9-DifferntEquations Page 259 Wednesday, February 4, 2004 12:51 PM
Differential Equations and Difference Equations 259analytically. Nevertheless, the statistics of chaos theory might still prove
to be meaningful.
The economy being a complex system, the expectation was that its
apparently random behavior could be explained as a deterministic cha-
otic system of low dimensionality. Despite the fact that tests to detect
low-dimensional chaos in the economy have produced a substantially
negative response, it is easy to make macroeconomic and financial
econometric models exhibit chaos.^6 As a matter of fact, most macroeco-
nomic models are nonlinear. Though chaos has not been detected in eco-
nomic time-series, most economic dynamic models are nonlinear in
more than three dimensions and thus potentially chaotic. At this stage
of the research, we might conclude that if chaos exists in economics it is
not of the low-dimensional type.PARTIAL DIFFERENTIAL EQUATIONS
To illustrate the notion of a partial differential equation (PDE), let’s
start with equations in two dimensions. A n-order PDE in two dimen-
sions x,y is an equation of the form∂
()
∂f ∂f
i
f
Fx y ,,------,------, , ... -------------------------------= 00 , ≤≤ ki, 0 ≤≤ in
∂x ∂y k –
∂
()
x∂
(ik)
yA solution of the previous equation will be any function that satisfies
the equation.
In the case of PDEs, the notion of initial conditions must be
replaced with the notion of boundary conditions or initial plus bound-
ary conditions. Solutions will be defined in a multidimensional domain.
To identify a solution uniquely, the value of the solution on some sub-
domain must be specified. In general, this subdomain will coincide with
the boundary (or some portion of the boundary) of the domain.Diffusion Equation
Different equations will require and admit different types of boundary
and initial conditions. The question of existence and uniqueness of solu-(^6) See W.A. Brock, W.D. Dechert, J.A. Scheinkman, and B. LeBaron, “A Test for In-
dependence Based on the Correlation Dimension,” Econometric Reviews, 15(3)
(1996); and W. Brock and C. Hommes, “A Rational Route to Randomness,” Econo-
metrica 65 (1997), pp. 1059–1095.