Chaotic Dynamics in Economics
Abstract: A new literature in the 1980s studied the possibility that endogenous cycles and irregular chaotic dynamics resembling stochastic fluctuations could be generated by deterministic, equilibrium models of the economy, in particular in overlapping generations models and in models with infinitely lived representative agents.
Keywords: Chaos, Endogenous Cycles, Dynamic Equilibrium Models
When a new literature in the 1980s showed that endogenous cycles and chaos can arise in equilibrium models in economics, it came as a surprise. The possibility of deterministic fluctuations, as opposed to fluctuations driven by exogenous stochastic shocks, had been noted in an earlier literature on business cycles, for example in the well-known multiplier-accelerator models, but not in equilibrium models of the economy with complete markets and no frictions. (See for example Frisch (1933) or Samuelson (1939). Yet deterministic fluctuations in equilibrium models with predictable relative price changes should be ruled out by intertemporal arbitrage. Such considerations led to the rejection of regular endogenous cycles in favor of models whose fluctuations are driven by stochastic shocks.
We can usually describe a dynamical system in discrete time as chaotic if it can generate cycles of every periodicity, where a sequence is of period n if
but
for
. In addition, this simple definition of chaos requires the existence of an uncountable number of initial x which give rise to bounded but aperiodic (not even asymptotically) sequences. For example the well-known hump-shaped function, 4x(1 - x) , when iterated, generates such chaotic dynamics. The kind of chaotic dynamics described above is usually referred to as topological chaos. If in addition we require that the set of initial conditions giving rise to aperiodic sequences are not simply uncountable but also have a positive (Lebesgue) measure, then we also have ergodic chaos. A useful sufficient condition to obtain topological chaos with a simple difference equation
, with f continuous and mapping a closed interval into itself, is the existence of some x such that
. Note that this condition will be satisfied if the difference equation has a solution of period three. A particularly interesting feature of some dynamic systems that are chaotic is their sensitive dependence on initial conditions: initial conditions that are arbitrarily close can generate sequences that tend to diverge over time. Thus, small measurement errors in initial conditions may cause large forecasting errors, which may explain some of the difficulties associated with business-cycle forecasting.
The aperiodic but bounded trajectories that characterize chaos and exhibit sensitive dependence on initial conditions cannot continue to diverge forever. They converge not to a point or a periodic cycle, but to a bounded chaotic or "strange" attractor. The dynamical system which induces the local separation and instability of the trajectories must eventually bend them back. The combination of local stretching and global folding generates the complex nature of the dynamics. Such dynamic behavior is in fact a familiar theme in economics that highlights the self-correcting nature of the economic system. Shortages create incentives for increased supply; dire necessities give rise to inventions as the invisible hand guides the allocation of resources. An equally familiar theme is that of instability: the multiplier interacts with the accelerator, leading to explosive or implosive investment expenditures; self-fulfilling expectations give rise to bubbles and crashes. In combination, these two themes suggest a non-linear system, somewhat unstable at the core, but effectively contained further out. The contribution of the new literature on chaotic dynamics starting in the early 80s has been to demonstrate the compatibility of endogenous irregular fluctuations with equilibrium dynamics in economics. For a very simple example of chaotic dynamics consider a simple overlapping generations model where each generation lives two periods.
The utility function of a generation born at t is , where
is consumption when young and
is consumption when old.
This generation faces a budget constraint , where
is the endowment when young,
is the endowment when old, and
is the rate of return on savings.
The first order condition to the problem of maximizing utility subject to the budget constraint, assuming interiority, yields . Here
&
denote derivatives of the utility function U with respect to the first and second arguments.
During each period t; market clearing requires that sum of the endowments of the young and the old add up to the sum of their consumptions: .
Now consider the quadratic ,
and
.
Substituting the first order condition into the budget constraint, and using the market clearing condition, the difference equation describing .
Techniques to empirically distinguish between data generated by non-chaotic stochastic systems and deterministic chaotic systems have been developed by physicists and mathematicians (see for example Ruelle and Eckmann (1985)). These techniques have been further refined into statistical tests for applications to economic data by Brock (1986), Brock, Dechert, LeBaron and Scheinkman (1996), among others. Very roughly, these methods exploit the idea that deterministic systems will generate trajectories that are of lower dimension than those generated by stochastic systems which have more scattered trajectories. The difficulty of empirically identifying chaos in high dimensional economic systems may be particularly important if chaotic dynamics is more likely to be manifested in disaggregated sectoral or industry data whose components, because of resource constraints or other scarcities, can move in ways that partially offset one another's cyclic or irregular movements. It would therefore be fair to say that at this point, while we know that standard dynamic equilibrium models with parameters calibrated to values often used in the literature may well generate chaotic dynamics, more definitive empirical evidence for chaos in economics has not yet been produced. While it may be instructive to set the theories of endogenous economic fluctuations in opposition to the theories of fluctuations driven by stochastic shocks, in practice it is more helpful to consider endogenously oscillatory dynamics as complementary to stochastic fluctuations. In certain environments it may make little difference if endogenous mechanisms by themselves generate regular and irregular persistent fluctuations, or whether they give rise to damped oscillations that are sustained by stochastic shocks. On the other hand, if the underlying equilibrium system is subject to distortions and there is room for stabilization policy, correctly identifying the source of the fluctuations becomes much more important. (See for example Benhabib, Schmitt-Grohe and Uribe (2002)). Furthermore recognizing the role of oscillatory dynamics may diminish our reliance on unrealistically large shocks to explain economic data, for example, in real business cycle theory.
Jess Benhabib, New York University, JEL Numbers: C0,E3,O41.
