q
The research for this paper was done as part of the project &Dynamics of nonlinear price adjustment processes' funded by the Netherlands Organisation for Scienti"c Research (NWO). Helpful comments by Cars Hommes, Leo Kaas and Claus Weddepohl are gratefully acknowledged.
*Tel.: 0031-20-525-4117; fax: 0031-20-525-4349.
E-mail address:[email protected] (J. Tuinstra) 24 (2000) 881}907
A discrete and symmetric price adjustment
process on the simplex
qJan Tuinstra
*
Department of Quantitative Economics and CeNDEF, Faculty of Economics and Econometrics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands
Accepted 30 April 1999
Abstract
We study the ta(tonnement process in an exchange economy with three commodities and three agents. We argue that for the multiplicative ta(tonnement process the simplex is a natural normalization rule. We analyse the in#uence of symmetry in preferences and endowments on the dynamics of the ta(tonnement process. Depending on this symmetry there are di!erent bifurcation routes to chaos. In particular, coexisting attractors may emerge when the symmetric equilibrium price vector becomes unstable. This coexistence of attractors is robust in the sense that it also exists for close-by asymmetric ta(tonnement processes. ( 2000 Elsevier Science B.V. All rights reserved.
JEL classixcation: D51; E30
Keywords: Ta(tonnement process; Chaotic dynamics; Bifurcations with symmetry
1. Introduction
In recent years there has been a renewed interest in the discrete ta(tonnement process. It has been shown that the ta(tonnement process does not always
converge to an equilibrium price vector and that it can exhibit periodic or chaotic behaviour (Bala and Majumdar, 1992; Day and Pianigiani, 1991; Goeree et al., 1998; Saari, 1985; Weddepohl, 1995). In this paper we concentrate on a two-dimensional discrete ta(tonnement process. Our approach di!ers from previous contributions in that we choose the simplex as a price normalization rule instead of taking some commodity as numeraire. We argue that the simplex is a natural price normalization rule since it is an invariant space of the multiplicative adjustment process, just as the unit sphere is an invariant space of the continuous time linear adjustment process.
We are interested in the dynamics of the ta(tonnement process for all possible economies. However, for a global analysis general equilibrium models have too many free parameters and therefore as a"rst step towards a systematic analysis, we focus on symmetric ta(tonnement processes. We apply the literature on dynamical systems with symmetries (see Golubitsky et al., 1988) to study these symmetric ta(tonnement processes. We "nd that di!erent routes to strange attractors can occur. Furthermore, the type of bifurcation that occurs depends upon the symmetry of the ta(tonnement process.
Since some economic models have a natural symmetry, our study of symmet-ric dynamical systems is important from a methodological viewpoint. Examples of symmetries arising naturally in economic models are the cyclical symmetry in overlapping generations models (see Tuinstra and Weddepohl, 1999) and the re#ectional symmetry in the cobweb model with heterogeneous beliefs and linear supply and demand curves (see Brock and Hommes, 1997).
After reviewing the ta(tonnement process in Section 2 we present our model and show results from numerical simulations in Section 3. In Section 4 we introduce symmetry. In Section 5 we look at the generic local bifurcations of the equilibrium price vector given a certain symmetry. Among other things we see that the so-called symmetry breaking bifurcations can occur. In Section 6 the global dynamics of a symmetric ta(tonnement process is studied. In Section 7 an asymmetric ta(tonnement process which is in some sense close to a symmetric process is analysed to illustrate the relevance of studying symmetric processes for processes that have no nontrivial symmetries. Section 8 summarizes. Proofs of the main results are given in the appendix.
2. The ta(tonnement process
We study a price adjustment process in a competitive exchange economy. Let there bemconsumers andncommodities. Each consumer takes prices as given and solves the optimization problem
max
xi1,2,xin
;
1Herex@denotes the transpose of the vector (or matrix)x. subject to his or her budget constraint
n
i(.) is a strictly quasi-concave, strictly monotonic utility function of
consumeri,x
ijthe amount of goodjconsumed by consumeri,pjthe price of
good j and w
ij the initial endowment of good j owned by consumer i. Let
xH
ij"xij(p,wi) be the utility maximizing demand for goodjby consumeri, where p"(p
1,2,pn)@ is the price vector and wi"(wi1,2,win)@ the vector of initial
endowments of consumeri.1
Aggregate excess demand for goodjis
z
The economy is in equilibrium at the equilibrium price vectorpHifz
j(pH)"0
for allj. The excess demand functions satisfy Walras'Law,+nj/1p
jzj(p)"0, and
are homogeneous of degree zero in prices, i.e.z
j(ap)"zj(p), ∀a'0. Di!
erenti-ating this last equality with respect to a at a"1 gives the Euler equation:
+nk/1p
kLzj(p)/Lpk"0.
It is commonly believed (for example expressed by the notion of thelaw of supply and demand) that there are economic forces that drive the economy towards equilibrium. We study the well-known ta(tonnement process which is the simplest way to model these equilibrating forces and which can serve as a"rst approximation to more realistic adjustment processes. This process can be summarized by the following two characteristics.
f Trade occurs if and only if all markets are in equilibrium.
f If the economy is not in equilibrium at a certain price vector, prices are
adjusted according to the following rule. If there is excess demand for a good its price goes up, if there is excess supply for a good its price goes down and if the market for a good is in equilibrium its price remains the same.
2For small values ofnthis assumption can be relaxed: forn"2 the ta(tonnement process is always stable and forn"3 a su$cient condition for stability of the ta(tonnement process dp/dt"z(p) is the so-called weak law of market demand, which states that+ni
/1(Lzi/Lpi)(p)(0 for allp(Keenan and Rader, 1985).
The following question arises: who adjusts the prices? In the perfect competi-tive economy described above all agents take prices as given in determining their optimal consumption. If the economy is not in equilibrium there are pro"t opportunities for economic agents since they can bene"t from charging a higher price for goods in excess demand. Therefore, it can be argued that perfect competition is only a good description of economic behaviour if the economy is in equilibrium (see Arrow, 1959). The traditional solution to this problem is the introduction of an auctioneer who sets prices. The ta(tonnement process now works as follows: the auctioneer states an arbitrary price vector and collects information on the aggregate excess demands given this price vector. He then adjusts the price vector according to the rule stated above. This process is repeated until an equilibrium price vector is found. In this interpretation the discrete adjustment process (4) seems to be the most appropriate way to describe the ta(tonnement process. Obviously the ta(tonnement process with an auction-eer is a very crude approximation of the adjustment processes in reality.
The continuous ta(tonnement process has been extensively studied in the literature (see Arrow and Hahn, 1971). The most important "nding is that a su$cient condition for global stability is that all goods are gross substitutes, that is: (Lz
j/Lpk)(p)'0, ∀j, ∀kOj,∀p.2 This condition also implies that the
equilibrium price vector is unique.
This is a rather severe condition. There are some well-known examples (e.g. Scarf, 1960) for which this condition does not hold and for which the continuous ta(tonnement process does not converge to an equilibrium price vector. Notice that the stability of the equilibrium price vector only depends on the aggregate demand functions and not on the speci"cation of the adjustment ruleF( . ).
For the discrete ta(tonnement process the situation is even worse. Stability of the competitive equilibrium is determined by the properties of the aggregate excess demand functions as well as the speci"c form of the adjustment rule.
3. The model
3.1. Price normalization
The ta(tonnement process (4) is a discreten-dimensional dynamical system. From the homogeneity of the aggregate excess demand functions we have that if
pHis an equilibrium price vector then so is apHfor any a'0. Instead of an equilibrium price vector we have a rayof equilibrium price vectors. None of these equilibria can be locally stable since every neighbourhood inRn`of such an equilibrium point contains other equilibrium points. The ta(tonnement process then seems to be ill-de"ned.
The ta(tonnement process can be restricted to ann!1 dimensional subspace ofRn`by imposing an extra condition on the prices. Such aprice normalization ruleg(p)"c, wherecis a constant, de"nes ann!1 dimensional manifold in
Rn`and has to be chosen in such a way that the intersection of this manifold with any ray of equilibrium price vectors consists of exactly one point. This intersec-tion point is the equilibrium price vector of the ta(tonnement process restricted to this manifold. Some well-known price normalization rules areg(p)"p
kfor
some "xed k,g(p)"+nj
/1pj and g(p)"+nj/1p2j, all with c"1, respectively,
corresponding to one good being chosen as numeraire, prices on the unit simplex and prices on the unit sphere. There is no economic motivation to choose a particular price normalization rule. In special cases some normaliz-ation rule seems obvious (e.g. money as a numeraire in a monetary economy), but no normalization rule is appropriate a priori.
Having chosen a normalization rule it subsequently has to be implemented. Assumeg(p)"ccan be rewritten asp
k"hk(p1,2,pk~1,pk`1,2,pn) for somek.
The ta(tonnement process restricted tog(p)"cthen becomes
p
j,t`1"pjt#Fj(pjt,zj(p1t,2,pk~1,t,hk( . ) ,pk`1,t,2,pnt)), jOk,
p
kt"hk(p1t,2,pk~1,t,pk`1,t,2,pnt).
(6)
So the dimension of the dynamical system is reduced by one ton!1. In fact thekth equation of motion has been deleted. From Walras'Law we have that if
n!1 markets are in equilibrium thenth market must also be in equilibrium, so an equilibrium price vector of (6) is also an equilibrium price vector of (4). The dynamics of the ta(tonnement process however depend crucially on this reduc-tion. Which equation is deleted obviously in#uences the dynamical system. The ta(tonnement process withp
1as numeraire may have di!erent stability
proper-ties than the process withp
nas numeraire. There is&goods discrimination'in the
sense that the price of one, arbitrarily chosen, good is treated di!erently than the prices of the other goods.
In some cases there is a nice solution to this last problem. If for some ta(tonnement process p
t`1"f(pt) there exists a price normalization rule g(.)
such thatg(f
1(p),f2(p),2,fn(p))"g(p1,p2,2,pn) then the ta(tonnement process
3This is not a restrictive assumption since we can always normalize the unit of goods such that the speeds of adjustment are equal across markets.
4For the sake of simplicity we have not imposed a nonnegativity constraint on the prices. If the speed of adjustment is not too large, sayj(j1, then for initial price vectors chosen not too close to the border of the simplex the ta(tonnement process stays on the simplex. Ifj'j1then for almost all initial price vectors the price of some good becomes negative in"nite time and the ta(tonnement process breaks down. However, all kinds of interesting dynamics can be observed forj(j1. motion of the ta(tonnement process are not independent. In fact, there is one redundant equation and any equation can be replaced by the normalization rule without a!ecting the dynamical system: the systems (6) and (4) are equivalent and no information is lost by imposingg(.)"c.
For the continuous time case such a normalization rule exists for the linear ta(tonnement process dp
i/dt"jzi(p(t)). We have: d/dt+pi2(t)"2+pi(t)dpi/dt"
2j+p
i(t)zi(p(t))"0, due to Walras' Law, so the linear ta(tonnement process
leaves the sphere invariant.
In the case of the discrete ta(tonnement process a similar price normalization rule exists for the so-called multiplicative ta(tonnement process
p
j,t`1"pjt(1#jzj(pt)), j"1,2,n, (7)
where j is the speed of adjustment and is assumed to be the same for all markets.3Summation of the prices gives (using Walras'Law)
n
Hence the multiplicative ta(tonnement process leaves the simplex invariant. It can be argued that (7) is a reasonable way to model price adjustments. It states that the rate of change of the price of a good is proportional to its excess demand and it is homogeneous of degree one in prices. From now on we shall use (7) to study the dynamics of price adjustments.4
3.2. Model specixcations
In order to analyse the global dynamics of our ta(tonnement process we have to specify utility functions and endowments. We will study an economy with three commodities and three consumers that have Constant Elasticity of Substi-tution (CES) utility functions, i.e. the utility function of consumeriis
The corresponding individual demand functions are
with p"1/(1!o) the (constant) elasticity of substitution. The individual de-mand functions are homogeneous of degree zero in pricespand in preference parametersa
i1, ai2 andai3and homogeneous of degree one in initial
endow-ments w
i. Under the assumption that aii'0 we can take, without loss of
generality,a
ii"1.
We assume that the initial endowments are of the following form:w
ij"1 if
i"jandw
ij"0 ifiOj. So each consumer supplies exactly one good and is the
sole supplier of that good. This restriction is made for computational simplicity and does not in#uence the qualitative results. The multiplicative price adjust-ment process now becomes
This is the dynamical system under study. The parameters of the model are
a
ij, p and j. We will study the behaviour of the dynamical system for given
preference parameters a
ij and substitution of elasticity p, as j increases. We
focus ona
ij3(0,1] andp3(0,1].
3.3. Some typical numerical simulations
Before analysing our ta(tonnement process in more detail we present some numerical results. We are interested in the long-run behaviour of our dynamical system: do prices settle down to an equilibrium price vector or is the discrete ta(tonnement process unstable and can prices#uctuate forever? To describe the long-run behaviour of the dynamical system we use the following de"nition (Guckenheimer and Holmes, 1983, p. 36).
De,nition1. A compact setAis called an attractorof a dynamical systemfif: (1) Ais invariant with respect tof:f(A)LA.
(2) Ais attracting, that is there is an open neighbourhoodUofAsuch that the trajectory of every initial statep3Uconverges toA.
Fig. 1. Attractors of the ta(tonnement process with rotational symmetry. (a)a12"a23"a31"1,
a13"a21"a32"34,p"34andj"3.11. (b)a12"a23"a31"1,a13"a21"a32"14,p"14and
j"8.45.
Fig. 2. Attractors of the ta(tonnement process with reflectional symmetry. (a) a12"a21"14,
a13"a23"a31"a32"12, p"101 and j"32. (b) a12"a21"201, a13"a23"a31"a32"14,
p"101 andj"44.2.
5To plot the pictures in two dimensions we have transformed the simplex inR3`into a triangle in
R2. This transformation is also used in the proof of Proposition 5 (in the appendix).
The simplest example of an attractor is a locally stable"xed point. Another simple example is an attracting periodic orbit. In that case we havefn(p)"pand
fk(p)Op,k(n. However, more complicated quasi-periodic attractors or at-tractors with a fractal structure can arise. This last type of atat-tractors is some-times calledstrange orchaotic since the dynamics on these attractors may be unpredictable and exhibitsensitive dependence on initial conditions.
Fig. 3. Attractors of the ta(tonnement process with rotational and re#ectional symmetry. (a)aij"1,
alli,j,p"101 andj"26. (b)a12"a13"a21"a23"a31"a32"201,p"101 andj"66.2.
for di!erent values of the preference parametersa
ij, the elasticity of substitution
pand the speed of adjustmentj.
Notice that the attractors have a certainsymmetry. In particular, the pictures in Fig. 1 have a rotation symmetry: rotating the pictures over an angle of
2
3presults in exactly the same pictures. The pictures in Fig. 2 have areyection
symmetry: re#ection of the pictures in Fig. 2 in the line withp
1"p2results in the
same pictures. The attractors in Fig. 3 have a rotation as well as a re#ection symmetry: the pictures are invariant both to a rotation over23pand to a re# ec-tion in the line withp
1"p2.
The prices move erratically across these chaotic attractors. The attractors show the long-run behaviour of the prices of the ta(tonnement process. From these simulations it is clear that for many parameter settings the ta(tonnement process will not settle down to an equilibrium price vector but will keep on
#uctuating forever. In the subsequent sections we will investigate bifurcations routes leading to these strange attractors. Symmetry will play an important role in characterizing these bifurcation routes.
4. Symmetry
The number of parameters of our model (10) is 20 (a
ij,wij,jandp). To study
the global dynamics of this model in a transparent manner we have to reduce this number. A"rst reduction is achieved by normalizing the preference para-meters, such that a
ii"1, and by taking endowmentswii"1 and wij"0 for
endowments and preferences. In Section 7 we discuss properties of close-by asymmetric cases. There is a large literature on symmetry in dynamical systems (see for example Golubitsky et al., 1988). We will apply concepts and results from this literature to our multiplicative ta(tonnement process.
Consider the following dynamical system on the two-dimensional simplex:
p
The simplex is invariant with respect to any permutation of prices, so the group of symmetries that is interesting to study for the ta(tonnement process on the simplex is the group of permutation matrices D
3"MM123,M132,M321,
123"Iis a trivial symmetry of all dynamical systems. Apart from
IandD
3there are four di!erent subgroups ofD3,MI,M132N,MI,M321N,MI,M213N
and MI,M
312,M231N. The"rst three of these are the so-calledZ2symmetries:
they correspond to a permutation of two of the three prices or equivalently a re#ection in one of the symmetry axes of the simplex. These symmetry axes are:
l
1"Mp3S2Dp2"p3N,l2"Mp3S2Dp1"p3N and l3"Mp3S2Dp1"p2N. The last
subgroup is theR
3symmetry and corresponds to a cyclical permutation of the
three prices, or equivalently a rotation over23p.
We can now de"ne the symmetry group of a price adjustment processfon the simplex as follows:Cf"MM3D
3DM"f"f"MN. By choosing suitable preferences
and endowments we will study a price adjustment process with aZ
2symmetry,
one with aR
3symmetry and a price adjustment process with both symmetries,
which is equivalent to symmetrygroupD
3. So we study ta(tonnement processes
with three di!erent symmetrygroups:Cf"MI,M
213N,Cf"MI,M312,M231Nand
Cf"D
3. These symmetry groups correspond respectively to ta(tonnement
pro-cesses that are invariant with respect to a re#ective, a cyclic or all possible permutations of prices.
De,nition2. Thexxed point subspace Fix(M) of a symmetryMis the set
The"xed point subspace of the symmetryMplays an important role in the bifurcations of the equilibrium price vector in a symmetric ta(tonnement process.
Fix(M) is a linear subspace that consists of all points of the simplex that are invariant under symmetryM.Fix(M) is a line ifMis a re#ectional symmetry and
sof(p)3 Fix(M). This means that anonlineardynamical system with a symmetry can have alinearinvariant space. This property will prove to be very useful when we study the bifurcations of ta(tonnement processes with a re#ectional symmetry. This property also implies that (1,1,1) is an equilibrium price vector of all ta(tonnement processes that have a rotational symmetry, sinceFix(M
312)"(1, 1, 1).
Now we want to impose symmetry on our economy with CES-utility func-tions. The following proposition establishes how we can construct a ta( tonne-ment process with a rotational and a re#ectional symmetry, respectively, if there are three agents and three commodities.
Proposition 3. Let ;,<:R3
f Consider an economy with three consumers with utility functions ;
1(x)"
;(x),;
2(x)";(M231x)and;3(x)";(M2231x)andvectors of initial
endow-mentsw
1"w,w2"M@231wandw3"(M@231)2w,respectively. Then the
corre-sponding multiplicative taLtonnement process has rotation symmetryM
231.
f Consider an economy with three consumers with utility functions ;
1(x)";(x), ;2(x)";(M213x) and;3(x)"<(x) andvectors of initial
en-dowmentsw
1"w,w2"M213wandw3"*,respectively.Then the
correspond-ing multiplicative taLtonnement process has reyectional symmetryM
213.
This result gives us a way to impose symmetry on our price adjustment process with CES utility functions. LetA"(a
ij) be the matrix of preferences. We then have
Corollary4. For an economy with three commodities and three agents with CES utility functions and initial endowmentsw
ii"1andwij"0foriOjwe have that the following types of preference matrices
6In this paper generic means with respect to the set of dynamical systems with a given symmetry group.
induce respectively a rotational,a reyectional and both kinds of symmetries of the multiplicative price adjustment process.
The attractors shown in Section 3.3 were obtained by using the matrices (13). The attractors in Figs. 1}3 obviously exhibit the same symmetry as the corre-sponding ta(tonnement processes. Rotating the pictures in Fig. 1 over23presults in exactly the same pictures and re#ection of the attractors in Fig. 2 in the line with p
1"p2 leaves the attractors invariant. For the pictures in Fig. 3 any
permutation of the prices can be executed without changing the picture. The symmetry of the ta(tonnement process seems to determine the symmetry of the resulting attractor.
Iffhas symmetryMandp
0converges to an attractorAthen obviouslyMp0
converges to an attractorMA. If they converge to the same attractor, that is if
MA"A, then this attractor can be said to have symmetryM. So we can de"ne the symmetrygroup of an attractor of a dynamical system with symmetrygroup Cf as RA"MM3C
fDMA"AN. For the attractors of Section 3.3 we have
RA"C
f. This might suggest that the attractor always has the same
symmetryg-roup as the dynamical system, but in general that need not be the case. We can haveMAOAand then we must have coexistence of attractors. These coexisting attractors are related by symmetry and are called conjugateattractors. In the sequel we shall encounter symmetry breaking bifurcations, where a single attractor breaks up into multiple coexisting conjugate attractors with a smaller symmetrygroup, and symmetry increasing bifurcations where coexisting conju-gate attractors merge into one single (symmetric) attractor.
5. Local bifurcation analysis
In this section we study which local bifurcations at the equilibrium price vector can occur in our ta(tonnement process. In Section 5.1 we show that the symmetry of the ta(tonnement process determines the form of the Jacobian matrix. In Sections 5.2}5.4 we study the implications of this result for the generic6 bifurcations that occur for the ta(tonnement processes with di!erent symmetries.
5.1. Symmetry and the Jacobian matrix
To study the local bifurcations of the equilibrium price vector that can occur in our ta(tonnement process we have to look at the eigenvaluesk
matrix J5((Lfi/Lpj)(p)) evaluated at the equilibrium price vector pH. These
eigenvalues depend on the parameters of the ta(tonnement process, so we have
k
i"ki(j,p,A) whereAis the matrix with preference parameters. The
equilib-rium price vectorpHis locally stable if all eigenvalues lie inside the unit circle and
pHis unstable when at least one eigenvalue lies outside the unit circle. If an eigenvalue crosses the unit circle, as a parameter varies, a bifurcation occurs. At such a bifurcation the behaviour of the ta(tonnement process changes. There are three di!erent ways in which an eigenvalue can cross the unit circle. At the bifurcation value an eigenvalue can be real and equal to#1, real and equal to
!1 or two eigenvalues can be complex conjugates with absolute value 1. These three di!erent cases correspond to di!erent kinds of bifurcations.
First we show that symmetry induces a special form of the Jacobian matrix evaluated in the (symmetric) equilibrium price vector.
Proposition5. The Jacobian of the two-dimensional taLtonnement process evaluated at a symmetric equilibrium has eigenvalues that are
f real if the taLtonnement process has aZ
2symmetry.In this case one eigenvector
lies in thexxed point subspace of theZ
2symmetry and the other eigenvector lies
perpendicular to thisxxed point subspace,
f complex conjugates if the taLtonnement process has anR
3symmetry,
f real and equal to each other if the taLtonnement process has both symmetries.
From this proposition it follows that the symmetry of the ta(tonnement process determines which type of primary bifurcation occurs as the symmetric equilibrium price vector loses stability. We now derive the eigenvalues of the Jacobian evaluated at the symmetric equilibrium for each symmetry. The Jacobian of our ta(tonnement process (7) evaluated in the equilibrium price vectorpHis
JH"I#j
A
pH1z
11(pH) pH1z12(pH) pH1z13(pH)
pH2z
21(pH) pH2z22(pH) pH2z23(pH)
pH3z
31(pH) pH3z32(pH) pH3z33(pH)
B
, (14)
where z
ij(p),Lzi(p)/Lpj. Notice that the matrix (pHizij(pH))i,j/1,2,3 has linear
dependent rows (+3
i/1pHizij(pH)"!zi(pH)"0 from Walras' Law) which
im-plies that one eigenvalue of JHis equal to 1 for all possible parameter values. This unit eigenvalue corresponds to the fact that every simplex is invariant under the multiplicative ta(tonnement process. Since we are only interested in the motion of the dynamical system on the simplex we can safely ignore this eigenvalue. The other two eigenvalues are
k1,2"1#1
2jMpH1(z11!z13)#pH2(z22!z23)$JDN,
D"[pH1(z
11!z13)!pH2(z22!z23)]2#4pH1pH2(z12!z13)(z21!z23).
Table 1
Eigenvalues of (7) given the symmetry group
Cf k
1,2
R
3"MI,M312N 1#12jM3z
11(pH)$(z12(pH)!z13(pH))J3iN Z
2"MI,M213N 1#jpH1Mz11(pH)!z13(pH)$(z12(pH)!z13(pH))N
D
3"MI,M312,M213N 1#32jz11(pH)
Under symmetry the expressions for these eigenvalues simplify considerably. These expressions are summarized in Table 1.
Notice that if C
f"R3 or Cf"D3 and z11(pH)'0 both eigenvalues lay
outside the unit circle for allj'0. Hence, in that case the equilibrium price vector is unstable for all positive values ofj.
In the following subsections we analyse which bifurcations occur for each of the symmetrygroups, as the speed of adjustmentjincreases.
5.2. Rotational symmetry and the Hopf bifurcation
In the previous subsection we found that the eigenvalues of the Jacobian of the ta(tonnement process with rotational symmetry (and no other sym-metry) evaluated at the symmetric equilibrium price vector pH"(1, 1, 1) are complex conjugates. The equilibrium price vector loses stability if these eigen-values cross the unit circle. Let j"*& denote the value of j such that
Dk1(j"*&)D"Dk2(j"*&)D"1. We then have the following.
Proposition 6. Assume that the taLtonnement process with symmetry group
Cf"MI,M
312Nhas a stable symmetric equilibrium price vector for j'0 small
enough and that z
12(pH)Oz13(pH). Furthermore assume that (ki(j"*&))rO1 for
r"1, 2, 3, 4.Then a Hopf bifurcation occurs at
j"*&"! 4z11(pH)
3[z
11(pH)]2#[z12(pH)!z13(pH)]2
. (17)
Forjclose toj"*&there exists an invariant closed curve.
Fig. 4. Primary bifurcations for the ta(tonnement process with di!erent symmetry groups. (a) Invariant closed curve created in Hopf bifurcation for the ta(tonnement process with cyclical symmetry,a"1,b"12,p"12andj"4.5. (b) Period two orbit o!the symmetry axis created in period doubling bifurcation for the ta(tonnement process with re#ectional symmetry, a"34,
b"c"12,p"101 andj"25. (c) Period two orbit on symmetry axis created in period doubling
bifurcation for the ta(tonnement process with cyclical symmetry,a"14,b"c"12,p"101 andj"25. (d) Six period two orbits created in symmetry breaking period doubling bifurcation in ta(tonnement process with cyclical and re#ectional symmetry,a"1,p"1001 andj"220. The period two orbit indicated byi (i"1, 2, 3) is the period two orbit on the symmetry axisl
i, the period two orbit indicated byiH(iH"1, 2, 3) is the period two orbit o!, but symmetric with respect to,l
i.
5.3. Reyection symmetry and the period doubling bifurcation
Consider the ta(tonnement process with symmetry groupC
f"MI,M213N. The
"xed point subspace of this Z
2 symmetry is the symmetry axis l3. From
7These two types of period doubling bifurcations correspond to pitchfork bifurcations off2. One of these pitchfork bifurcations is symmetry preserving (k2"!1) and the other one is symmetry breaking (k
1"!1).
system evaluated in a point in the "xed point subspace are real and have eigenvectors that lay perpendicular to the"xed point subspace and in the"xed point subspace, respectively.
The equilibrium price vector can only lose stability when the largest (in absolute value) eigenvalue goes through$1. If an eigenvalue goes through#1 asaddle-node,pitchforkortranscritical bifurcationoccurs (see e.g. Guckenheimer and Holmes, 1983). All these bifurcations in some way deal with the multiplicity of equilibria. Since the equilibrium price vectors of the ta(tonnement process correspond to the zeros of the system of aggregate excess demand functions, the location and multiplicity of these equilibrium price vectors is independent of the speed of adjustmentj. Therefore we know that these bifurcations cannot occur in our model asjvaries. When one of the eigenvalues goes through!1 a yip-or period doubling bifurcation occurs: at this bifurcation a period two orbit emerges. If this period doubling bifurcation occurs atj"j"*&then we have the following scenario. For j(j"*&the equilibrium price vector is stable and for
j'j"*&the equilibrium price vector is unstable. Forjclose toj"*&a period two orbit exists. Numerical observations suggest that this period two orbit exists for
j'j"*&and that it is stable.
Proposition7. Assume that the equilibrium pricevector of the taLtonnement process with symmetry group Cf"MI,M
213Nis stable for j'0 small enough and that
z
12(pH)Oz13(pH).Then we have
f the equilibrium loses stability through a period doubling bifurcation at
j"*&"min
G
! 2pH1(z
11(pH)!z12(pH))
,! 2
pH1(z
11(pH)#z12(pH!2z13(pH))
H
,
(18)
f the resulting period two orbit has the same symmetry group as the taLtonnement process.The period two orbit lies in l
3if z12(pH)(z13(pH)and it lies ow but
symmetric with respect tol
3whenz12(pH)'z13(pH).7
According to Proposition 7 there are two di!erent types of period two orbits withZ
2symmetry. Examples of these two period two orbits are shown in Fig. 4b
and c. Since the three prices have to sum up to 3 in every period an increase of one price has to be accompanied by a decrease of another price. Consider the period two orbitMp1,p2Nin Fig. 4b that lies o!l
3. This period two orbit has the
the"rst two prices#uctuate between the same values: whenp
1is high,p2is low
and vice versa. In this case p
2 has to adjust to the #uctuations in p1. This
happens whenz
12(pH)'z13(pH).
The period two orbit in Fig. 4c that lies inl
3has the property that prices of the
"rst two goods are equal to each other in every period: in one period they are both high (above their equilibrium value) and in the next period they are both low (below their equilibrium value). In this casep
3has to adjust to neutralise the
#uctuations inp
1(andp2). This happens whenz13(pH)'z12(pH). So the burden
of a change inp
1iscompletelycarried by the price of that good that is the closest
substitute to good 1 (at the equilibrium price vector).
5.4. D
3symmetry and the equivariant branching lemma
In the previous sections we saw that a rotational symmetry leads to a Hopf bifurcation and that a re#ectional symmetry leads to a period doubling bifurca-tion of the equilibrium price vector when it loses stability. We now want to study the case where the adjustment process has both a rotational and a re#ectional symmetry. We can then apply the equivariant branching lemma for period doubling (Chossat and Golubitsky, 1988, see the appendix) to arrive at the following result.
Proposition8. Assume that the symmetric equilibrium pricevectorpH"(1, 1, 1)of the taLtonnement process with rotational and reyectional symmetrygroup Cf"D
3"MI,M312,M213Nis stable forj'0small enough.pHundergoes a
peri-od doubling bifurcation asjgoes through
j"*&"! 4
3z
11(pH)
. (19)
At thisvalue ofjsix period two orbits emerge,three stable and three unstable. For everyxxed point subspacel
i, i"1, 2, 3there is a period two orbit onliand a period two orbit owl
i but with a reyectional symmetry with respect toli.
For j'j"*& six period two orbits exist, three of which are stable. Fig. 4d
shows the six period two orbits for a ta(tonnement process withD
3symmetry
(a"1,p" 1
100,j"220). In this case the three period two orbits o!the
8This particular bifurcation scenario corresponds to scenario 3 in Chossat and Golubitsky (1988). fact rotating each of these period two orbits over 23p gives one of the other period two orbits. Thus the symmetry of the three coexisting period two orbits is smaller than the symmetry of the equilibrium price vector. Therefore the bifurcation described in Proposition 8 is called asymmetry breakingbifurcation.
6. Global dynamics
In Section 5.4 we have seen that when the symmetric equilibrium price vector of a ta(tonnement process with both a re#ectional and a rotational symmetry loses stability three attracting and coexisting period two orbits emerge. Each of these new attractors only has a Z
2 symmetry. The symmetrygroup of the
attractor of the ta(tonnement process decreases from MI,M
213,M312Nto either
MI,M
213N,MI,M321NorMI,M132N. However, the attractors in Fig. 3 which come
from ta(tonnement processes with D
3 symmetry are fully symmetric. So as
jincreases at a certain point the symmetrygroup of the attractor has to increase again. This happens at asymmetry increasingbifurcation. Such bifurcations seem to be generic in ta(tonnement processes withD
3symmetry.
Consider one of the stable period two orbits that emerge after the symmetry breaking period doubling bifurcation. This period two orbit undergoes a bifur-cation route to chaos, resulting in a Z
2-symmetric strange attractor. By
sym-metry there are two other, conjugateZ
2-symmetric strange attractors. As the
speed of adjustment increases these attractors grow in magnitude. At a certain point these attractors withZ
2symmetry merge into one attractor which inherits
all symmetries of the conjugate attractors and therefore hasD
3symmetry again.
Fig. 5 illustrates such a bifurcation scenario.8After the symmetry breaking period doubling bifurcation there are three stable period two orbits (one of them is shown in Fig. 5a). The period two orbit undergoes a Hopf bifurcation and an invariant set consisting of two closed curves emerges. More important for the global dynamics is the emergence of a stable period six orbit in the neighbour-hood of each stable period two orbit. This period six orbit emerges through a saddle-node bifurcation of f6. Each of these stable period six orbits undergoes a Hopf bifurcation resulting in an attracting set consisting of six closed curves (Fig. 5b). From these six closed curves a two piece strange attractor is created (Fig. 5c). There are three of these strange attractors. If the speed of adjustment grows a little more a symmetry increasing bifurcation occurs and these three attractors merge into a large attractor withD
3symmetry (Fig. 5d).
This kind of bifurcation scenario (from an equilibrium price vector with
D
3 symmetry to three attractors with Z2 symmetry and then again to an
attractor withD
Fig. 5. Bifurcation scenario for ta(tonnement process withD
3symmetry,a"1,p"1001. (a)j"220,
(b)j"258, (c)j"260, (d)j"261.
D
3symmetry. However, Tuinstra (1999) gives examples where these symmetry
breaking and increasing bifurcation routes also occur for systems with only aZ
2orR3symmetry (see Tuinstra, 1999).
7. An asymmetric price adjustment process
D
3symmetry. Then, as the equilibrium price vector loses stability, three
coexist-ing attractcoexist-ing period two orbits emerge. Obviously, a price adjustment process with aD
3symmetry is very special. The question thus arises whether the analysis
of a special symmetric ta(tonnement process is useful in order to understand the global dynamics of general asymmetric ta(tonnement processes. In this section we show that nearby asymmetric ta(tonnement processes exhibit bifurcation routes to strange attractors very similar to the special symmetric bifurcation routes. Therefore, symmetric ta(tonnement processes can serve as an&organizing centre'in the analysis of the global dynamics in general ta(tonnement processes.
We study a numerical example with the following parameter values
A"
A
1 0.95 0.9
1 1 1
1 1 1
B
, p" 1
100 (20)
and initial endowments as before.
Notice that the multiplicative price adjustment process with these preferences has no nontrivial symmetry, but that it is close to a price adjustment process withD
3symmetry.
The equilibrium price vector is pH"(1.0177656, 1.0002052, 0.9820292) and the eigenvalues of the Jacobian evaluated at this equilibrium price vector are
k1"1!0.00982501jandk2"1! 1
100j, respectively. A period doubling
bifur-cation occurs at j"*&"200 resulting in a stable period two orbit. Only one period two orbit emerges. However, asjincreases to j4/1+228.2 the second iterate of the price adjustment process,f2, undergoes a saddle-node bifurcation and two additional period two orbits emerge, one stable and one unstable. As
j increases even more to j4/2+237.0 f2 undergoes a second saddle-node bifurcation again resulting in another stable and unstable period two orbit. Just as in the price adjustment process with D
3symmetry three stable period two
orbits coexist in the asymmetric ta(tonnement process. In this case however, these period two orbits are not created through a single bifurcation at a single value of the bifurcation parameterj, but through a series of three independent bifurcations at di!erent values ofj. Fig. 6 shows two bifurcation diagrams that were created with the software package LOCBiF (Khibnik et al., 1992). Fig. 6a shows the three stable period two orbits created in the symmetry breaking period doubling bifurcation for the ta(tonnement process with D
3 symmetry
(a
ij"1,p"1001 ). Fig. 6b shows three stable period two orbits and three
unsta-ble period two orbits created in a period doubling and two subsequent saddle-node bifurcations for the asymmetric ta(tonnement process described above.
Fig. 6. Bifurcation diagrams for symmetric and nearly symmetric ta(tonnement processes. Horizon-tal axis:13(2p
1#p2), vertical axisj. (a) Stable period two curves for ta(tonnement process with D
3symmetry, (aij"1,p"1001), created through symmetry breaking period doubling bifurcation. (b) Period two curves in a ta(tonnement process close toD
3symmetry (aij"1,a12"0.95,a13"0.9,
p"1001).
attractors forj"259. Asjincreases more the three coexisting attractors merge, just as in the symmetric case, to one large attractor. This attractor is shown in Fig. 7d forj"265.
This example illustrates that the bifurcation scenario of the price adjustment process with aD
3symmetry is useful in understanding the behaviour of a price
adjustment process that has no nontrivial symmetry but that is close to a price adjustment process withD
3symmetry.
8. Summary and conclusions
Price normalization plays an important role in the dynamics of the ta( tonne-ment process. The ta(tonnement process can be stable under one normalization rule and unstable under another. We have argued that for the multiplicative price adjustment process, using the simplex as a normalization rule is natural since the multiplicative price adjustment process leaves the simplex invariant.
Fig. 7. Attractors for ta(tonnement process close toD
3symmetry,aij"1,a12"0.95, a13"0.9,
p"1001. (a)}(c) Three di!erent coexisting attractors forj"259. (d) Unique attractor forj"265.
bifurcations where the symmetric equilibrium price vector becomes unstable and three coexisting stable period two orbits emerge.
example in Section 7 shows that the bifurcation scenario in a symmetric ta(tonnement process reveals information about possible bifurcation scenario's in general asymmetric ta(tonnement processes.
Appendix
In this appendix proofs of the main results are given.
Proof of Proposition3. Let the utility maximizing consumption bundle be given by
x"dU(p,w)"arg max x|R3`
M;(x)Dp@x4p@wN.
Consider the problem of maximizing;(Mx) subject to the budget constraint p@x4p@(M@w), whereM3MM
231,M213N. SinceM~1"M@, this problem is
equiv-alent with maximizing;(Mx) subject to (Mp)@Mx4(Mp)@w. The solution to this
last problem is easily seen to be
x"M@d(Mp,w).
f First consider M"M
231. Notice that M2231"M@231 and M3231"I. The
individual demand functions become x
k`1"(M@231)kdU(Mk231p,w), with
k"0, 1, 2. The aggregate excess demand functions z(p)"+3
i/1xi!+3i/1wi
can then be written as
z(p,w)"
A
dU1(p,w)#dU2(M2231p,w)#dU3(M
231p,w)!w1!w2!w3
dU1(M
231p,w)#dU2(p,w)#d3U(M2231p,w)!w1!w2!w3
dU1(M2231p,w)#dU2(M
231p,w)#dU3(p,w)!w1!w2!w3
B
.
It can be easily checked that we havez(M
231p,w)"M231z(p,w). This means
thatM
231is a symmetry of the aggregate excess demand functions.
f Now consider M"M
213. The demand functions are respectively
x
1"dU(p,w), x2"M213dU(M213p,w) and x3"dV(p,*) with dV2(p,*)"
dV
1(M213p,*) anddV3(M213p,*)"dV3(p,*). The aggregate excess demand
func-tions become
z(p,w)"
A
dU1(p,w)#dU2(M
213p,w)#dV1(p,*)!w1!w2!v1
dU1(M
213p,w)#dU2(p,w)#d1V(M213p,*)!w1!w2!v1
dU3(p,w)#dU3(M
213p,w)#dV3(p,*)!2w3!v3
B
.
We haveM
Now if the aggregate demand functions have symmetry M
231then for the
multiplicative price adjustment process we have
A
fand if the aggregate demand functions have symmetry M213 then for the multiplicative price adjustment process we have
A
fProof of Proposition5. Without loss of generality we can apply a linear trans-formation to the prices to transform the simplex inR3to a triangle in the plane. Letq"Tpwith
The price adjustment process in terms of these new variables becomes
q
Notice that T transforms the centre of the simplex to the origin and the
Mzcorresponds to a re#ection in theq2axis andMrcorresponds to a rotation over23p. With this transformation of variables it is easy to establish our result. LetMbe a symmetry ofh. Then by de"nition we haveMh(q)"h(Mq). Di! eren-tiating both sides with respect toqgives
MLh
Lq@(q)" Lh
Lq@(Mq)M.
Now ifqHis symmetric (that isMqH"qH) we have
MJH
h"JHhM.
So the Jacobian evaluated at the symmetric equilibrium commutes with the symmetries of the dynamical system. An easy calculation shows that the only 2]2 matrices commuting withMzand withMrareJzandJr, respectively with
Jz"
A
a 0
0 b
B
, Jr"A
a b
!b a
B
.The eigenvalues ofJzareaandbwith eigenvectorsla"(1, 0)@andlb"(0, 1)@, respectively, the eigenvalues ofJrare a$bi and therefore complex as long as
bO0. Note that if the dynamical system has both symmetries we have
JD
3"
A
a 0 0 a
B
"aI.The statements of the proposition follow. h
Proof of Proposition7. The eigenvalues are
k
1"1#jpH1(z11(pH)!z12(pH)),
k2"1#jpH1(z
11(pH)#z12(pH)!2z13(pH))
and the corresponding eigenvectorsl1andl2lay perpendicular tol
3and inl3,
respectively. If an eigenvalue goes through!1 a period doubling bifurcation occurs in the dynamical system restricted to the centre manifold (see Gucken-heimer and Holmes, 1983, p. 158). The centre manifold is locallyZ
2-symmetric
(Kuznetsov, 1995, Theorem 7.6). When k1"!1 the centre manifold lies tangent to l1 and therefore perpendicular to the"xed point subspace. When
k2"!1 the centre manifold coincides with the"xed point subspace. We have
k1(k2ifz
12(pH)'z13(pH) andk1'k2ifz12(pH)(z13(pH). h
¹heorem9 (Equivariant Branching Lemma for period doubling). Consider a dy-namical systemh:RnPRnwith a symmetry groupCh.Assume thatChacts absolute-ly irreducibabsolute-ly on Rn, that JH
h"!In and that dimFix(M)"1, for a subgroup
MLC
h=M$InN.Then generically,there exists a branch of period two points for
hbifurcating at thexxed point and tangent toFix(M)at thexxed point.IfMLC h, then the branch lies inFix(M).
Proof(Sketch). The symmetrygroupChis said to be actingabsolutely irreducible
onRnif the only matrices commuting with all symmetries ofChare multiples of the identity matrix. Then at a bifurcation all eigenvalues are equal to#1 or all eigenvalues are equal to!1. For a good understanding of the bifurcation that occurs when all eigenvalues are equal to !1, it is useful to discuss the case where all eigenvalues are #1 "rst. Recall that the "xed point subspace of a symmetry is an invariant space of the dynamical system. Now if h( . ) has a symmetry with a one-dimensional "xed point subspace then the search for
"xed points can be con"ned to this "xed point subspace. This gives the
Equivariant Branching Lemmawhich states that for every subgroup ofC
hthat has
a one-dimensional"xed point subspace there exists a branch of"xed points of
h bifurcating at the origin. This branch lies in this "xed point subspace (for a discussion of this theorem see Golubitsky et al. (1988, Chapter XIII)).
If all eigenvalues are equal to !1 the analysis is more complicated. The normal form does not only commute with the symmetry groupChbut also with the linear part of the dynamical system which is!I
n(Chossat and Golubitsky,
1988). Therefore the normal form commutes with Ch=M$I
nN. Now we can
apply the same reasoning as above, only we have to distinguish between subgroups with a one-dimensional"xed point subspace that are inC
hand ones
that are inC
h=M$InNbut not inCh. Therefore, there exists a branch of period
two points bifurcating at the "xed point and tangent to Fix(M), for each
M3C
h=M$InNwith dimFix(M)"1.
Proof of Proposition8. From Proposition 5 it follows that the symmetrygroup
D
3acts absolutely irreducible onR3`. Furthermore,D3=M$I3Ncontains six
subgroups that have a one-dimensional"xed point subspace. Three of these are also inD
3, namely the threeZ2symmetries with"xed point subspacesl1,l2and
l
3, respectively. The three other "xed point subspaces lH1,lH2 and lH3 can be
obtained by rotatingl
1,l2andl3over12p. Application of Theorem 9 then gives
the result. h
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