4 Three basic types of prices

 4.1. The N-income price system Let us examine the value of  a(P,X)  defined by eq. (10). As shown in § 3, there exists a vector X0³ 0, such that the ratio of aggregate material costs to income, that is a(P,X), is independent of the price vector; and analogously that there exists a vector P0³ 0, such that a(P,X)  is independent of the vector of total production X. In other words, the relation between total material costs and national income expressed in prices P0 will not change with any changes in the vector of production X. For proof, choose  n > 0, m = r = 0 in matrix  H(m,n,r)(I - A)-1 .  If we introduce values of the parameters into eq. (23), we obtain the following price relation:                                (1 + n*)A'P0  =  P0                                 (25) The price system given by equation (25) is called an N-income price system. (3)  N-income prices are proportional to material costs. In other words, the relation between income and material costs is the same for all prices and is equal to the parameter n*. The parameter n* is uniquely defined by eq. (25).                                0 £ n* £(1/la) - 1,                               where la is the maximum characteristic root of matrix A. The vector of  N-income prices, P0  is the characteristic vector of matrix A'. Note that the uniform relation of income to material costs n* depends only on matrix A and, therefore, cannot be influenced by changes in production vector X. From eqs. (25) and (26) it also follows that, at N-income prices,                   a(P0,X)   =   l/n*. Here, if matrix A is irreducible, P0  is positive and uniquely defined.

 4.2. The F-income price system Let us now examine value of b(P,X)  defined by eq. (11). From § 3 it follows that there exists a vector X0 > 0 such that the aggregate capital-output ratio b(P,X0)  is independent of the vector of prices P, and also that there exists a vector of prices P0 > 0 such that  b(P0, X) is independent of the production vector XTo prove this, set  m = n = 0 and   r>0 for matrix H(m,n,r)(I-A)-1 Since, by our assumptions, the matrix (I-A)-1B'  has a positive maximum characteristic root lb  we obtain from eq. (23), a price equation with m = n =  0                               A’P0   +  r*B' P0     =  P0                          (26) where                                    r*   =   1/lb A price system given by eq. (26) is called the F-income price system. As is now evident, the capital-income ratio is the same for all prices in the F-income price system. It depends only on the matrix of complex capital-output ratios and cannot therefore be influenced by changes in the production vector. From eqs. (21) and (22), it also follows that, in this case, the aggregate capital-income ratio is equal to the maximum characteristic root of the matrix of complex capital-income ratios; that is,                                   b(P0, X) = lb  =  1/ r*.  The vector of  F-income prices is a characteristic vector of the matrix (I-A)-1B' . If this matrix is irreducible, then P0 > 0  is uniquely defined.

 4.3. The value price system Analogously to the argument just presented, there exists a price vector   P0³ 0, such that  g(P0,X) is independent of production vector X.  Thus, the matrix of workers’ total consumption coefficients C(I-A)-1 would have a positive maximum characteristic root lc. Setting  r = n  =  0 , we obtain the price formula from eq. (23):                             A’P0   +  m*C' P0    =  P0                                    (27)                                    m*  =   1/lc The price system of eq. (27) is called a value system of prices in the labor theory of value sense of the expression. Value prices are proportional to total wage-costs and the income component of every price is proportional to direct wage-costs. The parameter  m*  now becomes the reciprocal of the maximum characteristic root of the matrix of workers’ total consumption coefficients. The vector of value prices is the characteristic vector of the matrix (I-A)-1C' For value prices:                            g(P0, X)   =  lc   =  1/ m*   and                        y(P0, X)   =  m* -  1    where y is the rate of surplus value as defined by eq. (13).All three types of prices are special limiting cases of a general three-channel price system that can be computed by setting two of the three parameters m*,n*,r* equal to zero, that is, the lower limit of condition (24), and the third parameter to its upper limit.

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