2.    Choice of the Price Model

The literature on theoretical price models computable from input-output coefficients is already very abundant. 2) Among many suggested price models, two, in particular, are most frequently discussed: “labor-value prices” and so-called “production prices”. These two models are used by Osiatynski in the theoretical section of his paper. They can be formulated in the following way:

"Labor Value Prices”

(la)       p'  =  p'A + (1 + m) wl' 

(1b)      p'c  =  w  

(1c)      p'b= 1

The so called “Production Prices” (Cost Prices) 

(2a)   p' = (1 + n) (p'A + wl’)      

(2b)   p'c  =  w      

(2c)   p'b   =  1

      where   A is the matrix (m x m) of input-output coefficients,
            l'   is the vector (1 x m) of labor—output coefficients. 3);
            c’ is the vector (1 x m) of goods consumed by a unit of labor,
            b'  is the arbitrary vector (1 x m) called numeraire,
            p'  is the vector (1 x m) of prices,
           w is the monetary wage rate (a scalar)
           m is the Marxian rate of surplus value (a scalar) , and
           n is the scalar representing the uniform rate of profit.

It is usually assumed that:
(a)    A, 1, c and b are given, 
(b)   A, c, 1 and b are semipositive (sometimes 1 is assumed to be strictly positive), 
(c)    A is productive, i.e. its dominant characteristic root is positive and less than one.

It can be seen from (la) that “labor-value prices” are equal to the sum of material costs (p'A), wage costs (wl'), and contain surplus which is proportional to wage costs. The “production prices” formula (2a) distributes surplus or profit proportionally to the sum of material and wage costs. The equations (1b) and (2b) guarantee that the real wages are independent of prices. They can be viewed as an expression of the “subsistence theory of wages”. Finally, (1c) and (2c) are numeraire equations which serve only to fix the level of prices, i.e. to normalize the vector p. The vector b does not influence relative prices, therefore, it can be selected arbitrarily or remain undetermined. If b is equal to c, then the unit wage rate becomes equal to one and can be omitted in equations (la) and (2a).

It follows from the above assumptions that there exists a vector p’>0 and scalars w> 0 and m>0 that are solutions of the equations (la), (lb), and (1c). This can be demonstrated in the following way. Let us first transform (la) into:

(3)        p' = (1 + m) w1’ (I-A) -1

 and then substitute for w, according to (lb):

 (4)        p' = (1 + m) p’cl' (I -. A) -1

 If we define the matrix of “workers’ consumption coeffïcients”4)  as:

 (5)        C     = cl’

then (4) can be rewritten as: 

(6)        p’ = (1 + m) p'C (I - A) -1

Equation (6) gives “labor value prices” as the left-hand characteristic vector and  1/(1 + m) as the characteristic root of the matrix C (I-A) -1. Because C (I-A) -1 is apparently semipositive, we can recall the Frobenius-Perron theorem which guarantees the existence of the semipositive vector associated with the nonnegative dominant characteristic root5)   lC(I A) -1 of the matrix C(I-A)-1.The rate of surplus value m is then determined by:

(7)              m  =  (1/ lC(I A) -1  ) - 1

from which it follows that   m > 0   if and only if   0 <   lC(I A) -1 < 1.  The sufficient - but not necessary - conditions for uniqueness and strict positivity of   p and positivity of  m and w are (i) either A is indecomposable or (ii) 1 is strictly positive.

We can proceed similarly with the “production prices”. Substituting for w in (2a) according to (2b), and using definition (5), we get:

(8)      p'  =  (1 + n) p'(A + C).

 The matrix A + C is apparently semipositive, therefore, there exists a semipositive characteristic vector associated with the dominant characteristic root  l (A+C).  The “rate of profit” is determined by

 (9)                 n   =   (1/l (A+C)) - 1

 and, therefore  n > 0  if and only if   0 < l (A+C) < l

  The sufficient condition for uniqueness and strict positivity of  p  is indecomposability of the matrix A + C.  It can be easily checked that if the dominant characteristic root of the matrix A + C is equal to one, the dominant characteristic root of the matrix C (I - A) -1  is, also equal to one. In that case, m = n = 0, and both formulas provide the same vector of prices, covering only material and wage costs and containing no surplus.

 The model (la) , (1b) , and (1c) corresponds fairly well to the labor theory of value, but the equations (2a), (2b), and (2c) are less satisfactory as a model for “production prices”. The main reason for this dissatisfaction is that the matrix A is usually assumed to contain the intermediate products only while the “production prices” should also depend on capital stock tight in production. All that is needed, is to add a matrix B of capital-output coefficients to the previously used technological parameters. The elements bij of the matrix B represent the quantity of the product i required as capital stock per unit of output of the product j. Matrix B may contain both “fixed” and “working” capital, i.e. buildings, machines, and equipment, as well as inventories of raw materials, unfinished products and finished goods, and, if we wish, the stock of cash and deposits on current accounts in banks, as well. Assuming a constant coefficient technology, the matrix B is identical to the matrix of investment coefficients which frequently appears in dynamic input-output models.

It follows from the definition of B that the product p'B represents a vector of capital-output ratios valued at prices p. “Production prices” should contain profit proportional to capital stock serving in each sector, therefore, the price model must assume the following form: 

True “Production Prices”

(l0a)                 p'   =   p'A  +  w1' +  r p'B  

(10b)                p'c  =  w

(10c)                p'b  =  1

A, 1, c, b, p, and c have the same meaning as previously  
B   is the semipositive matrix of capital-output ratios and  
r   is the uniform rate of profit ( a scalar).

   Substituting (l0b) for w into (l0a), and using definition (5), we get:  

  (1l)           p'   =   p'A  +  p'C +  r p'B        

Assuming that the dominant characteristic root of the matrix A + C is less than one, (11) can be transformed into:                

(12)                  p'  =   r p' B (I - A - C) -1  

  Because the matrix B (I - A - C) -1 is semipositive (according to our assumptions), it is certain that there exists a characteristic vector p ³ 0 associated with the dominant characteristic root of the matrix B(I-A-C)-1. The rate of profit   r   is apparently determined as a reciprocal value of that dominant characteristic root:  

(13)                  r   =    1/ l B (I - A - C) -1  

It is clear that  r is positive as long as the dominant characteristic root of the matrix A + C is less than one. The sufficient condition for the existence of a unique and strictly positive p is indecomposability of the matrix B (I - A - C) -1.

A careful analysis of Marxian texts would reveal that the concept of   “production prices” corresponds to the idea of general equilibrium prices under perfect competition, which are supposed to provide the same rate of profit per capital stock invested in each line of activity. There is nothing in Marxian theory that would suggest that equilibrium profits should be proportional to costs of production. It is therefore obvious that it is the model (10a), and not the model (2) , which truly corresponds to the Marxian concept of “production prices”. Based on the assumptions of our model, equation (10a) also loosely corresponds to Walrasian general equilibrium prices.  

The formula (2) resembles price-fixing practices of the East European centrally managed economies rather than general equilibrium prices of the market economies. This has been acknowledged in many theoretical and empirical studies in which prices, according to the formula (2a), are usually termed “cost prices” or “averaged-value prices”.  

The formula (10a) could raise three objections:

  Firstly, it may seem that the prices (10a) do not cover depreciation. This is true about the formulas (la) and (2a) as well. There are two possible ways of bystepping this problem; either matrix A is redefined to include a flows of investment goods needed to replace obsolete fixed capital, or the parameter r is redefined to cover gross profit including depreciation.  

Secondly, it may seem that the formula (10a) is not completely consistent with the Marxian conception of “production prices”, because profit is not charged on “variable plus constant” capital only. This is not so! The known confusion between “flows” and “stocks” in many Marxian texts, made some people to equate variable capital to the annual flow of wage costs (annual wage fund). But this is a. fallacy. “Variable capital” must be a “stock” variable. The only meaningful interpretation is to view “variable capital” as the average stock of cash, and bank deposits which a given firm must permanently hold in reserve in order to guarantee a smooth and regular payment of wages. In. any realistic case, this figure is much smaller than the annual wage fund. It should be noted that the stock of “variable capital” may easily be negative if wages are paid ex post in time intervals which are longer than the production cycle of the given firm. There is no particular reason for separating “variable capital” from other forms of “circulating capital” because both have to bear the same rate of profit. The matrix B in the model (10a) is the matrix of capital-output ratios and theoretically it should cover all forms of capital stock including “variable capital”.  

 Thirdly, it may seem that the formula (10a) is not realistic since it is not, ”operational”. No firm in a market economy calculates prices of its products according to such a formula. This is, of course, a misunderstanding. “Production prices” are not intended to be a calculation scheme for managers, but rather than “equilibrium” achieved in a competitive market economy through successive adjustments or “tatonements”.  

The formulae for labor-value prices (la) cost prices (2a) and production prices (10a) are likely to generate very dissimilar relative prices. This can be illustrated by the price vectors which were calculated from the Czechoslovak input-output table for 1966 (see the first three columns in table 1).  

Display Table 1

  in this window  

  in a new window  

  The fact that “production prices” correspond, theoretically, to general equilibrium prices in capitalist market economies under perfect competition does not necessarily mean that they are actually very close to real prices. Imperfect competitions may cause a considerable differentiation of profit rates among branches of production, and some underlying assumptions of input-output models may be violated. On the other hand, “production prices” may still be closer to real market prices than both labor-value prices and cost prices. The empirical calculations of  V. D. Belkin (see Table 2), indicate that the real prices in market economies may be closer to theoretical “production prices” than is the case in centrally-managed economies.

Display Table 2

  in this window 

  in a new window 





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:


Czech Republic


Kyn’s Publications


 American education


free hit counters
Nutrisystem Diet Coupons