible since lJ\ > 0 for all parameter values. Hence all trajectories converge either to the unique equilibrium or to a closed orbit. If the unique equilibrium is locally unstable, then there exists and open neighborhood U of the equilibrium such that the set S n 0 is itself invariant, where 0 represents the complement of U in R2. The limit sets of all trajectories not originating at the equilibrium must lie in S n 0, which contains no equilibrium point. Hence the limit set of all trajectories not originating at the equilibrium must, by the PoincarC&Bendixson Theorem (Arrowsmith and Place, 1982, p. 110) be closed orbits. 0 A.3. Proqf
qf Proposition
3
Let us define the function I/ = MC + EC, which measures the total wealth of chartists, with equities valued at fundamentals rather than current prices. The trading rule defined in Section 4 ensures that V remains non-negative along any trajectory of the system (8))(11). Furthermore, p = &fc + EC = (1 - eP)gc.
(43
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If p < 0, then Df > 0 so that E I 0. Similarly, if p > 0, then D’ < 0 so that J?’ 2 0. Together with (A2), these conditions imply that 6’ I 0. Since C’ is nonincreasing and bounded from below, it must be the case that as t tends to infinity, 9 goes to zero. This in turn implies that, in the limit, either p = 0 or hi’ = .E? = 0 (or both). The first of these conditions implies the second, so that the latter condition always holds in the limit. Hence trades cease in the limit and MC and EC converge to some magnitudes MC and EC respectively. To prove the proposition we need to establish that unless lim, _ m p = 0, R” = M and Ec = E, respectively (fundamentalist wealth vanishes in the limit). Suppose prices do not converge to fundamentals. We consider two cases separately: Case 1. Prices converge, but not to fundamentals. Let the limiting price be p # 0. Since prices converge, Ij = 0, which in turn implies from (8) and (9) that n = 0 and then DC = 0. If fundamentalist wealth is nonzero, then Df > 0, which implies that D( p, 0) > 0 and hence, from (8), that @# 0. This contradicts the hypothesis that prices converge. Hence, fundamentalist wealth must be zero. Case 2. Prices do not converge. Suppose that fundamentalist wealth is nonzero. Then D’ > 0 whenever p < 0 and D’ < 0 whenever p > 0. In order that trades do not occur, either chartist wealth must be zero, or x < 0 whenever p > 0 and 7c> 0 whenever p < 0. If chartist wealth is zero, then prices converge to fundamentals from Proposition 1, contradicting the hypothesis that they do not converge. If n < 0 whenever p > 0 and 7c> 0 whenever p < 0, then rj < 0 whenever p > 0 and 6 > 0 whenever p < 0. This too implies convergence of prices to fundamentals, contradicting the hypothesis. Hence fundamentalist wealth must be zero. C A.4. Proof qf Proposition 4
The Jacobean of the system (8)-(lo),
(12) is given by the following matrix, where
p = ,a(~, EC, MC) as in (7), D = D(p, 71,EC, MC) as in (3), T = T(p, r, EC, MC) as defined in Section 4, and a subscript i denotes the partial derivative of the function
with respect to its ith argument:
The trace of the above matrix is t = /3Dl + x(/?Dz - 1) + T3 + ePT, + p&l
From (3) we have
- 21.1).
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At the C-equilibrium p = rr = 0 and p = 1, and since g(0) = 0 and g’(0) = c, we have D, = 0 and D, = c. The derivatives & are not defined at all points since T is not differentiable everywhere. However, the derivatives T3 and T, do exist whenever p = 71= 0, since, from (1) and (2): dD’jaE’
= p2g(z) = p2g(0) = 0,
ZD’/aE’ = &3p = 0, dD’/8M’
= p3g(n) = pJg(0) = 0,
dD’/aM’
= /Qp = 0
Hence, T, = T, = 0 at the C-equilibrium. From (7) it is readily verified that p3 = l/(M + E) at the C-equilibrium, since p = 0. A sufficient condition for the local instability of the C-equilibrium is r > 0 which, since D, = 0, D, = c, and ,u = I, occurs whenever CY(/!JC - 1) - E/(M + E) > 0 or /?c > I + E/x(M + E), as required.
0
AS. Simulation details The simulation details are g(z) = 5 tanh(n/5), t, = 1, and E = 0.005.
E = 100, M = 200, c( = 3, /I = 3,
References Arrowsmith. D.K. and C.M. Place, 1982, Ordinary differential equations (Chapman and Hall, London). Beja. A. and M.B. Goldman, 1980, On the dynamic behavior of prices in disequilibrium, Journal of Finance 35,235-247. Blanchard, O.J. and M.W. Watson, 1982, Bubbles. rational expectations, and financial markets, in: P. Wachtel, ed., Crises in the economic and financial structure (Lexington Books. Lexington, MA). Chiarella, C., 1992, The dynamics of speculative behavior, Annals of Operations Research 37, 101-123. Conlisk, J., 1980, Costly optimizers versus cheap imitators, Journal of Economic Behavior and Organization 1, 275-293. Conlisk, J., 1988, Optimization cost, Journal of Economic Behavior and Organization 9, 213.-228. Day, R. and W. Huang, 1990, Bulls, bears and market sheep, Journal of Economic Behavior and Organization 14, 299-329. De Long, J.B., A. Shleifer, L. Summers and R. Waldman. 1990a, Noise trader risk in financial markets, Journal of Political Economy 98, 703-738. De Long, J.B., A. Shleifer, L. Summers and R. Waldman, 1990b, Positive feedback investment strategies and destabilizing rational speculation, Journal of Finance 45, 379-395. Evans. G.W. and G. Ramey, 1992, Expectation calculation and macroeconomic dynamics, American Economic Review 82, 207-224. Fama, E.F., 1989, Perspectives on’october 1987, or, What did we learn from the crash?, in: R.W. Kamphuis, R.C. Kormendi, and J.W.H. Watson, eds., Black Monday and the future of financial markets (Dow Jones-Irwin, Homewood, IL).
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Flood, R.P. and P.M. Garber, 1982, Bubbles, runs, and gold monetization, in: P. Wachtel, ed., Crises the economic and financial structure (Lexington Books, Lexington, MA). Frankel, J.A. and Froot, K.A., 1986, Understanding the US dollar in the eighties: The expectations chartists and fundamentalists. Economic Record, Special Issue on Exchange Rates and the Economy, 24438. Friedman, M.. 1953, The case for flexible exchange rates, in: Essays in positive economics (University Chicago Press, Chicago, IL). Guckenheimer, J. and P. Holmes, 1983, Nonlinear oscillations, dynamical systems, and bifurcations vector fields (Springer, New York). Sethi, R. and R. Franke, 1995, Behavioral heterogeneity under evolutionary pressure: Macroeconomic implications of costly optimization, Economic Journal 105, 5833600. Shiller, R.J., 1990, Market volatility (MIT Press, Cambridge, MA). Zeeman, E.C., 1974, On the unstable behavior of stock exchanges, Journal of Mathematical Economics I. 39949.
in of
of of