Sekerka - Kyn - Hejl:   Price Calculations



4. More Precise Price Formulae


If we want to substitute the general formulae of the above mentioned price types with more accurate formulae, indicating already the method of calculation, we must ask the Input-Output symbols for help.


Let us assume first that we have at our disposal a table detailed so far, that we can in each row express the production volume in a common natural unit and that it is logically justified to calculate prices of these units: let us introduce the following notation :

Qi ....... annual production volume of the j-th branch in natural units

pi ......... price of the production unit of the j-th branch        

Pi  .. annual production volume (gross production) of the j-th branch in current prices

It is evident that


             Pi = pi Qi                             (i =  1, 2, ….n)


let us denote further


Qij …. input of the i-th products needed to produce  Qi;

Vi ….  total  wages paid in the j-th branch during the year;

Zj……total gross profit  in the j-th branch during the year.

If we value the material inputs needed to produce the output of the j-th branch in their prices and then sum it up (i.e. if we perform Si pi Qij ), we obtain the total material cost of the j-th branch per year. If we add to this sum the wage costs and the gross profit of the year, we obtain the total annual value of output of the given branch. Then the following equation must hold:


            PjSi pi Qij  + Vj  + Zj           (j=1,2, …n)           (7)


If we now divide both sides of the equation by the total output in physical units  Qj we obtain:


            pjSi pi aij  + vj  + zj             (j=1,2, …n)             (8)




aij  =  Qij / Qj ….denotes the technical coefficients e.i. the input

                           of i-th product needed to produce one unit of output

                           of the j-th product.


vj  =  Vj / Qj …. denotes the direct wage costs per one unit

                           of the j-th  product


zj  =   Zj / Qj …. denotes the direct content of the gross profit in

                           one unit of the j-th  product


The material costs here are equivalent to the sum of their constituent components. At the same time the amount of costs depends on two distinct types of factors: on the technical coefficients  aij  which are given and not subject to changes(6) and on the prices varying in the process of computation. It is obvious from the formula (8) how it is possible to obtain the costs revaluation in the process of price changes.


Relation (8) represents a system of  n equations in which the following va­riables and coefficients appear:   n numbers pi expressing prices,     n2 technical coefficients, n direct costs vj  and  n numbers zj    expressing the direct contents of profit in the price. From what has been said of the above mentioned calculations follows that the technical coefficients and the direct wage costs are considered to be given constants; then there remains  2n variables -profits and prices. Therefore, if a certain system of numbers   zj  (j = 1, 2, … n) is chosen, only a certain system of numbers pj corresponds to it, a system which can be ascertained in solving the system (8). Here, of course, the problem of the existence and of the uniqueness of solution arises. In this place, however, we shall be satisfied with the statement that under current assumptions of the matrix coefficients aij a unique solution is to be expected.

Let us now start from the postulate that the system (8) should reflect the present price system. Then the n numbers zj   (i.e.. e. the vector of the direct contents of profits) should correspond to the true state of national economy. If we now want to obtain the redistribution of profits we must introduce a new vector of numbers zj into the system (8) in solving this system we obtain new prices pj, simultaneously the nominal evaluation of material costs will change.

Let us now correct our original assumption that we have a very detailed table classified in natural units. On the one hand, such a table is not at our disposal, on the other hand — even if such a table should exist the amount of computational work would be excessive. As we shall soon see, our previous considerations will be, with a small adjustment, valid even if we take the aggre­gate table (expressed in prices) as the basis of our computations.


Let us return to equation, (7) and let it be divided instead of by Qj by the volume of gross production Pj . Then we obtain


          1 =  Si a+ij  + v+j  + z+j              (j=1,2, …n)            (9)


where  a+ij , v+j, z+j  have analogical meanings as before, with the difference that they reflect the technical coefficients expressed in prices and the direct contents of wages and profits not in the natural units of production but in the money unit of production in respective branches.

The question remains, however, what is the possibility of using relation (9) for the price calculations? What happens when the vector of profit z changes? It is obvious that in such a case the sum on the right hand side has to change also and that therefore on the left-hand side a deviation from number one must occur, which is inevitably projected into the revaluation of the price costs. This implies that we must multiply the left side of the equation (9), and the coefficients aij  as well, by some indices reflecting the deviation from number one induced by the variations of numbers zj. If we denote these indices  p and  pi  respectively, we obtain a system equivalent to the system (8). [Let us notice that similarly we can consider (9) as an equivalent to the system (8) in the case that  pj = 1  is valid for all prices in (8).] The resulting numbers  pi  will represent the indices of changes of average price levels in different branches during the transition from the present prices to the value-, cost- and other prices. So e. g. if we obtain (see further) for branch "02 — coal mining and processing" the number 1,74, we can interpret it so that for introducing the value prices it is necessary to increase present prices in this branch by 74%,.


In the input — output models the matrix notation is generally used. If we translate the system (8) into the matrix notation we obtain


            p = p A + v + z                                                        (10)


p ... is the row vector of price indices .

A... is the technical coefficient matrix

v.... is the row vector of direct wage costs the row vector of the profit contents

Now we can formulate the matrix equations for individual price types and simultaneously seek a solution.


Labor Value price


         p = p A + v(1+ m)                                           (11)   


where m is a uniform rate of surplus for all branches of the economy.


Solving for p we obtain


         p = (1+ m) v( I – A) -1                                                       (12)


where  I is the identity matrix.

We can see from (12) that the labor value prices are proportional to the wage costs. The matrix  (I - A)-1 is the matrix of  "full (complex) input-output coefficients”; v (I - A)-1 is therefore a vector of "full wage or labor costs". The labor value price is therefore obtained by increasing the full labor costs proportionally to the uniform rate of surplus. The formula shows, among other things, one interesting feature, namely that the relative labor value prices are independent of the rate of surplus. The rate of surplus here determines the general price level only. This conclusion is valid for the labor value prices only. It will be shown -- in the section about the so called (n + 1)  condition — how can be the rate of surplus determined


Cost price


              p = (p A + v) (1 + r)                                      (13)


the solution for p is


              p = (1 + r)v [I - (1 + r) A] –1                                               (14)


From the resulting formula [compare (12)] the reason of the difference be­tween cost- and value price relations can be realized. Price relations vary with the magnitude changes of r


Production price

For the production price we must at first introduce the matrix of coefficients of the funds-intensity of production K, the element kij of which represent the quantity of the production funds of the i-th kind necessary to produce an unit of output in the j-th kind. Then it is possible to formulate a matrix equation for the production prices as follows:

                                               p=pA+v+rpK                                             (15)


If we know the matrix of technical coefficients, the matrix of direct funds-intensity coefficients, the vector of direct wage costs and if the rate of profit r is determined in some way (see further), then it is possible to derive from (15) the price vector


      p = v (I - A)-1 [I - rK (E - A)-1]-1                          (16)


Relative production prices are not independent of the rate of profit .


Two-channel price

 p = p A + (1 + m*) v + r*pK                             (17)

For the price vector then holds


     p = (1 +µ*)v(I - A)-1[I - r*K(I - A)-1 ]-1              (18)


The two-channel price, just as the production price, depends on two factors: full wage costs and full funds-intensity K(I - A)-1. So far as µ*> 0, it must be that r*< r  and between µ* and r* there is obviously such a relation that the increase of one is followed by the decrease of the other, which means that in two-channel relative prices the full wage costs have a greater weight and the full funds-inten­sity a smaller one than in the production price. It is evident that when µ* = µ and r*= 0 the two-channel price becomes the labor value type, and when r*= r and  = 0 it becomes the production price. It is therefore possible to see both the labor value and the production prices as special cases of the two-channel price. If we genera­lize further and allow also µ*< 0, then the "generalized" two-channel price can move even "over" the production price and approach the income price, which depends only on the full funds-intensity of production. When µ* = -1, the influence of wage costs on relative prices completely disappears and the rate of profit transforms into the uniform rate of income r*= d.


Income price


            p =  pA + dpK                                                                   (19)


As the equation is homogeneous in this case, the solution can be written as follows:


            p [I - dK (I - A) -1] = 0                                                     (20)


The condition for the existence of non zero vector of prices is 

                             [I - dK (I - A) -1] = 0

This relation determines the magnitude of the uniform rate of gross income. Equation (20) then determines only the relative prices and not the price level. To determine price level we need an additional (n + 1) condition. The income price depends exclusively on the full funds--intensity. In this sense it is possible to envisage both the value- and income price as two extreme cases of a general two-channel price, as the former depends exclusively on the full wage-costs and the latter on the full funds-intensity. The production price then is a certain "average" of both these types.








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