Time in Economic Theory - Regional Science Matters - page 65

sA ðÞ dA = dt ¼ mp ð ; A Þ ; ð 5 : 13 Þ

where s(A) represents the very large durability of infrastructure, indicating that s
(A) is a very small, positive number, possibly in the order of 0.01 or lower.
This implies that in the time frame of the other variables of this system dA/dt can
be set approximately equal to zero, most of the time (but not always). The fast and
slow processes will rarely be synchronized and the whole system will then go into a
period of creative destruction, eventually to come into rest at a new economic
structure.
We thus have a dynamic system:

fp,A dp = dt ¼ ð Þ , ð 5 : 14a Þ

to be solved for an equilibrium, i.e. with f ¼ 0, subject to the temporary constraint :

mp,A ð Þ 0, ð 5 : 14b Þ

where A* indicates a given level of infrastructure in all parts of the economy.
For systems of this kind we can apply Tikhonov's theorem (Sugakov 1998 ):
Assume a dynamic system of N ordinary differential equations, which can be
divided into two groups of equations. The first group consists of m fast equations,
the second group consists of m+ 1,

, N slow equations.
Tikhonov's theorem states that such a system has an equilibrium solution under
certain economically reasonable conditions:
For each position of the slow subsystem, representing the dynamics of infra-
structure, the fast general equilibrium market price subsystem has plenty of time to
stabilize. Such an approximation is called adiabatic . (For a proof see Sugakov
1998 )
In the very long run dA/dt cannot be assumed to be approximately equal to zero
and thus the infrastructure would have substantially changed. The structure of
prices and quantities of goods, as determined by f(p,A) could then cease to be as
well behaved as in the short term dynamics, given by (Eq. 5.12 ).
The system would in the very long term have all the bifurcation properties,
typical of non-linear, interactive dynamic systems. However, between periods of
change of the economic structure, there could be periods of stable General Eco-
nomic Equilibrium.
Most neoclassical economists have become skeptical about the possibility to
mathematically model the dynamics of economic systems. Modern mathematical
theory of dynamic systems supports this view. Chaos is the generic outcome of a
non-linear economic system if all interactive economic variables are moving on the
same time scale. General Equilibrium Theory, as formulated by e.g. Arrow and
Hurwisz ( 1957 ), Debreu ( 1959 ) and others, is thus in fact not general enough to be
expandable into a well behaved dynamic economic systems theory (and even less
into combined spatial and dynamic systems).

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