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Sekerka - Kyn - Hejl:  Price Systems Computable from Input-Output Coefficients

 

 

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 
 

 

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