Geoscience Reference
In-Depth Information
The larger the durability of any one product, ceteris paribus, the lower would be
the rate of interest and growth. A compensating reduction of the use of current
inputs is the only way of maintaining equilibrium rates of interest and growth, if any
goods durability is increased.
5.4
von Neumann and the Birth of Mathematical General
Equilibrium Economics
The models of growth shown in Eqs. (5.5) and (5.6) are special cases of the general
equilibrium theory formulated by the mathematicians John von Neumann (
1937
)
and AbrahamWald (
1936
, 1951). They developed the theory, when collaborating in
a Vienna colloquium in the 1930s on the mathematics of general equilibrium theory
as formulated by Walras (
1874
) and Cassel (
1918
, 1932). Wald proved the exis-
tence of a static general equilibrium and von Neumann proved the existence of a
dynamic general equilibrium of a growth model based on a simpler model,
formulated in Cassel's textbook. In Cassel's model the equilibrium rate of growth
is determined by the ratio of the savings ratio and the capital-output ratio. Von
Neumann proceeded to generalize this model into a theory of an economically
sustainable dynamic general equilibrium, based on his saddle point theorem, proved
in the 1920s.
He introduced time into his equilibrium growth theory in two ways:
First, he formulated the basic model in terms of discrete period dynamics.
Second, the durability of all products were introduced in an inverse form as
constant rates of depreciation between periods, which is consistent with the
assumption of a deterministic economic system, as shown by Lev and Theil (ibid).
Von Neumann assumed joint production in order to treat depreciation and
durability efficiently in his model, as for example in the process of making paper
in which wood, energy and machines are used as inputs at the start of the process. At
the end of the paper making process a
joint product
vector of outputs consisting of
paper, store of energy and of machines, which have depreciated and thus have
become smaller in capacity. Formally the model is given by Eq. (
5.7
).
q
T
B
q
T
A
α
Bp
ʲ
Ap
q
T
B
ð
5
:
7
Þ
ð
ʱ
A
Þ
p
¼
0
q
T
B
ð
ʲ
A
Þ
p
¼
0
q
0
;
p
0
Where
q
¼
vector of outputs
p
¼
vector of prices
ʱ¼
1 + rate of growth
ʲ¼
1 + rate of interest
A
¼
mn matrix of inputs
B
¼
mn matrix of outputs
