4.    Measurement at Artificial Prices or Artificially-Standardized Outputs.

Let us denote the ratio of aggregate material costs to national income as a      

(19)        a = p'Ax / p'(I - A) x    

Throughout the section 4, we shall assume that p and x are respectively the semipositive vectors of prices and total output such that    p'(I - A)x  ¹  0.   A is the matrix of input-output coefficients so that    p'Ax    is aggregate material costs, and    p' (I-A) x    is the aggregate value of final product or national income.

It is clear from (19) that   a   does not depend on the level of prices or on the scale of output, but it generally does depend on relative prices and on the structure of output. Suppose that the matrices   A   and vectors   x   are identical for two countries, then the ratios a will be different only if relative prices happen to be different. Similarly, for given p and A, the ratio   a   will change with changes in the structure of   x.

There exists a price vector   p   which makes   a   invariant to changes in   x.  It is easy to show that prices which have this property are “material-value prices”. The multiplication of both sides of (16a) by arbitrary x gives:

 (20)         p'x     =     p'Ax  +  n* p'Ax

It follows from (18), (19), and (20) that:

(21)    a    =    1/ n* =  lA /(1 - lA)     =    lA(I - A) -1

    The ratio of aggregate material costs to national income is equal to the dominant characteristic root of the matrix   A(I - A) -1  if valued at “material-value prices”. The dominant characteristic root of the matrix A does not depend on the vector x,  and it is, therefore, obvious that the ratio a valued at “material-value prices” is invariant to the output vector.

The same result can be obtained if the ratio a  is valued at arbitrary prices, but instead of  real output a certain artificial output vector x is used in formula (19). The vector x which makes a invariant to prices and equal to the dominant characteristic root of matrix   A(I-A)-1    is the right hand characteristic vector of  the matrix A. Let us call it “material-standardized output”.7) It is determined by the following equation:

 (22)             Ax    =    lx

After multiplying (22) by an arbitrary semipositive vector p, and making some rearrangements we get again:

(23)          a    =    lA /(1 - lA)     =    lA(I - A) -1

The dominant characteristic root of the matrix A (I - A) -1  is the best unbiased measure of a because it is invariant to both, prices p and outputs x. It can be obtained by measuring a either at arbi­trary prices on “material-standardized output” or at “material-value” prices on arbitrary output.

This leads us to an interesting and important conclusion: To make the macroeconomic ratios  a  of two countries mutually comparable, it is not necessary to recalculate statistical data of one country into prices of the other; rather, it is sufficient to express the ratio a for each country in its own artificially-computed “material-value prices”. Alternatively, the same result can be achieved by calculating a from artificially-computed “material-standardized output”. It ought to be stressed that neither “material-value prices”, nor “material-standardized output” are the same for two countries which have different technological matrices A.

Let   b  denote the ratio of aggregate capital stock to national income. Using previously introduced notation, aggregate capital stock may be written as  p'Bx,  where B is the matrix of capital-output coefficients. Definition of   b  is, therefore:

 (24)                   b    = p'Bx / p' (I - A) x

Clearly, the ratio b  is dependent on relative prices and on the structure of the output vector x. Analogically it can be demonstrated that there exists an artificial vector of prices which makes b invariant to the structure of output and that there exists an artificial vector of output which makes b invariant to prices.

The price vector having this property is the left-hand character­istic vector of the matrix   B(I-A)-1 or the vector of “capital-value prices” (see (14a) ). The output vector making b invariant to prices, is the right-hand characteristic vector of the matrix   (I-A)-1B   associated with its dominant characteristic root. We shall call it  “capital-standardized output”.

Equation (15) can be rearranged into:

 (25)                             p' (I-A)   =     r* p'B.

 If (25) is multiplied by an arbitrary vector x, then from (16) and (24) it follows that:

 (26)       b  =  1/ r*  =   l B(I-A) -1

 The aggregate capital-output ratio measured at “capital-value  prices” is equal to the dominant characteristic root of the matrix B(I-A)-1.

 "Capital-standardized output” is defined by:

(27)                                 (I - A)-1 Bx  =  l B(I-A) -1x

   This can be rearranged into:

(28)          Bx  =  l B(I-A) -1 (I - A) x

Multiplying (28) by an arbitrary price vector p and using definition (24), we get:

 (29)           b  =  1/ r*  =   l (I-A) -1 B

 As  l (I-A) -1 B  =   l B(I-A) -1 the capital-output ratios~ deter­mined by both (26) and (29), are identical. The dominant characteristic root of the matrix B(I-A)-1 is invariant both to prices and outputs. It may, therefore, be viewed as the best measure of the aggregate capital-output ratio.

For the international comparison of the aggregate capital - output ratios, it is not necessary to recalculate statistics into comparable prices, it suffices to measure b for each country at its own artificially computed “capital-value prices”. The same result can be obtained by measuring b on artificially—computed “capital-standardized outputs”.  

Let us now introduce a macroeconomic indicator g for measuring the share of wages in national income. The vector wl repre­sents wage costs per unit of output, therefore, the aggregate wage fund can be written as   wl'x. Using the “subsistence wage", equation and definition of the matrix C of “workers consumption coefficients” (see (5)), aggregate wage fund can be rewritten as p'Cx. The parameter g may, therefore, be defined by:         

(30)              g  =          p' Cx / p' (I-A) -1 x  

The fact that the vector of “labor-value prices” (6), makes g invariant to the structure of output x, can be easily checked. Similarly, the vector of “labor-standardized output” makes g invariant to prices. “Labor-standardized output” is the right-hand characteristic vector of the matrix (I  -A)-1C associated with its dominant characteristic root   l (I  -A)-1C 

    (31)           (I  -A)-1C x   =   l (I  -A)-1C x  

The multiplication of (6) by an arbitrary vector x or the multiplication of (31) by an arbitrary vector p and the use of (7) and of (30) lead to:

  (32)  g  =  1/(1 + m)   =   l  C(I  -A)-1  =   l (I  -A)-1C  

The dominant characteristic root of the matrix C(I-A)-1 is invariant to both prices and outputs, it is, therefore, the best measure of the share of wages in national income. It can be obtained either by calculating the ratio g at “labor—value prices” or by calculating it at arbitrary prices but on “labor—standardized output”. Suppose we wish to measure the “average rate of profit” f as a ratio of aggregate profits   p'(I - A - C)x to aggregate capital stock p'Bx:    

   (33)                                   f        =     p'(I - A - C)x/ p'Bx     

It would be natural to measure it in terms of  “production prices” (12) or on “production-standardized output”, which would be the right-hand characteristic vector of the matrix (I-A-C)-1B associated with its dominant characteristic root. It is apparent that in such a case: 

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

   Suppose on the contrary that we choose to measure f at “labor-value prices” and simultaneously on “capital-standardized outputs”. To see clearly the result, it will be useful to rearrange (33) into the following form:

(35)              f    =   [1 - p'Cx/p'(I-A)x] p'(I-A)x/p'Bx 

Because the vector p represents here “labor value prices” and x represents “capital-standardized output” (35) becomes:

(36)                       f  =  (1 - l C(I-A) -1)/ l B(I-A) -1  

 

   The values of f determined by (34) and (36) are generally not identical. For example, it is possible to derive the following values of discussed ratios from data in Table 1:

The share of wages in national income measured at “labor-value prices”

                g   =   1/(1 + m) =   1/2.71  =  .37

  The aggregate capital-output ratio measured at capital-value-prices:

              b  =   1/r* = 1/.42 = 2.34

 

The “average rate of profit” according to the formula (36):

             f    =   (1 - g)/ b  =   (1 - .37)/2.34  =  .26

We see, from the third column of Table 1 that the profit rate f measured at “production prices" was only .23.

It is also interesting to show the relationship between parameters b and g when both are measured at “production prices”. If (11) is multiplied by an arbitrary vector x, it can be rearranged into

    (37)           p'Cx/p'(I - A)x + r p'Bx/ p'(I - A)x = 1.

    With regard to (13) and to definitions (24) and (30), it must hold that:

      (38)            b = (1 - g) l B(I - A - C)-1  

 

The aggregate capital—output ratio is in this case a linear function of the share of wages in national income. The formula (38) shows that assuming constant real wages any changes in the structure of output, which increases the nominal share of wages in national income, leads inevitably to the decline of the aggregate capital—output ratio.

 

 

 

 

 

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