A probabilistic framework for hysteresis

Physica A 287 (2000) 577–586

www.elsevier.com/locate/physa

A probabilistic framework for hysteresis M. Grinfelda; ∗ , L. Piscitellib , R. Crossb

a Department

of Mathematics, University of Strathclyde, Glasgow G1 1XH, UK of Economics, University of Strathclyde, Glasgow G1 1XH, UK

b Department

Received 19 May 2000; received in revised form 27 June 2000

Abstract We introduce a probabilistic framework for hysteresis, a ubiquitous phenomenon in economic situations involving sunk costs. The framework is applied to a simple model of a rm’s market entry/exit decisions. We study the in uence of sunk cost size, width of the hysteretic loop, and c 2000 of availability of local information on wealth creation in a sector of the economy. Elsevier Science B.V. All rights reserved. PACS: 75.60; 05.45 Keywords: Hysteresis; Discrete dynamical systems; Investment theory

0. Introduction In a relatively short time, econophysics, that is, a systematic application of methods of statistical physics to economic problems, has had a signi cant impact on both the philosophy and practice of macroeconomic modelling. Essentially, the “econophysical approach” to economical phenomena starts with an analysis of econometric data. Once the properties of the statistics of the data are understood, a simpli ed model that reproduces the salient features of the data is sought, and its predictions and implications are considered in an iterative fashion. In the present paper we follow a dierent, more classical approach. We argue that the ubiquitous presence of sunk costs in economic activity implies that in response to changes in a macroeconomic control parameter, agents will adjust their behaviour hysteretically. However, there are severe methodological diculties in trying to nd an empirical basis for this statement. There are many reasons why clear-cut hysteresis at the micro-level is not simply re ected as hysteresis at the macro-level, e.g. such ∗

Corresponding author.

c 2000 Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 3 9 4 - 0

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as feedback loops, hidden variables, and noise in the system. Because of the obvious inability to conduct the controlled experiments with macroeconomic variables, testing for the existence of hysteresis in macroeconomic relationships requires the use of multivariate econometric techniques. Such techniques are currently being developed, for example, a procedure for uncovering hysteretic relations between two macroeconomic variables is presented in Refs. [1,2] use the cointegration techniques to show how UK unemployment is cointegrated with the hysteresis indices for the interest rates and oil prices in such a multivariate framework. Alternative approaches to testing for the existence of hysteresis in macroeconomic relationships are provided in Refs. [3–5]. Below, we shall discuss a number of approaches to modelling hysteresis at the level of an economic agent. The main question we pose is, what are the implications of hysteresis for dynamics in a closed-loop system? To discuss this question, we use for illustrative purposes a very simple model of rm market entry/exit decisions, which is an extension of the model suggested in Ref. [6]. The ideas of hysteresis, borrowed from the magnetics, have been current in economics for quite some time. The economic contexts in which they have been applied are far-ranging, including rm market entry/exit decisions, portfolio management, unemployment, inventory control, and saving strategies. For reviews of the literature see Refs. [3,7]. The crucial condition which makes the hysteresis framework applicable is the presence of xed costs of making adjustments, choosing a plan of action requires an investment that incurs such costs, some of which will be sunk in the sense that they cannot be recovered should the plan of action be reversed. We shall illustrate a typical application in rm theory. Consider a rm that is either inactive (state 0) or active and producing one unit of commodity (state 1). Obviously, the decision whether to produce or not depends on control parameters, such as the price of the commodity, the exchange rate, the overall level of demand in the economy and so on. It is a typical task of econometrics to work out what a relevant set of such controls is. For simplicity, we assume that there is only one such control parameter C taking positive values, in such a way (one can think about some measure of optimism regarding the future level of demand) that if C is low, a rm would prefer not to enter a market, and if C is high, it would prefer to stay in. Entering the market involves sunk costs due to plant purchase, the establishment of distribution networks, etc., these are lost if the market is exited. Therefore, a simple model of the rm’s activity would be in the form of a hysteron (in the nomenclature of Krasnosel’skii and Pokrovskii [8]), we postulate values CH ¿CL , such that if the rm was in state 0, and C¡CH , it will remain in state 0; if it were in state 0 and C¿CH it will jump to state 1. Similarly, if it is in state 1 and C¿CL , it will remain in state 1, while if C¡CL it will jump to state 0. We denote such an input–output operator by FCL ; CH (C). We denote the state of a rm at time t by St . The above simple model captures the discrete nature of decision making. The connection of sunk costs and the width of the hysteresis loop CH –CL is discussed in Ref. [9]. Hard empirical evidence on rms’ CH and CL values is dicult to obtain because rms’ exit and entry decisions involve issues of the commercial con dence.

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One exception is the survey evidence in Ref. [9], which revealed that the rms’ hurdle rates of return, i.e., the returns required before the investment projects are proceeded with, were far in excess of the cost of capital. This excess was very large to be explicable in terms of risk premia. It is also clear that rms often stay in market when, in response to, say, a deterioration in the exchange rate, they are no longer recouping the costs of the capital employed. In the absence of inside knowledge of the CH and CL triggers employed by rms, it is dicult to know a priori what values these “Rubicons” take. Given this, it is only after a decision to enter or exit the market is taken that one could say that the decision is consistent with CH or CL being set in a particular range of values. The probabilistic model suggested below allows us to confront this diculty. There are several ways of combining individual agents to give a macroscopic model. In Ref. [6] a Preisach-type model is used. In other words, if the total (considered constant) number of rms in a sector is M , and to ith rm corresponds the hysteron Fi (C) ≡ FCLi ; CHi (C), where CLi ; CHi are the thresholds for ith rm, we have that the number of active rms at time At+1 is given by At+1 =

M X

Fi (Ct+1 ) :

(1)

i=1

In addition to the diculty with the de nition of CH and CL , this type of model has the disadvantages that (a) there is no simple mechanism for asynchronous updating of the states of the agents and (b) it is not clear how to incorporate interaction among agents. There are two conceptually dierent types of model one can construct using the formalism of Eq. (1). One could consider models in which At has no in uence on Ct+1 . This is a typical situation in magnetics, where a slab of magnetic material is subjected to an applied magnetic eld. In economics a good example would be the in uence of the number of rms active in the export markets on the exchange rate, with the latter being determined by the capital ows. Here we are interested in closed-loop systems, in which Ct+1 =f(At ). Again, a typical example in magnetics would be an LCR circuit with an inductor having a saturable core [10]. Using this type of model in the context of market entry–exit decisions is motivated by taking Ct+1 to be the expected retail price ex of the commodity, pt+1 . The actual retail price satis es the laws of demand and supply through a price elasticity function f; pt =f(At ). A rational expectations assumption is ex that pt+1 =E(pt+1 | ’t ) = pt = f(At ), where ’t is the information set available at time t. This is the assumption we are making in the present paper; less nave assumptions are of course possible. A more exible approach, loosely based on the (zero temperature, zero relaxation time) mean- eld Ising model is proposed in Ref. [11]. In our context, if we let Sti be the state of unit i at time t, and set Rit =2Sti −1, this type of model would take the form    M M X X  1 + 1 sgn  (2) Jij Rjt − Âi + Ct+1  : At+1 = 2 2 i=1

i6=j

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Fig. 1. A Hysteron.

Here Âi ¿0 is a rm-dependent threshold, while Jij measure interactions among agents. Ref. [11] shows that this type of model has many properties in common with Preisachtype models, in particular, the memory wiping-out property. An easily recti able problem with this type of model is that it does not have hysteresis at the micro-level. The solution, also used, for example, in Ref. [12], is to modify Eq. (2) to read    M M X X  1 + 1 sgn  Jij Rjt + i Rit − Âi + Ct+1  : (3) At+1 = 2 2 i=1

i6=j

Here the term i Rit measures the “change-aversion” of a particular rm. Unfortunately, a complete speci cation of a model of this sort would involve arbitrarily xing, in addition to the (M − 1)2 coecients Jij ; 2M constants i and Âi . In this note we suggest further simplifying (3). Consider the hysteron of Fig. 1. It is de ned by two curves, the lower one,  and the upper one ÿ (which coincide for C 6∈ (CL ; CH )) One can interpret these curves as being graphs of certain probabilities depending on C;  is the graph of P(0; 1; C), the probability of moving into state 1 from state 0, while ÿ corresponds to P(1; 1; C), the probability of staying in state 1 when one is in state 1. This also allows us to de ne P(1; 0; C) = 1 − P(1; 1; C) and similarly P(0; 0; C). The hysteresis assumption then becomes simply that P(s; s; C)¿P(t; s; C)

(4)

for any pair of states s and t. Now, this approach clearly allows us to use any hysteresis loop in the context of decision making, under the proviso that all the functions satisfy X P(s; t) = 1 ∀ s ∈ S ; t∈S

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where S is the set of states (S = {0; 1} in the hysteron example above) and P(s; t; C) satisfy the hysteresis assumption (4). Of course, we could reverse the inequality in Eq. (4) (reverse the arrows in Fig. 1). There is no physical justi cation for that, but in economics such an antihysteretic assumption could correspond, say, to propensity to panic reactions, or non-extrapolative expectations. What we have here in essence is a Markov chain with state-dependent |S| × |S| transition matrix M (C); Mij (C) = P(si ; sj ; C), see Ref. [13] for a model of the stock market along the same lines. Obviously, the exact form of the elements of M depends on the control parameter(s) C. On the other hand, for making the kind of rough predictions economic modelling usually makes, one should just assume that, say for C measuring optimism, Mij (C) is monotone increasing in C if j¿i and monotone decreasing otherwise. The functional assumptions made on the transition probabilities are more transparent than the ones one of necessity makes while using the Preisach model. We think that this way of looking at hysteresis incorporates in a simple way, the give and take of a board meeting and in a sense makes it unnecessary to introduce the additional exogenous shocks. 1. A simple model In this section we reformulate the simple rm exit–entry model of Piscitelli et al. [6] in terms of the above concepts. As before, we consider a sector of industry containing a xed number M of rms. Of these, at time t; At are active, that is, in the market and produce one unit of commodity. The decision whether to enter, exit, or stay in the market at time t + 1 depends, as explained above, on the price of the commodity, pt , which is assumed to depend inversely on At . We de ne the number of inactive rms by It and let state 0 stand for being inactive and state 1, for active. If now we let Pij (p(At )); i; j = 0; 1 be the transition probabilities from state i to j given that the price of the commodity is p(At ), we have the following simple dynamical system:  T  T   P00 (p(At )) P01 (p(At )) It It+1 : = At+1 At P10 (p(At )) P11 (p(At )) Clearly, the assumptions of rational expectations, of a xed maximum number of rms and of units of commodity produced by a particular rm, as well as of reversible bankruptcy, etc., are unrealistic, our goal here is to understand the implications of hysteresis per se. Remembering that At + It = M and dividing by M , we arrive at a one-dimensional equation of evolution for xt = At =M : xt+1 = f(xt ) ≡ (1 − xt )P01 (p(xt )) + xt P11 (p(xt )) :

(5)

Since p0 (x)¡0, and the probabilities are increasing functions of their argument, the dynamics is very simple. For example, there is always a unique xed point. To see

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this, observe that the xed-point equation can be written as x(1 − P11 (p(x))) = (1 − x)P01 (p(x)) : The left-hand side is a monotone increasing function of x, while the right-hand side is monotone decreasing, hence uniqueness. We also have 0 0 (p(x))(1 − x) + P11 (p(x))x] : f0 (x) = P11 (p(x)) − P01 (p(x)) + p0 (x)[P01

(6)

Now, we can investigate the consequences of hysteresis and antihysteresis. 1.1. The antihysteretic case Here P11 (z)6P01 (z)

for all z ∈ [0; 1] :

and from this it is clear that f0 (x) is always negative. The second iteration of the map f is monotone increasing and it is easily shown that the economic consequences of antihysteresis are disastrous, as the system evolves towards a period 2 boom-bust dynamics. 1.2. The hysteretic case In this case P11 (z)¿P01 (z)

for all z ∈ [0; 1] :

The resulting map is no longer necessarily monotone decreasing, since now the difference P11 (p(x)) − P01 (p(x)) tends to increase the derivative, and there is now the possibility of the xed point being globally asymptotically stable, as in Fig. 2, where p(z) = 1=(2z + 1) and P01 (z) = 12 + 12 tanh 10(z − 0:7) and P11 (z) = 12 + 12 tanh 10(z − 0:4). These calculations show that in the deterministic case, hysteresis (as de ned in this paper) has a stabilizing in uence on the dynamics, and, naturally enough, antihysteresis has a destabilizing one. More generally, if we take P11 (z)=P01 (z), where the map f(x) is still monotone decreasing, and choose parameters such that the unique xed point is neutrally stable, it can be shown that introducing hysteresis stabilizes the system, while introducing antihysteresis destabilizes it. In electrical engineering, hysteresis is synonymous with energy loss; ecient transformers, for example, do not exhibit hysteresis. At the same time, there is a wide-spread agreement that the hysteretic circuits are less prone to disasters such as ferroresonance. The above analysis lends credence to the view that in economics as well as hysteresis has a stabilizing in uence. 1.3. A stochastic version of the model It is easy to modify the above model to allow for shocks in the price, as is done in Ref. [6]. In other words, we let the price at time t; pt , be the product of the

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Fig. 2. The map f of Eq. (5) in the hysteretic case.

Fig. 3. Dynamics of the stochastic model in the hysteretic case.

deterministic price (which depends on xt ), pdet (xt ) and a shock variable Xt , which is assumed to follow a random walk. A representative example of the dependence of xt on pt in the hysteretic case is presented in Fig. 3. Note the typical hysteresis loops and the small range of price variations. 1.4. Monte Carlo simulations Obviously, the above Markov chain formulation also allows us to perform Monte Carlo simulations, with both synchronous and asynchronous updating of states of all agents. Of these, the synchronous case is of more interest. Below we present results of typical runs for an economy sector consisting of a hundred rms. Here we incorporate the sunk costs directly, and compute the output yt generated by the sector up to time t

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Fig. 4. Dependence of output on the hysteresis loop width.

Fig. 5. Dependence of output on sunk cost size.

from yt+1 = yt + xt (pt − pprod )R − Cs (xt − xt−1 ) : Here pprod is production cost, R is some revenue conversion factor, and the last term is precisely the losses due to sunk costs Cs . Here we are assuming that the sunk and production costs are constant across the sector. A typical dependence of the revenue on the width of the hysteresis loop (again using tanh transition probabilities) is given in Fig. 4. In Figs. 4 and 5, pprod = 0:4 and R = 3000:0. The width of the hysteretic loop here is de ned as follows: w = x0 − x1 0 where xi maximizes Pi1 (x). In Fig. 4 Cs = 15. Note that all these runs are made for the hysteretic case; that is, there is a minimal size of a hysteresis loop for which the sector is at all viable. Similarly, varying only the sunk costs while keeping the rest of the parameters xed, we see in Fig. 5 how a slight increase in sunk costs (without the adjustment of the transition probabilities) can turn a sector inviable. Here w = 0:6.

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Fig. 6. In uence of local information on output.

1.5. Local information From the above, it seems plausible that the pro tability of a sector is an increasing function of the width of the hysteresis loop. Hence, any strategy that tends to increase the width will be bene cial to the sector. For example, suppose that a rm A that is not in the market uses checks before “ ipping a coin” to assess whether a randomly chosen rm B intends to be in the market. Again, if the intentions of B are not known, a fair estimate of its intention is whether it is in the market already. Suppose that if B is in the market, A discounts the pro tability of entrance by a factor , that is, as A’s decision is being made, the probability of it entering the market is P01 (pt =). Such a simple modi cation to the Monte Carlo protocol makes a sizable change to the pro tability, as seen from Fig. 6, where  = 1:1. 2. Conclusion Explicit inclusion of sunk costs makes many popular toy models of economic modelling, such as the El-Farol Bar problem [14] and the minority game [15] not directly relevant. It would be very interesting to see an equivalent theory developed for the case considered here. That this case is non-trivial can be seen from a game-theoretic setting involving just two rms. Clearly, if both of them are in the market which cannot support the two of them, it makes no sense for any of the rms to leave it, thus relinquishing all the pro ts to the rival and losing the sunk costs as well. We hope that the probabilistic cellular automata framework proposed in this note makes it reasonably easy to explore the realistic strategies in situations such as these. References [1] L. Piscitelli, R. Cross, M. Grinfeld, H. Lamba, A test for strong hysteresis, Comput. Econom. 15 (2000) 59. [2] R. Cross, J. Darby, J. Ireland, L. Piscitelli, Hysteresis and unemployment: some preliminary investigations, in: Unemployment Dynamics Workshop, Centre for Economic Policy Research, 1998.

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