5.     Bias Arising from Two-Level Systems of Prices

In the East European price systems the level of retail prices is usually considerably higher than is the level of wholesale prices, the difference being caused mainly by “turnover tax”. For the sake of simplification, we shall study only the case of the uniform rate of turnover tax, although we know that the real turnover taxes are greatly differentiated.

Let us retain the symbol p for the vector of “wholesale prices”, and define the vector of retail prices by the following relationship:


    (39)                        p*  =  (1 + d )p,


where the scalar  d is the uniform rate of turnover tax.

We shall further assume that turnover tax is imposed only on the commodities which are consumed by the population. This requires a redefinition of the “subsistence wage” equation:


    (40)                             w   =   p*'c   =   (1 + d ) p'c


If we assume that the only form of consumption is that which is paid out of wages, then the aggregate value of consumption at retail prices is equal to:


 p*'Cx     =    (1 +  d ) p'Cx,



of which  dp'Cx represents the total amount of turnover tax.

Let us denote the ratio of aggregate material costs to national income measured at two-level prices as  a* . Aggregate material costs are not affected by turnover tax, but national income is now equal to the value of final product in wholesale prices plus the whole volume of turnover tax, so that:


    (41)                      a*    =    p'Ax/[p'(I - A)x  +  d p'Cx]


Similarly, the aggregate capital-output ratio measured at two-level prices is:


    (42)                      b*    =    p'Bx/[p'(I - A)x  +  d p'Cx]


and the share of wages (or consumption) in national income is:


    (43)                      g*    =  (1 + d)p'Cx/[p'(I - A)x  +  d p'Cx]


If relative wholesale prices were not affected by the existence of turnover tax, then — for a given x —  a* and b* would be decreasing functions and  g*  an increasing function of d.

Turnover tax would cause the undervaluation of  a*   and  b*  and the overvaluation of  g*   as compared to these ratios measured at one-level prices. The economy would seem to be more efficient than it really is because it would seem to need less material and capital inputs per unit of output than it actually needs, and at the same time the share of consumption in national income would be artificially increased. The thing is not as simple, if turnover tax changes relative prices. In such a case, the previous conclusion might not be valid.

The impact of turnover tax can be best studied by using artificial prices. Let us begin with “labor-value prices”. The equations (la), (1b), and (1c) must be adjusted in order to provide for the difference between levels of retail and wholesale prices:

                        (44a)                    p'     = p'Ax + (1 + m)w l',

                        (44b)                    p*'     = (1 +  d) p'

                        (44c)                    p*'c    =  w

                        (44d)                   p*'b    = 1.


The numeraire (44d) fixes the level of retail prices, therefore, any increase in d must necessarily push the level of wholesale prices down. Instead of fixing the level of  retail prices, the numeraire equation can be used to fix the level of wholesale prices. Any increase in d would then push the level of retail prices up.

From (44a), (44b), (44c), and definition (5), it follows that:

                                  (45)            p'  =  (1 + m) (1 + d) p'C (I - A)-1.


It becomes immediately apparent that turnover tax does not influence relative prices in this case. Further:


(46)                 (1 + m) (1 + d)     =        1/lC(I-A)-1


If we assume m ³  0  then (46) gives the upper limit to the rate of turnover tax:


(47)                  dmax    =     1/lC(I-A)-1  - 1.



Because relative “labor-value-prices” are not affected by  changes in d , it follows directly from (40), (41) and (42) that both the ratio of material cost to national income a* and the aggregate capital-output ratio b* are biased downward, while the share of wages in national income g* is biased upward by the “two-level” system of “labor-value prices”.


In this case, it is quite easy to estimate the price bias in parameters a* , b*, and g*. If  p is the vector of “labor-value prices”, then it follows from (41), (42), and (43) that:


(48)                    a*  =  a/(1 + d lC(I-A)-1)


(49)                    b*  =  b/(1 + d lC(I-A)-1)


(50)                     g*  =  (1 +  d) lC(I-A)-1/ (1 + d lC(I-A)-1)


where   a  and  b  are  the respective ratios measured at one-level "labor-value prices".


Obviously, d > 0 implies a* < a , b* < b, and g* > g.  The price bias is at its maximum, when the rate of turnover tax reaches its upper limit (47). Then:


    (51)                    a*  =    a/ (2 - lC(I-A)-1)



    (52)                    b*  =    b/ (2 - lC(I-A)-1)


   (53)                     g*  =    1/ (2 - lC(I-A)-1)



The impact of turnover tax on the ratios a* ,b*, and g* valued at the artificial “capital-value-prices” and “material-value-prices” is also easy to trace. Clearly, relative prices (14a) und (16a) do not depend on wage rates, therefore, they cannot be affected by turnover tax imposed on consumer goods. Suppose that the numeraire equation:


(54)                      p'b        = (1 + d ) p'b = 1



is chosen instead of (14b) or (16b).

Valued at two-level “capital—value prices”:


(55)                                        a*  =  a/(1 +  dg)


(56)                                        b*  =  lB(I-A)-1/(1 +  dg)


(57)                                         g*  =  (1 + d)g/(1 +  dg)

where a and g  are the respective ratios measured at one-level “capital—value prices”.

Measured at two—level “material-value prices”:


     (58)                          a*  =  lA(I-A)-1/(1 +  dg)


     (59)                          b*  =  b/(1 +  dg)


     (60)                           g*  =  (1 + d)g/(1 +  dg)       


where b and g are the respective ratios measured at one-level “material—value prices”.

It is more difficult to analyze the impact of turnover tax on the ratios a*, b*  and  g* in “intermediate cases” between “labor-value”, “capital—value”, and “material-value prices” because in those cases relative prices are influenced by d.

The model of two-level “cost prices” consists of the following four equations:


     (61a)    p'  = (1 + v) (p'A + w1'),


     (61b)     p' =       (1 + d) p'


     (61c)     p'c     =   w,


     (61d)     p'b          1.




Substituting for w in (61a) according to (61b) and (61c)  and using definition (5) , we get:  


(62)         p = (1+ n )p' [A + ( 1+ d )C]



The “numeraire” (61d) has no influence on relative prices, and need not be taken into consideration here.

Because the matrix A + C is semipositive, we can immedi­ately conclude:

(a)      the parameter n is a nonincreasing (decreasing if A + C is indecomposable) function of  d;

(b)      changes in the parameter  d do affect relative prices except in very special and economically unrealistic cases;

(c)             the requirements n > 0 and  w > 0 impose upper and lower limits on d ;

(63)                         0   <  d   <  (1/lC(I-A)-1 ) - 1


(d)      if  d reaches its upper limit, then    n = 0 and “cost-prices” are transformed into “labor-value prices”.

Let us now study the impact of turnover tax  on the ratio   a* under the two-level “cost­ price” system. Because the relative prices do change with d it will be useful to use some artificially-standardized output” for measuring  a* . Naturally, “material-standardized output” will be the most appropriate output vector in this case.

It follows from (40) and (22) that


(64a)   d = 0        implies        a*  =  a  =   lA(I-A)-1



(64b)   0 < d <   (1/ lC(I-A)-1 ) - 1    implies   a*  =  lA(I-A)-1  /(1 + dg )



(64c) d = dmax = (1/ lC(I-A)-1 ) - 1  implies  a* = lA(I-A)-1  /(2 -lC(I-A)-1 )


g in (64b) is measured at wholesale “cost prices” and the notation g (d) shows that it is not constant with regard to d.

It is obvious that  lC(I-A)-1  £ 1 implies a** = a. An increase in d from its lower limit d = 0 to its upper limit  dmax = (1/ lC(I-A)-1 ) - 1   must depress the cost ratio a*  from  a  to  a**. It might be possible that this change is not monotonous. In small intervals, the increase in d may be offset by the decline of  g . With these possible exceptions the increase in d will cause a downward bias of  a*.

 Analogous conclusions can be reached about the capital output

ratio measured at two-level “cost prices” and on “capital­ standardized output”.

 From (41) we can read:


 (65a)       d = 0     implies     b* = b  =   lB(I-A)-1


 (65b)   0 < d <  (1/ lC(I-A)-1) - 1   implies   b* =   lB(I-A)-1/(1 + dg(d))



 (65c)  d = dmax = (1/lC(I-A)-1) - 1 implies  b*=b** = lB(I-A)-1/(2 -lC(I-A)-1 )


Similarly, if we measure the share of wages   g*  by using “labor-standardized output”, we get:


(66a)                 d = 0     implies     g* = g  =   lC(I-A)-1


(66b) 0 < d < (1/ lC(I-A)-1) - 1 implies   g*  =  [(1 + d)lC(I-A)-1]/( 1 + dlC(I-A)-1)


 (66c) d = dmax  =   (1/lC(I-A)-1) - 1    implies  b*  =  b**  =  1 /(2  - lC(I-A)-1 )


It is apparent from (66b) that   g* is monotonically increasing with the increase in d.

Let us turn to two-level “production prices”:

      (67a)                   p'    = p'A + wl' + r p'b

      (67b)                   p'    =  (1 + d)p'

      (67c)                   p'c   =   w

      (67d)                   p'b   = 1

The equations (67b) and(67c), and definition (5), can be used to transform (67a) into:


    (68)    p' = p'A + (1+ d) p'C + r pB


and if:   d  <  (1/ lC(I-A)-1 )  - 1



    (69)   p'   =   rp'B[ 1 - A - (1 + d)C]-1

It follows from (69) that:

(a) the rate of profit r is a nonincreasing (decreasing if A + C is indecomposable) function of the rate of turnover tax d:

(b) a change in d generally causes a change in relative prices;

(c) the limits for d are:

      0  < d   <   (1/lC(I-A)-1 )  - 1                                                                     

 (d)      when  d  reaches its upper limit r = 0, and “production prices” become “labor-va1ue prices".

The effect of turnover tax on ratios   a*, b*, and g* can be examined in the very same way as it was done in the previous case. Actually, all of the relationships (64) , (65) , and (66) remain valid.

The analysis in this section showed that the two-level system of prices, with turnover tax imposed on personal consumption, is very like to create a downward bias in the ratio of material cost to national income, as well as in the capital-output ratio.

On the contrary, we can expect the share of wages in national income to be biased upwards. Of course, the validity of these findings is limited because we have studied only the case of the uniform rate of turnover tax and only with the help of artificial prices. The existence of the bias can be indisputably proved in the case of three artificial systems of prices, namely “labor-value prices”, “capital-value prices”, and “material-value prices”. With respect to “cost prices” and “production prices”, the bias seems very probable, but we were forced to resort to artificially “standardized outputs” in order to demonstrate it. It is possible, although not very likely that if measured on some real output, the bias generated by turnover tax, could be partially or fully compensated by changes in relative prices. Therefore, it is useful to separate the impact of changes in relative prices, from the impact of turnover tax. This will be done in the next section.





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