Geoscience Reference
In-Depth Information
5.3
Durability, Depreciation of Capital and the Rate
of Economic Growth
The average durability of capital goods (i.e. all goods) of the economy is deter-
mined as the ratio between the value of capital as a stock and the value of
production as a flow and has the
dimension of time
(Hawkins
1948
; Hawkins and
Simon
1949
; Br
´
dy
1970
).
This implies that the average durability of the all goods (i.e. total capital) is
determined as the ratio of the aggregate value of capital to the aggregate value of
production. Under certainty, the optimal depreciation
ʴ
per year is 1/T. With an
uncertain durability with known mean value the optimal rate of depreciation (and
the implicit durability, T) is a constant fraction of the net asset value (Lev and Theil
1978
).
Assume that the outputs of production processes in the economy are subdivided
into currently used production and production for future use (i.e. investment).
Current production requires inputs in fixed proportions, represented by input-output
coefficients. Investment (I) is determined by an accelerator relation. I
Bgx, where
I and x are vectors, g is the rate of growth and the matrix B gives the fixed capital
requirements per unit of increase of production.
b
ij
divided by
a
ij
equals the
durability of the good,
T
i,
where
T
i
is the durability of good i. B
¼
T
0
A where T
0
¼
is a diagonal matrix of durability of goods (i
¼
1,
,n):
...
x
¼
Ax
þ
gBx
¼
ð
I
þ
gT
Þ
Ax
ð
5
:
5
Þ
;
The primal condition of a general equilibrium production structure and rate of
growth.
p
¼
pA
þ
rpB
¼
pA I
ð
þ
rT
Þ
;
ð
5
:
6
Þ
The dual price structure and rate of interest condition of a general growth
equilibrium.
where x
¼
production vector
p
¼
price vector
A
¼
n
n semi-positive input/output matrix
B
¼
n
n semi-positive capital/output matrix
¼
T'A.
g
¼
the maximal rate of growth at the general growth equilibrium
the minimal rate of interest at the general growth equilibrium
A unique equilibrium growth and interest rate with an associated pair of equili-
brium quantity and price vectors can be proved to exist [with the use of optimi-
zation procedures or the use of Perron's or Nikaido's theorems (Andersson and
Beckmann p. 26, pp. 204-206 and p.235 and Debreu and Herstein
1953
)].
Thus: The rate of interest equals the rate of growth in an economically sustain-
able equilibrium of this deterministic growing economy.
r
¼
