4. More Precise Price Formulae 
If we want to substitute the general
formulae of the above mentioned price types with more accurate formulae,
indicating already the method of calculation, we must ask the
InputOutput symbols for help. 
Let us assume first that we have at our
disposal a table detailed so far, that we can in each row express the
production volume in a common natural unit and that it is logically
justified to calculate prices of these units: let us introduce the
following notation : 
Q_{i}
_{....... }annual production volume of the jth branch in
natural units p_{i} _{......... }price of the production unit of the jth branch
P_{i}
.. annual production volume (gross production) of the jth branch in
current prices 
It is evident that
^{ } P_{i}
= p_{i} Q_{i}
(i.e. = 1, 2, ….n)
let us denote further
Q_{ij
}…. input of the ith products needed to produce Q_{i} V_{i}
…. total
wages paid in the jth branch during the year Z_{j}……total gross profit in the jth branch during the year. 
If we value the material inputs needed to produce
the output of the jth branch in their prices and then sum it up (i.e. if we perform
S_{i}
p_{i} Q_{ij
}),_{ }we obtain
the total material cost of the jth branch per year. If we add to this
sum the wage costs and the gross profit of the year, we obtain the total
annual value of output of the given branch. Then the following
equation must hold:
P_{j} = S_{i} p_{i} Q_{ij }+ V_{j} + Z_{j} (j=1,2, …n) (7)

If we now divide both sides of the
equation by the total output in physical units
Q_{j }
we obtain:
p_{j}
= S_{i}
p_{i} a_{ij
}+ v_{j}
+ z_{j}
(j=1,2, …n) (8)
where
a_{ij }= Q_{ij }/ Q_{j} ….denotes the technical coefficients e.i. the input of ith product needed to produce one unit of output of the jth product.
v_{j} = V_{j }/ Q_{j} …. denotes the direct wage costs per one unit of the jth product
z_{j} = Z_{j }/ Q_{j} …. denotes the direct content of the gross profit in
one unit

The material costs here
are equivalent to the sum of their constituent components. At the same
time the amount of costs depends on two distinct types of factors: on
the technical coefficients a_{ij}_{
}which are given and not subject to
changes^{(6)} and on the
prices varying in the process of computation. It is obvious from the
formula (8)
how it is possible to obtain the costs revaluation in the process of
price changes. 
Relation (8)
represents a system of
n
equations in which the
following variables and coefficients appear:
n
numbers
p_{i}
expressing prices,
n^{2}
technical coefficients,
n
direct costs v_{j}
and
n
numbers
z_{j
}expressing the direct contents of
profit in the price. From what has been said of the above
mentioned calculations follows that
the technical coefficients and the direct wage costs are considered
to be given constants; then there remains
2n
variables profits and prices. Therefore,
if a certain system of numbers z_{j
} (j
= 1, 2, …
n)
is chosen, only a
certain system of numbers p_{j}
corresponds to it, a system which can be ascertained in solving the
system (8).
Here, of course, the problem of the existence and of the
uniqueness of solution arises. In this place, however, we shall be
satisfied with the statement that
under current assumptions of the matrix coefficients
a_{ij}
a unique solution is to be expected. 
Let us now start
from the postulate that the system
(8)
should reflect the
present price system. Then the n
numbers z_{j}
(i.e.. e. the vector of the direct contents of profits) should
correspond to the true state of national economy. If we now want to
obtain the redistribution of profits we must introduce a new vector of
numbers z_{j
}into the system
(8)
in solving this system we obtain new prices
p_{j},
simultaneously the
nominal evaluation of material costs will change. 
Let us now correct our original
assumption that we have a very detailed table classified in natural
units. On the one hand, such a table is not at our disposal, on the
other hand — even if such a table should exist
the amount of computational work
would be excessive. As we shall soon see, our previous
considerations will be, with a small adjustment, valid even if we take
the aggregate table
(expressed in prices) as the basis of our computations. 
Let us return to
equation,
(7)
and let it be divided instead of by
Q_{j}
by the volume of gross production
P_{j
}.
Then we obtain
1 =
S_{i}
a^{+}_{ij
}+ v^{+}_{j}
+ z^{+}_{j}
(j=1,2, …n) (9)
where
a^{+}_{ij }
,_{ }
v^{+}_{j},
z^{+}_{j}
have analogical meanings as before, with the difference that
they reflect the technical
coefficients expressed in prices and the direct contents
of wages and profits not in the natural units of production but in the
money unit of production in respective branches. 
The question remains, however, what is the possibility of using
relation (9) for the price calculations? What
happens when the vector of profit z changes? It is
obvious that in such a case the sum on the right hand side has to change
also and that therefore on the lefthand side a deviation from number
one must occur, which is inevitably projected into the revaluation of
the price costs. This implies that we must multiply the left side of the
equation (9),
and the coefficients a_{ij
}as well, by some indices reflecting the deviation from
number one induced by the variations of numbers z_{j}._{
}If we denote these indices p_{j
}and p_{i}
respectively, we obtain a system equivalent to the system
(8). [Let us
notice that similarly we can consider (9)
as an equivalent to the system (8)
in the case that p_{j} = 1
is valid for all prices in (8).]
The resulting numbers p_{i}_{
} will represent the indices of changes of average price levels
in different branches during the transition from the present prices to
the value, cost and other prices. So e. g. if we obtain (see further)
for branch "02 — coal mining and processing^{"} the number 1,74,
we can interpret it so that for introducing the value prices it is
necessary to increase present prices in this branch by 74%,. 
In the input — output models the matrix
notation is generally used. If we translate the system (8) into the matrix
notation we obtain
p = p A + v + z
(10)
where
p
...
is the row vector of price indices .
A...
is the technical coefficient matrix v....
is the row vector of direct wage costs
z.....is the row vector of the profit contents 
Now we can formulate the matrix equations
for individual price types and simultaneously seek a solution. 
Labor Value price
p = p A + v(1+ m) (11) ^{ }
where
m
is a uniform rate of surplus for all branches of the economy.
Solving for p
we obtain
p = (1+
m)
v( I – A)^{ 1
}(12)
where I
is the identity matrix. 
We can see from (12)
that the labor value
prices are proportional to the wage costs. The matrix
(I  A)^{1}
is the matrix of "full (complex) inputoutput coefficients”;
v (I  A)^{1} is
therefore a vector of "full wage or labor costs^{"}. The labor
value price is therefore obtained by increasing the full labor costs
proportionally to the uniform rate of surplus. The formula shows, among
other things, one interesting feature, namely that the relative labor
value prices are independent of the rate of surplus. The rate of surplus
here determines the general price level only. This conclusion is valid
for the labor value prices only. It will be shown 
in
the section about the so called (n + 1) condition — how can be the rate
of surplus determined 
Cost price
p
=
(p
A + v)
(1 + r)
(13)
the solution for p is
p
=
(1^{ }
+ r)v
[I 
(1
+
r)
A] ^{
–1
}
(14)
From the resulting formula [compare (12)] the reason of the difference
between cost and value price relations can be realized. Price
relations vary with the magnitude changes of r 
Production price For the production price we must at first introduce the matrix of coefficients of the fundsintensity of production K, the element k_{ij}_{ } of which represent the quantity of the production funds of the ith kind necessary to produce an unit of output in the jth kind. Then it is possible to formulate a matrix equation for the production prices as follows: p=pA+v+rpK (15)
If we know the matrix of technical coefficients, the matrix of direct
fundsintensity
coefficients, the vector of direct wage costs and if the rate of profit
r
is determined in some way (see further), then it is possible to
derive from (15) the price vector
p = v
(I  A)^{1}
[I 
rK
(E  A)^{}^{1}]^{1}
(16)
Relative production prices are
not
independent of the rate of profit

Twochannel price p = p A + (1 + m*) v + r*pK (17)
For the price vector then holds
p
= (1
+µ*)v(I
 A)^{1}[I 
r*K(I
 A)^{1}
]^{1} (18)
The twochannel price, just as the production price, depends on two factors: full wage costs and full fundsintensity K(I  A)^{1}. So far as µ*> 0, it must be that r*< r and between µ* and r* there is obviously such a relation that the increase of one is followed by the decrease of the other, which means that in twochannel relative prices the full wage costs have a greater weight and the full fundsintensity a smaller one than in the production price. It is evident that when µ* = µ and r*= 0 the twochannel price becomes the labor value type, and when r*= r and = 0 it becomes the production price. It is therefore possible to see both the labor value and the production prices as special cases of the twochannel price. If we generalize further and allow also µ*< 0, then the "generalized" twochannel price can move even "over^{"} the production price and approach the income price, which depends only on the full fundsintensity of production. When µ* = 1, the influence of wage costs on relative prices completely disappears and the rate of profit transforms into the uniform rate of income r*= d. 
Income price
p = pA + dpK (19)
As the equation is homogeneous in this
case, the solution can be written as follows:
p
[I  dK
(I  A) ^{1}] = 0
(20)
The condition for the existence of non
zero vector of prices is
[I 
dK
(I  A) ^{1}] = 0
This relation determines the magnitude of the uniform rate of gross
income. Equation (20) then determines only the relative prices and not
the price level. To determine price level we need an additional (n + 1)
condition. The income price depends exclusively on the full
fundsintensity. In this sense it is possible to envisage both the
value and income price as two extreme cases of a general twochannel
price, as the former depends exclusively on the full wagecosts and the
latter on the full fundsintensity. The production price then is a
certain ^{"}average^{"} of both these types. 
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