IMPLICATIONS FOR TEACHING MACROECONOMICS OF …



IMPLICATIONS FOR TEACHING MACROECONOMICS OF COMPLEX DYNAMICS

J. Barkley Rosser, Jr.

Professor of Economics

MSC 0204

James Madison University

Harrisonburg, VA 22807 USA

Tel: 540-568-3212

Fax: 540-568-3010

Email: rosserjb@jmu.edu

[figures available upon request]

February, 1999

ABSTRACT:

The implications for how teach macroeconomics at the undergraduate level of the emergence of the multidisciplinary study of nonlinear complex dynamics are examined. A definition of complex dynamics is presented and a broad review of various applications in macroeconomics is made. Some particular implications are emphasized such as how complex dynamics raise serious doubts about the rational expectations assumption. Several models and approaches are suggested that can be used to make these ideas accessble to students.

Acknowledgments: The author wishes to acknowledge either the receipt of useful research materials or useful comments from the following, none of whom are responsible for any errors or misinterpretations contained in this paper: William A. Brock, David Colander, Paul Davidson, Richard H. Day, Steven N. Durlauf, Hans van Ees, Carla M. Feldpausch, Harry Garretsen, Roger Guesnerie, Richard P.F. Holt, Cars H. Hommes, Ted Jaditz, Mark Knell, Blake LeBaron, Marji Lines, Hans-Walter Lorenz, Benoit B. Mandelbrot, Tönu Puu, Roy Rotheim, and Wei-Bin Zhang.

IMPLICATIONS FOR TEACHING MACROECONOMICS OF COMPLEX DYNAMICS

INTRODUCTION

During the last several decades the study of nonlinear dynamical systems has led to a serious upheaval in many disciplines of thought as the implications for the ubiquitous possibility of complex dynamics of various sorts has swept across numerous fields of thought (Zeeman, 1977; Mandelbrot, 1983; Haken, 1983; Weidlich and Haag, 1983; Prigogine and Stengers, 1984; Gleick, 1987; Waldrop, 1992; Kauffman, 1993; Kiel and Elliott, 1996; Guastello, 1996). Unsurprisingly, this upheaval has spread to economics as well (Rosser, 1991; Day, 1995; Arthur, Durlauf, and Lane, 1997; Albin with Foley, 1998). So far, however, the discussion of this upheaval has remained at a relatively high level with only a few efforts to make them accessible to a broader audience (Baumol and Benhabib, 1989) or even to present them in graduate level textbooks (Blanchard and Fischer, 1989).

But students of economics, even at the undergraduate level, are increasingly aware of these developments from a variety of sources as discussion of them spreads out from many disciplines into the popular media and forms of popular expression. Examples of such mechanisms of transmission include the discussion of chaos theory in the popular movie, Jurassic Park, as well as the popularity of artwork based on fractal geometry, with such images even showing up on t-shirts worn by students. Thus, students today are much more prepared to think about the implications for economics of such ideas and concepts than was the case in the past. In effect they are more open to the broad paradigmatic shift that has occurred. But, exactly how much of it should be presented to them in the context of macroeconomics, especially at the undergraduate level, and which implications of it should be emphasized are clearly difficult and unresolved questions. They are the main topics of this paper.

The next several sections will review in a relatively non-technical way some of the recent developments in complex macroeconomic dynamics without attempting to be fully comprehensive. Such phenomena include discontinuities associated with nonlinear investment functions, the possibility of chaotic dynamics, discontinuous shifts or shocks implied by coordination failures of various sorts, forecasting difficulties arising from models with multiple equilibria and fractal basin boundaries, and the complexities in models with costly information. These discussions will be followed by a discussion of what elements of these ideas might actually be presented in macroeconomics textbooks.

Two very important implications from all of this stand out, although they may be disturbing to authors especially of undergraduate textbooks in macroeconomics. One is the possibility of multiple equilibria. The other is the extreme unreality of the assumption of rational expectations which has come to dominate presentations in many recent textbooks (Rosser, 1996a,b). That this assumption, now finally fully filtering into the undergraduate textbooks as the dominant approach, has become obsolete is most clearly seen by the recent defection of Thomas Sargent (1993), one of the early users of the idea in macroeconomics, from supporting it. Sargent’s defection is due to the influence of economists and others at the Santa Fe Institute studying nonlinear complex systems. Sargent now argues that in the face of complexity people behave according to bounded rationality, not rational expectations. Even if textbooks continue presenting models assuming rational expectations, it will increasingly behoove their authors to at least admit somewhere the limited and unrealistic nature of the assumption. Recognizing the apparent pervasiveness of nonlinear complex dynamics and multiple equilibria is one way of doing so.

WHAT ARE COMPLEX DYNAMICS?

Richard Day (1995) defines complex dynamics as those that for nonstochastic reasons do not converge to either a unique equilibrium point or to a periodic limit cycle or that explode. In macroeconomics this means there are endogenous oscillations of an irregular nature, not triggered by mere exogenous shocks as in the real business cycle models of New Classical macroeconomics. A necessary condition for such complex behavior is that the dynamical system as defined by its differential or difference equations contain some element of nonlinearity, although the existence of nonlinearity is not a sufficient condition for complexity, the simple exponential growth model being an example of a non-complex nonlinear system.

Although most undergraduate macro textbook presentations use linear models, it is clear that nonlinearities of many sorts pervade the real economy. Sources of such nonlinearities include increasing and decreasing returns to scale, strategic complementarities and externalities in production, threshold effects in information gathering, floors or ceilings to investment, nonlinearities in investment function lags, nonlinear consumption functions, non-monotonic labor supply functions, nonlinearities in liquidity preference functions, the well-documented presence of ARCH effects and kurtosis in the distributions of asset returns, and even possible nonlinearities in policy response functions of various sorts. The linear model is the peculiar special case, not the nonlinear model.

EARLY NONLINEAR MACRODYNAMIC MODELS

Curiously enough the idea of nonlinearity as a source of endogenous fluctuations in macroeconomic models initially arose during the Great Depression and its aftermath (Kalecki, 1935; Kaldor, 1940; Hicks, 1950; Goodwin, 1951), although the sorts of models developed at that time did not exhibit the full array of complex dynamics possible in such models studied later (Lorenz, 1989; Puu, 1989; Rosser, 1991; Zhang, 1991). Rather these models were used to show the possibility of relatively simple periodic cycles or other non-complex fluctuations other than simple convergence to a stable equilibrium point.[i] All relied upon ad hoc assumptions regarding the motives for the behavior implied by the models. Most were consistent with Keynes, even if they did not derive from his work directly.

One of the most studied of these models as a potential generator of complex dynamics is that of Kaldor (1940). Varian (1979) used catastrophe theory to show how a modified version could generate simple cycles.[ii] Later, Lorenz (1992) showed the possibility of chaotic dynamics, fractal basin boundaries, and non-chaotic dynamics on a strange attractor for variations on this model. Underlying all of these dynamics is Kaldor’s assumption that investment is a sigmoid function of national income that shifts up and down with the capital stock as depicted in Figure 1. Kaldor’s justification of this shape was that there would be a low propensity to invest at low GDP because of excess capacity and also at high GDP because of high interest costs and capacity constraints in the capital goods sector. This shows already the link between nonlinearity and the possibility of multiple equilibria and complex dynamics. The models of these other authors depend on variations of this formulation.

These models fell out of favor during the 1970s as the rational expectations movement made them seem like quaint antiques of no relevance to modern macroeconomics. However, as the realization that nonlinear dynamics undermine rational expectations has spread, these models have enjoyed a renaissance.

CHAOTIC DYNAMICS

Probably the most famous form of complex dynamics is deterministic chaos. The key idea of chaotic dynamics is that of sensitive dependence on initial conditions (SDIC), that a small change in initial conditions can lead to a large change in outcomes, also known as the “butterfly effect” from the image of a butterfly causing a hurricane on the other side of the world by flapping its wings (Lorenz, 1963). Although originally discovered in studies of celestial mechanics in the late 1800s (Poincaré, 1880-1890), the idea became formalized much later with a clear sufficiency condition being that the real parts of the Lyapunov exponents of a dynamical system are positive (Oseledec, 1968). These Lyapunov exponents are the solutions to

->

L = lim ln(2DFt(x)v2)/t, (1)

t->4

where F is a dynamical system, Ft(x) is the t-th iterate of F

starting from an initial condition x, D is the derivative, t

->

is time, and v is a direction vector. The more positive is

the greatest Lyapunov exponent, the more rapidly will the

local instability manifest itself in unforecastibility. Such a separation of outcomes locally from slightly different initial conditions is shown in Figure 2, with τ being time units after the initial condition.

Chaotic systems behave in a way that looks random while being generated deterministically. They are both bounded while also exhibiting a local instability that is associated with the SDIC characteristic. A famous and simple sufficiency condition for chaos is that a properly defined system exhibit a three-period cycle (Li and Yorke, 1975).

Many dynamical systems exhibit period-doubling bifurcations whereby as a control parameter is varied, the dynamical pattern goes from converging to a unique equilibrium to a two-period cycle, then a four-period cycle, and so forth, until an odd-numbered cycle appears at which point the system is chaotic. A famous simple example showing this behavior is the logistic difference equation studied by the ecologist Robert May (1976) in a paper which first suggested the possibility of chaotic economic dynamics. This is given by

xt+1 = μxt(k-xt), (2)

with μ being the control (or “tuning”) parameter, x being the variable, t being time, and k being some kind of upper limit, interpreted as a carrying capacity in ecological systems. For an economic model this might represent a limit to growth due to environmental or other constraints and this equation lies at the heart of many of the early models showing the possibility of chaotic economic dynamics (Day, 1982). The pattern of period-doubling bifurcations leading to chaos is depicted for this equation in Figure 3, with μc being the value beyond which chaotic dynamics emerge.

Fairly early (Benhabib and Day, 1981; Grandmont, 1985) it was shown to be theoretically possible to have a model with rational expectations, indeed perfect foresight, which could exhibit chaotic dynamics.[iii] Figure 4 depicts a phase diagram with real money supply in time t as μ on the horizontal axis and in t+1 as χ(μ) on the vertical axis for the Grandmont (1985) model when it is exhibiting chaotic dynamics. Both the Benhabib-Day and Grandmont models involve overlapping generations with chaotic dynamics arising from strong intertemporal income or wealth effects. In infinitely-lived representative agent models with a single good, a source of chaotic dynamics can be very low discount factors (high discount rates), with Mitra (1996) and Nishimura and Yano, 1996) indicating a least upper bound equal to (3-π5)/2. Most of these models also exhibit multiple equilibria.

Ironically these models imply the extreme difficulty of forming rational expectations as SDIC implies that small errors will cause divergent results. This suggests extreme difficulties in the learning process to form rational expectations. Grandmont (1985) and also Guesnerie (1993) have argued that this feeds into the broader problem of the inability to coordinate expectations and thus to the problem of coordination failure more generally. On the other hand models have been developed showing how government intervention in a macroeconomy might induce chaotic dynamics where none would otherwise exist (Dwyer, 1992; DeCoster and Mitchell, 1992).[iv]

Considerable debate exists regarding the empirical existence of chaotic economic dynamics. The preponderance of evidence leans against the existence of easily measurable such chaos, especially for macroeconomic aggregates (Brock and Sayers, 1988; Jaditz and Sayers, 1993; LeBaron, 1994), although some argue in the opposite direction for specific micro markets such as soybeans (Blank, 1991), milk (Chavas and Holt, 1993), and the S&P stock market index (Eldredge, Bernhardt, and Mulvey, 1993). Much of this controversy involves difficult econometric issues, especially surrounding diagnostics tests of estimates of Lyapunov exponents (Dechert, 1996).

Although other forms of complex dynamics may be more empirically realistic, chaotic dynamics are one that is fairly easily comprehended by students, especially given the publicity and intuitive explanations presented in such places as the film, Jurassic Park. Thus, the problems associated with complex dynamics more broadly might be introduced in a macroeconomics textbook more easily using this example than others, despite its likely rarity as an actual empirical economic reality.

OTHER FORMS OF COMPLEX DYNAMICS

Various other forms of complex dynamics can occur in nonlinear dynamical systems besides chaos. One of these is a strange attractor, long thought to be equivalent to chaotic dynamics, as indeed they often coincide. An attractor is any set of points toward which a dynamical system will asymptotically converge. A system may have multiple attractors, in which case each will have its own basin of attraction, a space within which if the system becomes located it will converge on that attractor. Basins of attraction are separated by basin boundaries. In the simplest of economic systems possessing a unique stable equilibrium, the equilibrium point is the attractor and the basin of attraction is the entire space.

However, if an attractor consists of many points that constitute a set that is odd or erratic in some sense, then the behavior of a dynamical system that is on or converging to such an attractor may exhibit odd or erratic patterns. A much studied such type is the strange attractor. Its “strangeness” is given by its dimensionality. Most people think of what is known as Euclidean dimensionality, sets that have dimensionalities of integer values and are very regular topologically, like points, lines, squares, or cubes. But, as Benoit Mandelbrot (1983, p. 1) remarked, “Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”

The idea that sets might have dimensionality not equal to integers also dates back into the last century to Georg Cantor. There is more than one such measure with a widely used one in econometrics being the Grassberger-Procaccia correlation dimension (Brock, Hsieh, and LeBaron, 1991, p. 16). It is

lim lnN(r)/lnr, (3)

r->0

with r being a radius and N(r) being the number of pairs of points in a trajectory within the radius r. If this number is not an integer then it is strange. Such non-integer dimensionality is labeled fractal by Mandelbrot (1983). Hence, we can say that a strange attractor has fractal dimensionality. Figure 5 shows a portion of the attractor known as the Rössler attractor (Peitgen, Jürgens, and Saupe, 1992, pp. 686-88) which has been studied in macroeconomic models (Goodwin, 1990).

Although many systems exhibiting chaotic dynamics also possess strange attractors, the Rössler system being one such, it is possible for a system to not exhibit SDIC and yet possess a strange attractor. Lorenz (1992) studied the behavior of a modified version of the Kaldor (1940) trade cycle model which exhibits such characteristics. His model exhibits transient chaotic behavior during its adjustment path, but is not chaotic in the long run.

This model of Lorenz also exhibits another form of complex dynamics, namely fractal basin boundaries, other examples studied by Rosser and Rosser (1996), Brock and Hommes (1997), and Soliman (1997). In a system with multiple equilibria, and hence multiple basins of attraction, the nature of the basin boundaries can become an important determinant of dynamic behavior, especially in the presence of noise. A system may neither by chaotic in the sense of SDIC or even have any strange attractors. The attractors might all be simple periodic cycles or even fixed points. But if the basin boundaries possess fractal dimensionality, the basins of attraction may be complexly intertwined with each other such that if there is noise in the system it can be easily kicked from one basin to another, each with very disparate attractors. This implies potentially very unstable and highly unpredictable behavior implying problems for forming rational expectations under such conditions. Figure 6 shows such boundaries for a dynamical system of a pendulum over three magnets, including blowups to smaller scales to highlight the complex intertwinings that can occur (Peitgen, Jürgens, and Saupe, 1992, p. 765).

EXTERNALITIES, MULTIPLE EQUILIBRIA, AND COORDINATION FAILURE

That externalities or strategic complementarities can generate multiple equilibria with the possibility of coordination failure leading to sub-optimal outcomes has been studied by Cooper and John (1988) in a static framework not involving any complex dynamics.[v] Allowing for more generalized increasing returns leads also to the idea of path dependence (Arthur, 1994) wherein random shocks at crucial points in time can lock a system into a suboptimal equilibrium path. These increasing returns may take the form of technological complementarities. In more computational framework such ideas lead to models of self-organizing economies (Krugman, 1996) that may exhibit multiple patterns of development depending on initial conditions and varying degrees of increasing returns or externalities. Another group that has pursued ideas of self-organization has been the Austrians (Hayek, 1967; Lavoie, 1989), which parallels the work on out-of-equilibrium thermodynamic phase transitions by the Brussels School (Prigogine and Stengers, 1984).

One variation on this has been to explicitly model the process of communication and interaction between agents from the perspective of the interacting particle systems (IPS) models of statistical mechanics (Brock, 1993; Durlauf, 1993). In Brock’s (1993) formulation agents optimize in a framework with noise where their utilities interact and where they choose from a discrete choice set (often identified as “optimistic” or “pessimistic”) with an intensity of choice element that determines their willingness to switch their discrete choices, that is to change their general attitudes. In a simple formulation with the discrete choice set being only (-1, 1), the discrete choice parameter for own-behavior over errors being h, J representing the sum of utilities of an agent’s neighbors which proxies their degree of interaction, β represents an “intensity of choice” parameter, and m is the average own-utility of other agents, the Nash social welfare optimum solution is given by

m = tanh(βJ + βh), (4)

where tanh is the hypertangent. This bifurcates at βJ = 1 with two nonzero solutions m(-) = -m(+), arising for βJ > 1. Such a solution is depicted in Figure 7.

Such a model implies that a continuous variation in either the degree of interaction (J), which can be viewed as a proxy for the degree of coordination, or in the intensity of choice (β) can lead to a discontinuous phase transition equivalent to a change in the state of matter such as the freezing or boiling of water. In macroeconomic terms this would imply either a sudden improvement in the growth rate of the economy or a sudden collapse. Rosser and Rosser (1997) use this model to analyze divergent macroeconomic performances in transitional economies. Other applications can be found in Arthur, Durlauf, and Lane (1997).

Another category of models that explicitly lay out the interrelationships between firms and agents in the form of a lattice structure. The structure is regularly shocked by noise that triggers reverberations and reactions throughout the lattice. Because of the possibility for effects to either bottleneck or accumulate through the system, the distribution of outcomes is skewed relative to the distribution of the noisy input. Such models are known as sandpile models or models of self-organized criticality, and the skewed outcomes can be viewed as representing the equivalent of “avalanches” in the context of literal sandpiles. Such a system has been studied in a macroeconomic context by Bak, Chen, Scheinkman, and Woodford (1993) with their lattice structure of the economy depicted in Figure 8.

LEARNING, INFORMATION, AND COMPLEX DYNAMICS

Ironically as intermediate textbooks have increasingly reflected the rational expectations movement of the 1970s, higher level macroeconomic analysis has moved away from this to consider problems of learning and imperfect information in a world without rational expectations (Heiner, 1989; Marcet and Sargent, 1989; Woodford, 1990). Much of this work studies convergence to rational expectations equilibria, with some of these convergence processes resulting in complex dynamics.[vi]

One approach to this problem has been carried out at the Santa Fe Institute using computational methods that follow the classifier model of Holland (1992) and incorporate ideas of inductive learning. A specific model using these ideas is Arthur, Holland, LeBaron, Palmer and Tayler (1997). It involves numerous agents buying assets, with each agent following a rule that evolves according to the success or failure of the rule. This system leads to a self-organizing evolutionary outcome with a stable distribution of wealth, even as individual agents move up or down within the distribution. Rules depend on bits of information and new rules can emerge according to a genetic algorithm.

Simulations of this system show that sometimes it converges on a fundamental equilibrium. However, with a more complex environment, divergent behavior by agents emerges. A common pattern is for the system to remain near the unique fundamental equilibrium for extended periods of time, but then for bull or bear deviations to appear for shorter periods of time. The analogy to self-fulfilling investment behavior generating business cycles in a macroeconomic model is obvious.

Brock (1997) and Brock and Hommes (1997) present analytical counterparts to these kinds of models. In their systems agents can either use informationally cheap but potentially destabilizing simple rule of thumb behaviors or informationally expensive but potentially stabilizing and even rational expectations behaviors. When they are far from equilibrium they use the informationally expensive strategy and move toward the equilibrium. However, the closer they get the less incentive they have to use the informationally expensive strategy and the more likely they are to switch to destabilizing but informationally cheap strategies. This leads to oscillations that can easily become highly complex.

Brock (1997) characterizes this as the prediction paradox, of which the “index paradox” is a special case. If everyone in a stock market invests in an index fund, then it will pay someone to do securities analysis to find mispriced securities. If everyone is doing securities analysis, then it will pay to avoid these costs by investing in an index fund. The result may be complex oscillations.

Although these models have focused on financial markets, they can be extended to broader macro models. One way is to focus on investment. Another involves analysis of labor market behavior. Rosser and Rosser (1996) use such an approach to study complex adjustments in labor markets in transitional economies.

CONCLUSIONS

We have seen that modern macroeconomies are rife with nonlinearities that imply the likely existence of both multiple equilibria and various forms of complex dynamics. The latter in particular undermines the currently predominant assumption of rational expectations that is used widely in macroeconomics textbooks. These problems do not suggest that a solution is to revert to Old Keynesian neoclassical synthesis type models either. Although textbooks are likely to continue to be dominated by these approaches, it would behoove textbook writers to acknowledge these issues. The question then is, which of these are worth talking about, in what ways, and to what extent?

The view of this author is that despite its likely rare existence in actually existing economies the model that can probably be most easily taught to students and which highlights the essential issues is that of chaotic dynamics. Besides the widespread publicity given to the idea, it has certain intuitive aspects as well as intellectually stimulating elements that make it especially suitable for heuristic purposes. It can easily be shown from the case of the nonlinear multiplier-accelerator model. That sensitive dependence on initial conditions profoundly undermines rational expectations is a pretty straightforward story to tell.

Of the other models discussed above the other one that I see as having some possible use in textbooks, although perhaps more at the graduate rather than the undergraduate level, is the idea of an oscillation arising from a conflict between the use of competing rules or adjustment mechanisms. With the continuing spread of computers in classrooms and the use of simulation models students might even be able to study such models themselves more directly, if not of the complexity of those currently under study at the Santa Fe Institute. Not unrelated here is the possibility of doing classroom experiments with such learning rules, where students might see the evolution of strategies occurring in their interactions with each other.

Thus, although probably the majority of the developments in complex nonlinear dynamics are unlikely to be suitable for inclusion in macroeconomics textbooks in the near future, some of these things can and should be said to students somehow.

As regards the ultimate effect of these developments upon economic theory, we can do worse than to contemplate the following comments of Steve Smale (1977, p. 95):

“How did Relativity Theory respect classical mechanics? For one thing Einstein worked from a very deep understanding of the Newtonian theory. Another point to remember is that while Relativity Theory lies in contradiction to Newtonian theory, even after Einstein, classical mechanics remains central to physics. I can well imagine that a revolution in economic theory could take place over the question of dynamics, which would both restructure the foundations of Walras and leave the classical theory playing a central role.”

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ENDNOTES

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[i]. Probably the most famous model of this period that showed such fluctuations was the linear multiplier-accelerator model of Samuelson (1939a). He recognized the possibility of nonlinearity in the consumption function, but did not consider the implications of this in depth (Samuelson, 1939b), although later it would be realized that this could imply complex dynamics (Blatt, 1983). This author has personally found using the Samuelson model to be a good way to introduce the idea of endogenous fluctuations to undergraduate students. It is relatively simple then to note how nonlinearities can lead to more complex results than those initially presented by Samuelson.

[ii]. Catastrophe was the first major form of nonlinear complexity seriously studied in economics (Rosser, 1991; Guastello, 1996), but fell out favor in reaction to the fad for it in the 1970s. Ironically some of the grounds for the reaction now look silly. Thus, Zeeman=ðs (1974) catastrophe theorelook silly. Thus, Zeeman’s (1974) catastrophe theoretic model of stock market crashes was criticized for having heterogeneous agents, some lacking rational expectations (Zahler and Sussman, 1977). Models of financial markets with such heterogeneous agents are now commonplace (Black, 1986; De Long, Shleifer, Summers, and Waldmann, 1990).

[iii]. Boldrin and Woodford (1990) review rational expectations equilibrium models that generate chaotic dynamics. Such models without rational expectations are legion (Rosser, 1990, 1991, 1996a,b). Bullard and Butler (1993) review policy implications for both kinds.

[iv]. Grandmont (1986) particularly exhibits optimism that governments can “control chaos” through interventionist macro policies. Kaas (1998) presents an updated version of this argument drawing on more recent developments in chaos theory. However all such efforts require a degree of knowledge of the economy that the presence of chaotic dynamics would make almost impossible to acquire.

[v]. Colander (1995, 1998) sees coordination failure with multiple equilibria as central to Post Walrasian macroeconomics. It is economic institutions that reduce the number of possible outcomes and create and maintain stability of the macroeconomy according to this perspective.

[vi]. A curious twist on this is the literature on “learning to believe in chaos” (Hommes and Sorger, 1998) in which agents converge through learning on a simple rule of thumb that mimics an underlying chaotic dynamic. This follows the idea of the “self-fulfilling mistake” (Grandmont, 1998).

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