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 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 capital-output coefficients expressed in physical units; element Kij of this matrix expresses the stock of capital of i-kind necessary for the production of a unit of output in the j-sector during a given unit of time; the l vector of labour; lj is the volume (number of man-hours) of labour necessary for the production of a j-kind 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 = lwdiag.              (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. The coefficients according to which profits are distributed into prices by the various principles mentioned above may in principle differ in various sectors of the economy. But insofar as there do not exist serious factual reasons for setting differentiated coefficients, it seems much more reasonable to set uniform coefficients as a point of departure. Ricardo, Marx, Walras, Dmitriyev, von Neumann, Morishima, and Sraffa9), whose work the theory of price types directly follows, in most cases presupposed uniform coefficients. Let us first derive a general formula, a so-called three-channel 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                             z1 = npA.                                  (13)     (2)    In the second channel profits are distributed proportionally to wage costs,                             z2 = mv.                                      (14)     (3)    In the third channel profits are distributed proportionally to capital stock                             z3 = rpK.                                  (15) The sum of all three channels gives us total of profits, present in wholesale prices:                         z = z1+z2+z3                             (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. z1 + z3  =  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:                               p = (1 + n)pA + (1 + m)v + rpK.              (17)       The solution for p gives us the following general formula for a three-channel price:       p = (1 + m)v [I -  (1 + n)A]-1{I - rK[I - (1 + n)A] -1}-1.      (18) This formula expresses the dependence of wholesale prices on the technological characteristics of the economic system A, K and v, and on the three parameters n, m and r. If we also take into consideration that in order to find a system of retail prices we must also know the vector of the turnover tax rate d, we can briefly describe the whole price system in the following way:      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 three-channel price (18) is a general formula, which includes special cases of simpler types of prices—so called two-channel 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 capital-output coefficients. The parameters m and r are specific kinds of ‘weights’ by which full wage cost and full capital-output 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 capital-­output 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 two-channel price. There exist three possible types of two-channel prices. We shall investigate especially the following two types: By an N-two-channel 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:       p = (1 + m)v [I -  (1 + n)A]-1    (20)       By an F-two-channel 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 two-channel 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 N-two-channel 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 F-two-channel prices depend on v, A, and K. In this type of price two elements are evident: full wage costs and the full capital-output ratios. The parameter p gives the weight, by which the full capital-output ratios influence prices. In both types of two-channel price the parameter m influences only the level of prices and not price relations. Each of the two-channel price formulae contains, as special cases, ‘elementary’ types of prices. If, in formula (20), for an N-two-channel 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:       p = (1 + m)v(I - A)-1.          (22)       In a one-level 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 so-called 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)       The number n in this case expresses a uniform rate of rentability i.e. the ratio of  profits to total costs. (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 N-income price. Take (17) and put m =  -1 and r = 0. This is a formula for N-income price, from which we can obtain   p[I—(1 + n)A] = 0.          (24) The number n in this case evidently expresses a uniform rate of income in relation to material costs. An N-income 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 N-income 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 cost-price formula can be arbitrarily chosen, provided n ³ 0. With N-income prices the situation is somewhat different. Here the parameter ³ cannot be set arbitrarily. The condition for the existence of a non-zero p vector, according to equation (24), is the following:       | I - (1 + n)A|  =  0. 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 non-negative and non-zero matrix, there exists a non-negative 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. N-income 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 F-two-channel 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)       (2)    If we write m = 0, we obtain a so-called production price (in the sense of the term used in the third volume of Marx’s Das Kapital), where profits are distributed on the basis of a uniform rate of profit in relation 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 so-called F-income 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 N-two-channel price, as well as an F-two-channel price. Value price thus forms a sort of bridge between N- and F-two-channel prices. Income prices form extreme cases at the other end. It is evident from what has been said that the relative  F-income prices are independent of wage costs and are given by the full capital-output 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 capital-output 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 F-two-channel 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 F-two-channel price. There is of course a difference, in that for m > 0 the size of  r in a two-channel 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  F-income prices are independent of wage costs and are given by the full capital-output 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 capital-output 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 F-two-channel 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 F-two-channel price. There is of course a difference, in that for m > 0 the size of  r in a two-channel 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 F-income price we may again note that 1/r    is the characteristic root of a matrix of full capital-output 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 F-income price we may again note that 1/r    is the characteristic root of a matrix of full capital-output 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.

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