5. Two-channel prices

 If, in the equation for three-channel prices (23), we set one of three parameters m*,n*,r* equal to zero, we obtain a two-channel type of price. We distinguish three variants of two-channel prices

5.1. N-two-channel prices

N-two-channel prices are obtained if we set
r* = 0.

P0  = (1 + n*) AP0 + m*CP0.                  (28)

If we choose  n* arbitrarily from the interval [0; (l/la) - 1), we can write

P0  =  m*[I -  (1 + n*) A’]-1CP0                (29)

The parameter
m* is thus uniquely defined as the reciprocal of the maximum characteristic root, and the vector of prices P0 as the characteristic vector of the matrix  [I - (1+n*)A’]-1C’. It is thus possible  to consider m* as a function of n*. From the theory of non-negative matrices it follows that dm*/dn* £ 0, and in cases of irreducibility, dm*/dn* < 0.

Similarly, it can be shown that the choice of parameter
m* from the interval [0; 1/lc) uniquely defines n*.

Value prices and N-income prices are evidently limiting cases of N-two-channel prices. A certain ‘middle case’ is also of some interest. If we write m* = 1 + n
*
we obtain the following price formula:

P0 = (1 + n*)(A’ + C’)P0.        (30)

These prices cover material and wage costs and profit proportional to costs. We shall call them
cost prices. Note that n* > 0, that is, there exists positive profit only if the maximum characteristic root of matrix  A’ + C’ is less than unity. With cost prices P0 is given by eq. (30),

f(P0, X) =  n*

where f is the ratio of total profits to total costs as defined by (14), and 1 + n*  is the reciprocal of the maximum characteristic root of matrix    A’ + C

 5.2. F-two-channel prices F-two-channel prices are obtained as follows: in eq. (23), set n* = 0.              P0    =   A'P0  + m*C'P0  +   r*B'P0                 (31) Choose r* arbitrarily from the interval  [0; 1/lb ). We can then write        P0  =  m*[I -  r* (I - A)-1B' ]-1  (I - A)-1C' P0          (32) The parameter m* is thus uniquely defined as the reciprocal of the maximum characteristic root, and the vector P0  as the characteristic vector, of the matrix                               [I -  r* (I - A)-1B' ]-1  (I - A)-1C' . Now m* is a function of the parameter  r* such that   dm*/dr*  £ 0  and in case of  irreducibility dm*/dr* < 0. Similarly, the reverse can be shown; by choice of  m* from the interval   [0; 1/lc ) r* is uniquely defined. It is evident that value prices and F-income prices are limiting cases of the F-two-channel type. We obtain an interesting ‘middle case’ if we set  m* = 1. If the characteristic root of the matrix A + C  is less than unity we can write                      P0    =   r*[I - (A' + C')']-1 B'P0                  (33) These prices cover material and wage cost and contain profit in proportion to capital. We call them production prices.Note in equation (31) that if the maximum characteristic root of matrix  A+C  is unity and m* = 1, then r* = 0 and we would have value prices. Then prices would just equal the sum of material and wage costs and profit would be zero. It can easily be shown that. in a system of production prices P0   defined by equation (33), d(P0, X) =  r* where d(P0, X) is the average rate of profit as defined by eq. (15) and r*   is the reciprocal of the maximum characteristic root of  matrix                    [I - (A' + C')']-1 B'.

 5.3. D-two-channel prices D-two-channel prices are obtained if we set  m* = 0 in eq. (23).                   P0    =   (1 +  n*)A'P0  +  r*B'P0                     (23)  This type of prices is less interesting from the point of view of economic interpretation. However, it can be shown that  n*   is a function of  r*  such that  dn*/dr*  £  0

 OK Economics was designed and it is maintained by Oldrich Kyn. To send me a message, please use one of the following addresses: okyn@bu.edu --- okyn@verizon.net This website contains the following sections: General  Economics: Economic Systems: Money and Banking: Past students: http://econc10.bu.edu/okyn/OKpers/okyn_pub_frame.htm Czech Republic Kyn’s Publications American education