EQUILIBRIUM PRICE BIFURCATION IN WALRAS PRICE FORMATION MODEL WITH DELAYS. Natalia K. Obrosova

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1 EQUILIBRIUM PRICE BIFURCATION IN WALRAS PRICE FORMATION MODEL WITH DELAYS Natalia K. Obrosova Dorodniyn Computin Centre of the Russian Aademy of Siene Vavilov st.40, 999 Mosow GSP-, Russia Abstrat: The author investiates bifuration of equilibrium prie in prie formation model of Walras type with delays, finds the stability limits for equilibrium prie and ives their eonomi interpretation. The Hopf's bifuration theorem for the lass of differential equations with delay is proved. It is shown that in the ase of nonessential ommodity market a periodi orbit emeres or dies after loss of stability. The results of the investiation allow usin this model as a blok of a maroeonomi model for desribin endoenous rises ineption. Copyriht 005 IFAC Keywords: system analysis eonomi systems differential equations stability domains stability limits. INTRODUCTION The problem of modelin eonomi rises is well known in mathematial eonomy. However no satisfatory model desribin the emerene of suh rises has been offered. To analyze the reason of this fat onsider known methods of eonomy desription. A desription of eonomi phenomenon beins with seletion of harateristi times of proesses. While onstrutin eonomi models, it is usually onsidered that all proesses are divided into slow (prodution reoranization, modifiation of onsumer's preferenes) and fast (prie formation proesses), aordin to Marshall s hypothesis of times separation. Slow-time proesses are studied by maroeonomi theory. The modifiation of industrial apaities, prodution tehnoloies and onsumer's preferenes are maroeonomi proesses whih hane the struture of demand and supply. Thus, the form of demand and supply funtions hanes in slow time. While onstrutin maroeonomi models it is usually onsidered that fast proesses are in equilibrium state. Properties and stability of this equilibrium are onsidered in miroeonomi theory. Usually maro and miro models are not onneted with eah other. Miroeonomi equilibrium models in fast time desribe the results of interations between demand and supply in the market on harateristi times, when preferenes of onsumers, prodution apaities and tehnoloies are onsidered to be a onstant. Therefore, in fast proesses supply and demand funtions are known and depend on time impliitly. Under these onditions demand for the oods and their supply are adjusted to demand and supply pries only. Let eonomi proesses be separated under their harateristi times. Let x be a vetor of slow variables (maro variables) and y be vetor of fast variables (miro variables). Then the shemati representation of eonomi system is the followin Tyhonov's system: dx = f ( xy, ) dy ε ( ) = xy,, ()

2 where ε is a small parameter. The first equation () desribes slow proesses, and the seond - fast proesses (Petrov, et al., 996). Let the left part of the seond equation of () be equal to zero. Then () turns to the followin system: dx = f ( xy, ) ( xy, ) = 0. () Aordin to the Tyhonov s theorem, the solutions of () approximate solutions of () when the dy stationary solution y( x ) of ε = ( x, y ) is asymptotially stable. Substitutin the stationary solution y( x ) to the first equation in () makes it lear that the eonomi proess is desribed by the equation dx f ( ( )) = x, y x. (3) The model (3) represents the eneral sheme of lassial maroeonomi model. The idea of modelin of emerin rises onsists in onstrution of maroeonomi model based on miro desriptions. The model should desribe stability loss of fast relaxation proesses by modifiation of slow parameters. As a result a ompliated prie formation dynami is obtained so that eonomi aents an t foreast their ativity. Thus, the obtained model of emerin eonomi rises reflets atual proesses of eonomi development more authentially. The problem in this ase is to onstrut a prie formation model, in whih equilibrium prie loses stability due to the hane of slow parameters. The majority of maroeonomi models desribe dynami of ross domesti produt. Thus the onsistent prie formation model should desribe salar quantity - prie index. Grandmont (985) and Shananin (99) suested prie formation models in disrete time. Shananin (99) onsidered the disrete analo of elementary prie formation model of Walras type pn-c(p n- ) p =, where p is a prie of ommodity, n (p n- ) C( p ) is a demand funtion, ( p ) is a supply funtion. Shananin (99) analyzed the equilibrium prie stability and ave eonomi interpretation of stability loss for this model. Namely, the equilibrium prie loses stability when industrial apaity exeeds some ritial value. Also it was shown that a sequene of doublin period bifurations appears in the system after stability loss. So when apaities are lare enouh the behavior of the system beomes haoti and eonomi aents annot foreast their ativity reliably (overprodution risis). However, disrete models have essential defet. The time step in disrete models is not interpreted in any way. Besides, it is diffiult to use disrete prie formation models as a part of a maroeonomi model desribed in ontinuous time. Therefore, it is neessary to build the model in ontinuous time, whih inherits the properties of disrete model. The simplest model of Walras type, whih is a diret analo of disrete prie formation model, is not appropriate here, beause under usual assumptions about supply and demand funtions (i.e. demand dereases with rowth of the prie, and supply inreases with rowth of the prie) the equilibrium prie in this model is always stable. The elementary model of Walras type has the followin form dp C( p) ( p) = χ p, (4) ( p) where parameter χ > 0 has dimension and time haraterizes the speed of market reation to prie hane. The reason of instability appearane in disrete model is the impliit inertia in onsumer's and produer's reation to prie hanes. There are two ways of inertia modelin in ontinuous models. By the first way, suested by Lorenz (989), the inertia is modeled as follows: supply doesn't diretly depend upon prie, but supply's derivative does. Thus, in this model prie formation proess is desribed by a system of differential equations. However, parameters have unsatisfatory eonomi interpretation. In this paper another way is used. Delays in onsumer's and produer's reation are added to the model of Walras type. They reflet real market proesses and have ood eonomi interpretation. For example, produer's delay may be interpreted as harateristi time of prodution yle.. MODEL DESCRIPTION Consider homoeneous ommodity market. Aordin to the traditions, whih date bak to Marshall and Walras, we shall onsider the followin assumptions to be true: a) at every moment of time the ommodity is sold at the sinle prie p; b) onsumer's and produer's behavior is desribed by demand funtion C( p ) and supply funtion ( p ) respetively. Moreover, typial time of demand and supply funtions hanin is onsiderably larer than typial time of prie hanin. It means that these funtions do not expliitly depend on time. Under these assumptions the model of Walras type with delays has the followin form:

3 dp C( p( t τ )) ( p( t τ )) = χ p() t, ( p( t τ )) (5) where oeffiient χ > 0 has dimension and time haraterizes the speed of market reation to prie hane, τ 0 and τ 0 are onstant onsumer's and produer's delays respetively. Assume that ) the equilibrium prie exists in the model, i.e. equation C( p) = ( p) has a solution p = p, ) C( p ) and ( p ) are n-time differentiable ( n 9 ) in a neihborhood of p, 3) demand funtion C( p ) is steadily dereasin and supply funtion ( p ) is steadily inreasin, 4) funtions C( p ) and ( p ) are bounded for p INVESTIGATION OF EQUILIBRIUM PRICE STABILITY LIMITS Stability limits for the prie formation model (5) are found in terms of supply elastiity p d = and demand elastiity ( p) dp p= p p dc = alulated at equilibrium C( p) dp p= p prie p. These parameters are primary harateristis of prodution tehnoloy and onsumer's demand struture. The stability analysis is made with the help of Lyapunov method of system linearization. Chane of () variables θ τχ, θ τχ, xt ( ) ln p t = = =, time p salin and linearization in the neihborhood of x = 0 in (5) leads to the followin linear equation: where, 0. dx() t = xt ( θ) xt ( θ), (6) There are several problems in applyin the Lyapunov method for equations with delays. In this ase the harateristi equation is an analyti funtion and the investiation of real part sin of its root is not simple problem. This investiation is done in this paper for three ases: produer's delay only, onsumer's delay only and equal delays of both produer and onsumer. In order to analyze the sin of real part of the harateristi equation roots the followin lemma is proved. z Lemma. Quasipolynomial ϕ( z) = z+ ae θ with a, θ > 0 has no roots in the riht half-plane Re z > 0 if and only if 0 < aθ < π /. With the help of Lemma the followin proposition is proved. Proposition. In the ase of equal onsumer's and produer's delays, i.e. τ = τ = τ, the equilibrium prie p * is asymptotially stable if <, and π unstable if >, where = τχ. Analysis of the linearization (6) allows provin the followin results. Proposition. Consider the ase of produer's delay only, i.e. τ = 0, τ > 0. ) In the ase 0< < the equilibrium prie is asymptotially stable, if τχ < aros, and unstable if τχ > aros. ) In the ase > the equilibrium prie is asymptotially stable. Proposition 3. Consider the ase of onsumer's delay only, i.e. τ > 0, τ = 0. ) In the ase 0< < the equilibrium prie is asymptotially stable if τχ < aros, and unstable if τχ > aros. ) In the ase > the equilibrium prie is asymptotially stable. The stability limits obtained in Propositions -3 have an expliit parametri form on the plane ( vu, ), where v =Θ, u =Θ, and τχ if τ = 0, Θ= τχ if τ = 0, τχ if τ = τ = τ. (7)

4 v produer's delay only onsumer's delay only equal onsumer's and produer's delays α sν ( p) = M, p (8) where M is ross industrial apaity, α = + γ, µ where γ 0 is apaity rowth rate, µ 0 is the rate of apaity retirement. Parameter sν haraterizes the industry, moreover p > sν (the ondition of the best industry tehnoloy not to be unprofitable). The onsumer's demand is desribed by the model funtion u Fi.. Stability limits of equilibrium prie in the prie formation model (5). The shemati form of stability limits is shown on Fiure. In ases of produer's delay only and equal produer's and onsumer's delays the stability domain is under the orrespondin stability bound. In the ase of onsumer's delay only the stability domain is above the orrespondin stability bound. Analysis of stability limits shows that the equilibrium loses stability when the supply elastiity exeeds the ritial value in the ases of equal delays and produer's delay only. In the ase of onsumer's delay only there exists onstraint from below on supply elastiity. The advantae of the prie formation model (5) in omparison with the disrete models is the possibility to investiate the dependene between stability areas and delays values. Aordin to eonomist's apprehension, inreasin of delay value leads to stability areas dereasin. 4. ECONOMIC INTERPRETATION OF STABILITY LIMITS Consider an example that allows ivin an eonomi interpretation of the obtained results. Let the supply and demand funtions have an expliit form that shows their dependene from slow maroeonomi parameters. In the followin example the funtions of form suested by Shananin (99) is used. Supply funtion is based on the Houthakker-Johansen model of manufature and has the form β p C( p) = C, s ν (9) where onstant C is demand harateristi, β is a deree of ommodity neessity. For supply and demand funtions (8), (9), the supply elastiity alulated at equilibrium prie is M = α( ), where = and demand C( p ) elastiity is = β. Parameter haraterizes ratio between industrial apaities and equilibrium onsumer's demand. In eonomi theory, eonomi rises ineption is usually onneted with hane of the parameter.it follows from Propositions - that in the ases of both equal delays of produer and onsumer and produer's delay only the equilibrium prie loses stability when industrial apaity (parameter ) exeeds some ritial value. This fat aords to eonomists' opinion about auses of overprodution rises ineption. In the ase of onsumers delay only (Proposition 3) the onstraint from below on value appears. In this ase, the stability loss is the result of demand's inreasin, for example, in onsequene of investment ativity inreasin. The analysis of stability limits shows that inreasin of the industry rowth rate leads to dereasin of stability funds ("eonomy overheatin"). Dependene of the stability areas from onsumer's elastiity is obtained. In the example above, the onsumer's elastiity dereases when ommodity neessity deree β inreases. The stability area rows when inreases in the ase of produer's delay only and delines in ases of onsumer's delay only and equal produer's and onsumer's delays. Thus, the stability limits dependene on slow maroeonomi parameters and their eonomi interpretation are obtained.

5 5. ANALYSIS OF EQUILIBRIUM PRICE BIFURCATION TYPE In order to analyze the equilibrium prie bifuration in the model (5), the Hopf's bifuration theorem for the stationary solution xt ( ) 0 is proved in lass of differential equations with delay dx = H( x(), t x( t τ) ), τ > 0. (0) Let the funtion H ( xy, ) fulfil the followin onditions: ) H ( xyis, ) n-time differentiable ( n 9 ); ) for either ontinuous differentiable funtion yt ( ), t [0, τ ] the equation (0) with x( t τ ) = y( t) has a bounded solution on [0, τ ] ; H( x, y) 3) u = τ 0 and x (0,0) (0,0) H( x, y) v = τ 0. y The linearization of (0) arees with the linearization (6) in the ase of θ = 0 with auray to denotations and time sale. Thus, it follows from Proposition that in the ase of u = 0 the solution x = 0 of (0) is asymptotially stable if v < v and unstable if v > v, where π v =. () In the ase of u > 0 it follows from Proposition that the stability limit v of stationary solution x = 0 an be found from the equation v u = aros u v v. () π Theorem. Let u > 0 and u. When the 3 3 stationary solution x = 0 of (0) loses stability (i.e. parameter v inreases and rosses throuh the stability limit v found from ()) a periodi orbit emeres or dies in the system (0). The proof of Theorem onsists of four main steps. For bifuration analysis with the help of steps method (Bellman and Cooke, 963) the initial equation (0) is redued to disrete dynami system xn+ ( t) =Ψ µ ( xn()) t in funtional spae C [0,]. The mappin Ψµ : C [0,] C [0,] belons to C p - lass ( p 9 ) in zero neihborhood of the spae C [0,]. Parameter µ = v v. In this ase the fixed point x = 0 of the dynami system orresponds to the stationary solution x = 0 of (0). Now investiate the bifuration by µ in the ase of fixed u. On the seond step, it is proved that the entral manifold theorem onditions (Lanford, 97) are true for the infinity-dimensional mappin Ψ µ. The proof of this fat needs the linearization of Ψ µ in zero neihborhood with the help of impliit funtion theorem and spetrum analysis of obtained linear operator A µ. The analysis of operator A µ shows that the entral manifold theorem an be applied to the mappin ( Ψ µ, µ ): C [0,] R C [0,] R. With the help of the mentioned theorem the initial infinitydimensional problem is redued to two dimensional. On the third step, the Hopf's theorem (Marsden and MCraken, 976) is applied to the obtained two-dimensional mappin. From the Hopf's theorem follows that two time ontinuously differentiable one-dimensional manifold emeres or dies in the system when parameter µ inreases and rosses throuh zero. The final step is to prove that a periodi orbit aords to the obtained one-dimensional manifold. To prove this result the irle diffeomorphisms theory (Denjoy's theorem and rotation number definition) (Niteki, 97) is used. The same method allows analyzin the type of bifuration in the ase of u = 0. It is proved that in this ase the stron resonane is observed in the system (0) when parameter v inreases and rosses throuh the stability limit v (found from ()). With the help of the Theorem the stability loss type of equilibrium prie in the prie formation model (5) is analyzed. Theorem. Let τ = 0, τ > 0, > 0 and π. A periodi orbit emeres or dies in 3 3χτ the system (5) when the equilibrium prie p loses stability. The analoous result is proved in the ase of τ > 0, τ = 0. Computational experiments show that in the ase of τ = 0, τ > 0 a periodi orbit emeres in the model. It is proved that in the ase of equal produer's and onsumer's delays, there is the stron resonane in the system (5) when equilibrium prie loses stability. In this ase omputational experiments also show that a periodi orbit emeres in the system (5).

6 6. CONCLUSION At the first lane the obtained results do not solve the problem of eonomi rises ineption modelin. It should seem that eonomi aents may predit self-behavior takin into aount the periodi market behavior. The hane of supply desription makes this predition possible for the produers. But empirial data, for example the well known "pi yle" in Great Britain, shows that eonomi aents an't follow periodi prie flutuations (Coase and Fowler, 937). This fat ives apoloy to produer's desription in the model (5). Shananin, A.A. (99). Ob ustojhivosti rinohnih mehanismov. Matematiheskoje modelirovanije, 3,, 4-6. The results of the present work allow usin the prie formation model of Walras type with delays as a blok of a maroeonomi model for desribin endoenous rises ineption. ACKNOWLEGMENTS This work was finanially supported by the Russian Siene Support Fund, by the proram for State Support of Leadin Russian Sientifi Shools (projet no. NS ) and by the Russian Foundation for Basi Researh (projet nos , ). REFERENCES Bellman, R. and K.L. Cooke (963). Differential- Differene Equations. Aademi Press, London. Coase, R.H. and Fowler R.F. (937). The Pi-Cyle in Great Britain: An Explanation. Eonomia, 4, 3, Grandmont, J.-M. (985). On endoenous ompetitive business yles. Eonometria,53, 5, Lanford, O.-E. (97). Bifuration of periodi solutions into invariant tori: The work of Ruelle and Takens. Leture Notes in Mathematis, 3, Lorenz, H.-W. (989). Nonlinear Dynamial Eonomis and Chaoti Motion. Spriner- Verla, New York-Berlin. Marsden, J.E., M. MCraken (976). The Hopf bifuration and its appliations. With ontributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Gukenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Sheter, D. Shmi, and S. Smale. Spriner-Verla, New York-Berlin. Niteki, Z. (97). Differentiable Dynamis: an Introdution to the Orbit Struture of Diffeomorphisms. The MIT Press, Cambride. Petrov, A.A., I.G. Pospelov and A.A. Shananin (996). Opit matematiheskoo modelirovaniya eonomiki. Eneroatomisdat, Mosow.