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Let it be: chaotic price instability can be beneficial
Chaos, Solitons and Fractals 18 (2003) 745–758 www.elsevier.com/locate/chaos
Let it be: chaotic price instability can be beneficial Akio Matsumoto
*
Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan Accepted 21 January 2002 Communicated by Y. Aizawa
Abstract This study investigates an economic implication of chaotic fluctuations generated by a discrete price adjustment process. For this purpose, it uses a pure exchange model with two goods and two consumers in which chaotic price fluctuations can arise. In order to reveal some statistical properties of such price dynamics, this study constructs a density function of chaotic trajectory, calculates a long-run average utility, and then compares it with the utility corresponding to a stationary state. The following two results are analytically as well as numerically demonstrated: (1) chaotic price dynamics can be beneficial for one consumer and harmful to the other consumer and (2) the whole economy is possibly better off along chaotic fluctuations than at a stationary state in the long-run. Further, it is shown that the second result is sensitive to the social judgement on the ranking of consumersÕ utilities. These results imply the possibility that chaotic fluctuations can be preferable to a stationary state. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction In the recent literature, it has been demonstrated that a discrete t^ atonnement process may be chaotic; see Day and Pianigiani [3], Day [4], Weddepohl [13] and Mukherji [8] for a one-dimensional process, and Goeree et al. [5] and Tuinstra [11] for a multi-dimensional process. Yet, a salient feature of the t^ atonnement process is that the market price mechanism (in other words, an auctioneer) adjusts the price in response to an excess of demand over supply, and actual transactions take place only when an equilibrium price is attained. A chaotic t^ atonnement, in spite of the working of the price adjustment, does not converge to an equilibrium price but fluctuates around it while remaining bounded. The chaotic t^atonnement thus has a ‘‘crux’’ that no actual transactions are carried out because no convergence is brought about. Persistent continuation of market disequilibrium is apparently against the spirit of competitive market adjustment and thus thought as an unfavorable phenomenon in the traditional economics. There are at least two ways to deal with this crux: one is to stabilize or control chaos; the other is to admit intermediate trades in the middle of the price adjustment process. In fact, research in the former direction has already begun; Bala et al. [1] (BMM henceforth) construct an exchange economy, in which the logistic map adjusts the price, and demonstrate a possibility of controlling a chaotic t^atonnement in such a way that a chaotic price trajectory can be led to an arbitrarily small neighborhood of a competitive equilibrium price. In this study, we take the latter approach. In particular, we derive a basic model from BMM and consider chaos in the non-t^ atonnement process in their setting. Since trade occurs in a non-clearing market, a consumer may be better off or worse off than he could be at the equilibrium price, depending on the prevailing market price. So our primary concern is to investigate whether, from the long-run point of view, the gain from the chaotic price instability can dominate the loss. Alternatively put, we are interested in answering the following question: can chaos be preferable to a stationary state in a disequilibrium economy? *
Tel.: +81-426-74-3351; fax: +81-426-74-3425. E-mail address: [email protected] (A. Matsumoto).
0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00005-5
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This is essentially the same question as Waugh [12] asked more than half a century ago, the question of whether consumers can benefit from price instability. He provided a positive answer but only under the limited circumstances. Although it has been believed in the traditional economics that a stationary state is preferred to irregular motions, it has not yet been verified whether or not appropriate welfare measure can take on better value on chaotic trajectories than in the stationary state. It is thus imperative to investigate whether or not chaos is beneficial for consumers in a disequilibrium economy in which demand is unequal to supply. 1 The main purpose of this study is to demonstrate a possibility that a long-run average utility taken on chaotic paths can be preferable to a utility taken in the stationary state. It is for this purpose that the present study analyzes an exchange economy with two goods and two consumers in which price trajectories exhibit chaotic fluctuations. In doing so, this study will shed light on the nature of disequilibrium dynamics that has been neglected in the traditional economics. This study is organized as follows. Section 2 outlines a pure exchange model constructed in BMM and derives a dynamic system of the model. Section 3 describes our non-t^ atonnement process. Section 4 analytically and numerically shows the main result of this study that chaotic fluctuations can be preferable to a stationary state in the long-run. Section 5 provides an intuitive explanation for the source of the main result. Section 6 provides conclusions and a pointer to future research. 2. Pure exchange economy In this section, we present a discrete-time price adjustment process. Since our study follows BMM, we recapitulate the basic elements of their analysis and then proceed to our own investigation. 2.1. Piecewise-continuous price adjustment process Consider a pure exchange economy in which there are two consumers, A and B, and two goods, x and y. p and q are prices of goods x and y, respectively. We adopt two simplifying assumptions. First, q is chosen as a numeraire price so that q ¼ 1. Second, the fixed endowments of goods in the economy are initially allocated in such a way that consumer A starts off with an endowment xA ¼ ð1; 0Þ and consumer B starts off with an endowment xB ¼ ð0; 1Þ. Inspecting price p available at a market of good x, each consumer determines his optimal choice so as to maximize his utility subject to the budget constraint. Entering the markets, consumers exchange some of these goods with each other. Preference of consumer A is represented by a utility function of the Leontief type uA ðx; yÞ ¼ minff ðxÞ; ayg;
ð1Þ
where f ðxÞ ¼
8 > < x x2 > :1 4
1 if 0 6 x 6 ; 2 1 if x > ; 2
and a > 0 is a preference parameter. AÕs optimal (gross) demands are given by 8 1 > < xA ðpÞ ¼ ap and yA ðpÞ ¼ pð1 apÞ if 0 6 p 6 ; 2a 1 1 1 > : xA ðpÞ ¼ and yA ðpÞ ¼ if p P : 2 4a 2a
ð2Þ
It is known that price changes for the Leontief preference have no substitution effect. Consequently a change in demand for good x caused by a change in price is due entirely to the income effect. Consumer B has a utility function of the quasi-linear type uB ðx; yÞ ¼ gðxÞ þ y;
ð3Þ
1 Boldrin and Montrucchio [2] show that a chaotic trajectory can be an admissible solution to discounted dynamic optimization problems. Their study implies that equilibrium trajectories can be chaotic, and chaos can be consistent with optimal behavior. Our approach differs from theirs in one significant point. We are concerned with chaos in a disequilibrium economy whereas they are concerned with it in an equilibrium economy in which demand equals supply.
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where ( gðxÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffi b ð1 xÞ3 if 0 6 x 6 1; 3 0 if x > 1;
and b > 0 is a preference parameter. BÕs optimal (gross) demands are given by 8 pffiffiffi p2 p3 < and yB ðpÞ ¼ 1 p þ if 0 6 p < b; xB ðpÞ ¼ 1 b b pffiffiffi : xB ðpÞ ¼ 0 and yB ðpÞ ¼ 1 if p P b:
ð4Þ
It is known that a shift in income causes no change in demand for good x in the case of quasi-linear preference. Thus the entire change in demand x caused by a change in price is due to the substitution effect. The extreme allocation of initial endowment (i.e., xA ¼ ð1; 0Þ and xB ¼ ð0; 1Þ) ensures that consumer A is a supplier of good x and consumer B is a demander. As shown in (2) and (4), supply for good x is kinked at p ¼ 2a1 while demand is pffiffiffi kinked at p ¼ b. Thus the supply function and the demand function are piecewise-continuous and spelled out as
1 1 2 SðpÞ ¼ max 1 ap; and DðpÞ ¼ max 1 p ; 0 : ð5Þ 2 b Let ZðpÞ be the excess demand, i.e., ZðpÞ ¼ DðpÞ SðpÞ. The standard price adjustment process in discrete time is then given by
1 1 ;0 ; ð6Þ /ðpt Þ ¼ max½pt þ hZðpt Þ; 0 ¼ max pt þ h max 1 pt2 ; 0 max 1 apt ; b 2 where h is an adjustment speed and a positive constant and pt is the price at period t. The process takes explicit account of the non-negative constraint on price and is also piecewise-continuous. The stationary price is defined by p ¼ /ðpÞ. Solving the equation for p generates two solutions, a trivial solution, p ¼ 0, and a non-trivial solution, p ¼ pE 6¼ 0 which is rffiffiffi 8 b 1 > > < pE ¼ ab 6 ; if 2 r 2affiffiffi rffiffiffi ð7Þ > > : pE ¼ b if 1 < b: 2 2a 2 For the local stability of pE , the eigenvalue of the dynamic system (6) evaluated at the stationary point must be less than unity in absolute value; the eigenvalue, in the case of this one-dimensional system, is equal to the slope of /ðpt Þ. 2 Even if the stationary state loses its local stability, it is shown to be globally stable in the sense that all price trajectories are bounded; roughly speaking, the adjustment process (6) generates excess demand for lower prices and excess supply for higher prices and thus induces price trajectories to bounce back to a neighborhood of the stationary state whenever trajectories move far away from the stationary point. Fig. 1 presents a two-parameters bifurcation diagram that is a picture of bifurcation response of the adjustment process to changes in parameters a and b where a fixed adjustment coefficient h is assumed. The diagram has been produced using the pattern recognition algorithm developed by Onozaki et al. [9]. 3 The diagram is produced using a 600 600 grid of a and b where h ¼ 6, 4 a 2 ½1h ; 4h and 2 b 2 ½0:01; h2 . Different colors in the diagram indicate different regions of stable, periodic or aperiodic behavior. The dark region labelled S1 shows the attractor of non-trivial fixed point (of order 1). The dark region labelled S2 represents parameter combinations for which price is mapped to zero (or becomes negative without the non-negative constraint). The white region indicates aperiodic behavior involving chaos. The other colors indicate the periodic attractors of up to 16 different periods. The diagram shows that /ðpt Þ can generate various dynamics ranging from stable convergence to chaotic fluctuations, depending on parameter values specified.
2
/0 ðpt Þ ¼ 1 ah for pE ¼ ab, and /0 ðpt Þ ¼ 1 p2hffiffiffiffi for pE ¼ 2b
qffiffi b 2.
3 After a transient time of 5000 periods, the adjustment process is further iterated 200 times for every parametric combination. The difference between a generated trajectory and a periodic trajectory with period-n is calculated. If the maximum difference in absolute value becomes less than 0.00001, it is determined that the trajectory possesses a period n. 4 Although this is the value which BMM use in their numerical simulations and will be used in our own simulations later, the qualitative nature of the bifurcation diagram does not depend on the particular value of h.
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Fig. 1. Two-parameter bifurcation diagram where h ¼ 6.
A simultaneous variation of two different parameters is considered in Fig. 1. If we fix one parameter and choose the other as a bifurcation parameter, we can have a familiar bifurcation diagram with respect to one parameter value. For h2 constant values of parameters, b ¼ 36 and h ¼ 6, increasing a from 2h to 3h along the lower dotted line in Fig. 1 implies the emergence of the period-doubling bifurcation cascade to chaos as illustrated in Fig. 2a. Further, for another values of 2 bð¼ 3h8 Þ, increasing a from 1h to 4h along the upper dotted line exhibits the period-doubling and period-halving bifurcation as illustrated in Fig. 2b. 2.2. Logistic price adjustment process Due to the piecewise-continuity, /ðpt Þ takes on various profiles according to various combinations of parameters. In the following, to investigate statistical properties of chaotic dynamics, we confine our analysis to a case in which the excess demand function is smooth and the resultant adjustment function has a single-peaked profile. pffiffiffi If the price is restricted to be less than or equal to the minimum of those kinked prices (i.e., p 6 min½2a1 ; b ), then the total excess demand for p is smooth and given by ZðpÞ ¼ ap
p2 : b
ð8Þ
It has an inverted U-shaped profile since the income effect dominates the substitution effect for small p, and the substitution effect dominates the income effect for large p. The non-trivial solution of ZðpÞ ¼ 0 is pE ¼ ab. Substituting (8) into (6) yields a single-peaked price adjustment function h ptþ1 ¼ ð1 þ ahÞpt pt2 : b
ð9Þ
We make the following assumption in order to apply this dynamics to a class of economies parameterized by a single parameter l.
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Fig. 2. Two bifurcation diagrams with respect to a for different values of b; b ¼ 19 in (a) and b ¼ 32 in (b).
Assumption 1. ð1 þ ahÞ ¼ bh ¼ l Under this assumption, the dynamic system (9) is represented by the logistic map with the parameter l ptþ1 ¼ Hðpt Þ lpt ð1 pt Þ:
ð10Þ
l can be a proxy of the strength for the nonlinearity included in the adjustment process. It is well known that the logistic map can generate periodic as well as aperiodic dynamics according to values of the parameter l, when the stationary state is locally unstable. Now we are in a position to clarify the parameter configuration under which the logistic map can be well defined as an appropriate price adjustment process. First of all, Assumption 1 implies that one of three parameters, a, b and h, are not independent. In the following, we let b be a dependent parameter b¼
h : 1 þ ah
ð11Þ
Second, it is required that l is not greater than four in order to prevent Hðpt Þ from being negative. This requirement is, under Assumption 1, spelled out as ð12Þ p ffiffiffi Lastly it is also required that the parameter values are so defined that p 6 min½2a1 ; b pfor ffiffiffi all p. Since the maximum value of Hðpt Þ is unity provided l 6 4, the requirement is ensured if values of both 2a1 and b are not less than unity, that is, if ah 6 3:
a6
1 2
and
hP
1 : 1a
ð13Þ
All price trajectories generated by Hðpt Þ stay within the unit interval for any combination of parameters that satisfy the inequality conditions (12) and (13). Since the slope of Hðpt Þ evaluated at the stationary point pE ¼ ab is H0 ðpE Þ ¼ 1 ah; the stationary state is locally stable if the slope is less than unity in absolute value (i.e., 0 < ah < 2). Price converges monotonically to pE if 0 < ah < 1 holds, and cyclically if 1 < ah < 2. It is locally unstable if the slope is equal to or greater than unity i.e., 2 6 ah. To sum up, the price adjustment process makes the stationary state locally unstable but every trajectory remains within a finite distance from the stationary state for a and h in the following set, U ,
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9 pt +1 <0
U:Unstable Region
αθ =3
6
αθ =2
Stable Region
(1−α)θ =1
β <1
0 α
1/ 3
1/ 2
Fig. 3. Stable and unstable regions.
U¼
ða; hÞj2 6 ah 6 3;
0 < a6
1 ; 2
where (11) holds. A dark gray region in Fig. 3 is a part of the unstable set U . Suppose h ¼ 6. Then increasing a from 13 to 1 along the dotted line in Fig. 3 is identical with increasing l from 3 to 4 under Assumption 1 so that the period2 doubling bifurcation to chaos occurs along this line as illustrated in Fig. 2a. It is further verified that b > 1 for any ða; hÞ 2 U .
3. Transaction–consumption process Our model indicates that the economy can generate complex dynamics if the stationary state is unstable. In this section, we describe what consumers can afford via intermediate transactions in the disequilibrium market. The dynamic process involving the intermediate transactions is known as the non-t^ atonnement process. Here we call our process a transaction–consumption process to emphasize its distinct feature that, as described shortly, not only intermediate transactions but also actual consumption of goods are allowed in the middle of adjustment. In the so-called non-t^atonnement process, it is assumed that exchange transactions can take place during the process of price adjustment. As a result, the ex-ante stock of goods held at the beginning of the period differs from the ex-post stock of goods held at the end of the period. Having the resultant stock of the two goods, each consumer inspects the price available at the beginning of the next period, decides how much he wants to buy and sell, and then enters the markets again. Since the total stock of goods in the economy is re-distributed in the non-t^ atonnement process from time to time and, accordingly, consumerÕs purchasing power varies, the excess demand function changes from time to time. Thus the main concern in the non-t^atonnement process is related to the question of whether a dynamic system that allows for the price adjustment and the quantity adjustment associated with allocations of goods can reach a stationary state. 5 We also consider the price dynamics in the non-t^ atonnement process, but our main concern is different; we will see what welfare or benefit consumers can get in the non-t^ atonnement process when the price fluctuates chaotically. To this end, we start with to make two assumptions: the first assumption is that the transacted amounts of goods are actually consumed by the end of a period, and, the second is that exactly the same stock of endowment is allocated to each consumer at the beginning of the next period. These two assumptions assume away the redistribution effect of the
5
See Takayama [10] for a brief survey of the stability of the non-t^atonnement process.
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excess demand function and make us confine our analysis to a case in which there is a continuous flow of goods going to each consumer. 6 Therefore the adjustment function (10) that does not involve the redistribution effect can be relevant for the dynamic analysis of the transaction–consumption process. The sequence of time periods in our process is the following. At the beginning of a period, each consumer receives his endowment and learns the price of good x. Seeing how much his endowment is worth, he determines demand and supply so as to maximize his utility. In the middle of the period, consumers make their purchases and sales, although, for an arbitrary price of p, there is no guarantee that supply equals demand. To make intermediate transactions at such a disequilibrium price, we will introduce a transaction rule that takes disequilibrium phenomenon into account. After transactions are completed in markets, each consumer takes the amount of traded goods home from the markets and actually ends up consuming it. At the end of the period, the consumer evaluates what level of satisfaction he enjoys through consumption, and the price is adjusted according to Hðpt Þ if supply is not equal to demand. At the beginning of the next period, the new price is announced and the same endowment is given to each consumer. The process is repeated in the same way in subsequent periods. In the non-t^atonnement process, a transaction rule is necessary to make actual transactions in a disequilibrium market. Following the received wisdom in disequilibrium theory, we assume the min-rule or the short-side rule, according to which the actual transaction is the minimum of supply and demand in the market. x ¼ min½D; S ;
ð14Þ
where the bar, i.e. ‘‘-’’ over the variable means actual transaction, and S ¼ 1 ap is supply of good x and D ¼ 1 b1 p2 is demand for good x. The equilibrium price, pE , is the one where the demand curve crosses the supply curve. For 0 < p < pE , demand for good x is greater than supply so that consumer A is on the short-side of the market. In consequence, consumer A has more demand than he can handle. Consumer A, therefore, can sell his desired amount (i.e., x ¼ S) and can consume 1 S of good x. Since net transactions must be identically balanced in the market, consumer B, who is on the long-side of the market, is rationed and able to afford to buy x. Thereby, gross demands of good x, which are realized at the going price, are xA ¼ ap
and xB ¼ 1 ap:
ð15Þ
Actual transactions for gross demands of good y are determined through the budget constraints, yA ¼ pð1 xA Þ for consumer A and yB ¼ 1 pxB for consumer B, yA ¼ pð1 apÞ and yB ¼ 1 pð1 apÞ:
ð16Þ
On the other hand, for pE < p < 1, the situation is reversed: demand is greater than supply so that consumer B is on the short-side and able to realize his desired transaction (i.e., x ¼ D) while consumer A is on the long-side and rationed. As h b ¼ 1þah under Assumption 1, gross transactions are given by 1 þ ah 2 1 þ ah 2 xA ¼ p and yA ¼ p 1 p for consumer A ð17Þ h h and xB ¼ 1
1 þ ah 2 p h
1 þ ah 2 and yB ¼ 1 p 1 p for consumer B: h
ð18Þ
These transacted amounts of goods are consumed and then determine the actual level of each consumerÕs utility at the end of the period. Inserting (15)–(18) into utility functions in (1) and (3) yields indirect utility functions, UA ðpÞ uA ðxA ; yA Þ and UB ðpÞ uB ðxB ; yB Þ. Since one of the consumers is rationed according to whether the prevailing price is greater or less than the equilibrium price, his indirect utility function is piecewise-continuous in p. For consumer A, it is 8 L < UA ðpÞ apð1 apÞ if 0 < p 6 pE ; 1 þ ah 2 UA ðP Þ ¼ ð19Þ H if pE < p 6 1; p : UA ðpÞ ap 1 h
6 In this study which is a starting point for a more complete study, we adopt these convenient assumptions for two reasons. First, it makes the formidable mathematical problem simpler and manageable. Second, such a simplification enables us to derive rigorous results and economically interesting insights. Needless to say, since the redistribution effect is the main feature of the non-t^atonnement process, a further study taking account of the effect is necessary.
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and for consumer B, it is 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 hðapÞ3 > < U L ðpÞ þ 1 pð1 apÞ if 0 < p 6 pE ; B 3 1 þ ah UB ðpÞ ¼ > > 1 þ ah 3 > : UBH ðpÞ 1 p þ p if pE < p 6 1; 3h
ð20Þ
where L (respectively H) means that the prevailing price is lower (respectively, higher) than equilibrium price pE . For p ¼ pE , the market reaches equilibrium where both consumers can trade desired amounts. Since consumer A can realize his desired amount if p < pE , the first equation of (19) evaluated at pE determines the desired level of consumer AÕs utility. Similarly, since consumer B can realize his desired trade if p > pE , the second equation of (20) evaluated at pE determines the desired level of consumer BÕs utility. 1 þ ah 2 pE UAE apE ð1 apE Þ and UBE 1 pE 1 ð21Þ 3h 4. Long-run average behavior In this section, we present two numerical examples concerning the long-run average behavior of consumers. Setting l ¼ 4 in the first example and l 6¼ 4 in the second example, we investigate the first example analytically and the second numerically in the following three ways. First, we calculate each consumerÕs utility taken at the stationary state which we call the stationary utility. Second, we calculate the long-run average utilities taken along a chaotic trajectory. Third, we compare those long-run average utilities with the stationary utilities. In doing so, we show a possibility that a longrun average utility can be larger than the stationary utility. 4.1. Analytical result for l ¼ 4 The stationary utility is derived as follows. Under Assumption 1, l ¼ 4 implies 1 þ ah ¼ 4 and bh ¼ 4 so that h and b can be written in terms of a, h¼
3 a
and b ¼
3 : 4a
ð22Þ
Inserting the stationary price, pE ¼ ab, which is equal to 34, into indirect utility functions defined in (21) yield the stationary utilities, UAE ðaÞ ¼
3 að4 3aÞ and 16
UBE ðaÞ ¼
1 ð4 þ 3aÞ: 16
ð23Þ
It is observed that UAE ðaÞ and UBE ðaÞ are increasing, and UBE ðaÞ > UAE ðaÞ for a 6 12. It has been demonstrated that the logistic map Hðpt Þ exhibits ergodic chaos if l ¼ 4 (i.e, ah ¼ 3). 7 Let fHt ðp0 Þg1 t¼0 be a chaotic trajectory of price from an initial condition p0 , and fUj ðHt ðp0 ÞÞg1 t¼0 be the corresponding trajectory of consumer j0 s utility associated with the price trajectory. The mean value of the utility can be a basic statistical property of the chaotic price dynamics. Although it is impossible to calculate an average of infinite elements, the Mean Ergodic Theorem of Birkhoff–von Neuman 8 indicates that, if Hðpt Þ is ergodic, the time average of utility evaluated along a chaotic trajectory converges to its space average, namely, the mean value of utility evaluated on the domain of the price Z T 1 1 X Uj ðHt ðp0 ÞÞ ¼ Uj ðpÞUðpÞ dp for almost all p 2 I; ð24Þ lim T !1 T I t¼0 where UðpÞ is an invariant density function of chaotic trajectory. While it is, in general, difficult to construct the densities for smooth maps, it has been known that the explicit form of the density function for the logistic map with l ¼ 4, as shown in Day [4], is given by 1 UðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : p pð1 pÞ 7 8
ð25Þ
HðpÞ : I ! I where I is the unit interval is ergodic chaos if there is only one absolutely continuous invariant ergodic measure on I. See Day [4] and references there (p. 142) for this theorem.
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We can make a statistical characterization of the long-run average behavior of consumers by calculating the right-hand side of (24) instead of the left-hand side. Indeed, taking account of the piecewise-continuity of the indirect utility function, we observe that the long-run average utility, denoted as U j for j ¼ A; B, is defined by Z pE Z 1 Uj ¼ UjL ðpÞUðpÞ dp þ UjH ðpÞUðpÞ dp: ð26Þ 0
pE
Choosing a as an independent variable and other parameters, b and h, dependent variables, which satisfy (22), we investigate the long-run average behavior for different values of a. The long-run average utility of consumer A is, after inserting (19) into (26), calculated as pffiffiffi að16pð9 7aÞ 27 3aÞ : ð27Þ U A ðaÞ ¼ 288p The utility level of consumer A summarizes his value judgement on the allocation of goods. The stationary utility, UAE ðaÞ of (23), represents consumer AÕs preference in the stationary state, and the long-run average utility, U A ðaÞ of (27), represents his preference on the chaotic trajectory from the long-run point of view. The difference between these utilities is pffiffiffi að27 3a þ 2pð36 25aÞÞ 1 > 0 for 0 < a 6 : ð28Þ UAE ðaÞ U A ðaÞ ¼ 288p 2 The inequality of (28) indicates that consumer A prefers the stationary state to the non-stationary state in the sense that he can get a higher utility at the stationary state. By the same token, substituting (20) into (26) and integrating the resultant equation give the long-run average utility of consumer B, pffiffiffi 432p þ ð256p 255 3Þa : ð29Þ U B ðaÞ ¼ 864p Then the difference between the stationary utility and the long-run average utility is pffiffiffi 216p þ ð94p 255 3Þa 1 UBE ðaÞ U B ðaÞ ¼ < 0 for 0 < a 6 : 864p 2
ð30Þ
The direction of the inequality in (30), which runs in opposite to that of the inequality in (28), implies that consumer B can get a higher utility on average on the chaotic trajectory than at the stationary state. Thus, it can be said that, contrary to consumer A, consumer B prefers the non-stationary state to the stationary state. These results concerning the individual characteristics of chaotic dynamics are summarized as the First property of chaotic non-t^ atonnement price adjustment process. The First property. In the transaction–consumption process in which h ¼ 3a and b ¼ 4a3 hold, the long-run average utility of consumer A is less than the stationary utility, U A ðaÞ < UAE ðaÞ while the long-run average utility of consumer B is greater, U B ðaÞ > UBE ðaÞ for 0 < a 6 12. Admitting the possibility that the market disequilibrium is beneficial for consumer B and harmful for consumer A, we may ask the obvious question: Is a disequilibrium allocation of goods preferable from the perspective of the whole economy, namely, the distribution of welfare among all people in the economy? To answer this question, we need a social preference that provides a way to rank different distributions of utility among consumers. For this purpose, we assume the following; (1) social preference can be constructed by adding up the individual utilities and (2) the resulting sum can be used as an indicator of social utility. Our focus will be on a special example of a social welfare function that is the sum of the individual utility functions i.e., a classical utilitarian or Benthamite welfare function. The social welfare over the stationary allocation, WE and the one over the non-stationary allocation, W are given by X X UiE ðaÞ and W ¼ U i ðaÞ: ð31Þ WE ¼ i
i
Each can be thought to represent social judgement on each allocation. Adding (28) and (30) yields the welfare difference pffiffiffi 3 3að85 þ 27aÞ 2pð75a2 61a þ 108Þ 1 WE W ¼ < 0 for 0 < a 6 : ð32Þ 864p 2
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Fig. 4. Convergence of numerically calculated average utility to the theoretical value.
This result is summarized as the Second (statistical) property of chaotic non-t^ atonnement price adjustment process. The Second property. A whole economy is possibly better off along chaotic trajectories than at a stationary state in the long-run if the welfare function is classical utilitarian, U A ðaÞ þ U B ðaÞ > UAE ðaÞ þ UBE ðaÞ for 0 < a 6 12. 4.2. Simulation results for l 6¼ 4 We have shown that chaotic trajectories can be preferable to the stationary state when l ¼ 4. However, when l 6¼ 4, explicit forms of densities for chaotic trajectories are unable to be constructed, and thus the long-run properties are not able to be derived with the help of the Mean Ergodic Theorem. In this section, taking a numerical approach, we perform simulations in order to confirm that our theoretical results still hold even when l 6¼ 4. In particular, we directly calculate the time average of consumersÕ utilities over the first T periods of chaotic trajectory, uA ¼
t¼T 1 X UA ðHt ðp0 ÞÞ T t¼1
and
uB ¼
t¼T 1 X UB ðHt ðp0 ÞÞ T t¼1
and then compare these with the corresponding stationary utilities. Before proceeding, we need to check whether a finite average can approximate the long-run average. Fig. 4 illustrates how finite averages ‘‘converge’’ to the long-run averages. In Fig. 4, we set a ¼ 12 which, under (22), leads to b ¼ 32 and h ¼ 6 and, therefore, imply l ¼ 4. In order to better indicate the quality of the convergent process, we have drawn a dotted horizontal line at U A in Fig. 4a and at U B in Fig. 4b where these stationary utilities are calculated as pffiffiffi pffiffiffi 176p 27 3 5ð224p 51 3Þ ’ 0:139856 and U B ¼ ’ 0:566789: UA ¼ 1152p 1728p Further, we have drawn two solid lines, one of which is separated upward from the dotted line by 0.006, and the other separated downward by 0.006. 9 Fig. 4a illustrates three different trajectories which start with different initial prices, p0 ¼ 0:1 (red trajectory), p0 ¼ 0:6 (green trajectory) and p0 ¼ 0:8 (blue trajectory), respectively. It shows that the finite average utilities of consumer A approach the long-run average as number of periods used in the simulation increases. Fig. 4b shows the simulation results for consumer B in which three different trajectories are seen to approach to U B . Although irregularity of consumer BÕs trajectories seems stronger, it may be safe to assume that the finite average ‘‘converges’’ to the theoretical value of the long-run average utility if the number of iterations is over 1000 i.e., uA ’ U A and uB ’ U B if T > 1000. Now we are ready to conduct numerical experiments to check whether the theoretical results summarized as the First and Second properties hold when l 6¼ 4. Setting h ¼ 6 and b ¼ 4a3 , the finite averages of utility are computed for various values of a; these are obtained from the average of T ¼ 1000 for 200 points on interval ½13 ; 12 . The corresponding stationary utilities are also obtained from (23). Calculating the difference between the stationary utility and the finite
9 0.006 is the value which BMM used to check the convergence of their controlled trajectory. When a deviation of a trajectory from a stationary price (in absolute value) is less than this number, they conclude that the trajectory is successfully controlled and has reached the stationary price.
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Fig. 5. Long-run average utilities with respect to a: numerical simulations.
average utility, we plot 200 points of differences versus a and connect these points together to construct welfare beneficial curves as illustrated in Fig. 5. There, the red curve is the beneficial curve for consumer A while the blue curve for consumer B. Further, the difference between the sum of the stationary utility and the sum of the average utility (green ) is also depicted. Since the vertical distance indicates the difference between the stationary utility and curve marked w the finite-average utility, a positive value means that stationary utility is higher than the finite average utility, and a negative value means that the average utility is higher. It is fairly clear by looking at the these welfare curves that these simulations yield qualitatively the same result as the theoretical analysis; at least one of consumers can benefit from the price instability and the other harmed (i.e., First property) but the average social welfare is higher on the chaotic trajectories (i.e., Second property). In the middle of Fig. 5, we re-produce the period-doubling bifurcation diagram depicted in Fig. 2a, in which the height is appropriately adjusted.
5. Statistical properties reconsidered In the first half of this section, we give intuitive explanations for two characteristics of chaotic price dynamics, the First and Second properties, which the analysis so far has revealed. In the latter half, we provide the Third property of chaotic price dynamics, the property which clarifies a dependency of the Second property on a construction of social preference. According to the First property, consumer A is worse off and consumer B is better off when an intermediate transaction is allowed in the middle of the price adjustment. This property is graphically confirmed in Fig. 6 where the indirect utility functions (blue curves), the stationary level of utility (dotted lines) and the density functions are illustrated for a ¼ 12. 10 As can be seen in Fig. 6a, the indirect utility function of consumer A is increasing for p < pE , decreasing for p > pE and reaches its maximum for p ¼ pE . That is, UAE P UA ðpÞ for all p, which leads to the first result (28). To explore the dominance of the long-run average utility of consumer B over the stationary utility, we discuss the specific situation, only for the sake of simplification, in which price dynamics is characterized by regular alternating behavior with respect to the stationary price, that is, the situation where if a price in period t is less (respectively, greater) than pE , then it is mapped to a point greater (respectively, less) than pE in period t þ 1. Since the indirect utility defined in (20) is a decreasing function of p and kinked at p ¼ pE as depicted in Fig. 6b, the utility level also alternates with respect to the stationary utility level, that is, if a utility level in one period is greater (respectively, less) than the 10
The height of the density function is appropriately adjusted.
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A. Matsumoto / Chaos, Solitons and Fractals 18 (2003) 745–758 UA
(a)
utility
(b)
UAE
UBE
0
pE
1
p
0
pE
1
p
Fig. 6. Indirect utility, stationary utility and density function.
stationary level, then the utility level in the next period is less (respectively, greater) than the stationary level. Let U2 be the average utility of consumer B for a two-period iteration fpt ; Hðpt Þg i.e., U2 ¼ 12 ðUB ðpt Þ þ UB ðHðpt ÞÞÞ. Then the difference between U2 and UBE is given by U2 UBE ¼
1 UB ðpt Þ UBE þ UB ðHðpt ÞÞ UBE : 2
Suppose pt < pE in period t. 11 Fig. 6b indicates UB ðpt Þ UBE > 0 and UB ðHðpt ÞÞ UBE < 0 so that the sign of the sum is ambiguous. It will be positive if pt is closer to zero and Hðpt Þ is closer to pE , i.e., the absolute value of the first term is greater than that of the second term, and negative if pt is closer to pE and Hðpt Þ is closer to unity. Roughly speaking, the long-run average utility is the average of such a U2 over the entire interval of p. Since a density function of a price trajectory describes how often it visits every measurable set in the trapping interval, it depends on the shape of the density function whether the long-run average utility is larger or smaller than the stationary utility. The dominance of the long-run average utility over the stationary utility is quite plausible as seen from the shape of the density of the logistic map illustrated in Fig. 6b. The length of a subinterval where UB ðpt Þ UBE > 0 holds is much longer than the length of a subinterval where UB ðHðpt ÞÞ UBE < 0 so that U B > UBE in the long-run as (32) indicates. According to the second property, the intermediate trade raises the economic well-being as a whole because the gain of consumer B exceeds the loss of consumer A. We illustrate the utility possibilities frontier of two consumers in Fig. 7. There, WE is the set of utilities associated with the stationary allocation of goods while W is the set of utilities associated with the long-run average allocation. Sets of finite averages of consumersÕ utilities, which are calculated for T ¼ 1; 2; . . . ; 100, are plotted as blue dots and seen to be scattered around the long-run average level after finite iterations. The downward sloping solid line passing through the point WE is an iso-welfare curve of the Benthamite welfare function, the curve which depicts those distributions of utility that have the same welfare as the stationary welfare. We can observe the Second property from Fig. 7 i.e., W has a higher social welfare than WE as it is located above the iso-welfare curve. This observation implies that there are many ways to construct a welfare function which can generate the Second property. At the same time, it also implies that the Second property is sensitive to a choice of a social welfare function. In fact, while using the product of utilities as a welfare function generates qualitatively the same result as the Benthamite welfare function, the minimax or Rawlsian social welfare function, W R ¼ minfUA ; UB g, apparently does not have the Second property. 12 Further, it can be seen in Fig. 7 that WE is a Pareto efficient allocation and W is a Pareto improving allocation as the former is on the frontier and the latter is in the utility possibility set. Consequently, we can find a welfare function for which WE has a maximum welfare, since the utility possibility set is, as illustrated, convex in our pure exchange economy. 13 Thus the next question that naturally raises is: To what extent can the Second property hold? To answer this question, we make a slight generalization of the Benthamite welfare function and take the weighted sum of consumersÕ utilities as a social welfare,
11
The same results will be obtained if we suppose pt > pE . R The Rawlsian social welfare at the stationary state is WER ¼ UAE and the long-run average welfare is W ¼ U A . The inequality of R (28) leads to WER > W . 13 There is no way that both consumers can be benefit in our economy with convex structure. However, as pointed out by Kirman and Guesnerie, there might be a possibility that both consumers are happy in an economy with a non-convex utilities possibility set. 12
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Fig. 7. Utility frontier curve.
e ¼ aA UA þ aB UB ; W
ðaA ; aB Þ > 0:
e passes through two points, W and We can find a set of weight coefficients, aA and aB , for which the iso-welfare line of W WE , as depicted as a dotted line in Fig. 7. Using this dotted line for reference, it can be said that W is more (respectively, less) preferable to WE if a iso-welfare line that passes through point WE is flatter (respectively, steeper). The slope of a iso-welfare line is determined by the ratio of a aA =aB . The weight coefficient indicates the importance of consumerÕs utility to the social welfare and represents the social judgement between consumersÕ utilities. Taking account of these characteristics of coefficients, we summarize this result as the Third property of chaotic dynamics. The Third property. If the social judgement attaches more importance to welfare of consumer A (i.e., aA > aB ), then the stationary social welfare is preferable to the long-run average welfare so that the Second property does not hold. If the social judgement attaches more importance to welfare of consumer B (i.e., aA < aB ), then the longrun average welfare is preferable to the stationary social welfare so that the Second property holds. As stated in Section 1, we have two affordable choices to deal with chaotic fluctuations; the first choice is controlling chaos and the second is admitting the intermediate transactions. The Third property just stated is useful to detect which choice is effective in attaining a higher social welfare. BMM have already demonstrated that the stationary state is attainable with the help of control method. Therefore, if the social judgement over-weights the utility of the worse-off consumer along the chaotic trajectory in the overall social welfare, then some form of market intervention aiming to stabilize or control price fluctuations is effective in improving social welfare. On the other hand, if the social judgement weights the utility of better-off consumer, then the market adjustment mechanism with intermediate transactions is effective in improving the social welfare.
6. Concluding remarks This study presents a price adjustment process in discrete time which generates chaotic motions. To detect an economic meaning of chaotic motion, it focuses on a case in which actual transactions take place at non-stationary
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prices. The long-run average utility of consumer is calculated by applying the Mean Ergodic Theorem and is compared with the stationary utility. Our main results are summarized as three properties of chaotic price dynamics: (1) it is analytically as well as numerically demonstrated that consumers can benefit from chaotic price instability in the sense that the long-run average utility can be larger than the stationary utility; (2) it is also demonstrated that the long-run average of the Benthamite social welfare along a chaotic trajectory can be greater than the social welfare taken at the stationary state and (3) the benefit of price instability as a whole has a ‘‘social-judgement-dependency’’, according to which the Second property holds if the judgement is favorable for welfare of a better-off consumer along chaotic trajectory. These results suggest a possibility that chaotic (disequilibrium) motions can be preferable to the stationary (equilibrium) state. In a future research, it may be worthwhile to check whether there is some way to make both consumers better off.
Acknowledgements An early version of this paper was presented at the International Conference on Progress in Nonlinear Science, Nizhny Novgorod, Russia, July 2–6, 2001, the Annual Conference of the Society for Chaos Theory in Psychology and Life Science, Madison, Wisconsin, USA, August 3–6, 2001, and the Conference of NEW (New Economic Windows; New Paradigms for the New Millennium), Salerno, Italy, September 13–15, 2001. Helpful comments by Roger Guesnerie, Cars Hommes, Alan Kirman, Tamotsu Onozaki and an anonymous referee of this journal are gratefully acknowledged. The financial support from Chuo University (Chuo University Grant for Special Research) is also acknowledged.
References [1] [2] [3] [4] [5] [8] [9] [10] [11] [12] [13]
Bala V, Majumdar M, Mitra T. A note on controlling a chaotic t^atonnement. J Econ Behav Organ 1998;33:411–20. Boldrin M, Montrucchio L. On the indeterminacy of capital accumulation paths. J Econ Theory 1996;40:26–39. Day R, Pianigiani G. Statistical dynamics and economics. J Econ Behav Organ 1991;16:37–83. Day R. Complex economic dynamics. Cambridge: MIT Press; 1994. Goeree J, Hommes C, Weddepohl C. Stability and complex dynamics in a discrete t^atonnement model. J Econ Behav Organ 1998;33:395–410. Mukherji A. A simple example of complex dynamics. Econ Theory 1999;14:741–9. Onozaki T, Sieg G, Yokoo M. Complex dynamics in a cobweb model with adaptive product adjustment. J Econ Behav Organ 2000;41:101–15. Takayama A. Mathematical economics. Dryden Press; 1974. Tuinstra J. A discrete and symmetric price adjustment process on the simplex. J Econ Dyn Contr 2000;24:881–907. Waugh F. Does the consumer benefit from price instability?. Quart J Econ 1944;58:602–14. Weddepohl C. A cautious price adjustment mechanism: chaotic behavior. J Econ Behav Organ 1995;27:293–300.
Let it be: chaotic price instability can be beneficial Akio Matsumoto
*
Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo 192-0393, Japan Accepted 21 January 2002 Communicated by Y. Aizawa
Abstract This study investigates an economic implication of chaotic fluctuations generated by a discrete price adjustment process. For this purpose, it uses a pure exchange model with two goods and two consumers in which chaotic price fluctuations can arise. In order to reveal some statistical properties of such price dynamics, this study constructs a density function of chaotic trajectory, calculates a long-run average utility, and then compares it with the utility corresponding to a stationary state. The following two results are analytically as well as numerically demonstrated: (1) chaotic price dynamics can be beneficial for one consumer and harmful to the other consumer and (2) the whole economy is possibly better off along chaotic fluctuations than at a stationary state in the long-run. Further, it is shown that the second result is sensitive to the social judgement on the ranking of consumersÕ utilities. These results imply the possibility that chaotic fluctuations can be preferable to a stationary state. Ó 2003 Elsevier Science Ltd. All rights reserved.
1. Introduction In the recent literature, it has been demonstrated that a discrete t^ atonnement process may be chaotic; see Day and Pianigiani [3], Day [4], Weddepohl [13] and Mukherji [8] for a one-dimensional process, and Goeree et al. [5] and Tuinstra [11] for a multi-dimensional process. Yet, a salient feature of the t^ atonnement process is that the market price mechanism (in other words, an auctioneer) adjusts the price in response to an excess of demand over supply, and actual transactions take place only when an equilibrium price is attained. A chaotic t^ atonnement, in spite of the working of the price adjustment, does not converge to an equilibrium price but fluctuates around it while remaining bounded. The chaotic t^atonnement thus has a ‘‘crux’’ that no actual transactions are carried out because no convergence is brought about. Persistent continuation of market disequilibrium is apparently against the spirit of competitive market adjustment and thus thought as an unfavorable phenomenon in the traditional economics. There are at least two ways to deal with this crux: one is to stabilize or control chaos; the other is to admit intermediate trades in the middle of the price adjustment process. In fact, research in the former direction has already begun; Bala et al. [1] (BMM henceforth) construct an exchange economy, in which the logistic map adjusts the price, and demonstrate a possibility of controlling a chaotic t^atonnement in such a way that a chaotic price trajectory can be led to an arbitrarily small neighborhood of a competitive equilibrium price. In this study, we take the latter approach. In particular, we derive a basic model from BMM and consider chaos in the non-t^ atonnement process in their setting. Since trade occurs in a non-clearing market, a consumer may be better off or worse off than he could be at the equilibrium price, depending on the prevailing market price. So our primary concern is to investigate whether, from the long-run point of view, the gain from the chaotic price instability can dominate the loss. Alternatively put, we are interested in answering the following question: can chaos be preferable to a stationary state in a disequilibrium economy? *
Tel.: +81-426-74-3351; fax: +81-426-74-3425. E-mail address: [email protected] (A. Matsumoto).
0960-0779/03/$ - see front matter Ó 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0960-0779(03)00005-5
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This is essentially the same question as Waugh [12] asked more than half a century ago, the question of whether consumers can benefit from price instability. He provided a positive answer but only under the limited circumstances. Although it has been believed in the traditional economics that a stationary state is preferred to irregular motions, it has not yet been verified whether or not appropriate welfare measure can take on better value on chaotic trajectories than in the stationary state. It is thus imperative to investigate whether or not chaos is beneficial for consumers in a disequilibrium economy in which demand is unequal to supply. 1 The main purpose of this study is to demonstrate a possibility that a long-run average utility taken on chaotic paths can be preferable to a utility taken in the stationary state. It is for this purpose that the present study analyzes an exchange economy with two goods and two consumers in which price trajectories exhibit chaotic fluctuations. In doing so, this study will shed light on the nature of disequilibrium dynamics that has been neglected in the traditional economics. This study is organized as follows. Section 2 outlines a pure exchange model constructed in BMM and derives a dynamic system of the model. Section 3 describes our non-t^ atonnement process. Section 4 analytically and numerically shows the main result of this study that chaotic fluctuations can be preferable to a stationary state in the long-run. Section 5 provides an intuitive explanation for the source of the main result. Section 6 provides conclusions and a pointer to future research. 2. Pure exchange economy In this section, we present a discrete-time price adjustment process. Since our study follows BMM, we recapitulate the basic elements of their analysis and then proceed to our own investigation. 2.1. Piecewise-continuous price adjustment process Consider a pure exchange economy in which there are two consumers, A and B, and two goods, x and y. p and q are prices of goods x and y, respectively. We adopt two simplifying assumptions. First, q is chosen as a numeraire price so that q ¼ 1. Second, the fixed endowments of goods in the economy are initially allocated in such a way that consumer A starts off with an endowment xA ¼ ð1; 0Þ and consumer B starts off with an endowment xB ¼ ð0; 1Þ. Inspecting price p available at a market of good x, each consumer determines his optimal choice so as to maximize his utility subject to the budget constraint. Entering the markets, consumers exchange some of these goods with each other. Preference of consumer A is represented by a utility function of the Leontief type uA ðx; yÞ ¼ minff ðxÞ; ayg;
ð1Þ
where f ðxÞ ¼
8 > < x x2 > :1 4
1 if 0 6 x 6 ; 2 1 if x > ; 2
and a > 0 is a preference parameter. AÕs optimal (gross) demands are given by 8 1 > < xA ðpÞ ¼ ap and yA ðpÞ ¼ pð1 apÞ if 0 6 p 6 ; 2a 1 1 1 > : xA ðpÞ ¼ and yA ðpÞ ¼ if p P : 2 4a 2a
ð2Þ
It is known that price changes for the Leontief preference have no substitution effect. Consequently a change in demand for good x caused by a change in price is due entirely to the income effect. Consumer B has a utility function of the quasi-linear type uB ðx; yÞ ¼ gðxÞ þ y;
ð3Þ
1 Boldrin and Montrucchio [2] show that a chaotic trajectory can be an admissible solution to discounted dynamic optimization problems. Their study implies that equilibrium trajectories can be chaotic, and chaos can be consistent with optimal behavior. Our approach differs from theirs in one significant point. We are concerned with chaos in a disequilibrium economy whereas they are concerned with it in an equilibrium economy in which demand equals supply.
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where ( gðxÞ ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 pffiffiffi b ð1 xÞ3 if 0 6 x 6 1; 3 0 if x > 1;
and b > 0 is a preference parameter. BÕs optimal (gross) demands are given by 8 pffiffiffi p2 p3 < and yB ðpÞ ¼ 1 p þ if 0 6 p < b; xB ðpÞ ¼ 1 b b pffiffiffi : xB ðpÞ ¼ 0 and yB ðpÞ ¼ 1 if p P b:
ð4Þ
It is known that a shift in income causes no change in demand for good x in the case of quasi-linear preference. Thus the entire change in demand x caused by a change in price is due to the substitution effect. The extreme allocation of initial endowment (i.e., xA ¼ ð1; 0Þ and xB ¼ ð0; 1Þ) ensures that consumer A is a supplier of good x and consumer B is a demander. As shown in (2) and (4), supply for good x is kinked at p ¼ 2a1 while demand is pffiffiffi kinked at p ¼ b. Thus the supply function and the demand function are piecewise-continuous and spelled out as
1 1 2 SðpÞ ¼ max 1 ap; and DðpÞ ¼ max 1 p ; 0 : ð5Þ 2 b Let ZðpÞ be the excess demand, i.e., ZðpÞ ¼ DðpÞ SðpÞ. The standard price adjustment process in discrete time is then given by
1 1 ;0 ; ð6Þ /ðpt Þ ¼ max½pt þ hZðpt Þ; 0 ¼ max pt þ h max 1 pt2 ; 0 max 1 apt ; b 2 where h is an adjustment speed and a positive constant and pt is the price at period t. The process takes explicit account of the non-negative constraint on price and is also piecewise-continuous. The stationary price is defined by p ¼ /ðpÞ. Solving the equation for p generates two solutions, a trivial solution, p ¼ 0, and a non-trivial solution, p ¼ pE 6¼ 0 which is rffiffiffi 8 b 1 > > < pE ¼ ab 6 ; if 2 r 2affiffiffi rffiffiffi ð7Þ > > : pE ¼ b if 1 < b: 2 2a 2 For the local stability of pE , the eigenvalue of the dynamic system (6) evaluated at the stationary point must be less than unity in absolute value; the eigenvalue, in the case of this one-dimensional system, is equal to the slope of /ðpt Þ. 2 Even if the stationary state loses its local stability, it is shown to be globally stable in the sense that all price trajectories are bounded; roughly speaking, the adjustment process (6) generates excess demand for lower prices and excess supply for higher prices and thus induces price trajectories to bounce back to a neighborhood of the stationary state whenever trajectories move far away from the stationary point. Fig. 1 presents a two-parameters bifurcation diagram that is a picture of bifurcation response of the adjustment process to changes in parameters a and b where a fixed adjustment coefficient h is assumed. The diagram has been produced using the pattern recognition algorithm developed by Onozaki et al. [9]. 3 The diagram is produced using a 600 600 grid of a and b where h ¼ 6, 4 a 2 ½1h ; 4h and 2 b 2 ½0:01; h2 . Different colors in the diagram indicate different regions of stable, periodic or aperiodic behavior. The dark region labelled S1 shows the attractor of non-trivial fixed point (of order 1). The dark region labelled S2 represents parameter combinations for which price is mapped to zero (or becomes negative without the non-negative constraint). The white region indicates aperiodic behavior involving chaos. The other colors indicate the periodic attractors of up to 16 different periods. The diagram shows that /ðpt Þ can generate various dynamics ranging from stable convergence to chaotic fluctuations, depending on parameter values specified.
2
/0 ðpt Þ ¼ 1 ah for pE ¼ ab, and /0 ðpt Þ ¼ 1 p2hffiffiffiffi for pE ¼ 2b
qffiffi b 2.
3 After a transient time of 5000 periods, the adjustment process is further iterated 200 times for every parametric combination. The difference between a generated trajectory and a periodic trajectory with period-n is calculated. If the maximum difference in absolute value becomes less than 0.00001, it is determined that the trajectory possesses a period n. 4 Although this is the value which BMM use in their numerical simulations and will be used in our own simulations later, the qualitative nature of the bifurcation diagram does not depend on the particular value of h.
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Fig. 1. Two-parameter bifurcation diagram where h ¼ 6.
A simultaneous variation of two different parameters is considered in Fig. 1. If we fix one parameter and choose the other as a bifurcation parameter, we can have a familiar bifurcation diagram with respect to one parameter value. For h2 constant values of parameters, b ¼ 36 and h ¼ 6, increasing a from 2h to 3h along the lower dotted line in Fig. 1 implies the emergence of the period-doubling bifurcation cascade to chaos as illustrated in Fig. 2a. Further, for another values of 2 bð¼ 3h8 Þ, increasing a from 1h to 4h along the upper dotted line exhibits the period-doubling and period-halving bifurcation as illustrated in Fig. 2b. 2.2. Logistic price adjustment process Due to the piecewise-continuity, /ðpt Þ takes on various profiles according to various combinations of parameters. In the following, to investigate statistical properties of chaotic dynamics, we confine our analysis to a case in which the excess demand function is smooth and the resultant adjustment function has a single-peaked profile. pffiffiffi If the price is restricted to be less than or equal to the minimum of those kinked prices (i.e., p 6 min½2a1 ; b ), then the total excess demand for p is smooth and given by ZðpÞ ¼ ap
p2 : b
ð8Þ
It has an inverted U-shaped profile since the income effect dominates the substitution effect for small p, and the substitution effect dominates the income effect for large p. The non-trivial solution of ZðpÞ ¼ 0 is pE ¼ ab. Substituting (8) into (6) yields a single-peaked price adjustment function h ptþ1 ¼ ð1 þ ahÞpt pt2 : b
ð9Þ
We make the following assumption in order to apply this dynamics to a class of economies parameterized by a single parameter l.
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Fig. 2. Two bifurcation diagrams with respect to a for different values of b; b ¼ 19 in (a) and b ¼ 32 in (b).
Assumption 1. ð1 þ ahÞ ¼ bh ¼ l Under this assumption, the dynamic system (9) is represented by the logistic map with the parameter l ptþ1 ¼ Hðpt Þ lpt ð1 pt Þ:
ð10Þ
l can be a proxy of the strength for the nonlinearity included in the adjustment process. It is well known that the logistic map can generate periodic as well as aperiodic dynamics according to values of the parameter l, when the stationary state is locally unstable. Now we are in a position to clarify the parameter configuration under which the logistic map can be well defined as an appropriate price adjustment process. First of all, Assumption 1 implies that one of three parameters, a, b and h, are not independent. In the following, we let b be a dependent parameter b¼
h : 1 þ ah
ð11Þ
Second, it is required that l is not greater than four in order to prevent Hðpt Þ from being negative. This requirement is, under Assumption 1, spelled out as ð12Þ p ffiffiffi Lastly it is also required that the parameter values are so defined that p 6 min½2a1 ; b pfor ffiffiffi all p. Since the maximum value of Hðpt Þ is unity provided l 6 4, the requirement is ensured if values of both 2a1 and b are not less than unity, that is, if ah 6 3:
a6
1 2
and
hP
1 : 1a
ð13Þ
All price trajectories generated by Hðpt Þ stay within the unit interval for any combination of parameters that satisfy the inequality conditions (12) and (13). Since the slope of Hðpt Þ evaluated at the stationary point pE ¼ ab is H0 ðpE Þ ¼ 1 ah; the stationary state is locally stable if the slope is less than unity in absolute value (i.e., 0 < ah < 2). Price converges monotonically to pE if 0 < ah < 1 holds, and cyclically if 1 < ah < 2. It is locally unstable if the slope is equal to or greater than unity i.e., 2 6 ah. To sum up, the price adjustment process makes the stationary state locally unstable but every trajectory remains within a finite distance from the stationary state for a and h in the following set, U ,
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9 pt +1 <0
U:Unstable Region
αθ =3
6
αθ =2
Stable Region
(1−α)θ =1
β <1
0 α
1/ 3
1/ 2
Fig. 3. Stable and unstable regions.
U¼
ða; hÞj2 6 ah 6 3;
0 < a6
1 ; 2
where (11) holds. A dark gray region in Fig. 3 is a part of the unstable set U . Suppose h ¼ 6. Then increasing a from 13 to 1 along the dotted line in Fig. 3 is identical with increasing l from 3 to 4 under Assumption 1 so that the period2 doubling bifurcation to chaos occurs along this line as illustrated in Fig. 2a. It is further verified that b > 1 for any ða; hÞ 2 U .
3. Transaction–consumption process Our model indicates that the economy can generate complex dynamics if the stationary state is unstable. In this section, we describe what consumers can afford via intermediate transactions in the disequilibrium market. The dynamic process involving the intermediate transactions is known as the non-t^ atonnement process. Here we call our process a transaction–consumption process to emphasize its distinct feature that, as described shortly, not only intermediate transactions but also actual consumption of goods are allowed in the middle of adjustment. In the so-called non-t^atonnement process, it is assumed that exchange transactions can take place during the process of price adjustment. As a result, the ex-ante stock of goods held at the beginning of the period differs from the ex-post stock of goods held at the end of the period. Having the resultant stock of the two goods, each consumer inspects the price available at the beginning of the next period, decides how much he wants to buy and sell, and then enters the markets again. Since the total stock of goods in the economy is re-distributed in the non-t^ atonnement process from time to time and, accordingly, consumerÕs purchasing power varies, the excess demand function changes from time to time. Thus the main concern in the non-t^atonnement process is related to the question of whether a dynamic system that allows for the price adjustment and the quantity adjustment associated with allocations of goods can reach a stationary state. 5 We also consider the price dynamics in the non-t^ atonnement process, but our main concern is different; we will see what welfare or benefit consumers can get in the non-t^ atonnement process when the price fluctuates chaotically. To this end, we start with to make two assumptions: the first assumption is that the transacted amounts of goods are actually consumed by the end of a period, and, the second is that exactly the same stock of endowment is allocated to each consumer at the beginning of the next period. These two assumptions assume away the redistribution effect of the
5
See Takayama [10] for a brief survey of the stability of the non-t^atonnement process.
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excess demand function and make us confine our analysis to a case in which there is a continuous flow of goods going to each consumer. 6 Therefore the adjustment function (10) that does not involve the redistribution effect can be relevant for the dynamic analysis of the transaction–consumption process. The sequence of time periods in our process is the following. At the beginning of a period, each consumer receives his endowment and learns the price of good x. Seeing how much his endowment is worth, he determines demand and supply so as to maximize his utility. In the middle of the period, consumers make their purchases and sales, although, for an arbitrary price of p, there is no guarantee that supply equals demand. To make intermediate transactions at such a disequilibrium price, we will introduce a transaction rule that takes disequilibrium phenomenon into account. After transactions are completed in markets, each consumer takes the amount of traded goods home from the markets and actually ends up consuming it. At the end of the period, the consumer evaluates what level of satisfaction he enjoys through consumption, and the price is adjusted according to Hðpt Þ if supply is not equal to demand. At the beginning of the next period, the new price is announced and the same endowment is given to each consumer. The process is repeated in the same way in subsequent periods. In the non-t^atonnement process, a transaction rule is necessary to make actual transactions in a disequilibrium market. Following the received wisdom in disequilibrium theory, we assume the min-rule or the short-side rule, according to which the actual transaction is the minimum of supply and demand in the market. x ¼ min½D; S ;
ð14Þ
where the bar, i.e. ‘‘-’’ over the variable means actual transaction, and S ¼ 1 ap is supply of good x and D ¼ 1 b1 p2 is demand for good x. The equilibrium price, pE , is the one where the demand curve crosses the supply curve. For 0 < p < pE , demand for good x is greater than supply so that consumer A is on the short-side of the market. In consequence, consumer A has more demand than he can handle. Consumer A, therefore, can sell his desired amount (i.e., x ¼ S) and can consume 1 S of good x. Since net transactions must be identically balanced in the market, consumer B, who is on the long-side of the market, is rationed and able to afford to buy x. Thereby, gross demands of good x, which are realized at the going price, are xA ¼ ap
and xB ¼ 1 ap:
ð15Þ
Actual transactions for gross demands of good y are determined through the budget constraints, yA ¼ pð1 xA Þ for consumer A and yB ¼ 1 pxB for consumer B, yA ¼ pð1 apÞ and yB ¼ 1 pð1 apÞ:
ð16Þ
On the other hand, for pE < p < 1, the situation is reversed: demand is greater than supply so that consumer B is on the short-side and able to realize his desired transaction (i.e., x ¼ D) while consumer A is on the long-side and rationed. As h b ¼ 1þah under Assumption 1, gross transactions are given by 1 þ ah 2 1 þ ah 2 xA ¼ p and yA ¼ p 1 p for consumer A ð17Þ h h and xB ¼ 1
1 þ ah 2 p h
1 þ ah 2 and yB ¼ 1 p 1 p for consumer B: h
ð18Þ
These transacted amounts of goods are consumed and then determine the actual level of each consumerÕs utility at the end of the period. Inserting (15)–(18) into utility functions in (1) and (3) yields indirect utility functions, UA ðpÞ uA ðxA ; yA Þ and UB ðpÞ uB ðxB ; yB Þ. Since one of the consumers is rationed according to whether the prevailing price is greater or less than the equilibrium price, his indirect utility function is piecewise-continuous in p. For consumer A, it is 8 L < UA ðpÞ apð1 apÞ if 0 < p 6 pE ; 1 þ ah 2 UA ðP Þ ¼ ð19Þ H if pE < p 6 1; p : UA ðpÞ ap 1 h
6 In this study which is a starting point for a more complete study, we adopt these convenient assumptions for two reasons. First, it makes the formidable mathematical problem simpler and manageable. Second, such a simplification enables us to derive rigorous results and economically interesting insights. Needless to say, since the redistribution effect is the main feature of the non-t^atonnement process, a further study taking account of the effect is necessary.
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and for consumer B, it is 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 hðapÞ3 > < U L ðpÞ þ 1 pð1 apÞ if 0 < p 6 pE ; B 3 1 þ ah UB ðpÞ ¼ > > 1 þ ah 3 > : UBH ðpÞ 1 p þ p if pE < p 6 1; 3h
ð20Þ
where L (respectively H) means that the prevailing price is lower (respectively, higher) than equilibrium price pE . For p ¼ pE , the market reaches equilibrium where both consumers can trade desired amounts. Since consumer A can realize his desired amount if p < pE , the first equation of (19) evaluated at pE determines the desired level of consumer AÕs utility. Similarly, since consumer B can realize his desired trade if p > pE , the second equation of (20) evaluated at pE determines the desired level of consumer BÕs utility. 1 þ ah 2 pE UAE apE ð1 apE Þ and UBE 1 pE 1 ð21Þ 3h 4. Long-run average behavior In this section, we present two numerical examples concerning the long-run average behavior of consumers. Setting l ¼ 4 in the first example and l 6¼ 4 in the second example, we investigate the first example analytically and the second numerically in the following three ways. First, we calculate each consumerÕs utility taken at the stationary state which we call the stationary utility. Second, we calculate the long-run average utilities taken along a chaotic trajectory. Third, we compare those long-run average utilities with the stationary utilities. In doing so, we show a possibility that a longrun average utility can be larger than the stationary utility. 4.1. Analytical result for l ¼ 4 The stationary utility is derived as follows. Under Assumption 1, l ¼ 4 implies 1 þ ah ¼ 4 and bh ¼ 4 so that h and b can be written in terms of a, h¼
3 a
and b ¼
3 : 4a
ð22Þ
Inserting the stationary price, pE ¼ ab, which is equal to 34, into indirect utility functions defined in (21) yield the stationary utilities, UAE ðaÞ ¼
3 að4 3aÞ and 16
UBE ðaÞ ¼
1 ð4 þ 3aÞ: 16
ð23Þ
It is observed that UAE ðaÞ and UBE ðaÞ are increasing, and UBE ðaÞ > UAE ðaÞ for a 6 12. It has been demonstrated that the logistic map Hðpt Þ exhibits ergodic chaos if l ¼ 4 (i.e, ah ¼ 3). 7 Let fHt ðp0 Þg1 t¼0 be a chaotic trajectory of price from an initial condition p0 , and fUj ðHt ðp0 ÞÞg1 t¼0 be the corresponding trajectory of consumer j0 s utility associated with the price trajectory. The mean value of the utility can be a basic statistical property of the chaotic price dynamics. Although it is impossible to calculate an average of infinite elements, the Mean Ergodic Theorem of Birkhoff–von Neuman 8 indicates that, if Hðpt Þ is ergodic, the time average of utility evaluated along a chaotic trajectory converges to its space average, namely, the mean value of utility evaluated on the domain of the price Z T 1 1 X Uj ðHt ðp0 ÞÞ ¼ Uj ðpÞUðpÞ dp for almost all p 2 I; ð24Þ lim T !1 T I t¼0 where UðpÞ is an invariant density function of chaotic trajectory. While it is, in general, difficult to construct the densities for smooth maps, it has been known that the explicit form of the density function for the logistic map with l ¼ 4, as shown in Day [4], is given by 1 UðpÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : p pð1 pÞ 7 8
ð25Þ
HðpÞ : I ! I where I is the unit interval is ergodic chaos if there is only one absolutely continuous invariant ergodic measure on I. See Day [4] and references there (p. 142) for this theorem.
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We can make a statistical characterization of the long-run average behavior of consumers by calculating the right-hand side of (24) instead of the left-hand side. Indeed, taking account of the piecewise-continuity of the indirect utility function, we observe that the long-run average utility, denoted as U j for j ¼ A; B, is defined by Z pE Z 1 Uj ¼ UjL ðpÞUðpÞ dp þ UjH ðpÞUðpÞ dp: ð26Þ 0
pE
Choosing a as an independent variable and other parameters, b and h, dependent variables, which satisfy (22), we investigate the long-run average behavior for different values of a. The long-run average utility of consumer A is, after inserting (19) into (26), calculated as pffiffiffi að16pð9 7aÞ 27 3aÞ : ð27Þ U A ðaÞ ¼ 288p The utility level of consumer A summarizes his value judgement on the allocation of goods. The stationary utility, UAE ðaÞ of (23), represents consumer AÕs preference in the stationary state, and the long-run average utility, U A ðaÞ of (27), represents his preference on the chaotic trajectory from the long-run point of view. The difference between these utilities is pffiffiffi að27 3a þ 2pð36 25aÞÞ 1 > 0 for 0 < a 6 : ð28Þ UAE ðaÞ U A ðaÞ ¼ 288p 2 The inequality of (28) indicates that consumer A prefers the stationary state to the non-stationary state in the sense that he can get a higher utility at the stationary state. By the same token, substituting (20) into (26) and integrating the resultant equation give the long-run average utility of consumer B, pffiffiffi 432p þ ð256p 255 3Þa : ð29Þ U B ðaÞ ¼ 864p Then the difference between the stationary utility and the long-run average utility is pffiffiffi 216p þ ð94p 255 3Þa 1 UBE ðaÞ U B ðaÞ ¼ < 0 for 0 < a 6 : 864p 2
ð30Þ
The direction of the inequality in (30), which runs in opposite to that of the inequality in (28), implies that consumer B can get a higher utility on average on the chaotic trajectory than at the stationary state. Thus, it can be said that, contrary to consumer A, consumer B prefers the non-stationary state to the stationary state. These results concerning the individual characteristics of chaotic dynamics are summarized as the First property of chaotic non-t^ atonnement price adjustment process. The First property. In the transaction–consumption process in which h ¼ 3a and b ¼ 4a3 hold, the long-run average utility of consumer A is less than the stationary utility, U A ðaÞ < UAE ðaÞ while the long-run average utility of consumer B is greater, U B ðaÞ > UBE ðaÞ for 0 < a 6 12. Admitting the possibility that the market disequilibrium is beneficial for consumer B and harmful for consumer A, we may ask the obvious question: Is a disequilibrium allocation of goods preferable from the perspective of the whole economy, namely, the distribution of welfare among all people in the economy? To answer this question, we need a social preference that provides a way to rank different distributions of utility among consumers. For this purpose, we assume the following; (1) social preference can be constructed by adding up the individual utilities and (2) the resulting sum can be used as an indicator of social utility. Our focus will be on a special example of a social welfare function that is the sum of the individual utility functions i.e., a classical utilitarian or Benthamite welfare function. The social welfare over the stationary allocation, WE and the one over the non-stationary allocation, W are given by X X UiE ðaÞ and W ¼ U i ðaÞ: ð31Þ WE ¼ i
i
Each can be thought to represent social judgement on each allocation. Adding (28) and (30) yields the welfare difference pffiffiffi 3 3að85 þ 27aÞ 2pð75a2 61a þ 108Þ 1 WE W ¼ < 0 for 0 < a 6 : ð32Þ 864p 2
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Fig. 4. Convergence of numerically calculated average utility to the theoretical value.
This result is summarized as the Second (statistical) property of chaotic non-t^ atonnement price adjustment process. The Second property. A whole economy is possibly better off along chaotic trajectories than at a stationary state in the long-run if the welfare function is classical utilitarian, U A ðaÞ þ U B ðaÞ > UAE ðaÞ þ UBE ðaÞ for 0 < a 6 12. 4.2. Simulation results for l 6¼ 4 We have shown that chaotic trajectories can be preferable to the stationary state when l ¼ 4. However, when l 6¼ 4, explicit forms of densities for chaotic trajectories are unable to be constructed, and thus the long-run properties are not able to be derived with the help of the Mean Ergodic Theorem. In this section, taking a numerical approach, we perform simulations in order to confirm that our theoretical results still hold even when l 6¼ 4. In particular, we directly calculate the time average of consumersÕ utilities over the first T periods of chaotic trajectory, uA ¼
t¼T 1 X UA ðHt ðp0 ÞÞ T t¼1
and
uB ¼
t¼T 1 X UB ðHt ðp0 ÞÞ T t¼1
and then compare these with the corresponding stationary utilities. Before proceeding, we need to check whether a finite average can approximate the long-run average. Fig. 4 illustrates how finite averages ‘‘converge’’ to the long-run averages. In Fig. 4, we set a ¼ 12 which, under (22), leads to b ¼ 32 and h ¼ 6 and, therefore, imply l ¼ 4. In order to better indicate the quality of the convergent process, we have drawn a dotted horizontal line at U A in Fig. 4a and at U B in Fig. 4b where these stationary utilities are calculated as pffiffiffi pffiffiffi 176p 27 3 5ð224p 51 3Þ ’ 0:139856 and U B ¼ ’ 0:566789: UA ¼ 1152p 1728p Further, we have drawn two solid lines, one of which is separated upward from the dotted line by 0.006, and the other separated downward by 0.006. 9 Fig. 4a illustrates three different trajectories which start with different initial prices, p0 ¼ 0:1 (red trajectory), p0 ¼ 0:6 (green trajectory) and p0 ¼ 0:8 (blue trajectory), respectively. It shows that the finite average utilities of consumer A approach the long-run average as number of periods used in the simulation increases. Fig. 4b shows the simulation results for consumer B in which three different trajectories are seen to approach to U B . Although irregularity of consumer BÕs trajectories seems stronger, it may be safe to assume that the finite average ‘‘converges’’ to the theoretical value of the long-run average utility if the number of iterations is over 1000 i.e., uA ’ U A and uB ’ U B if T > 1000. Now we are ready to conduct numerical experiments to check whether the theoretical results summarized as the First and Second properties hold when l 6¼ 4. Setting h ¼ 6 and b ¼ 4a3 , the finite averages of utility are computed for various values of a; these are obtained from the average of T ¼ 1000 for 200 points on interval ½13 ; 12 . The corresponding stationary utilities are also obtained from (23). Calculating the difference between the stationary utility and the finite
9 0.006 is the value which BMM used to check the convergence of their controlled trajectory. When a deviation of a trajectory from a stationary price (in absolute value) is less than this number, they conclude that the trajectory is successfully controlled and has reached the stationary price.
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Fig. 5. Long-run average utilities with respect to a: numerical simulations.
average utility, we plot 200 points of differences versus a and connect these points together to construct welfare beneficial curves as illustrated in Fig. 5. There, the red curve is the beneficial curve for consumer A while the blue curve for consumer B. Further, the difference between the sum of the stationary utility and the sum of the average utility (green ) is also depicted. Since the vertical distance indicates the difference between the stationary utility and curve marked w the finite-average utility, a positive value means that stationary utility is higher than the finite average utility, and a negative value means that the average utility is higher. It is fairly clear by looking at the these welfare curves that these simulations yield qualitatively the same result as the theoretical analysis; at least one of consumers can benefit from the price instability and the other harmed (i.e., First property) but the average social welfare is higher on the chaotic trajectories (i.e., Second property). In the middle of Fig. 5, we re-produce the period-doubling bifurcation diagram depicted in Fig. 2a, in which the height is appropriately adjusted.
5. Statistical properties reconsidered In the first half of this section, we give intuitive explanations for two characteristics of chaotic price dynamics, the First and Second properties, which the analysis so far has revealed. In the latter half, we provide the Third property of chaotic price dynamics, the property which clarifies a dependency of the Second property on a construction of social preference. According to the First property, consumer A is worse off and consumer B is better off when an intermediate transaction is allowed in the middle of the price adjustment. This property is graphically confirmed in Fig. 6 where the indirect utility functions (blue curves), the stationary level of utility (dotted lines) and the density functions are illustrated for a ¼ 12. 10 As can be seen in Fig. 6a, the indirect utility function of consumer A is increasing for p < pE , decreasing for p > pE and reaches its maximum for p ¼ pE . That is, UAE P UA ðpÞ for all p, which leads to the first result (28). To explore the dominance of the long-run average utility of consumer B over the stationary utility, we discuss the specific situation, only for the sake of simplification, in which price dynamics is characterized by regular alternating behavior with respect to the stationary price, that is, the situation where if a price in period t is less (respectively, greater) than pE , then it is mapped to a point greater (respectively, less) than pE in period t þ 1. Since the indirect utility defined in (20) is a decreasing function of p and kinked at p ¼ pE as depicted in Fig. 6b, the utility level also alternates with respect to the stationary utility level, that is, if a utility level in one period is greater (respectively, less) than the 10
The height of the density function is appropriately adjusted.
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A. Matsumoto / Chaos, Solitons and Fractals 18 (2003) 745–758 UA
(a)
utility
(b)
UAE
UBE
0
pE
1
p
0
pE
1
p
Fig. 6. Indirect utility, stationary utility and density function.
stationary level, then the utility level in the next period is less (respectively, greater) than the stationary level. Let U2 be the average utility of consumer B for a two-period iteration fpt ; Hðpt Þg i.e., U2 ¼ 12 ðUB ðpt Þ þ UB ðHðpt ÞÞÞ. Then the difference between U2 and UBE is given by U2 UBE ¼
1 UB ðpt Þ UBE þ UB ðHðpt ÞÞ UBE : 2
Suppose pt < pE in period t. 11 Fig. 6b indicates UB ðpt Þ UBE > 0 and UB ðHðpt ÞÞ UBE < 0 so that the sign of the sum is ambiguous. It will be positive if pt is closer to zero and Hðpt Þ is closer to pE , i.e., the absolute value of the first term is greater than that of the second term, and negative if pt is closer to pE and Hðpt Þ is closer to unity. Roughly speaking, the long-run average utility is the average of such a U2 over the entire interval of p. Since a density function of a price trajectory describes how often it visits every measurable set in the trapping interval, it depends on the shape of the density function whether the long-run average utility is larger or smaller than the stationary utility. The dominance of the long-run average utility over the stationary utility is quite plausible as seen from the shape of the density of the logistic map illustrated in Fig. 6b. The length of a subinterval where UB ðpt Þ UBE > 0 holds is much longer than the length of a subinterval where UB ðHðpt ÞÞ UBE < 0 so that U B > UBE in the long-run as (32) indicates. According to the second property, the intermediate trade raises the economic well-being as a whole because the gain of consumer B exceeds the loss of consumer A. We illustrate the utility possibilities frontier of two consumers in Fig. 7. There, WE is the set of utilities associated with the stationary allocation of goods while W is the set of utilities associated with the long-run average allocation. Sets of finite averages of consumersÕ utilities, which are calculated for T ¼ 1; 2; . . . ; 100, are plotted as blue dots and seen to be scattered around the long-run average level after finite iterations. The downward sloping solid line passing through the point WE is an iso-welfare curve of the Benthamite welfare function, the curve which depicts those distributions of utility that have the same welfare as the stationary welfare. We can observe the Second property from Fig. 7 i.e., W has a higher social welfare than WE as it is located above the iso-welfare curve. This observation implies that there are many ways to construct a welfare function which can generate the Second property. At the same time, it also implies that the Second property is sensitive to a choice of a social welfare function. In fact, while using the product of utilities as a welfare function generates qualitatively the same result as the Benthamite welfare function, the minimax or Rawlsian social welfare function, W R ¼ minfUA ; UB g, apparently does not have the Second property. 12 Further, it can be seen in Fig. 7 that WE is a Pareto efficient allocation and W is a Pareto improving allocation as the former is on the frontier and the latter is in the utility possibility set. Consequently, we can find a welfare function for which WE has a maximum welfare, since the utility possibility set is, as illustrated, convex in our pure exchange economy. 13 Thus the next question that naturally raises is: To what extent can the Second property hold? To answer this question, we make a slight generalization of the Benthamite welfare function and take the weighted sum of consumersÕ utilities as a social welfare,
11
The same results will be obtained if we suppose pt > pE . R The Rawlsian social welfare at the stationary state is WER ¼ UAE and the long-run average welfare is W ¼ U A . The inequality of R (28) leads to WER > W . 13 There is no way that both consumers can be benefit in our economy with convex structure. However, as pointed out by Kirman and Guesnerie, there might be a possibility that both consumers are happy in an economy with a non-convex utilities possibility set. 12
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Fig. 7. Utility frontier curve.
e ¼ aA UA þ aB UB ; W
ðaA ; aB Þ > 0:
e passes through two points, W and We can find a set of weight coefficients, aA and aB , for which the iso-welfare line of W WE , as depicted as a dotted line in Fig. 7. Using this dotted line for reference, it can be said that W is more (respectively, less) preferable to WE if a iso-welfare line that passes through point WE is flatter (respectively, steeper). The slope of a iso-welfare line is determined by the ratio of a aA =aB . The weight coefficient indicates the importance of consumerÕs utility to the social welfare and represents the social judgement between consumersÕ utilities. Taking account of these characteristics of coefficients, we summarize this result as the Third property of chaotic dynamics. The Third property. If the social judgement attaches more importance to welfare of consumer A (i.e., aA > aB ), then the stationary social welfare is preferable to the long-run average welfare so that the Second property does not hold. If the social judgement attaches more importance to welfare of consumer B (i.e., aA < aB ), then the longrun average welfare is preferable to the stationary social welfare so that the Second property holds. As stated in Section 1, we have two affordable choices to deal with chaotic fluctuations; the first choice is controlling chaos and the second is admitting the intermediate transactions. The Third property just stated is useful to detect which choice is effective in attaining a higher social welfare. BMM have already demonstrated that the stationary state is attainable with the help of control method. Therefore, if the social judgement over-weights the utility of the worse-off consumer along the chaotic trajectory in the overall social welfare, then some form of market intervention aiming to stabilize or control price fluctuations is effective in improving social welfare. On the other hand, if the social judgement weights the utility of better-off consumer, then the market adjustment mechanism with intermediate transactions is effective in improving the social welfare.
6. Concluding remarks This study presents a price adjustment process in discrete time which generates chaotic motions. To detect an economic meaning of chaotic motion, it focuses on a case in which actual transactions take place at non-stationary
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prices. The long-run average utility of consumer is calculated by applying the Mean Ergodic Theorem and is compared with the stationary utility. Our main results are summarized as three properties of chaotic price dynamics: (1) it is analytically as well as numerically demonstrated that consumers can benefit from chaotic price instability in the sense that the long-run average utility can be larger than the stationary utility; (2) it is also demonstrated that the long-run average of the Benthamite social welfare along a chaotic trajectory can be greater than the social welfare taken at the stationary state and (3) the benefit of price instability as a whole has a ‘‘social-judgement-dependency’’, according to which the Second property holds if the judgement is favorable for welfare of a better-off consumer along chaotic trajectory. These results suggest a possibility that chaotic (disequilibrium) motions can be preferable to the stationary (equilibrium) state. In a future research, it may be worthwhile to check whether there is some way to make both consumers better off.
Acknowledgements An early version of this paper was presented at the International Conference on Progress in Nonlinear Science, Nizhny Novgorod, Russia, July 2–6, 2001, the Annual Conference of the Society for Chaos Theory in Psychology and Life Science, Madison, Wisconsin, USA, August 3–6, 2001, and the Conference of NEW (New Economic Windows; New Paradigms for the New Millennium), Salerno, Italy, September 13–15, 2001. Helpful comments by Roger Guesnerie, Cars Hommes, Alan Kirman, Tamotsu Onozaki and an anonymous referee of this journal are gratefully acknowledged. The financial support from Chuo University (Chuo University Grant for Special Research) is also acknowledged.
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