Chaotic PDF Free Download

Agriculrural
PII:
SO308-521X(97)00003-6
Vol. 55. No. 3, pp. 445459, 1997 C 1997 Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0308-521X/97 $17.00 +O.OO Sysrems,
ELSEVIER
Modelling Chaotic Behaviour in Agricultural Prices Using a Discrete Deterministic Nonlinear Price Model Zsuzsanna Bacsi Pannon
University
of Agricultural
(Received
Sciences, Keszthely Hungary
23 September
1996; accepted
8360, Keszthely,
16 December
DePk F. u. 16,
1996)
ABSTRACT In economic modelling, a frequent problem is that the generally used deterministic equilibrium models cannot describe the ‘random-looking ’ oscillations and irregular motions often observed in real time series. Nonlinear chaotic models o#er a way to produce this behaviour without the introduction of stochastic elements. Recently, many publications have dealt with chaos in economic processes, but the majority of them used rather d#icult mathematical functions to model these processes. Furthermore, chaotic behaviour emerged most often in parameter ranges that are [email protected] to interpret as economically meaningfiil values. The present paper analyses the behaviour of a discrete deterministic non-linear model of supp1.v and demand of a single product with many producers on the market. N producers operate in the market of a single product; their production technology is described by a quadratic cost function. They produce the amount which maximises their gain considering an expected market price in the next time period. The demand function is d(t) = D*p(t) + d, a linear function of the p(t) real market price, and the real market price is the value which balances the total supply from the N producers and the total demand in the market. This linear model is made a piecewise linear one by putting a lower and an upper limit on the expected prices and the real market price. The sensitivity of the model to a wide range of negative values of D is tested. As the value of D increases, the model produces various types qf steady state behaviour, such as equilibrium point, periodic behaviour with increasing periods, ‘quasiperiodic-like’ behaviour, period-3 cycle, and chaotic behaviour. The above model showed that a deterministic price model without any stochastic component is fully capable of producing irregular oscillations, fluctuations often observed in real price series. Furthermore, it was 445
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Z. Bacsi
possible to achieve this behaviour with a simple, piecewise linear cobweb model, with parameters that have straightforward economic meaning, in realistic parameter ranges. Ifparameters have sound meaningful values, then chaotic models can direct the attempts made at price stabilization by describing clearly which is the parameter range the decision maker has to avoid in order to reach a stable or regular behaviour. 0 1997 Published by Elsevier Science Ltd
MODELLING
ECONOMIC
TIME SERIES
A frequent problem in modelling time series in economics is that the generally used smooth equilibrium models produce a much stabler behaviour than is experienced in real time series. While the deterministic models used show a kind of stable behaviour, the real data show sudden bursts and extreme fluctuations which are difficult to explain with the model. Economists frequently face a situation when some element of the economy, e.g. prices, inflation, investment figures, etc., behave irregularly, and steps ought to be taken to stabilize this irregular behaviour. In macroeconomic modelling, several attempts were made at stabilizing time series showing undesired oscillations or fluctuations, but the parameter values the models suggested to lead to stabilization often caused results totally opposite to what was expected. Several attempts were made to incorporate exogenous shocks or stochastic elements into the models to create similar time sequences as the observed data, but the theoretical foundation of these additions to the models is rather weak (Gabisch & Lorenz, 1987; Benhabib, 1992). Recently, much significant work has been done to create fully deterministic models that are capable of describing the cyclic behaviour and the irregular motions of economic time series. In the last few decades chaos theory, a developing branch of mathematical theory has been used in economic modelling with encouraging results.
CHAOTIC
MODELS AND THEIR APPLICATION
IN ECONOMICS
Chaotic models are fully deterministic models which produce ‘randomlooking’, but bounded, trajectories which do not converge anywhere, but show non-periodic fluctuations in time. A typical feature of chaotic models is that they are very sensitive to minor changes in initial values or parameters. This latter fact means that, as small measurement errors in estimating the parameters or initial values of the model are unavoidable in practice, the behaviour of the model might be completely different from the behaviour of
Modelling chaotic behaviour in agricultural prices
447
the process the modeller intended to describe. Because of this, chaotic models are generally not suitable for forecasting but, with a full analysis of their behaviour, one can identify the types of possible behaviour in a wide range of parameter values. In economic modelling, a frequent problem is that the generally used deterministic equilibrium models cannot describe the ‘random-looking’ oscillations and irregular motions often observed in real time series. Nonlinear chaotic models offer a way to produce this behaviour without the introduction of stochastic elements. It is relatively easy to create models showing chaotic dynamics. Collett and Eckmann (1980) show that any iterative relationship in which the curve of x(t + 1) as a function of x(t) is ‘hill-shaped’, can produce this type of behaviour with suitable parameters. They give a clear and detailed description of the importance of chaotic models in economics. In economic modelling, the application of chaotic models has gained much attention. Burton (1993) demonstrated the appearance of chaos in an agricultural commodity model based on the so called logistic model, which is one of the most well-known chaotic models. Gabisch and Lorenz (1987) show several macroeconomic models which are based on the above mentioned logistic model, and are capable of chaotic behaviour. In macroeconomics, the well-known cobweb model can be used to describe the dynamics of prices in a fully competitive environment. As Chiarella (1988) shows, ‘this model also produces chaotic behaviour at certain parameter values. However, the majority of chaotic models have used rather complicated mathematical functions, which are difficult to interpret from an economic viewpoint. Also, chaotic behaviour was experienced with parameter values that are rather far from realistic values. The present paper attempts to demonstrate that a very simple cobweb model, made piecewise linear by adding a lower and an upper bound to a linear market price model, and with very straightforward economic meaning, is fully capable of describing chaotic behaviour. Similar approaches have been used in electronics, in circuit theory, and a somewhat similar idea is pursued by Simonovits (1990) in analysing investment decisions, but the approach is new to modelling market behaviour and price formation in agriculture. Chaotic models are usually characterised by a set of indicators. The most important ones of these are: The Lyapunov-spectrum, which is the indicator of the speed of divergence of trajectories (solutions) in time, starting from nearby initial conditions. In the case of stable models, the values of the Lyapunov spectrum are all negative, for periodic solutions it has zero and negative values, while for chaotic models the largest Lyapunov-exponent has to be positive, indicating an exponential rate of divergence. The relevant mathematical literature
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Z. Bacsi
describes methods to calculate the Lyapunov-spectrum, or to estimate the largest Lyapunov exponent, see eg. Holden (1990), or Ramsey et al., (1990). Another indicator of chaotic behaviour is the so called capacity dimension, or fractal dimension. This value indicates the relative amount of information needed to describe the geometrical position of a point in the trajectory in comparison to the unit interval. For stable trajectories, the capacity dimension is an integer value, eg. for a trajectory having a fixed point limit set it is 0, for a quasiperiodic limit set (the sum of K periodic trajectories with incommesurable frequencies) it equals K, and for a strange, chaotic limit set it is a non-integer value. Thus a non-integer capacity dimension value is another indicator for the presence of chaos in the model analysed. For calculating the capacity dimension, the so-called box counting algorithm is used. For the exact definition and calculation methods see Holden (1990), or Ramsey et al., (1990). The onset of chaos can be visualised in a bifurcation diagram. This diagram shows the points of the limit set as the function of a chosen parameter. From the bifurcation diagram, it is easy to identify the qualitative behaviour of the model depending on the value of the chosen parameter. Bifurcation points are the parameter values where this qualitative behaviour changes, eg. from a fixed point a periodic limit set, or from a periodic limit set a chaotic one is born.
THE MATHEMATICAL
MODEL
The present paper analyses the behaviour of a discrete deterministic nonlinear model of supply and demand of a single product with many producers in the market. The model is based on the production model of Szidarovszky and Molnar (1994) as below: N Peck, t> 4%
t)
B(k), b(k), c(k)
denotes the number of producers present in the market : denotes the market price expectation of the k-th producer in time t : denotes the amount of the product produced by the k-th producer in time t : are the parameters of the cost-function of the k-th producer, where the cost function is as follows: :
C(k, t) = x’(k, t) * B(k) + x(k, t) * b(k) + c(k) where C(k, t) is the total cost of the k-th producer associated with x(k, t) amount produced
Modelling chaotic behaviour in agricultural prices
449
denotes the market price at time t denote the parameters of the demand function, which is: d(t) = p(t) * D + d : the total amount demanded of the product at time t
p(t)
:
D, d
:
The model assumes that each producer produces the amount that maximizes the expected gain, that is, the difference of the expected price income and the associated cost of production. x(k, t) = arg max (pe(k, t) * x(k, t) - C(k, t))
which gives: x(k, t) = (‘pe(k, t) - b(k))/(2 * B(k)) k = 1,2, .. . . IV Then the real market price will be the equilibrium price for which the total demand d(t) equals the total amount of product supplied by the N producers. From these, the p(t) time series can be calculated as the output of the model.
The behaviour of the model depends on how the producers estimate the expected price. Three versions of price estimation are discussed: I, Static model. 2. Adaptive model: 3. Extrapolative model:
pe(k, t) = p(t - 1) pe(k, t) = alpha(k) *p(t - l)+ (1 - alpha(k)) * pe(k, t - 1) pe(k, t) = alpha(k) *p(t - I)+ (1 - alpha(k)) * p(t - 2)
where 0 =< alpha(k) =< 1 for
k = 1,2, .., N
One slight modification was made to the original model, which changed the linear structure into a piecewise linear one; that is, a lower and an upper limit were specified for the pe(k,t) and p(t) values. pmax P(t) = P(t) i pmin
if if if
p(t) > pmax pmin =< p(t) =< pmax p(t) < pmin
Z. Bacsi
450
In the analysis, the above model was used to describe the dynamics of agricultural prices, and the range of parameters was chosen accordingly. Advantages of the model: The model is capable of handling many producers with different cost functions. Through the utilisation of expected prices, the model is capable of handling the fact that producers have to decide about the amount to be produced well before they would know the real market price. The cost function utilised fits to the theory of diminishing returns, and is suitable to deal with both the direct production costs and fixed costs (including, eg. the costs of storage). The model, as will be shown later, is capable of producing similar fluctuations and sudden ‘irregular’ behaviour as is often experienced in real observed time series of prices, which the majority of equilibrium models used in economics fail to describe. Stability of the model In Szidarovszky and Molnar (1994), the conditions of the global asymptotic stability for the non-limited model are given as the following: Static case:
D<-
Ckc,.Z..N (l/(2 * B(W))
Adaptive case: D <
-
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. 2..N @?-WWW
-
2 * alpha(W)
* W)))
Extrapolative case: K4HW
c
k=l )2,N ((1
-
-
WC2
* W)));
2 * alpWW(Wk)))
In the following analysis the model will be restricted to the case of two producers (N = 2). After some simple algebra it can be shown that if the model shows unstable behaviour, then the following must hold for any of the price estimation models:
Modelling chaotic behaviour in agricultural prices
451
( (D) 1* 1min{B(k), k = 1, ..N) I=< 1, where ) . ( denotes the absolute value.
THE CHAOTIC
BEHAVIOUR
OF THE MODEL
The model was tested with two producers with slightly differing values for the cost function parameters (assuming slightly different cost structure and production technology), and it was assumed that both producers use the same type of price estimation, though the alpha(k) values were slightly different. As, in the above section, it was shown that the stability of the model is directly related to the magnitudes of the D and B(k) values, it was reasonable to choose one of these in trying to find unstable behaviour in the model. In the present analysis, D was chosen as a variable to which the sensitivity of the model was tested, for two reasons: ??
??
Using statistical data finding exact values for B(k) is rather difficult, as information about the cost structure of individual producers is often scarce. Also, B(k) values can change frequently if the producers modify their production technology, which most probably affects their costs as well. From the above, it seemed more meaningful to analyse whether, with an arbitrary B(k) parameter set, the change in D may or may not lead to unstable behaviour in the price series. The value of D may reflect external effects, such as effects of foreign markets and trade agreements, or unfavourable weather conditions in the production season leading to increased demand. Besides, governments often try to stabilize the situation of agricultural producers through influencing the behaviour of consumers. The analysis of the model sensitivity to the change in the value of D can lead to the assessment of the effectivity of these strategies.
Using time series of observed Hungarian price and production data, a regression estimation was carried out to find a reasonable range of parameters to be used in the analysis. Time series of wheat and maize production and market prices were used to estimate the B(k) and b(k) values and the D and d parameters of the demand function. The values of alpha(k) were chosen arbitrarily - though further analysis showed that their value was not crucial in the qualitative behaviour of the model. The model was set with the following parameters: (units are: price: Hungarian Ft, amount of product: tons)
452
N=2 B(1) = 0.01453 b(I) = -1.789 alpha(l) =0.8 d= 552 660 D = -200 to -0.001 pmax = 80 000 time: t = O-2000
Z. Bacsi
B(2) = 0.01500 b(2) = -1.789 (the same as b(1)) alpha (2) = 0.9 initial value: p(0) = 14 500 pmin = 5000
These values were derived by analysing potato production costs and price series in the wholesale market between 1991 and 1995. The upper and lower limits were chosen somewhat arbitrarily but, again, the real market price observations were considered as guidelines. With these parameters the stability conditions given by Szidarovszky and Molnar (1994) were calculated precisely to find the relevant stability limit values for D. Then the parameter range was chosen for D that covers values in the stable region and in the unstable region as well. for the adaptive case and for the extrapolative case:
D > =-50.21, D > =-47.31
The parameter range examined in the analysis is D = (-100, -0 . 01) for both the adaptive and the extrapolative model types. The static case is a special case of the adaptive model with alpha(k) = 1, so the further analysis will be restricted to the adaptive and the extrapolative cases. The steady state behaviour is simulated by running the model from t = 1 to 2000 and then from t =2001 to 2050. The last 50 points are plotted against the corresponding D-value to estimate the behaviour of the trajectory in the long run. Adaptive case:
Figure 1 shows the bifurcation diagram for the adaptive case for the values of D from -50 to -0.01. For D values smaller than -50, the limit set is always one point, which means that the model converges with time. Then, at D=-50.21, the unique limit point disappears and two limit points are born. At first, these two points are very close to each other at around the value 5000, but they separate quickly; one of them, the upper branch grows and the other remains at the same value as before the bifurcation point. At the value of D = -8.3, the upper branch of the limit set bifurcates again and, within a small interval of D values, a series of bifurcations occur, while in the lower range the p = 5000 values still remain limit points, too.
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0
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I
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-7.78
t D
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I
D = 0.
I f
*=
-7.52
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From -8.3 lo -7.13, a chaotic-looking area appears. As D increases in this range the limit points become a set of nearly continuous-looking bands, between about p = 24 000 and p = 5000 (Fig. 2). From D z-7.23 to D [email protected], in the remaining range of analysis, the chaotic behaviour disappears and the limit set contains only two limit points,
454
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To illustrate the instability of the prices in the chaotic range of D, the modelled price time series is shown in Fig. 3, for D = -7.35 and t = I to 500. It can be seen that, as time increases, the price values fluctuate with increasing distance up to a point, then they fall close to each other and the whole pattern repeates itself. It is worth pointing out, as it is difficult to identify from the chart, that nearly every second value in the time series is pmin, while the other values fluctuate between a lower and a higher value. It is important to notice that, while the pattern of the time series looks similar in repeated time intervals, they are never the same, and the length of the similar-looking interval also changes. As time grows, there appear time intervals where the price fluctuation is small - except for the pmin values - and nearly neglectable, then sudden bursts of extreme values appear both above and below this ‘nearly stable’ range. If we assume that the model describes reality more or less correctly, and measure time in weeks, then the above means that, without any exogenous reason, the price behaves relatively stably, oscillating between a fixed low and a slightly changing high value, for about 50 weeks (nearly one year), then again, with no exogenous reason, it starts to fluctuate wildly. Afler a time, the price settles again without any forced change in the economic environment, and this pattern is repeated with changing lengths of stability and instability.
Modelling chaotic behaviour in agricultural prices
455
Extrapolative case: A similar analysis was carried out for the extrapolative case. Figures 4, 5 and 6 show similar results. In the chaotic-looking range, the Lyapunov exponent was calculated for both the adaptive and for the extrapolative case to describe sensitivity of the steady state behaviour to the choice of initial conditions. As was mentioned Bifurcation
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above, in chaotic systems, the largest value of this exponent is positive, meaning an exponential rate of divergence of trajectories starting from nearby conditions. Also, the capacity dimension of the chaotic limit set was estimated, which is a non-integer value, as is typical for the dimension of fractals.
Adaptive model, D = -7.35 Extrapolative model, D = -9.34
Largest Lyapunov exponent
Capacity dimension
0.1557 0.140
0.9618 0.9352
Figure 7 shows real weekly potato prices collected from the Budapest wholesale market. The upper part of the chart gives the real maximum and minimum prices, the lower part shows the average price after some corrections for inflation. It can be seen that, though no attempt at fitting the model to real price series has yet been made, the real price series also show irregular, chaotic-like motion in the same value range as in the case of the models, though the different fluctuation pattern indicates the need for further analysis.
451
Modelling chaotic behaviour in agricultural prices
Potato prices
1 min
max
80 60
O!
I
I
0
50
250 150 200 100 weeks (1991, 1st week- 1994.49th week)
2
Potato prices (adjusted) 50 45 40 --
35 30 25 20 15 10
51 0
Fig. 7. Observed minimum
I
I
I
I
I
250 50 100 150 200 weeks (1991 1st week - 1995.49th week)
300
wholesale market prices for potatoes - upper figure: actual maximum market prices; - lower figure: average prices adjusted for inflation.
and
Z. Bacsi
458
CONCLUSIONS
-
THE UTILISATION
OF CHAOTIC
MODELS
The above model showed that a deterministic price model without any stochastic component is fully capable of producing irregular oscillations, fluctuations often observed in real price series. It is true that, due to the extreme sensitivity to initial conditions and parameter values, the chaotic models cannot easily be used for predictions but, instead, they have a great advantage: they can identify the parameter ranges where the process modelled is unstable. This way, chaotic models can direct the attempts made at stabilization by describing clearly which is the parameter range the decision makers have to avoid in order to reach a stable or regular behaviour. The present model is one of the simplest non-linear model types which, while corresponding to the relevant economic theory, can produce similar motions as often observed in reality. Further work is being carried out to compare the model behaviour to long time series of real agricultural prices. Further investigations were made to test whether the model shows the same kind of sensitivity to the change of other parameters (e.g. d in the demand function, or B(k) in the costs functions), and it was found that the unstable, fluctuating behaviour is far from being infrequent. An extreme sensitivity was also detected to the lower and upper bounds on price (pmin and pmax). Assuming that the model described the price behaviours in the home market, the boundary values may reflect the impact of world market prices on the home market. Thus the sensitivity to these will also have significant implications. A logical continuation to the present study is to see whether the price series provided by the model fit to real observed price series, and to test whether, in observed time series, a similar pattern of irregular behaviour can be generated to what the model produces for parameter values set accordingly. Considering the sensitivity detected in the model, the efforts have to be directed in the future towards identifying reasonable parameter values for the cost and the demand functions.
ACKNOWLEDGEMENTS The present paper is a part of the PhD study of the author, carried out at the Operations Research Department of the Eotvos Lorand University of Sciences, under the supervision of Dr Bela Vizvari. The computer program for calculating the adaptive and extrapolative time series and bifurcation sequences was written in BorlandC++ by the present author. The capacity dimension calculation program was written by the author of the paper in BorlandC ++ following a suggestion of B. Vizviri on the calculation method.
Modelling chaotic behaviour in agricultural prices
459
The largest Lyapunov exponents were estimated using the MTRCHAOS and MTRLYAP software by M. T. Rosenstein (1993, Dynamical Research, 15 Pecunit Street, Canton, MA, USA).
REFERENCES Baumol, W. J. and Benhabib, J. (1989) Chaos: significance, mechanism and economic applications. Journal of Economic Perspectives, 3(l), Winter, 77-105. Benhabib, J. (ed) (1992) Cycles and chaos in economic equilibrium. Princeton University Press, Princeton. Brock, W. A. and Sayers, C. L. (1988) Is the business cycle characterized by deterministic chaos? Journal of Monetary Economics, 22, 71-90 (1988), reprinted in Benhabib (ed) (1989), pp. 374-393. Burton, M. (1993) Some illustrations of chaos in commodity models. Journal of Agricultural Economics 44(l), 38-50. Chiarella, C. (1988) The cobweb model, its instability and the onset of chaos. Economic Modelling, October 1988, 376384. Collett, P. and Eckmann, J. P. (1980) Iterated maps on the interval as d~namiral systems Birkhauser, Basle. Gabisch, G. and Lorenz H.-W. (eds) (1987) Business cycle theory. Lecture Notes in Economics and Mathematical Systems No. 283. Springer-Verlag, Berlin Heidelberg. Holden, A. V. (ed) (1986) Chaos. Manchester University Press, Manchester. Ramsey, J. B., Sayers, C. L. and Rothman, P. (1990) The statistical properties of dimension calculations using small data sets: some economic applications International Economic Review, 31, 991-1020. reprinted in Benhabib (ed) (1989), pp. 394428. Rosenstein, M. T. (1993) The MTRCHAOS and MTRLYAP software packages. Dynamical Research, 15 Pecunit Street, Canton, MA, USA. Simonovits, A. (1990) Beruhazisi hatirciklusok a szocialista gazdasigban (Investment limit cycles in the socialist economy. In Hungarian, with English summary), K6zgazdasdgi Szemle, 37, 114331156. Szidarovszky, F. and Molnar, S. (1994) Adaptiv es extrapolativ becslesek egy specialis diszkret dinamikus termeliii-fogyasztoi modellben (Adaptive and extrapolative estimations in a special discrete dynamic producer-consumer model - in Hungarian, with English summary). Stigma, 25(4), 221-227. Vizviri, B. (1995) Personal communication on the dynamical behaviour of a bounded system, and on a computation method of capacity dimension.
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