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
A particularly simple and transparent case, which is moreover of immediate relevance to scissor crises, arises when the number of sectors is reduced to two and
when, additionally, the nature of sectors andor variables is redefined to make the elements in the main diagonal of the generator matrix disappear, thus converting the
matrix W into
W =
b
1
a
2
This makes the eigenvalues, w and v directly computable from first principles and allows Eq. 2 to be expressed entirely in terms of the generators and efficiency
changes. Indeed, w can be immediately shown to be a
2
b
1
, the square root of the product of the off-diagonal elements, since the characteristic determinant
à Ã
à a
2
b
1
b
1
a
2
a
2
b Ã
à Ã
evidently vanishes. By the same token v will be o
1
o
2
a
1
b
1
. Hence the ratio vw emerges as
o
1
o
2
or e
1
e
2
. Accordingly, as Eq. 2 and Eq. 5 teach us, the now unique change in value-price is from wb
1
to vb
1
, and the change in the relative value-price of sector 1 becomes
p
12
p
12
= vb
1
} wb
1
= vwb
1
b
1
= e
2
e
1
e
2
= e
2
e
1
6a This betokens a rise in the relative value-price of sector 1 if e
1
B e
2
, i.e. if sector 1 increases its efficiency by less than sector 2. In fact the less dynamic sector in terms
of efficiency is compensated by an improvement in its terms of trade, just as the more dynamic sector suffers a virtual tax by a deterioration.
Moreover, if the improvement takes the form of a secular growth converting e
1
and e
2
into expg
1
t and expg
2
t, respectively, the secular rise in the value-price will be measured by
expg
2
texpg
1
t or exp[
1 2
g
2
− g
1
t], in fact by a growth-rate of one half the physical efficiency lag of the sector in question. Conversely the leading
sector will suffer a relative price fall of one half its physical efficiency lead. If, on the other hand, the sectoral efficiencies change in opposite directions e
1
B 1 B e
2
, i.e. g
1
is equal to a rate of decline, d
1
, the resulting change in value-price of the growing sector will be exp[
1 2
g
2
− d
1
], showing a gain in the relative value price of the declining sector equal to one half the sum of the absolute values of the efficiency changes, i.e.
their arithmetic mean, while the efficiency-gaining sector will be penalised by a loss in its terms of trade of an equal amount.
This result, in spite of the highly stylised form in which R has been obtained, may not be without interest, the more so since it applies to any two-fold division of the
economy, whether agriculture and industry, heavy and light industry, material production and services traditional and modern sector, or any of the divisions felt
to be structurally important to a given economy.
While smooth secular rises in the generators thus produce similarly smooth, but mitigated movements in relative value-prices, other types of movements may produce
aggravated, or even explosive movements in them. Suppose for instance the generators show movements of a sinusoidal type through time as instanced by
e
l
= cos t and e
2
= sin t, having values bounded above and below. The corresponding
value movements would then be sin tcos t=
tan t which increases without limit
as time approaches p2, leaving ample scope for disparities in value-prices that could produce opposing movements such as might engender scissor crises.
7. Specific instances of surplus-levelling value-prices
