Mode-locking in a forced business cycle

NORTH- HOLLAND

Mode-Locking in a Forced Business Cycle ERIK

REIMER

LARSEN

and CHRISTIAN

HAXHOLDT

ABSTRACT Recently, interest in nonlinear dynamics in economics and other sciences has grown rapidly. Mode-locking is a typical phenomenon that can occur in systems where several oscillatory processes interact. For linear systems, the principle of superposition applies. However, as soon as nonlinear interactions become significant, this principle ceases to be valid, and two or more oscillatory modes will tend to adjust to one another to produce a "locked" solution in which one mode performs precisely q cycles each time the other mode performs p cycles, with p and q being integers. In this study we discuss mode-locking in the context of the well-known Goodwin business cycle. We demonstrate how a simple model of this cycle when perturbed by a sine wave can produce mode-locking along with the associated phenomena of a devil's staircase and so-called Arnol'd tongues. © 1997 Elsevier Science Inc.

Introduction This study is c o n c e r n e d with the p r o b l e m of describing c o m p l e x d y n a m i c p h e n o m ena in m a c r o e c o n o m i c systems. This a r e a of r e s e a r c h has a t t r a c t e d a rapidly g r o w i n g interest since the mid-1970s w h e n it b e c a m e k n o w n that v e r y simple n o n l i n e a r systems can exhibit e x t r e m e l y c o m p l i c a t e d d y n a m i c s [1, 2]. A c t u a l e c o n o m i c t i m e series are s e l d o m c h a r a c t e r i z e d by the regular d y n a m i c s typical of linear systems. Instead, various types of irregularity in the f o r m of large-scale fluctuations of d i f f e r e n t f r e q u e n c i e s are o b s e r v e d . T h e q u e s t i o n t h e r e f o r e arises to w h a t d e g r e e the fluctuations are the result of r a n d o m e x o g e n o u s events, and to w h a t d e g r e e t h e y m a y be ascribed to s o m e kind of internally g e n e r a t e d , e x p l a i n a b l e a l t h o u g h c o m p l i c a t e d dynamics. C o n s i d e r i n g the phase shifts and lags that can be o b s e r v e d in the i n t e r n a l relations b e t w e e n v a r i o u s e c o n o m i c variables, d y n a m i c p h e n o m e n a s e e m m a n i f e s t in m a c r o e c o n o m i c systems. A t the s a m e time, the significant variations that m a n y e c o n o m i c variables exhibit r e l a t i v e to their m e a n values suggest that m o s t e c o n o m i c systems are i n h e r e n t l y nonlinear. M o d e - l o c k i n g (also k n o w n as e n t r a i n m e n t or f r e q u e n c y - l o c k i n g ) was originally o b s e r v e d in physics and electronics. In r e c e n t years, h o w e v e r , it has b e c o m e the subject of r e s e a r c h in m a n y o t h e r areas as well. In physics the c o n c e p t s h a v e b e e n applied, for instance, to J o s e p h s o n J u n c t i o n s [3], charge density w a v e s [4], and f o r c e d R a y l e i g h ERIK REIMER LARSEN is Marie Curie Fellow at the University of Bologna, Italy. CHRISTIAN HAXHOLDT is Assistant Professor of Management Science and Statistics at the Copenhagen Business School, Denmark. Address reprint requests to Christian Haxholdt, Department of Management Science and Statistics, Copenhagen Business School, Julius Thomsens Plads 10, DK-1925 Frederiksberg C, Denmark. Technological Forecasting and Social Change 56, 119-130 11997) © 1997 Elsevier Science Inc. All rights reserved. 655 Avenue of the Americas, New York, NY 10010

0040-1625/97/$17.00 PII S0040-1625(97)00004-8

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B6nard convection [5]. Theoretical analyses have been performed for a driven van der Pol oscillator [6] and for the forced Brusselator [7]. In biological realm, mode-locking has been observed in experiments with chicken hearts cells [8], the human heart [9], and in connection with pulsatile insulin secretion [10]. In economics it has been shown how long waves can lock with other cycles of the macroeconomic system [11], and how different sectors of the economy might interact with one another to produce an overall periodic motion [12]. Mode-locking is a universal phenomenon that arises in all systems where two or more oscillatory modes interact. Already in the 1930's, Schumpeter suggested that there might be a system in the timing of different cycles in the economy: "We now go on to postulate that each Kondratieff 1 should contain an integral number of Juglars 2 and each Juglar an integrated number of Kitchins 3. The warrant for this is the nature of the circumstances which give rise to multiplicity" [13]. However, Schumpeter did not provide any explanation for how this phenomenon could arise. Forrester [14] made similar observations about the timing of cycles in different industries and with his engineering background referred to this phenomenon as entrainment. In the present study we discuss how Schumpeter's and Forrester's observation can be explained in terms of the modern theory of nonlinear dynamics. For this purpose we consider a simple model of self-sustained business cycles. By forcing the model with an external sine wave signal, we show how mode-locking can arise and at the same time illustrate the potential of modern nonlinear analysis in economics.

Goodwin Model The model that we consider in this study is Goodwin's business cycle model [15]. Although more recent models exist that reflect today's state of the art in economics [16], we have chosen Goodwin's model as it is a well-known and relatively simple model. Any model that describes the business cycle as a self-sustained oscillation will give rise to similar results (see [17] for other examples). Before Goodwin, most economists assumed that linear structural relations could form the basis for a theory of business cycles. Goodwin emphasized that the linear approach represented an oversimplified special case and was insufficient for a more general theory of oscillations in the economy. To get a better understanding, one had to use nonlinear models. This had several advantages: it allowed for the occurrence of sustained oscillations, and it provided the possibility of having a different time periods for "upturn and downturn" in the economy. Goodwin presented three different models of nonlinear business cycles. We do not go into details here but describe only the final version. The model is a nonlinear accelerator-multiplier. The consumption function is of the traditional Keynesian type with an extra term that depends negatively on the growth in income. The desired stock of capital is proportional to the level of income, and the investment function is piecewise linear. In general, the investment is equal to lagged changes in the desired stock of capital. For very large changes in the desired stock of capital, the function is truncated so that investment is restricted by a lower or an upper bound. These bounds can be interpreted as arising from adjustment costs, resource constraints, capacity restrictions of the investment goods industry, etc. Income is equal The Kondratieff cycle or long wave has a length of 40 to 60 years : The Juglar or investment cycle has a length of 7 to 11 years. The Kitchin or inventory cycle has length of 3 to 5 years.

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to the sum of consumption and investment. The model takes the form of a second order differential equation [15]: 2 -- - ( ( ~ + (1 - ~,)0)z + + ( ~ )

-

(1 -

~,)z

(1)

or0 where g is the deviation of income from it's steady-state value, (~ is the factor by which changes in income affect consumptions, c( is the marginal propensity of consume, 0 is a lag variable, and (I) (~) is the investment function. In the original version, the investment function is a linear function with ceiling and floor, i.e., a nondifferentiable function. This is clearly an advantage for the analytical treatment but can cause numerical problems in more detailed computer simulations. However, we have replaced the nonlinear investment function with a similar but differentiable function represented by: ~(~) = v arctan (z) + ¢.

(2)

'IT

Here v, and ¢ are scaling parameters chosen to fit the function to a graph similar to the one used in Goodwin's original model. In this case v = 12 and ~ = 3. For the other parameters we have used the values suggested by Goodwin. They may not be the most appropriate parameters for today's economy, but our concerns are general with the dynamic properties of the model. The parameters are: (r = 0.5, (x = 0.6, and 0 = 1. The single major modification we have made is to change the autonomous investment demand by introducing a periodic variation given as: d ( t ) = 1 + A sin(2;t),

(3)

where A is the amplitude, P the period of the forcing sine wave, and t is time. A can be seen as the factor that determines the strength of the interaction between the forcing and the internally generated cycle. The modified model is then given as: z = - ( ~ + (1 - ~)0)£ + d ( t ) ~ ( £ ) or0

-

(1 - c0z

(4)

can be interpreted as a cyclical variation in investment arising, for instance, from seasonal factors or from variations in the interest rate. A n investment cycle with a period between 7 and 11 years is referred to as a medium wave and corresponds, perhaps, to the life cycle of machinery and other short-lived investment goods. After World War II, the average length of these cycles may have decreased to approximately 5 to 7 years [17]. This indicates that the parameters may no longer be appropriate for today's economy, but the model still provides insights into how cycles can be generated in the economy. d(t)~(z)

Mode-Locking For linear systems, the principle of superposition applies. Different modes can therefore exist and develop independently. The response of a linear system to an external disturbance will be the same regardless of the phase of the modes already excited. In nonlinear systems, on the other hand, different modes will interact, and the behavioral characteristics of one mode may depend upon the phase of another mode.

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Modern theory of nonlinear dynamics also suggests that the different cyclic modes may entrain with one another through the process of mode-locking. In the presence of another oscillatory mode, a given mode may speed up its rate of change in certain phases of the other mode to slow down in other phases. The combination of nonlinearities and self-reinforcing processes in macroeconomic systems therefore imposes important modifications to the conventional linear interpretation of economic time series. When oscillatory modes with similar frequencies interact they tend to adjust to one another such that their periods become precisely the same. The classic example of such behavior is the synchronization of the rotational motion of the moon to its orbital motion, which is why the same half of moon always is seen from Earth. Other wellknown examples are the synchronization of the circadian rhythm of most living organisms to the shift between day and night, the synchronization of clocks hanging on the same wall, and synchronization of menstrual cycles between women living in close contact. Synchronization is a universal phenomenon in nonlinear systems, and the same process is likely to be involved both in the interaction between individual companies and the coupling between the various sectors of the overall economic system [14]. Most likely, synchronization therefore plays an important role in the genesis of each of the macroeconomic cycles. However, synchronization is only one manifestation of a much more general phenomenon known as frequency-locking or entrainment [18-21]. In nonlinear systems, an oscillatory mode contains various harmonics, and two modes may synchronize whenever a harmonic of one mode is close to a harmonic of the other. As a result, nonlinear oscillators tend to lock to one another such that one oscillator completes precisely p cycles each time the other oscillator completes q cycles, with p and q being integers. To understand the phenomenon of mode-locking in more detail, let us first consider the case where two oscillators with different periods exist in the same system without interaction. The trajectory of the system may be represented as a curve on a torus. For particular parameter values, namely when two periods are commensurate, the trajectory forms a closed orbit, and the motion is periodic. In general, however, the periods will be incommensurate (their ratio is irrational), and the trajectory will fail to close to itself. In this case, the total motion is quasiperiodic, and the trajectory gradually covers the entire surface of the torus. Thus, quasiperiodicity is characterized by the fact that the motion never repeats itself, i.e., each cycle is unique and the cycles may look irregular as for a chaotic motion. In contrast to deterministic chaos, however, a quasiperiodic motion is not sensitive to the initial conditions, since a small change in these conditions only shifts the whole trajectory and does not alter the trajectory profoundly [22]. With stronger coupling, the interaction of the two cycles can also produce chaotic behavior [21].

Mode-Locking in the Goodwin Model Figure 1 shows the effect of mode-locking in the Goodwin model. Here, we have plotted the temporal variation of z over a period of 200 years. For the first 50 years the model is simulated with the parameters given in the previous section, and there is no external forcing signed (A = O). Under these circumstances, the model produces a simple limit cycle with a period of 8.5 years. At time = 50 years, the variable investment function is switch on with a period P = 6.5 years and A = 0.2. After transient so short that it cannot be seen in the figure, the model finds a new steady-state. The period of the internally generated cycle is now 24 years, which corresponds to precisely four

MODE-LOCKING IN A FORCED BUSINESS CYCLE

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external cycles. However, each cycle can be viewed of as taking 8 years (24/3), which might still be considered the period of the internal cycle. The internal cycle has changed its period with 6% to get in step with the external cycle in a 3:4 mode-locked solution. Although the change in the actual period is small, the change in the appearance of the endogenous cycle is significant. Before, there was a simple limit cycle. After introduction of the external cycle, the internal cycle has changed both amplitude and form significantly. Figures 2A-C show three different phase-plots of the model. Figure 2A is a "simple" 1:4 solution. There is one maximum of z as compared with four maxima in the external forcing d. As the solution is periodic, the phase plot is a closed curve. Figure 2B shows a 5:6 mode-locked solution, obtained for A = 0.3 and P = 7.2 years. The phase plot is now more complicated, but it is still a closed curve. Finally, Figure 2C shows a quasiperiodic solution to the model obtained for A = 0.1 and P = 6.3 years. For a quasiperiodic solution, the trajectory in phase space is not closed and may be compared to a ball of yarn. While the simulation is continued, the band generated by the solution will spread slowly as each cycle misses the former by a small amount. Modeqocking occurs around all rational ratios between the two periods. It may be characterized by a so-called winding number W = p/q, with p,q E N. The interval in which the mode is mode-locked tends to increase with increasing amplitude A. The range of a mode-locked interval also depends on the winding number. If the ratio between the two cycles is simple, such as 1:1 or 1:2, the mode-locking will be more pronounced than mode-locking between more complicated periods such as 6:11. In Figure 3, the observed winding number W for the steady-state solution is plotted against the forcing period P. The figure was obtained by performing a large number of

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simulations with the Goodwin model using different values of P while the amplitude A was kept constant at 0.15. The structure depicted in this plot is known as a devil's staircase [23]. In accordance with the above discussion, the most pronounced "steps" are those with simple winding numbers. Between these simple winding numbers more complicated solutions can be found. It is always possible to find a new rational number between any given two rational numbers. Another observation can be made by looking at the staircase: the most pronounced step (i.e., the largest) between two given steps (or winding numbers) will be determined by the sum of the numerators and the denominators of the two, e.g., the largest step between 1:1 and 1:2 is 2:3. A devil's staircase is a fractal structure that shows self-similarity under magnification. Self-similarity means that independently of the scale, the same structure will always be found. Figure 3 also contains a magnification of the interval between the 1:3 and 1:2 mode-locked solutions. It is clear how the structure of the staircase is repeated. The name devil's staircase can be seen as conveying the impossible task of trying to "walk up" this complicated structure of infinite many steps. Lyapunov

Exponents,

Arnord

Tongues,

and Lock-In

In the practical analysis of a nonlinear, dynamic system, calculation of the Lyapunov exponents is know to be extremely useful. Named after the Russian mathematician A. Lyapunov, these exponents indicate whether the steady-state solution of the system is a stable equilibrium point, a periodic or quasiperiodic orbit, or a chaotic or hyperchaotic solution.

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The basic idea behind calculating Lyapunov exponents is to measure the long-term average divergence or convergence of nearby trajectories. A drawback of the method is that even relatively small models, such as the forced Goodwin model, require relatively large amounts of computer time to make reliable estimates. The method used to calculate the Lyapunov exponents in this study is based on the work of Wolf [24, 25]. The idea is to track the long-term development of a small n-sphere of initial conditions as a function of time. The center of the n-sphere follows a particular trajectory defined by a certain set of initial conditions, while points on the surface of the n-sphere are located on (neighboring) trajectories. As time passes, this n-sphere will evolve into a n-ellipsoid because of the expansions and contractions in different directions of the phase space. The i'th Lyapunov exponent is then defined in terms of the growth rate of the i'th principal axis of this n-ellipsoid. Table 1 shows the connection between the types of stationary behavior and the Lyapunov exponents for a two-dimensional model with external forcing. If the largest Lyapunov exponent is positive, the system is chaotic. For this reason, it is only necessary to calculate the largest Lyapunov exponent to determine the behavior of the model for a given parameter combination (assuming that one wants to establish whether or not the system is chaotic). Note that the sum of all Lyapunov exponents must always be negative for a dissipative system. Figure 4 shows the largest Lyapunov exponent for the forced Goodwin model. In this case A = 0.15, and the Lyapunov exponent is shown as a function of the driving period P. It is important to realize that the Lyapunov exponents contain considerable more structure than one can see on Figure 4. Much of the finer structure has disappeared

MODE-LOCKING IN A FORCED BUSINESS CYCLE

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TABLE 1 Connection between Behavior and Lyapunov Exponents for a Two-Dimensional Model with External Forcing

Lyapunov exponents (-, -, (0, , (0, 0, (+, 0,

Behavior

-,) ,) ) )

Stable Periodic Quasiperiodic Chaotic

due to the finite resolution of the horizontal scan. There are no chaotic solutions in the model, but a number of quasiperiodic solutions (i.e., two Lyapunov exponents are zero). If Figure 4 is compared to the devil's staircase in Figure 3, the area where the largest Lyapunov exponent is negative coincides exactly with the steps of the staircase. To get an even better overview of the model behavior than that provided by the devil's staircase, further simulations were performed. If both the forcing period P and the amplitude A are changed, it is possible to see how mode-locking changes in this two-dimensional parameter space. Figure 5 shows an example of this. A plot like Figure 5 is known as an Arnol'd tongues diagram [20]. The hatched areas in the figure show where the mode-locked solutions occur; the white areas represent areas with quasiperiodic or more complicated mode-locking behavior. For vanishing forcing amplitude, no mode-locking can occur. However, the points from where the A r n o l ' d tongues "start" are exactly those at which a rational relationship exists between

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the uncoupled internal and external periods. The devil's staircase at A = O would only consist of points, representing all the rational numbers. As A increases, the model starts to mode-lock and the tongues becomes wider. In the forced Goodwin model, the 1:2 mode-locked solution dominates. Up to A = 0.2, all the tongues in the plot grow; for A greater than 0.2, the 1:2 mode-locked solution takes over and forces the other tongues to decrease as A increases. The width of the tongues, i.e., the range of mode-locked solutions, can not continue to grow. At some point, the mode-locked solutions will fill up the whole interval. This is the critical coupling strength of the system at which a complete devil's staircase exists (entrainment occurs for all period ratios). Mode-locking is a long-term steady-state behavior. What happens when the steadystate behavior is excited by random events from outside the system? Each external event will drive the model away from the steady-state. As soon as this happens, the model will start to return to this long-term steady-state. The time it takes the system to reach the steady-state can be characterized by a time constant, ~, which is equal to the magnitude of the absolute reciprocal of the largest Lyapunov exponent [11]. The more negative the largest Lyapunov constant, the shorter the time to reach the longterm steady-state of the mode-locked solution. The time constant is given as: = x1

(5)

and can be seen as an expression of the strength of the entrainment, i.e., the smaller -r, the faster is the mode-locking. The constant 7 is known as the lock-in time.

MODE-LOCKING IN A FORCED BUSINESS CYCLE

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Figure 6 shows an example of the lock-in time as a function of the forcing period P. As before A is 0.15. There is a clear connection between this plot and the Arnol'd tongue diagram on Figure 5. The shortest time constant is around 2.5 years. The lockin plot shows that in most cases the lock-in time will be more than 10 years, i.e., more than a cycle time for the model. Conclusions

We have shown how a simple nonlinear dynamic model, such as the Goodwin business cycle model, can produce very complicated behavior if it is perturbed with a periodic external disturbance. In this case we have used a simple business cycle model perturbed by an external sine wave to show how mode-locking might occur. However, other nonlinear models capable of producing a limit cycle could have yielded similar results. Forrester [14] suggested that mode-locking could explain why different sectors in the economy experience the business cycle at the same time ro why indeed there is a coherent business cycle at all. Schumpeter [13] suggested that there was a fixed integral relationship between different cycles int he economy. However, neither Schumpeter nor Forrester were able to give a theoretical explanation of how this was possible. This study outlined how suggestions like these can be supported using concepts from modern nonlinear dynamics. This work was supported in part by the Danish Social Science Research Council. The authors thank J. S. Thomsen for developing the program used to calculate L yapunov exponents and E. Mosekilde for comments on a preliminary version of the article.

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References I. May, R. M.: Simple Mathematical Models with Very Complex Dynamics, Nature 261, 459467 (1976). 2. Feigenbaum, M. J.: Quantitative Universality for a Class of Nonlinear Transformations, J. Star. Phys. 19, 25-52 (1978). 3. Bohr, T., Bak, P., and Jensen, M. H.: Transition to Chaos by Interaction of Resonances in Dissipative Systems (1I): Josephson Junctions, Charge-Density Waves and Standard Maps, Physical Review A 30, 1070-1081 (1984). 4. Bak, P., Bohr, T., and Jensen, M. H.: Mode-Locking and the Transition to Chaos in Dissipative Systems, Physica Scripta 9, 50-58 (1985). 5. Glazier, J., Jensen, M. H., Libchaber, A., and Stavans, J.: Structure of Arnol'd Tongues and the f (alfa) Spectrum for Period Doubling Experimental Results, Physical Review A 34, 1621-1624 (1986). 6. Parlitz, U., and Lauterborn, W.: Period-Doubling and Devil's Staircase in the Driven Van der Pol Oscillator, Physical Review A 36, 1429-1434 (1987) 7. Knundsen, C., Sturis, J., and Thomsen, J. S.: Generic Bifurcation Structure of Arnol'd Tongues in the Forced Oscillator, Physical Review A 44, 3503-3510 (1991). 8. Glass, L., Shrier, A., and Belair, J.: Chaotic Cardiac Rhythms, in Chaos. A.V. Holden, ed., Manchester University Press, 1986 pp. 237-256. 9. Glass, L., and Mackey, M. C.: From Clocks to Chaos. Princeton University Press, Princeton, NJ, 1988. 10. Sturis, J., van Cauter, E., Blackman, J. D., and Polonsky, K. S.: Entrainment of Ultradian Pulses of Insulin Secretion by Oscillatory Glucose Infusion, Journal of Clinical Investigation 87, 493-445 (1991). 11. Mosekilde, E., Larsen, E. R., Thomsen, J. S., and Sterman, J. D.: Nonlinear Mode-lnteraction in the Macroeconomy, Annals of Operations Research 32, 185-215 (1992). 12. Haxholdt, C., Kampmann, C., Mosekilde, E., and Sterman, J. D.: Mode-Locking and Entrainment of Endogenous Economic Cycles, System Dynamics Review 11,177-198 (1995). 13. Schumpeter, J. A.: Business Cycles. MacGraw-Hill, New York, 1939. 14. Forrester, J. W.: Growth Cycles, De Economist 125, 525-543 (1977). 15. Goodwin, R. M.: The Nonlinear Accelerator and the Persistency of Business Cycles, Econometrica 19(1), 1 17 (1951). 16. Semmler, W., ed.: Business Cycles: Theory and Empirical Methods. Kluwer Academic Publishers, New York, 1994. 17. Gabish, G., and Lorenz, H. W.: Business Cycle Theory. Springer Verlag, Berlin, 1989. 18. Arnol'd V. I.: Small Denominators. I Mapping of the Circumference onto Itself. American Mathematical Society Transactions 46, 213-284 (1965). 19. Guevara, M. R., Glass, L., and Shrier, A.: Phase-Locking, Period-Doubling Bifurcation and Irregular Dynamics in Stimulated Cardiac Cells, Science 214, 1350-1353 (1981). 20. Jensen, M. H., Bak, P., and Bohr, T.: Transition to Chaos by Interaction of Resonance in Dissipative Systems (I): Circle Maps, Physical Review B 30, 1960-1981 (1984). 21. Larsen, E. R., Morecroft, J., Mosekilde, E., and Thomsen, J. S.: Devil's Staircase and Chaos from Macroeconomic Mode Interaction, Journal of Economic Dynamics and Control 17, 759-769 (1993). 22. Thompson, J. M. T., and Stewart, H. B.: Nonlinear Dynamics and Chaos. John Wiley and Sons, UK, 1986. 23. Mandelbrot, B. B.- Fractals: Forms, Change and Dimension, Freeman, San Francisco, CA, 1977. 24. Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A.: Determining Lyapunov Exponents from a Time Series, Physica 16D 285-317 (1985). 25. Wolf, A.: Quantifying Chaos with Lyapunov Exponents, in Chaos. A. V. Holden ed., Manchester University Press, UK, 1986, pp. 273-290. Received 19 October 1995; accepted 21 January 1997