Sekerka - Kyn - Hejl:  A Model for Planning of Prices




Our discussion up to this point has usually presupposed that the vector of average wages w is given so that we have been able to consider the vector of wage costs as a technological characteristic of the need for labour for a given output. Since price calculations are only looking for new prices to a given physical structure of output, our model must also define the relation between the wage level and the level of retail prices in such a way that the real final distribution of output between personal consumption on the one hand and social consumption and investment on the other, is not influenced.

It is possible to define this relation in two ways: (1) so that during calculation the level of wage-rates and the level of retail prices do not change; (2) so that during calculation we make sure that the level of wage rates moves in accordance with changes in the level of retail prices and thus average real wages remain constant.

In this paper we have chosen the first alternative. In all of the formulae (with the exception of income prices) mentioned in part 3, wage cost has been of a given and unchanging magnitude. From this, then, stems the demand for the level of new (calculated) retail prices to be the same as the level of old retail prices.


If we now return to the formulation of equation (19) which sets a price system we see that we have:

n equations for wholesale prices, n equations for retail prices and one condition (keeping the level of retail prices constant), we have then 2n + 1 equations;

n unknown wholesale prices, n unknown retail prices, n unknown turnover tax rates and three unknown parameters m, n, r. We have then 3n + 3 unknowns.

This means that it is possible to set all the turnover tax rates and two parameters arbitrarily; the size of the third parameter will be the result of the solution of the system of equations.  

We may simplify our calculations by choosing a price system with a uniform turnover tax rate; in such a case the number of degrees of free­dom is reduced to three. Of the four parameters m, n, r and d we may set three and calculate the fourth from the system of equations. If, for instance, we arbitrarily set m, n, and r, then the size of d or the distance between the levels of wholesale and retail prices is also given.

From an economic point of view the case where d and two of the remaining parameters are set, is of special interest. Choice of d will designate the distance between the level of wholesale and retail prices and the choice of two parameters gives us the type of price.  


We shall investigate only the case where d = 0 (i.e. a one-level price system), without a differentiated turnover tax rate. In such a case p = p and the number of unknowns and equations can be lower by n.

Let us write

xs        for the vector of personal consumption priced by the old retail prices.

qs        for the vector of personal consumption in the old wholesale prices.*

j = (1, 1, ..., 1) is the n-dimensional vector of unities.

p       will in this case represent the vector of transformation indices from the old wholesale prices to the new ones.  

The wish to keep the level of retail prices (under conditions of having a single-level system) can be formulated in the following way:


                                     pqs'  =  jxs'  = (1 + d)jqs           (28)


Relation (28) forms the (n + 1)st equation of the price model. It will prove advantageous to adjust it in the following manner:


                                            pb’ = 1,                                (29)  



                     b’ =  b*'/(1+d)                and   b*’ =  qs'/jqs'           (30)

d being here the average turnover tax rate in the old price system.

The b* vector has an evident economic interpretation, since it expresses the relative shares of outputs of individual sectors in total personal consumption (in the old wholesale prices). Vector b keeps relations of b* but has a different level. The difference between the levels of  b and b* is due to the necessity of keeping balance between retail prices and wages.  


We shall now demonstrate the solution of a model with (n+1)st con­dition on the example of the basic types of prices. Let us return to equa­tion (17) (where we presuppose p = p), and let us choose:

(1)     n = 0, r = 0. By this we have chosen the value type of price. Keeping the (n+1)st condition in mind it must now hold true that:


                            p = pA + (1 + m)v,      pb’ = 1.                   (31)


From (31) we may derive: p = pA + pb’v(1 + m), and further


                            p[I - (1 + m)b'v(I - A)-1 = 0                         (32)  


It is now evident that 1/(1 + m) is the characteristic root of the [b'v(I - A)-1]' matrix. The coefficients, which form the b'v matrix, have the following economic meaning: they express the deliveries (in old wholesale prices) of i-kind products for the consumption of workers in the j-sectors which is necessary for a unit of production. Here we are pre­supposing that the structure of consumption by workers is the same in all sectors. This b’v matrix is somewhat similar to the matrix of technological input coefficients A which also expresses the necessary consumption of i-kind products for the production of a unit of j-kind of output. This does not however pertain to the direct consumption of these products in the technological process, but to consumption, which reproduces the labour used up in production. The b’v(I -  A)-1 matrix can thus be interpreted as the matrix of complex workers’ consumption. The rate of surplus value m is then the function of the characteristic root of this matrix, in a system of value prices.  


To obtain m it is possible to find a simple method of calculation. From (31) it follows that:  


                                        1/(1 + m)  = v(I - A)-1b’.                            (33)  


(2)               n = 0, m = 0. By this we have chosen a production type of price, 

whose equation with the (n + 1)st condition may be written as follows:



                            p  =  pA + v + rpK,    pb’ =  1. (34)

and thus                  


                              p = pA +    pb’v + rpK.         (35)


Let us introduce the matrix A* = A + b’v, which we shall call an aug­mented matrix of productive consumption according to Morishima. We can now write equation (35) in the following manner:


                                  p[I - rK(I - A*)-1]  =  0,                                                 (36)


where 1/r is the characteristic root of matrix [K(I - A*)-1]’.

(3)        m = n, r = 0. By this we have chosen a cost type of price whose equation, taking the (n + 1)st condition into account, is the following:


p = (1 + n)pA + (1 + n)v,     pb’ = 1;                   (37)


from which it follows that

           p = (1 + n)p(A + b’v)  

and thus                               p[I - (1 + n)A*]   =   0.                        (38)


In a system of cost prices 1/(1 + n) is the characteristic root of matrix


From the point of view of economic interpretation it is interesting to compare equations (36) and (38) with equations (27) and (24). We see some similarity between cost price and N-income price on the one hand and between production price and F-income price on the other. The only difference between them is that in income prices we have a pure A matrix, while in the cost and production prices an augmented matrix  A*  = A + b’v  appears.






OK Economics was designed and it is maintained by Oldrich Kyn.
To send me a message, please use one of the following addresses: ---

This website contains the following sections:

General  Economics:

Economic Systems:

Money and Banking:

Past students:

Czech Republic

Kyn’s Publications

 American education

free hit counters
Nutrisystem Diet Coupons