3. ALTERNATIVE PRICE FORMULAE 
We may now attempt to formulate equations for different types of
prices. Let us suppose that the following indicators of the economic system are given:
the A matrix of input coefficients expressed in physical units;
the K matrix of capitaloutput coefficients expressed in physical units; element K_{ij }of this matrix expresses the stock of capital of ikind necessary for the production of a unit of output in the jsector during a given unit of time; the l vector of labour; l_{j} is the volume (number of manhours) of labour necessary for the production of a jkind unit of output. 
If we know the w vector of average wages in the sectors (which is supposed to be constant) we can also express the labour coefficients by the vector of coefficients of wage costs v = lw_{diag}. (12) 
We are looking for a ‘rational price formula’, which would assign some price system to the technological characteristics of the economic system designated by A, K, and v.
We shall investigate primarily one class of formula, which sets prices in such a way that they cover material cost and distribute income in some uniform manner into all prices. Income may be distributed according to the following principles:
(a) in proportion to material costs;
(b) in proportion to wage costs;
(c) in proportion to stock of capital.

From the class of formula mentioned above, we are interested mainly in those which cover not only material, but also wage costs, and distribute profit instead of income according to certain principles. 
Let us first derive a general formula, a socalled threechannel price in which profits are distributed according to all three principles at the same time, i.e. by the following three channels:
(i) In the first channel profits are distributed proportionally to material costs
z_{1 }=
npA. (13)
(2) In the second channel profits are distributed proportionally to wage costs,
z_{2 }=
mv. (14)
(3) In the third channel profits are distributed proportionally to capital stock
z_{3 }=
rpK. (15) 
The sum of all three channels gives us total of profits, present in wholesale prices:
z = z_{1}+z_{2}+z_{3} (16)
n, m, r are the parameters of the model. We assume
n
³ 0,
r
³ 0. If
m =  1, then the whole income is distributed through channels 13 and 15, i.e. z_{1 }+ z_{3 } = v + z. We assume
m
³ 1
but it should be understood that if 0
³ m ³ 1 there is no guarantee that prices will cover wage costs. 
Setting (13), (14), (15) and (16) into (4) we obtain:
The solution for p gives us the following general formula for a threechannel price:
p = (1 + m)v [I  (1 + n)A]^{1}{I  rK[I  (1 + n)A] ^{1}}^{1}. (18)

p = F(A,K,v, n, m,
r), p = G(A, K, v, n, m, r, d).

By choosing parameters
n, m,
r, and d we can then make the price system a definite one, or in other words for a given A, K and v we can choose specific price relations and levels.
The formula for a threechannel price (18) is a general formula, which includes special cases of simpler types of prices—so called twochannel prices, as well as elementary types of prices such as value price, production price, income price and cost price. 
Formula (18) looks complicated, but its composition is advantageous from the point of view of economic interpretation. This economic interpretation is especially evident if we choose
n = 0. In such a case we see that besides the parameters, prices depend only on two factors: v(I  A)^{1 } the vector of full wage costs coefficients, K(I  A)^{1} the matrix of full capitaloutput coefficients. 
The parameters
m and r are specific kinds of ‘weights’ by which full wage cost and full capitaloutput ratios enter into the setting of a price.
If we investigate the possibility of
n
¹ 0, we then come to the conclusion that the parameter
n does nothing more than change the weight by which direct material costs A influence full wage costs and full capitaloutput ratios. 
From equation (18) we see that the parameter
m influences only the level of prices, but not their relations, while parameters
n and
r influence both the level of prices and relative prices. 
If in equation (18) we make any of the parameters equal to zero, we obtain a so called twochannel price. There exist three possible types of twochannel prices. We shall investigate especially the following two types:
By an Ntwochannel price,^{?)} we mean a price which keeps the channel proportionate to material costs and the channel proportionate to wage costs. Putting
r = 0, in (18) we obtain the following formula:
By an Ftwochannel price ^{?)} we mean a price which keeps the channel proportionate to wage costs and the channel proportionate to capital.
Putting
n = 0 we obtain the following:
p = (1 + m)v [I  A]^{1}{I  rK[I  A] ^{1}}^{1} (21)

From equations (20) and (21) it is evident that in both types of twochannel price there can exist different price relations and price levels for the same technological circumstances, depending on how we choose the various parameters. In a Ntwochannel price, relative prices will depend wholly on v, A and the parameter
n. At the same time relative prices are in effect equal to the relations between hypothetical full wage costs, which would exist if technological consumption in all sectors was (1 +n) times larger than it is in reality. 
On the other hand the relative Ftwochannel prices depend on v, A, and K. In this type of price two elements are evident: full wage costs and the full capitaloutput ratios. The parameter p gives the weight, by which the full capitaloutput ratios influence prices.
In both types of twochannel price the parameter
m influences only the level of prices and not price relations. Each of the twochannel price formulae contains, as special cases, ‘elementary’ types of prices. 
If, in formula (20), for an Ntwochannel price we put
(1)
n = 0, we obtain a value price (in the labour theory of value sense of the word), in which profits are distributed only by one channel, proportionally to wage costs. The formula for a value price reads as follows:
In a onelevel price system the number m has an evident economic interpretation—it expresses the rate of surplus value. 
(2) Similarly if we put
m =
n, we obtain a socalled cost price, in which profits are distributed in proportion to total costs (the sum of material and wage costs). The formula for a cost price reads as follows:
p = (1 +
n)v{I  (1 +
n)A]^{1}. (23)

(3) If we distribute total income (v + z) into prices in proportion to material costs, we obtain a type of price, which we shall call an Nincome price. Take (17) and put
m = 1 and
r = 0. This is a formula for Nincome price, from which we can obtain

The number
n in this case evidently expresses a uniform rate of income in relation to material costs. An Nincome price also means that the relative share of income in all prices is the same. From formulae (22), (23), and (24) it is evident that: (a) the relations between value prices are determined only by the relations between full wage costs, (b) the relations between cost prices are equal to relations between value prices which would exist if direct consumption of material were (1 +
n) times higher than it is in reality, and (c) the relations between Nincome prices depend only on the properties of the A matrix. 
With value prices the parameter
m has no influence on price relations, which means that the value price formula gives us the same price relations whatever the price level is. This does not hold true for the cost price, where it is evident that the size of parameter
n influences the price level as well as the relations between prices. We assume that parameter
n in a costprice formula can be arbitrarily chosen, provided
n
³ 0. 
With Nincome prices the situation is somewhat different. Here the parameter
³ cannot be set arbitrarily. The condition for the existence of a nonzero p vector, according to equation (24), is the following:

It is evident that 1/(1 +
n) is the characteristic root of the A’ matrix and p’ is the corresponding characteristic vector. Since A’ is nonnegative and nonzero matrix, there exists a nonnegative characteristic vector p’, which corresponds to a positive, real, characteristic root with maximum absolute value.* Since from the economic point of view a solution giving negative prices has no meaning, we can come to the conclusion that price relations are given by equation (24). But this equation does not give us the level of prices, which may be set arbitrarily, or rather derived from the (n + 1)st condition. 
Nincome prices have some surprising features, from the point of view of their economic interpretation: their relations depend only on the properties of the A matrix, and are the same, regardless of the vector of total output q; the rate of income
n is given independently of the manner in which the national income is distributed between income realized by the enterprises and income realized through the turnover tax.
Let us recall that from the theory of Leontief type models we know that the maximum characteristic root of an A matrix has one more interesting economic interpretation: it expresses the ‘productivity’ of the system characterized by matrix A. If this characteristic root is equal to 1 (as is the case in a closed model Aq’ = q’), this means that total output is equal to input. The smaller the characteristic root is, the more productive is the system, the greater is its net efficiency or final output. We thus come to the conclusion, that the number v in a system of income prices expresses a certain ‘average’ productivity of the economic system or the average relative surplus of final output over material input. 
From formula (21), for an Ftwochannel price we can deduce three elementary types of prices:
(i) If we write r = 0 we obtain a value type price
p = (1 +
m)v(I—A)^{1}.
(25)
to capital: p = v(I  A)^{1}[I  K(I  A)^{l}]^{1}. (26)

In a production price then the
r parameter expresses an average rate of profit.
(3) If we write
m = 1, we obtain a socalled Fincome price, which can be determined by the following equation:
p[I  rK(I  A)^{1}] = 0 (27)
The number
r n this case expresses a uniform rate of income in relation to capital.
We have already mentioned the properties of a value price. Let us note that value price represents an extreme case of an Ntwochannel price, as well as an Ftwochannel price. Value price thus forms a sort of bridge between N and Ftwochannel prices. Income prices form extreme cases at the other end. 
It is evident from what has been said that the relative Fincome prices are independent of wage costs and are given by the full capitaloutput ratios. Production prices are then a sort of ‘mix’ of relative value prices and relative income prices. The weight of the influence of the full capitaloutput ratios on the relative production prices depends on the size of parameter
r Let us further note that for the same
r the relative production prices and the relative Ftwochannel prices are the same. If the same
r is given, then from the point of view of price relations there is no important difference between a production price and an Ftwochannel price. There is of course a difference, in that for
m
>
0 the size of
r in a twochannel price cannot achieve the maximum possible size of
r in the production price, if the level of wholesale prices is not to be above the level of retail prices.
It is evident from what has been said that the relative Fincome prices are independent of wage costs and are given by the full capitaloutput ratios. Production prices are then a sort of ‘mix’ of relative value prices and relative income prices. The weight of the influence of the full capitaloutput ratios on the relative production prices depends on the size of parameter
r Let us further note that for the same
r the relative production prices and the relative Ftwochannel prices are the same. If the same
r is given, then from the point of view of price relations there is no important difference between a production price and an Ftwochannel price. There is of course a difference, in that for
m
>
0 the size of
r in a twochannel price cannot achieve the maximum possible size of
r in the production price, if the level of wholesale prices is not to be above the level of retail prices.
With the Fincome price we may again note that 1/r
is the characteristic root of a matrix of full capitaloutput ratios K(I  A) ^{1}. The number
r here expresses the average ‘productivity’ or efficiency of capital and is independent of the q vector, and of the proportion in which the national income is distributed between the income of enterprises and turnover taxes. With the Fincome price we may again note that 1/r is the characteristic root of a matrix of full capitaloutput ratios K(I  A) ^{1}. The number r here expresses the average ‘productivity’ or efficiency of capital and is independent of the q vector, and of the proportion in which the national income is distributed between the income of enterprises and turnover taxes. 



