their derivation from a singular matrix, they will all give the same proportional results as Aa, thus all obeying Eq. 1 which establishes proportionality with p.
This fact may be expressed more concisely by the equation
p = ki
r
W 2
implying relativities p
1
p
2
= A
r
A
s
, where k is a proportionality factor, i
r
, the rth row of the unit vector, and r may be any number from 1 to n. Eq. 2 enables us
to convert the market prices of all commodities and of any indicator or aggregate derived from them into the equivalent measure in value prices, but does so only up
to a proportionality factor. To establish an absolute conversion we must in addition ‘scale’ the elements of p to ensure the invariance of any chosen indicator or
aggregate in the face of this conversion. The most convenient aggregate from this point of view would seem to be the national income py, where y is the column
vector of deliveries to final purposes, requiring k of Eq. 2 to be derived from the supplementary equation
ki
r
W
0 y=y 3
This ensures that the weighted average of the conversion factors in p is equal to unity, and therefore any element of it greatersmaller than unity can justifiably be
interpreted as the undervaluationovervaluation of the corresponding commodity by the market.
5. Technological change
We now wish to enquire how the relative value-price p
i
p
j
or 6
ij
will be affected by technical progress or some change which will bring about changes in all
generators, thus
transforming the
matrix
W
into a
new matrix
V= Á
à Ã
à Ä
a
1
b
1
..... g
1
a
2
b
2
..... g
2
. .
..... .
a
n
b
n
..... g
n
 Ã
à Ã
Å with eigenvalue v and left-hand eigenvector p. The use of Greek symbols corre-
sponding to the Latin parameters of W saves us from the tedium of repeating the previous derivation and enables us to write the new relative value prices p by simple
analogy as
p
r
p
s
= y
rs
= F
r
F
s
4 where the F
0 stand for the co-factors of a in V. If the various generators involved in the above formula are to be expressed in terms of proportionate changes in
efficiency in the sense of i imparting a productivity gain as an input into all its uses and ‘overall’ y the elements of matrix V have to be replaced as follows:
a
1
= a
1
e
1
, a
2
= a
2
e
1
, a
3
= a
3
e
1
, etc., b
1
= b
1
e
2
, b
2
= b
2
e
2
, b
3
= b
3
e
2
, etc., and v = wy 5
or, in terms of reciprocal efficiencies o
1
and h a
1
= a
1
o
1
, a
2
= a
2
o
1
, etc., b
1
= b
1
o
2
, b
2
= b
2
o
2
, etc., and v = wh In general, therefore, when efficiency changes vary between sectors and their
input utilisation, each element of the co-factors A and f will grow at a different rate, thus producing changes in relative value-prices which are crucially dependent-
on the exact configuration of sizes within W in relation to these rates. The most that can be said about these changes, therefore, is that they are encapsulated in the
formula
p
r
p
s
} p
r
p
s
= F
r
F
s
} A
r
A
s
6 where the As and Fs are, respectively, the first-column co-factors of the generator
matrices W and V before and after the efficiency gains, i.e.
W = Á
à Ã
à Ä
− w
b
1
c
1
.. n
1
a
2
− w
c
2
.. n
2
. .
. .
. a
n
b
n
c
n
.. −
w Â
à Ã
à Å
and
V= Á
à Ã
à Ä
− wh
b
1
o
2
c
1
o
3
.. n
1
o
n
a
2
o
1
− wh
c
2
o
3
.. n
2
o
n
. .
. .
. a
n
o
1
b
n
o
2
c
n
o
3
.. −
wh Â
à Ã
à Å
The growth factors of the price relatives Eq. 6 are in fact measured by the elements of what may concisely be described as the ‘Hadamard Ratio’
4
of the adjoining characteristics of the generators. It can be seen from Eq. 6 that in a
world of n sectors the results will be of a degree of complexity and irregularity which does not allow any intuitively helpful statement to emerge, and to that extent
the changes in technological efficiency cannot be said to carry any general implica- tions for movements in value-prices, scissor-like or otherwise, unless the precise
nature of the generator matrix is specified, though some conclusions may be possible in the case of radically reduced or highly aggregated models. As a
theoretical boundary condition it is perhaps worth mentioning that in the hypothet-
4
The expression is here introduced by analogy with the ‘Hadamard product’ of two matrices, say V
and W, being defined as the matrix whose elements are the pairwise products of the elements in the same positions as the parallel v’s and w’s.
ical and obviously unrealisable situation where changes in efficiency are uniform in all sectors, say e6, and equal to the overall efficiency gain h, the co-factors, being
determinants of order n − 1 in which each element is multiplied by o, would imply that all value-prices would increase by a factor of o
n − 1
, i.e. reduced in the uniform proportion e
n − 1
, in fact that relative value-prices would remain unaffected by the efficiency gains as defined in Eq. 5.
6. The two-sector case
