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Inflation and relaxation to equilibrium in a complex economic system
Pergamon 0960-0779(94)00291-6
Chaos, Solitons & Fractals Vol. 6, pp. 455~170, 1995 Elsevier Science Ltd Printed in Great Britain 0960-0779/95 $9.50 + .00
Inflation and Relaxation to Equilibrium in a Complex Economic System
R.P.J. PERAZZO, t* S.L. REICH,* J. SCHVARZER'wand M.A. VIRASORO § Centro de Estudios Avanzados,Univ. de Buenos Aires Uriburu 950, (1114) Buenos Aires,Argentina Depto. de Fisica,Lab. T A N D A R , C N E A Libertador 8250, (1429) Buenos Aires, Argentina ¶ Centro de Investigacionesde la Sociedad y el Estado Uriburu 950, (1114) Buenos Aires,Argentina § Dipartimento di Fisica,Universita di R o m a 1 "La Sapienza" Piazzale Aldo Moro 2, Roma, Italy Abstract We picture an economic system as a network of agents that produce goods and buy and sell them to each other. The dynamic of the system is described by simple behavioural equations that describe the change of prices and production rates of each agent. The model, albeit schematic naturally contains a monetary policy and also shows the influence of the changes in relative prices to trigger and propagate inflation.
1
Introduction
The orthodox approach to microeconomics generally emphasizes that economic systems display a good deal of stability and homeostasis. Large natural catastrophes or important social conflicts such as wars are absorbed by the economy showing little or no change in the expected values of some indicators not too much time after their occurrence. To a good degree of approximation, the economy looks like a coherent whole in which apparently disaggregated entities adjust their behaviours to each other. Economic laws are therefore supposed to explain this equilibrium. The first attempt in this direction was made by Walras more than a century ago [1], but it is only recently that Arrow and Debreu were able to give a rigorous proof in simple models incorporating perfect competition. In such a scheme it is possible to show that there exists a set of prices, one for each commodity, such that supply and demand balance with each other (theory of General Competitive Equilibrium [9.]). Assuming perfect competition, negative feedbacks due to local convexity of demand functions and decreasing returns on the margin, the Debreu-Arrow's theory is able to characterize equilibrium and its stability. Yet this approach does not try to answer the more dynamical question of how a system actually evolves towards .or off equilibrium. Even more appropriate generalizations of this approach [3] are known to lead to chaotic regimes and therefore never approach equilibrium. If we relax the assumptions of perfect competition or negative feedback theoretical evidence has been gathered that demonstrate the occurrence of self reinforcing processes, multiple equilibria [4] and virtuous or vicious cycles [5]. In this note we will develop a simple dynamical model under different assumptions from those of the GCE. By construction it has a region of stable equilibrium but we study its dynamics outside it. The model is supposed to simulate faithfully only the short term fluctuations and certain rapid processes like inflation. We therefore leave aside entire factors in the economy: in our model there is no technological change, no innovation and savings do not carry interest rates. Furthermore rates of 455
456
R.P.J. PERAZZOet al.
production are supposed to be far both from the capacity limit and from the threshold of minimum production. Therefore costs are approximated by linear functions of production. There are different theories about the origin of inflation ( see e.g.[6]) and we expect that our model will allow to examine some of them. It should however be clear that the real cause of an episode of inflation can only be determined through a historic analysis of the economic data [7]. In the model, inflation can be caused by what is essentially an exogenous monetary policy but also by the spontaneous convergence of independent economic agents applying simple price fixing rules. Both mechanisms appear to be independent from each other. Even under a neutral monetary policy an external shock to the system in equilibrium provokes a more or less long adjustment period during which all prices increase. The length of this period depends on a parameter of the economic system that measures its global efficiency. In the presence of continuous, random, arbitrarily small shocks this period becomes infinite. The model consists of a set of interacting economic agents that produce and consume goods, buying and selling them to each other. Each agent produces a different good in a monopolistic environment. We may imagine the agent representing a sector of the economy described at some intermediate level of aggregation. Although competition is excluded from the present model, part of the demand is elastic and governed by a utility function. Consistent with the short term range of the model, the agents have to react to a changing environment with simple, suboptimal, robust rules. They do not have access to global information and therefore must take decisions on the basis of demand, cost of production and stock level. As a consequence, each agent fixes the price and the rate of production according to some behavioral rules that are part of the assumptions of the model. In the next section we present the basic constraint equations of the economic variables of the agents of the system. The time evolution results from the superposition of their individual decisions. These are schematized by a pair of differential (or difference) equations: one for the price and the other one for the volumes of production for each agent(sect.3). The constraint equations determine all the other variables except for a global variable corresponding to the total amount of money used in all the transactions during the next time period. An additional dynamic equation has to be introduced that describes the time evolution of this quantity. It is amusing to notice that this quantity is equal to the Monetary Mass multiplied by the speed of exchange. Therefore under the standard hypothesis [7] that the speed is constant in time, the new dynamics can be interpreted as an external monetary policy that has to be imposed on the system. Under these dynamic equations, the system has a region of neutral equilibrium in the space of prices. This feature is discussed in sect.4. In sect.5 we discuss the evolution of the system when perturbed off equilibrium. This can be done in analytic form in few schematic situations but in general one has to use numerical experiments as presented in sect.6. We finally summarize the conclusions in sect.7.
2
A n e c o n o m i c s y s t e m of interacting agents.
We consider a system composed by n economic agents. At each instant of time, the i - t h agent is assumed to produce N~ (i = 1,2,3...n) units of a single type of good. At the same time the agent sells N~' units of the same good, at a price Pi. In order to produce a single unit of its good, the i-th agent must purchase Cid units of the j-th good. The Cij are therefore non negative technical coefficients, with Ci,i -- 0, that describe the trade pattern arising from production. These are assumed to be constant in time. We also assume that each agent has a (positive) amount Si of goods in stock in order to absorb differences between the production schedule N~' and the purchase orders N~'. Thus
dS~ at
- N ~ - JV~
(1)
The net amount of money )~i that the i-th agent receives is W i = N~pi - N~ ~_, C,,jp i J
(2)
Inflation and relaxation to equilibrium in a complexeconomic system
457
Each agent devotes all his budget to consume Mj(i) units of the good j, thus
Wi = ~ Mj(i)py J
(3)
The subjective preferences of each agent are described by a utility function/41. The actual values of Mi(i ) are obtained by maximizing b/i subject to the condition 3 of spending all the available budget. As explained in the introduction savings, with its characteristic increased future returns, are not included in the model. On the other hand labour, with its peculiar incidence on production costs, is not distinguished from any other good. We do this for simplicity and discuss its consequences on the next sections. For the sake of concreteness we make the following ansatz for the utility function:
12i = ~_, a i J M)-a( i)/(1 - fl)
(4) J The coefficients Gis, with Gi,i = 0 are non negative constants that describe the relative preference of the j - t h good by the i-th agent. In order to insure that the utility function is a lion decreasing function of the M-quantities, the supplementary condition has to be imposed that/3 < 1 If L/~ is maximized with respect to the Mi(i ) subject to the condition 3 a set of n Lagrange multipliers N/~ have to be introduced in terms of which the actual consumption menu of each agent is expressed as Uj(i) = N~ai,~pff (5) with a = - 1 / f l (a < - 1 ) . From 5 We can interpret the quantities N/~ with i = 1,2, ...n as gauging the overall consumption of the i - t h agent. The role played by the exponent a is to prevent a consumption menu heavily oriented towards expensive goods. Combining 5, 3 and 2 a first constraint can be written that expresses the balance of money at each node of the network of economic agents:
g.~pi = N .n,~ CidPj + g~ ~_, Gidpj (=+') J J
(6)
A second constraint equation can be written that expresses that the sales of each agent are devoted by the rest to either produce or consume: N7
°
J
3
The
evolution
of the
J
system.
The state of the economic system is fully specified by the four time dependent vectors, Pi, N~i, N~ and N~. The economic agents are assumed to adjust only their prices and rates of production. The vectors N~ and N~ can not be changed freely because they depend on the others through the constraint equations 6 and 7. The matrices Cid and Gid are kept constant. As stressed in the introduction this amounts to neglecting the effect of technical innovations consistent with limiting the analysis to short time processes. We also disregard effects of economy of scale. We make the following ansatz for the relative changes of price and production rates:
1 dp, p, d t
N?, - N; +
-
1 dN~ _ N~ dt
(8)
_-=Si_ 1) -- N/" Ai( s~ - BiN~Nf
(9)
with
Zi(Si,lri)--1 +
dGi
0(~-)
[O(Si-
S°)O(~ri-~ci°) -
1]
.
(10)
458
R.P.J. PERAZZO et al.
In the preceding equations 0(z - x0) = 1 if z > x0 and zero otherwise stands for the Heaviside function. ~/i, pi, Ai a n d / 3 / are dimensionless constants that characterize the "behaviour" of each agent. 7ri stands for the markup made by the i-th agent that is defined as 7r, -
Pi
(11)
Ej 5'~,jp~
and 7rio is an agent dependent parameter representing the minimum markup that he can tolerate. The second term in 8 therefore corresponds to a markup policy. Each agent figures out its costs of production at current market prices and increases the price if p/ is less than 7ri° times the cost. The relative correction of the price is a finite amount pl. The first term in 8 is simply the law of offer and demand: the price is increased whenever the stock goes down and viceversa. The Z/ function that appears as a factor introduces an interaction between the two terms: the agent does not decrease prices when the rate of production is larger than the demand if stocks or markups are too low. As far as eq.9 is concerned, the role played by S °, corresponds to the reference stock of goods that the i-th agent chooses to keep. The first term of 9 is a linear adjustment of the production rate to keep the pace of the purchase orders. The second term acts as a further instantaneous adjustment to keep the rate of production close to the sales. In addition this term tends to damp unrealistic sustained oscillations of the production rate. In order to interpret that one of the agents, say t h e / - t h one, represents the labor of the whole system, the corresponding values of N~ and N~t should be taken respectively to be the labour that is being used and that is being offered in the system. One has further to assume that there exists some chronic unemployment to absorb the situations in which N~ > NJ'. Since the stock St is meaningless the first term of the l-th equation 9 is eliminated by setting At = 0. Our conclusions depend also on assuming that the labor aggregate will fix prices as in equation 8. The model can apply if, for instance, trade unions are sufficiently strong so that they can prevent salaries from getting out of phase with prices. Otherwise labour will have to be distinguished in a non trivial way. Eqs. 8 and 9 are written in the continuous time limit. It is a trivial exercise to convert them into finite difference equations. In this latter case one should also specify in which order the variables are updated. In the numerical experiments described in sect.6 the updating is simultaneous. Once pi and N/' are determined at t + 1 the 2n constraint equations 6 and 7 have to be solved to derive the vectors N/~ and N~'. However these equations are not independent. This signals the fact that the system has still an undetermined degree of freedom. If the evolution is assumed to take place in discrete time steps, an inspection of the constraint equations shows that the quantity
(12)
Jtd(t) = ~ N.~(t)pi(t) i
is not determined at time t + 1. This quantity corresponds to the total amount of money that will be exchanged during the next trade session and is unknown at the beginning of the session. We add therefore a dynamical equation: N~Ct)piCt) = Y~ N~(t - 1)pi(t i
1) + 6(t)
(13)
i
Equation 13 can be interpreted as reflecting the "monetary policy" prevalent in the economic system. We parameterize g in the following way:
1 riM=b+ a ~
- ~ d---i-
-~
dp,
. N~' dt
(14)
In this parameterization b = 0 and a = 1 reflects a neutral monetary policy in the sense that money is made available to the system at the same rate at which prices are increased, b > 0, a > 1 or b = 0, a > 1 represent an expansion of the monetary base.
Inflation and relaxation to equilibrium in a complex economic system
4
459
Equilibrium
The dynamic equations 8, 9 and 14 provide the equilibrium conditions. The vector of prices will remain constant if N~' = N~' = N (~q) and ri _> ri °. The first condition implies that all stocks will remain constant while the second defines a region of neutral stability in the space of prices. The constancy of the production rates require in addition that Si = S °. Equilibrium also requires jk4 to be constant (i.e. b=O).
Condition for constant stocks implies that at equilibrium the vectors p ~ ) and Ni(~) are respectively right and left eigenvectors of the same positive definite matrix with the same eigenvalue equal to unity.
pl
:
N} "q) =
A i.jpj
(15)
E N}e')aia
(161
~
with N.~(eq) G" "-~eql , ,~ Ai,~ = Cid + -'~ g(ieq) ,,,pj
(17)
The condition that a'i > ri o requires that the vector p~q) belongs to a region in the n-dimensional space of prices defined by pi ~ ri o ~ Ci,jpj (18) Clearly if the minimum markups ri o or the costs Cid are too high, this region may collapse to zero volume and finally disappear. The economic system will becomes intrinsically unstable. But, more in general, we will show in the next section that if the volume of the region bounded by the n inequalities 18 is too small, the system will relax more slowly to equilibrium. This volume characterizes the "efficiency" of the economic system. By analysing how this volume is restricted by each agent, one could discover possible "bottlenecks" in the economy.
5 Dynamics of relative prices In this section we solve the dynamic equation 8 in the approximation that the quantities yi(N/' - N~') are negligible. This may occur, for instance, if production adjusts instantaneously to sales. Under these circumstances l?rices decouple from production and we can discuss the effect of the markup policy. If p~eqj belongs to the equilibrium region, ,kp~~q) also belongs to the same region. This scale invariaace is due to an arbitrariness in the choice of the money unit. To study the dynamics in terms of relative prices we project it onto the hyperplane P defined by ~ i pi = 1 by changing variables from pi to zi = P l / ~ j pj. The dynamic of the xi is constrained to the plane P by a Lagrange multiplier/~ that is determined at every moment by the condition ~ j d ~ j / d t = O. The dynamics then reduces to: 1 d~i _ pie(ri o ~ Ci,j~j - ~i) - ~ agi dt j
(19)
The evolution of prices corresponds to a motion of a point X in the hyperplane P. A geometrical argument can be made to prove that under the dynamic induced by eq.19, the point X will approach the stability region. We assume that the dynamics takes place in discrete times and that the agents update their prices sequentially. For the sake of clarity we discuss the case of only three agents. The generalization to more dimensions calls for no conceptual complications (see fig. 1).
460
R.P.J. PERAZZO et aL
SPACE OF RELATIVE PRICES I~XAMPLE OF THREE ECONOMIC AGENTS
(o,o,1)= ,p,ps)=(x
(0,1,0)=
(P~,P2,Ps)=
agent operating below threshold markup STABILITYREGION , ~ ' ~
(Xl'x2'xS)
RELAXATIONTOEQUILIBRIUM (finite corrections,sequentialupdating)
FIGURE 1
The region of the positive octant of the space of prices is projected onto a triangle on the plane P at whose vertices only one of the xi is equal to one and the others are zero. Each of the planes defined by eqs. 18 intersects P along the straight lines Lb. Each of these limits a region of acceptable prices (closer to the i-th vertex). The three lines define a smaller triangle that corresponds to a stability region in which equilibrium is neutral with respect to the (relative) prices. Assume that originally the point X is outside such region and therefore one or two agents have an unacceptably low markup at current prices. Assume that the i-th agent is the first to update his price. The point X will then be displaced by a finite amount to a new position approaching Lb. This is because the displacement of X takes place along the line that joins its previous position with the i-th vertex, lying at the other side of Li. The updating of any other agent operating at unacceptably low prices produces the same effect, further approaching the point X to other line Lj. In the limit in which the values of all the pi tend to zero, the successive displacements will drive the point X continuously towards the border of the stability region where it will stop moving. There are circumstances in which the outcome of this adjustment of relative prices may not bring the system to an equilibrium. This happens when the stability region is small when measured in terms of the price corrections pi. This corresponds to a scenario in which all the agents claim high markups and introduce large price corrections. When this happens the point X can skip the stability region during the successive price corrections. Instead of equilibrium the system will approach a cycle in
Inflation and relaxation to equilibrium in a complex economic system
461
which several groups of agents take turns forcing the others to operate below an acceptable markup. A steady, permanent inflation with an (average) exponential increase in prices will occur. But even when the price corrections are infinitesimal (for instance in the continuum limit) a small stability region may be indirectly the cause of continuous inflationary bursts. Our system has been up to here assumed to be unrealistically isolated. Much more reasonable is to suppose that in addition to the set of variables explicitly considered there may be many other (exogenous) factors acting on the system. If the latter are small and uncorrelated their effect can be easily estimated by imagining the system immersed in a thermal bath. The picture is then similar to a Brownian motion with a drift force: 1 dz, _ p,O(,~o ~ • i dt
C , , ~ j - ~,) - ~ + e(t)
(20)
with e(t) a random variable with white noise:
,(t)
=
0
e(t)e(t')
=
2 T 6 ( t - t')
(9.1)
In this case it is obvious that the relative price vector will spend a percentage of the total time outside the stability region producing a proportional rate of inflation. As the dynamics are now stochastic, the steady regime will be characterized by an equilibrium distribution. This can be explicitly calculated in the more symmetric situation in which all the Cid, Pl and 7ri are taken to be equal. Then the center of the stability region will correspond to the vector zi ° with all components equal to 1In. If the stability region and T are small we can linearize the force in the right hand side of 20 to obtain the Langevin equation (see e.g. [8]), dzi e~
07-I
-
~ ~,,~-~-~j + ~(t)
(221
3
where 7[ = ~_, ~j°(Tr - mj)pO(Tr - mj) J
(23)
and 7~ is the projector on the plane P: Pls :
&s - n~i°xJ °
(24)
In deriving 22 we have used the fact that g
0 ( ~ - ~,) :
~ , [(~ - =,)o(~ -
=,)]
(25)
where 7r lumps together the average value of the Cid and the markups a'i. If eq. 22 holds, the probability that the system will visit the neighborhood IIidzl of a point (Zl, z2, ..., z,~) during its evolution can statistically be described through the Gibbs distribution:
p(=~, ~ , . . . , =,,) = ~ - ' ~ / ~ ( ~
=, - 1).
(28)
i
Eq. 26 indicates that even if the system is sitting within the stability region, the effect of the fluctuations is to produce an exponential tail of the distribution leaking outside that region. This corresponds to a finite probability that the system will live temporarily there, and consequently display a steady rate of inflation.
R.P.J. PERAZZOet aL
462
6
Numerical experiments
The dynamic equations 8, 9 and 14 can only be explored in its most general form by numerical experimentation. In this section we report a few illustrative scenarios, some of which can be related to the cases analyzed in the preceding section. We use 15 and 16 to construct the matrices Ci,j , Gi,j and a possible equilibrium configuration. We generate two random matrices (with zero diagonal) whose elements are independent, uniformly distributed in [0, 1] variables. We normalize one of them so that its highest eigenvalue is 1. The corresponding eigenvector is identified with a possible p~'q).We then multiply each matrix element by a factor smaller than 1. This defines the matrix Ci,j such that:
(27)
pl °q) > i
while the second matrix is identified with Gi,i. We then determine the v e c t o r solving equation 15. We finally solve the homogeneous system 16 to derive N[~q). The vector p~eq)/~-~.~ Cidp~.~q) is, by definition, larger than one. We then define a possible set of ri o such that:
ie(e'q)/i(ieq)
1< ~
0
P!¢q) < v" ~..-('q) z.,j v,,~gj
(28)
The reference stocks S ° that are needed for eqs. 8 and 9 are defined randomly. Once constructed, the system is set off equilibrium and its evolution is determined integrating 8, 9 and 14. We have assumed that the evolution takes place in discrete time steps. In each trade session all the agents of the system update their prices and production rates and the required global indicators can be calculated such as the average price, rate of production, stocks, etc. In each moment we also calculate the fraction of the agents that are operating below their acceptable thresholds. For simplicity we have set all constants r/i, pi, Ai and Bi independent of i. We took a = - 2 . In all our simulations we have considered a system composed by 10 agents.
6.1
Relaxation
to equilibrium
with
a neutral
monetary
policy.
In these experiments equilibrium is perturbed by letting one arbitrary agent to double his price. Since all others find that their production costs have gone up, their current markups are too low and they are forced to correct their prices. The evolution is shown in figs.2 and 3. In all cases we have set a = 1 and b = 0 in eq.14 , therefore the amount of money in the system changes only at the same pace as the prices. Fig.2 shows the process of relaxation to equilibrium. The average price grows exponentially as long as there are agents with unacceptable markups. Evolution proceeds roughly as outlined in the preceding section, successively cutting down the number of agents operating below threshold. Equilibrium is finally reached in less than 24 units of time. The difference between Fig 2 and 3 are a sizable increase in the value of r/, that corresponds to a greater influence of offer and demand in the mechanism of price updating, and a smaller value of B that corresponds to a lower damping. The process of relaxation to equilibrium turns out to be partially impaired because as suggested by the discussions of the preceding section once the equilibrium region has been reached, the fluctuations induced by other degrees of freedom forces the system to scape temporarily from it. This is seen to happen several times whenever there are periods in which a decrease of stocks has caused an increase of prices. Inflationary bursts tend to disappear because the fluctuations in stocks and production rates are damped away.
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The average rate of production and average stocks are seen to oscillate. The period of the oscillations is determined by the value of A. Periods in which the rates of production grow, are seen to occur whenever the agents need to replenish stocks because they have fallen below the average reference value.
6.2
Inefficient
economic
systems.
The scenario of fig.4 represents a less efficient system in which the stability region is too small as compared to the price corrections. The monetary policy is also neutral as in the preceding subsection. A large value of 7/makes the prices more heavily dependent on ~small) stock fluctuations that cause the system to be driven away from the stability region. Once the system has approached the stability region, is driven away from it and an uninterrupted inflationary process is produced by the markup policy that operates from that moment on. An average of roughly 20 % of the agents struggle to operate at acceptable prices. In spite of the fact that stocks and production rates reach a stable value, the system engages in a cyclic behaviour in which different groups of agents force the others to operate below the acceptable threshold markup. 6.3
Effect
of monetary
restrictions.
Although our model has not been built in such a way as to describe correctly the simultaneous effect of a monetary policy and a price push-up, it is amusing to analyze some of the scenarios discussed above with an active monetary policy tending to keep the amount of money of the market constant in spite of any increase in prices. In Fig. 5 it is displayed the evolution of the system with the same parameters as in fig 2 but setting a = 0. The initial convergence is kept in most of its essential features but the system is less efficient in rela0dng to equilibrium because several times is driven outside the stability region. A greater inflation is produced and production rates are drastically diminished. What actually happens is that in order to keep A4 constant with increasing prices, the number of traded goods diminishes and the production rates follow this trend in order not to accumulate stocks. To some extent we believe this "recessive" effect to be overemphasized by the construction of the model. When sales are decreasing but markups are low our agents simply add two terms in the balance to decide whether to increase or decrease prices. This is not perhaps the correct interaction between the two fundamental forces. For instance, if the decrease in sales is steady but slow, our agents prefer to push prices up and decrease production. In the long run this will lead to decreasing profits. It is conceivable that realistic agents will pay more attention to the trend in time and would eventually modify their markup policy. 6.4
Monetary
expansion
from
equilibrium.
A last situation is presented in fig.6 in which the initial equilibrium is perturbed only by setting b > 0, i.e. by deliberately pumping money into the market. The coefficient a in eq. 14 is set equal to 1 not to introduce a competing effect. Prices grow but never get out of the stability region because relative prices do not change. Sales have to grow to match the increase of the amount of money in the market. Consumption and production rates therefore grow at the same pace to match that behavior and stocks are kept essentially constant.
7
Conclusions.
The model presented above, albeit schematic and unrealistic in many of its assumption, is nevertheless able to retain many relevant stylized facts found in real inflationary situations. The more elementary yet remarkable feature of the model is the importance of the change in relative prices as one of the basic mechanisms for triggering and propagating in time an inflationary process.
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The second point that is worth to stress is that the monetary policy turns out to be naturally contained as an indeterminacy of the set of coupled microeconomic equations describing the trading pattern of the interacting agents of the system. This indeterminacy allows us to paxameterize the dynamic in time of the total amount of money used in the system. As fax as a neutral monetary policy is concerned the model seems to account for several different scenarios depending on the "efficiency" of the system. The latter depends on the general compatibihty of the expected markups and costs of production. A more "efficient" economy has a larger region of relative prices that accommodate all the operators. In this case an external shock is easily absorbed by a relaxation to a perhaps different equilibrium set of relative prices. Inflation here appears simply as a temporary perturbation. With less "efficiency" the relaxation may be impaired and the equilibrium never reached. In spite of the fact that there exists a stabihty region of finite volume, the system wanders around it without stabilizing. This situation gives rise to lasting periods of exponential inflation in which the burden of operating below acceptable prices, shifts periodically among different groups of agents of the system. This behavior may be caused either by big price increases as compared with the small range of compatible relative prices or by small stochastic noise. We have also seen that our model incorporates inflation due to expansion of the monetary base even when there is no turbulence in relative prices. We also find that a mild increase in the amount of money in the system, when produced in a situation of equilibrium produces what are known as the "beneficial" effects of inflation, i.e. an expansion of consumption and production. We have also discussed active monetary policies simultaneous with relative price shocks although we are aware that our model might be misleading in this situation.
8
Acknowledgments
This presentation of the model (perhaps not final) was shaped during the 2nd Workshop on "The Economy as an evolving Complex System" held at the Santa Fe Institute, in Santa Fe, NM. We thank all participants and specially the organizers K.J.Arrow and P.W.Anderson for sharing with us their inside information on the glories and miseries of Microeconomic Theory. In particular we would hke to thank: J. Scheinkman for an interesting discussion on the monetary theory of inflation and G. Dosi for anticipating to us his explanation of the emergence of markups policies in noisy environments.
References [1] L.Walras,"Elements d'Economie Politique Pure". Lausanne, Corbaz, 1874. [2] K.J.Arrow and G.Debreu, Econometrica, 22, 1954, 265-290, "Existence of Equihbrium for Competitive Economy"; K.J.Arrow, H.D.Block and L.Hurwicz, Econometrica, 27, 1959, 82-109, "On the Stability of the Competitive Equilibrium II". [3] M.Boldrin, "Persistent Oscillations and Chaos in Economic Models: Notes for a Survey" in The Economy as a Complex Evolving System, Santa Fe Institute, Studies in the Sciences of Complexity, Vol. V, P.W.Anderson, K.J.Arrow and D.Pines, Eds., Addison Wesley Publ.Co. 1988. [4] B.Arthur, "Self Reinforcing Mechanisms in Economics" in The Economy as a Complex Evolving System, Santa Fe Institute, Studies in the Sciences of Complexity, Vol. V, P.W.Anderson, K.J.Arrow and D.Pines, Editors. Addison Wesley Publ.Co. 1988. [5] G.Hurd,'Macrodynamics and Return to Scale", The Economic Journal 96, 1986, 191-198.
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[6] F.Machlup, Review of Economics and Statistics, 42, 1960, 125-139, "Another view of Cost-push and Demand-pull Inflation"; J.H.G.Olivera, Oxford Economic Papers, 16, 1964, 321-332, "On Structural Inflation and LatinAmerican 'Structuralism'"; F.Hirsh and J.Goldthrope (Eds.)"The political economy of inflation", Harvard University Press, Cambridge U.S.A., 1978. [7] M. Friedman and A.Schwartz, "A Monetary History of the United States, 1867-1960" Study for the National Bureau of Economic Research. Princeton University Press, Princeton N.J. 1963 [8] R.Balescu, "Equilibrium and Non Equilibrium Statistical Mechanics", John Wiley and Sons, N.Y.
(1975); G.Parisi, "Statistical Field Theory", Addison Wesley Publ.Co.Inc. N.Y.(1988)
Chaos, Solitons & Fractals Vol. 6, pp. 455~170, 1995 Elsevier Science Ltd Printed in Great Britain 0960-0779/95 $9.50 + .00
Inflation and Relaxation to Equilibrium in a Complex Economic System
R.P.J. PERAZZO, t* S.L. REICH,* J. SCHVARZER'wand M.A. VIRASORO § Centro de Estudios Avanzados,Univ. de Buenos Aires Uriburu 950, (1114) Buenos Aires,Argentina Depto. de Fisica,Lab. T A N D A R , C N E A Libertador 8250, (1429) Buenos Aires, Argentina ¶ Centro de Investigacionesde la Sociedad y el Estado Uriburu 950, (1114) Buenos Aires,Argentina § Dipartimento di Fisica,Universita di R o m a 1 "La Sapienza" Piazzale Aldo Moro 2, Roma, Italy Abstract We picture an economic system as a network of agents that produce goods and buy and sell them to each other. The dynamic of the system is described by simple behavioural equations that describe the change of prices and production rates of each agent. The model, albeit schematic naturally contains a monetary policy and also shows the influence of the changes in relative prices to trigger and propagate inflation.
1
Introduction
The orthodox approach to microeconomics generally emphasizes that economic systems display a good deal of stability and homeostasis. Large natural catastrophes or important social conflicts such as wars are absorbed by the economy showing little or no change in the expected values of some indicators not too much time after their occurrence. To a good degree of approximation, the economy looks like a coherent whole in which apparently disaggregated entities adjust their behaviours to each other. Economic laws are therefore supposed to explain this equilibrium. The first attempt in this direction was made by Walras more than a century ago [1], but it is only recently that Arrow and Debreu were able to give a rigorous proof in simple models incorporating perfect competition. In such a scheme it is possible to show that there exists a set of prices, one for each commodity, such that supply and demand balance with each other (theory of General Competitive Equilibrium [9.]). Assuming perfect competition, negative feedbacks due to local convexity of demand functions and decreasing returns on the margin, the Debreu-Arrow's theory is able to characterize equilibrium and its stability. Yet this approach does not try to answer the more dynamical question of how a system actually evolves towards .or off equilibrium. Even more appropriate generalizations of this approach [3] are known to lead to chaotic regimes and therefore never approach equilibrium. If we relax the assumptions of perfect competition or negative feedback theoretical evidence has been gathered that demonstrate the occurrence of self reinforcing processes, multiple equilibria [4] and virtuous or vicious cycles [5]. In this note we will develop a simple dynamical model under different assumptions from those of the GCE. By construction it has a region of stable equilibrium but we study its dynamics outside it. The model is supposed to simulate faithfully only the short term fluctuations and certain rapid processes like inflation. We therefore leave aside entire factors in the economy: in our model there is no technological change, no innovation and savings do not carry interest rates. Furthermore rates of 455
456
R.P.J. PERAZZOet al.
production are supposed to be far both from the capacity limit and from the threshold of minimum production. Therefore costs are approximated by linear functions of production. There are different theories about the origin of inflation ( see e.g.[6]) and we expect that our model will allow to examine some of them. It should however be clear that the real cause of an episode of inflation can only be determined through a historic analysis of the economic data [7]. In the model, inflation can be caused by what is essentially an exogenous monetary policy but also by the spontaneous convergence of independent economic agents applying simple price fixing rules. Both mechanisms appear to be independent from each other. Even under a neutral monetary policy an external shock to the system in equilibrium provokes a more or less long adjustment period during which all prices increase. The length of this period depends on a parameter of the economic system that measures its global efficiency. In the presence of continuous, random, arbitrarily small shocks this period becomes infinite. The model consists of a set of interacting economic agents that produce and consume goods, buying and selling them to each other. Each agent produces a different good in a monopolistic environment. We may imagine the agent representing a sector of the economy described at some intermediate level of aggregation. Although competition is excluded from the present model, part of the demand is elastic and governed by a utility function. Consistent with the short term range of the model, the agents have to react to a changing environment with simple, suboptimal, robust rules. They do not have access to global information and therefore must take decisions on the basis of demand, cost of production and stock level. As a consequence, each agent fixes the price and the rate of production according to some behavioral rules that are part of the assumptions of the model. In the next section we present the basic constraint equations of the economic variables of the agents of the system. The time evolution results from the superposition of their individual decisions. These are schematized by a pair of differential (or difference) equations: one for the price and the other one for the volumes of production for each agent(sect.3). The constraint equations determine all the other variables except for a global variable corresponding to the total amount of money used in all the transactions during the next time period. An additional dynamic equation has to be introduced that describes the time evolution of this quantity. It is amusing to notice that this quantity is equal to the Monetary Mass multiplied by the speed of exchange. Therefore under the standard hypothesis [7] that the speed is constant in time, the new dynamics can be interpreted as an external monetary policy that has to be imposed on the system. Under these dynamic equations, the system has a region of neutral equilibrium in the space of prices. This feature is discussed in sect.4. In sect.5 we discuss the evolution of the system when perturbed off equilibrium. This can be done in analytic form in few schematic situations but in general one has to use numerical experiments as presented in sect.6. We finally summarize the conclusions in sect.7.
2
A n e c o n o m i c s y s t e m of interacting agents.
We consider a system composed by n economic agents. At each instant of time, the i - t h agent is assumed to produce N~ (i = 1,2,3...n) units of a single type of good. At the same time the agent sells N~' units of the same good, at a price Pi. In order to produce a single unit of its good, the i-th agent must purchase Cid units of the j-th good. The Cij are therefore non negative technical coefficients, with Ci,i -- 0, that describe the trade pattern arising from production. These are assumed to be constant in time. We also assume that each agent has a (positive) amount Si of goods in stock in order to absorb differences between the production schedule N~' and the purchase orders N~'. Thus
dS~ at
- N ~ - JV~
(1)
The net amount of money )~i that the i-th agent receives is W i = N~pi - N~ ~_, C,,jp i J
(2)
Inflation and relaxation to equilibrium in a complexeconomic system
457
Each agent devotes all his budget to consume Mj(i) units of the good j, thus
Wi = ~ Mj(i)py J
(3)
The subjective preferences of each agent are described by a utility function/41. The actual values of Mi(i ) are obtained by maximizing b/i subject to the condition 3 of spending all the available budget. As explained in the introduction savings, with its characteristic increased future returns, are not included in the model. On the other hand labour, with its peculiar incidence on production costs, is not distinguished from any other good. We do this for simplicity and discuss its consequences on the next sections. For the sake of concreteness we make the following ansatz for the utility function:
12i = ~_, a i J M)-a( i)/(1 - fl)
(4) J The coefficients Gis, with Gi,i = 0 are non negative constants that describe the relative preference of the j - t h good by the i-th agent. In order to insure that the utility function is a lion decreasing function of the M-quantities, the supplementary condition has to be imposed that/3 < 1 If L/~ is maximized with respect to the Mi(i ) subject to the condition 3 a set of n Lagrange multipliers N/~ have to be introduced in terms of which the actual consumption menu of each agent is expressed as Uj(i) = N~ai,~pff (5) with a = - 1 / f l (a < - 1 ) . From 5 We can interpret the quantities N/~ with i = 1,2, ...n as gauging the overall consumption of the i - t h agent. The role played by the exponent a is to prevent a consumption menu heavily oriented towards expensive goods. Combining 5, 3 and 2 a first constraint can be written that expresses the balance of money at each node of the network of economic agents:
g.~pi = N .n,~ CidPj + g~ ~_, Gidpj (=+') J J
(6)
A second constraint equation can be written that expresses that the sales of each agent are devoted by the rest to either produce or consume: N7
°
J
3
The
evolution
of the
J
system.
The state of the economic system is fully specified by the four time dependent vectors, Pi, N~i, N~ and N~. The economic agents are assumed to adjust only their prices and rates of production. The vectors N~ and N~ can not be changed freely because they depend on the others through the constraint equations 6 and 7. The matrices Cid and Gid are kept constant. As stressed in the introduction this amounts to neglecting the effect of technical innovations consistent with limiting the analysis to short time processes. We also disregard effects of economy of scale. We make the following ansatz for the relative changes of price and production rates:
1 dp, p, d t
N?, - N; +
-
1 dN~ _ N~ dt
(8)
_-=Si_ 1) -- N/" Ai( s~ - BiN~Nf
(9)
with
Zi(Si,lri)--1 +
dGi
0(~-)
[O(Si-
S°)O(~ri-~ci°) -
1]
.
(10)
458
R.P.J. PERAZZO et al.
In the preceding equations 0(z - x0) = 1 if z > x0 and zero otherwise stands for the Heaviside function. ~/i, pi, Ai a n d / 3 / are dimensionless constants that characterize the "behaviour" of each agent. 7ri stands for the markup made by the i-th agent that is defined as 7r, -
Pi
(11)
Ej 5'~,jp~
and 7rio is an agent dependent parameter representing the minimum markup that he can tolerate. The second term in 8 therefore corresponds to a markup policy. Each agent figures out its costs of production at current market prices and increases the price if p/ is less than 7ri° times the cost. The relative correction of the price is a finite amount pl. The first term in 8 is simply the law of offer and demand: the price is increased whenever the stock goes down and viceversa. The Z/ function that appears as a factor introduces an interaction between the two terms: the agent does not decrease prices when the rate of production is larger than the demand if stocks or markups are too low. As far as eq.9 is concerned, the role played by S °, corresponds to the reference stock of goods that the i-th agent chooses to keep. The first term of 9 is a linear adjustment of the production rate to keep the pace of the purchase orders. The second term acts as a further instantaneous adjustment to keep the rate of production close to the sales. In addition this term tends to damp unrealistic sustained oscillations of the production rate. In order to interpret that one of the agents, say t h e / - t h one, represents the labor of the whole system, the corresponding values of N~ and N~t should be taken respectively to be the labour that is being used and that is being offered in the system. One has further to assume that there exists some chronic unemployment to absorb the situations in which N~ > NJ'. Since the stock St is meaningless the first term of the l-th equation 9 is eliminated by setting At = 0. Our conclusions depend also on assuming that the labor aggregate will fix prices as in equation 8. The model can apply if, for instance, trade unions are sufficiently strong so that they can prevent salaries from getting out of phase with prices. Otherwise labour will have to be distinguished in a non trivial way. Eqs. 8 and 9 are written in the continuous time limit. It is a trivial exercise to convert them into finite difference equations. In this latter case one should also specify in which order the variables are updated. In the numerical experiments described in sect.6 the updating is simultaneous. Once pi and N/' are determined at t + 1 the 2n constraint equations 6 and 7 have to be solved to derive the vectors N/~ and N~'. However these equations are not independent. This signals the fact that the system has still an undetermined degree of freedom. If the evolution is assumed to take place in discrete time steps, an inspection of the constraint equations shows that the quantity
(12)
Jtd(t) = ~ N.~(t)pi(t) i
is not determined at time t + 1. This quantity corresponds to the total amount of money that will be exchanged during the next trade session and is unknown at the beginning of the session. We add therefore a dynamical equation: N~Ct)piCt) = Y~ N~(t - 1)pi(t i
1) + 6(t)
(13)
i
Equation 13 can be interpreted as reflecting the "monetary policy" prevalent in the economic system. We parameterize g in the following way:
1 riM=b+ a ~
- ~ d---i-
-~
dp,
. N~' dt
(14)
In this parameterization b = 0 and a = 1 reflects a neutral monetary policy in the sense that money is made available to the system at the same rate at which prices are increased, b > 0, a > 1 or b = 0, a > 1 represent an expansion of the monetary base.
Inflation and relaxation to equilibrium in a complex economic system
4
459
Equilibrium
The dynamic equations 8, 9 and 14 provide the equilibrium conditions. The vector of prices will remain constant if N~' = N~' = N (~q) and ri _> ri °. The first condition implies that all stocks will remain constant while the second defines a region of neutral stability in the space of prices. The constancy of the production rates require in addition that Si = S °. Equilibrium also requires jk4 to be constant (i.e. b=O).
Condition for constant stocks implies that at equilibrium the vectors p ~ ) and Ni(~) are respectively right and left eigenvectors of the same positive definite matrix with the same eigenvalue equal to unity.
pl
:
N} "q) =
A i.jpj
(15)
E N}e')aia
(161
~
with N.~(eq) G" "-~eql , ,~ Ai,~ = Cid + -'~ g(ieq) ,,,pj
(17)
The condition that a'i > ri o requires that the vector p~q) belongs to a region in the n-dimensional space of prices defined by pi ~ ri o ~ Ci,jpj (18) Clearly if the minimum markups ri o or the costs Cid are too high, this region may collapse to zero volume and finally disappear. The economic system will becomes intrinsically unstable. But, more in general, we will show in the next section that if the volume of the region bounded by the n inequalities 18 is too small, the system will relax more slowly to equilibrium. This volume characterizes the "efficiency" of the economic system. By analysing how this volume is restricted by each agent, one could discover possible "bottlenecks" in the economy.
5 Dynamics of relative prices In this section we solve the dynamic equation 8 in the approximation that the quantities yi(N/' - N~') are negligible. This may occur, for instance, if production adjusts instantaneously to sales. Under these circumstances l?rices decouple from production and we can discuss the effect of the markup policy. If p~eqj belongs to the equilibrium region, ,kp~~q) also belongs to the same region. This scale invariaace is due to an arbitrariness in the choice of the money unit. To study the dynamics in terms of relative prices we project it onto the hyperplane P defined by ~ i pi = 1 by changing variables from pi to zi = P l / ~ j pj. The dynamic of the xi is constrained to the plane P by a Lagrange multiplier/~ that is determined at every moment by the condition ~ j d ~ j / d t = O. The dynamics then reduces to: 1 d~i _ pie(ri o ~ Ci,j~j - ~i) - ~ agi dt j
(19)
The evolution of prices corresponds to a motion of a point X in the hyperplane P. A geometrical argument can be made to prove that under the dynamic induced by eq.19, the point X will approach the stability region. We assume that the dynamics takes place in discrete times and that the agents update their prices sequentially. For the sake of clarity we discuss the case of only three agents. The generalization to more dimensions calls for no conceptual complications (see fig. 1).
460
R.P.J. PERAZZO et aL
SPACE OF RELATIVE PRICES I~XAMPLE OF THREE ECONOMIC AGENTS
(o,o,1)= ,p,ps)=(x
(0,1,0)=
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agent operating below threshold markup STABILITYREGION , ~ ' ~
(Xl'x2'xS)
RELAXATIONTOEQUILIBRIUM (finite corrections,sequentialupdating)
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The region of the positive octant of the space of prices is projected onto a triangle on the plane P at whose vertices only one of the xi is equal to one and the others are zero. Each of the planes defined by eqs. 18 intersects P along the straight lines Lb. Each of these limits a region of acceptable prices (closer to the i-th vertex). The three lines define a smaller triangle that corresponds to a stability region in which equilibrium is neutral with respect to the (relative) prices. Assume that originally the point X is outside such region and therefore one or two agents have an unacceptably low markup at current prices. Assume that the i-th agent is the first to update his price. The point X will then be displaced by a finite amount to a new position approaching Lb. This is because the displacement of X takes place along the line that joins its previous position with the i-th vertex, lying at the other side of Li. The updating of any other agent operating at unacceptably low prices produces the same effect, further approaching the point X to other line Lj. In the limit in which the values of all the pi tend to zero, the successive displacements will drive the point X continuously towards the border of the stability region where it will stop moving. There are circumstances in which the outcome of this adjustment of relative prices may not bring the system to an equilibrium. This happens when the stability region is small when measured in terms of the price corrections pi. This corresponds to a scenario in which all the agents claim high markups and introduce large price corrections. When this happens the point X can skip the stability region during the successive price corrections. Instead of equilibrium the system will approach a cycle in
Inflation and relaxation to equilibrium in a complex economic system
461
which several groups of agents take turns forcing the others to operate below an acceptable markup. A steady, permanent inflation with an (average) exponential increase in prices will occur. But even when the price corrections are infinitesimal (for instance in the continuum limit) a small stability region may be indirectly the cause of continuous inflationary bursts. Our system has been up to here assumed to be unrealistically isolated. Much more reasonable is to suppose that in addition to the set of variables explicitly considered there may be many other (exogenous) factors acting on the system. If the latter are small and uncorrelated their effect can be easily estimated by imagining the system immersed in a thermal bath. The picture is then similar to a Brownian motion with a drift force: 1 dz, _ p,O(,~o ~ • i dt
C , , ~ j - ~,) - ~ + e(t)
(20)
with e(t) a random variable with white noise:
,(t)
=
0
e(t)e(t')
=
2 T 6 ( t - t')
(9.1)
In this case it is obvious that the relative price vector will spend a percentage of the total time outside the stability region producing a proportional rate of inflation. As the dynamics are now stochastic, the steady regime will be characterized by an equilibrium distribution. This can be explicitly calculated in the more symmetric situation in which all the Cid, Pl and 7ri are taken to be equal. Then the center of the stability region will correspond to the vector zi ° with all components equal to 1In. If the stability region and T are small we can linearize the force in the right hand side of 20 to obtain the Langevin equation (see e.g. [8]), dzi e~
07-I
-
~ ~,,~-~-~j + ~(t)
(221
3
where 7[ = ~_, ~j°(Tr - mj)pO(Tr - mj) J
(23)
and 7~ is the projector on the plane P: Pls :
&s - n~i°xJ °
(24)
In deriving 22 we have used the fact that g
0 ( ~ - ~,) :
~ , [(~ - =,)o(~ -
=,)]
(25)
where 7r lumps together the average value of the Cid and the markups a'i. If eq. 22 holds, the probability that the system will visit the neighborhood IIidzl of a point (Zl, z2, ..., z,~) during its evolution can statistically be described through the Gibbs distribution:
p(=~, ~ , . . . , =,,) = ~ - ' ~ / ~ ( ~
=, - 1).
(28)
i
Eq. 26 indicates that even if the system is sitting within the stability region, the effect of the fluctuations is to produce an exponential tail of the distribution leaking outside that region. This corresponds to a finite probability that the system will live temporarily there, and consequently display a steady rate of inflation.
R.P.J. PERAZZOet aL
462
6
Numerical experiments
The dynamic equations 8, 9 and 14 can only be explored in its most general form by numerical experimentation. In this section we report a few illustrative scenarios, some of which can be related to the cases analyzed in the preceding section. We use 15 and 16 to construct the matrices Ci,j , Gi,j and a possible equilibrium configuration. We generate two random matrices (with zero diagonal) whose elements are independent, uniformly distributed in [0, 1] variables. We normalize one of them so that its highest eigenvalue is 1. The corresponding eigenvector is identified with a possible p~'q).We then multiply each matrix element by a factor smaller than 1. This defines the matrix Ci,j such that:
(27)
pl °q) > i
while the second matrix is identified with Gi,i. We then determine the v e c t o r solving equation 15. We finally solve the homogeneous system 16 to derive N[~q). The vector p~eq)/~-~.~ Cidp~.~q) is, by definition, larger than one. We then define a possible set of ri o such that:
ie(e'q)/i(ieq)
1< ~
0
P!¢q) < v" ~..-('q) z.,j v,,~gj
(28)
The reference stocks S ° that are needed for eqs. 8 and 9 are defined randomly. Once constructed, the system is set off equilibrium and its evolution is determined integrating 8, 9 and 14. We have assumed that the evolution takes place in discrete time steps. In each trade session all the agents of the system update their prices and production rates and the required global indicators can be calculated such as the average price, rate of production, stocks, etc. In each moment we also calculate the fraction of the agents that are operating below their acceptable thresholds. For simplicity we have set all constants r/i, pi, Ai and Bi independent of i. We took a = - 2 . In all our simulations we have considered a system composed by 10 agents.
6.1
Relaxation
to equilibrium
with
a neutral
monetary
policy.
In these experiments equilibrium is perturbed by letting one arbitrary agent to double his price. Since all others find that their production costs have gone up, their current markups are too low and they are forced to correct their prices. The evolution is shown in figs.2 and 3. In all cases we have set a = 1 and b = 0 in eq.14 , therefore the amount of money in the system changes only at the same pace as the prices. Fig.2 shows the process of relaxation to equilibrium. The average price grows exponentially as long as there are agents with unacceptable markups. Evolution proceeds roughly as outlined in the preceding section, successively cutting down the number of agents operating below threshold. Equilibrium is finally reached in less than 24 units of time. The difference between Fig 2 and 3 are a sizable increase in the value of r/, that corresponds to a greater influence of offer and demand in the mechanism of price updating, and a smaller value of B that corresponds to a lower damping. The process of relaxation to equilibrium turns out to be partially impaired because as suggested by the discussions of the preceding section once the equilibrium region has been reached, the fluctuations induced by other degrees of freedom forces the system to scape temporarily from it. This is seen to happen several times whenever there are periods in which a decrease of stocks has caused an increase of prices. Inflationary bursts tend to disappear because the fluctuations in stocks and production rates are damped away.
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The average rate of production and average stocks are seen to oscillate. The period of the oscillations is determined by the value of A. Periods in which the rates of production grow, are seen to occur whenever the agents need to replenish stocks because they have fallen below the average reference value.
6.2
Inefficient
economic
systems.
The scenario of fig.4 represents a less efficient system in which the stability region is too small as compared to the price corrections. The monetary policy is also neutral as in the preceding subsection. A large value of 7/makes the prices more heavily dependent on ~small) stock fluctuations that cause the system to be driven away from the stability region. Once the system has approached the stability region, is driven away from it and an uninterrupted inflationary process is produced by the markup policy that operates from that moment on. An average of roughly 20 % of the agents struggle to operate at acceptable prices. In spite of the fact that stocks and production rates reach a stable value, the system engages in a cyclic behaviour in which different groups of agents force the others to operate below the acceptable threshold markup. 6.3
Effect
of monetary
restrictions.
Although our model has not been built in such a way as to describe correctly the simultaneous effect of a monetary policy and a price push-up, it is amusing to analyze some of the scenarios discussed above with an active monetary policy tending to keep the amount of money of the market constant in spite of any increase in prices. In Fig. 5 it is displayed the evolution of the system with the same parameters as in fig 2 but setting a = 0. The initial convergence is kept in most of its essential features but the system is less efficient in rela0dng to equilibrium because several times is driven outside the stability region. A greater inflation is produced and production rates are drastically diminished. What actually happens is that in order to keep A4 constant with increasing prices, the number of traded goods diminishes and the production rates follow this trend in order not to accumulate stocks. To some extent we believe this "recessive" effect to be overemphasized by the construction of the model. When sales are decreasing but markups are low our agents simply add two terms in the balance to decide whether to increase or decrease prices. This is not perhaps the correct interaction between the two fundamental forces. For instance, if the decrease in sales is steady but slow, our agents prefer to push prices up and decrease production. In the long run this will lead to decreasing profits. It is conceivable that realistic agents will pay more attention to the trend in time and would eventually modify their markup policy. 6.4
Monetary
expansion
from
equilibrium.
A last situation is presented in fig.6 in which the initial equilibrium is perturbed only by setting b > 0, i.e. by deliberately pumping money into the market. The coefficient a in eq. 14 is set equal to 1 not to introduce a competing effect. Prices grow but never get out of the stability region because relative prices do not change. Sales have to grow to match the increase of the amount of money in the market. Consumption and production rates therefore grow at the same pace to match that behavior and stocks are kept essentially constant.
7
Conclusions.
The model presented above, albeit schematic and unrealistic in many of its assumption, is nevertheless able to retain many relevant stylized facts found in real inflationary situations. The more elementary yet remarkable feature of the model is the importance of the change in relative prices as one of the basic mechanisms for triggering and propagating in time an inflationary process.
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The second point that is worth to stress is that the monetary policy turns out to be naturally contained as an indeterminacy of the set of coupled microeconomic equations describing the trading pattern of the interacting agents of the system. This indeterminacy allows us to paxameterize the dynamic in time of the total amount of money used in the system. As fax as a neutral monetary policy is concerned the model seems to account for several different scenarios depending on the "efficiency" of the system. The latter depends on the general compatibihty of the expected markups and costs of production. A more "efficient" economy has a larger region of relative prices that accommodate all the operators. In this case an external shock is easily absorbed by a relaxation to a perhaps different equilibrium set of relative prices. Inflation here appears simply as a temporary perturbation. With less "efficiency" the relaxation may be impaired and the equilibrium never reached. In spite of the fact that there exists a stabihty region of finite volume, the system wanders around it without stabilizing. This situation gives rise to lasting periods of exponential inflation in which the burden of operating below acceptable prices, shifts periodically among different groups of agents of the system. This behavior may be caused either by big price increases as compared with the small range of compatible relative prices or by small stochastic noise. We have also seen that our model incorporates inflation due to expansion of the monetary base even when there is no turbulence in relative prices. We also find that a mild increase in the amount of money in the system, when produced in a situation of equilibrium produces what are known as the "beneficial" effects of inflation, i.e. an expansion of consumption and production. We have also discussed active monetary policies simultaneous with relative price shocks although we are aware that our model might be misleading in this situation.
8
Acknowledgments
This presentation of the model (perhaps not final) was shaped during the 2nd Workshop on "The Economy as an evolving Complex System" held at the Santa Fe Institute, in Santa Fe, NM. We thank all participants and specially the organizers K.J.Arrow and P.W.Anderson for sharing with us their inside information on the glories and miseries of Microeconomic Theory. In particular we would hke to thank: J. Scheinkman for an interesting discussion on the monetary theory of inflation and G. Dosi for anticipating to us his explanation of the emergence of markups policies in noisy environments.
References [1] L.Walras,"Elements d'Economie Politique Pure". Lausanne, Corbaz, 1874. [2] K.J.Arrow and G.Debreu, Econometrica, 22, 1954, 265-290, "Existence of Equihbrium for Competitive Economy"; K.J.Arrow, H.D.Block and L.Hurwicz, Econometrica, 27, 1959, 82-109, "On the Stability of the Competitive Equilibrium II". [3] M.Boldrin, "Persistent Oscillations and Chaos in Economic Models: Notes for a Survey" in The Economy as a Complex Evolving System, Santa Fe Institute, Studies in the Sciences of Complexity, Vol. V, P.W.Anderson, K.J.Arrow and D.Pines, Eds., Addison Wesley Publ.Co. 1988. [4] B.Arthur, "Self Reinforcing Mechanisms in Economics" in The Economy as a Complex Evolving System, Santa Fe Institute, Studies in the Sciences of Complexity, Vol. V, P.W.Anderson, K.J.Arrow and D.Pines, Editors. Addison Wesley Publ.Co. 1988. [5] G.Hurd,'Macrodynamics and Return to Scale", The Economic Journal 96, 1986, 191-198.
470
R.P.J. PERAZZOet al.
[6] F.Machlup, Review of Economics and Statistics, 42, 1960, 125-139, "Another view of Cost-push and Demand-pull Inflation"; J.H.G.Olivera, Oxford Economic Papers, 16, 1964, 321-332, "On Structural Inflation and LatinAmerican 'Structuralism'"; F.Hirsh and J.Goldthrope (Eds.)"The political economy of inflation", Harvard University Press, Cambridge U.S.A., 1978. [7] M. Friedman and A.Schwartz, "A Monetary History of the United States, 1867-1960" Study for the National Bureau of Economic Research. Princeton University Press, Princeton N.J. 1963 [8] R.Balescu, "Equilibrium and Non Equilibrium Statistical Mechanics", John Wiley and Sons, N.Y.
(1975); G.Parisi, "Statistical Field Theory", Addison Wesley Publ.Co.Inc. N.Y.(1988)