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



3. The general three-channel price system


Values of a(P,X) ,b(P,X), and g(P, X), defined in 2, depend on the price vector P and the production vector X. From definitions (4)-(12), it follows that, for any arbitrary, non-zero constants h, w, 

a(hP,wX) = a(P,X)
b(hP,wX) = b(P,X)
g(hP,wX) = g(P,X)

Thus, we see that these values depend on relative prices and relative quantities, that is, on relations pi/pj xi/xj (i, j = 1, 2, ...n) where n is the order of matrix A. They do not, however, depend on the price or production levels.



Let us now ask whether it is possible to find a vector of production X, such that the values of a, b, and  g or some linear combinations of them are independent of prices; and so find a price vector P, such that the values of a, b, and  g  or linear combinations of them are independent of X

Consider the matrix

H(m,n,r)   =    nA + rB + mC              (16)

where m,n,r are non-negative parameters, at least one of which is not zero. Under these assumptions,  

                                H(m,n,r)(I - A)-1    0. 

Thus, there exists a characteristic root  l0(m,n,r) > 0 and a characteristic vector Y0 (
m,n,r)  0 such that

H(m,n,r)(I - A)-1Y0  =   l0Y0 .                     (17)

From eq. (1)    X0 = (I - A)-1 Y0,     and     H(m,n,r)X0  =   l0Y0 .               
Then, for any vector 
P 0 such that   P'Y  > 0, 

na(P,X0) + rb(P,X0) + mg(P, X0) =  l0 .                 (19)





Therefore, for any three non-negative parameters m,n,r of which at least one is greater than zero, there exists a vector of total production X0  0 such that eq. (19) holds independently of the price system chosen, provided that P' Y0 > 0. Note that if the matrix H(m,n,r)(I-A)-1 is irreducible, then the vector X0 is positive and uniquely determined.

For the characteristic root  l0, there exists a characteristic vector
P0 = P0(m,n,r) 0 (in case of irreducibility P0 > 0), such that

                         (I - A')-1H'(
m,n,r)P0 =   l0P0                                  (20)

This means that for any vector Y 0, such that Y'P0 > 0, 

(P0,X) + rb(P0,X) + mg(P0, X) =  l0 .                  (21)


For each group of three non-negative parameters m,n,r with at least one greater than zero, there exists a non-negative price vector, such that eq. (21) holds independently of the production vector, provided that P'0Y>0. 

P0 is the characteristic vector of the matrix (I-A')-1H'(m,n,r)  and if this matrix is irreducible, it is positive and unique. Because we have assumed l0 > 0, we can define the following parameters

m*l0-1m,   n* =   l0-1n,   r* =   l0-1r.        (22)

From eq. (20), then

P0    =   (1 +  n*)A'P0  +  r*B'P0  + m*C'P0        (23)

Clearly, A'P0 , B'P0, and
C'P0 are material costs, capital, and wages necessary for one unit of production at P0 prices.




Thus, prices P0 cover costs of materials and form income as a sum of three components: the first is proportional to cost of materials, the second to capital, and the third to wages. Prices defined by eq. (23) are therefore called three-channel prices.

For parameters 
m, n, r we have assumed only non-negativity, and that at least one of them is positive. Otherwise, they may be chosen arbitrarily. Parameters m*,n*,r* are, however, mutually bound by the condition that the maximum characteristic root of the matrix (1+n*)A'+r*B'+m*C' is unity. This means that the values of all three parameters n*, m* and  r* cannot be chosen quite arbitrarily.

First of all, the following limits exist for these parameters:

0 n* ((1/la) - 1),  0 r* (1/lb),  0 m* (1/lc);         (24)



where la is the maximum characteristic root of matrix A;
lb is the maximum characteristic root of matrix B(I-A)-1;
lc is the maximum characteristic root of matrix C(I-A)-1.
If two of the three parameters are chosen, then the value of the third is given by eq. (23).
In specifying 
n*, m* and  r* we can:

(a) choose the values of 
m, n, r arbitrarily in the desired relations;

(b) find the maximum characteristic root l0, and on the basis of eq. (22), transform
m, n, r to m*, n*, r*. They will then be within the limits given by condition (24).

Alternatively, we can choose two of them and compute the third from eq. (23). 




The fact that the two chosen parameters lie within the limits given by eq. (24) is not, however, a sufficient condition for the non-negativity of the third parameter. We shall return to this problem later. 
Note, finally, that if 
la < 1, then at least one of the parameters m*, n*, r*
must be greater than zero. This follows from the inequality

A   (1 + n*)A'  +  r*B'  +  m*C'

and the condition that the maximum characteristic root of the matrix  

(1 + n*)A' + r*B' + m*C'


is unity.






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