On International Comparisons in

On International Comparisons in Artificial Prices

Oldrich Kyn

Forschungsbericht No. 15, May 1974

Wiener Institut fur Internationale Wirtschaftsvergleiche

1. Introduction

It is a well-known problem that macroeconomic indicators calculated for different countries are not directly comparable if the original statistical data are expressed in different prices. This problem can be neglected only if there is a good reason to believe that the relative prices of both countries were reasonably similar. In such a case, it is sufficient to find a “correct” conversion rate between the two currencies.

Should the economic systems of the countries which are being compared, be known to contain very dissimilar mechanisms for price determination, adjustments for the differences in relative prices are also needed. It may be quite difficult to accomplish this task, however, if the actual differences in the relative prices are not known. Therefore, some short-cut method is usually applied. For example, one of the approaches sometimes used in comparisons of capitalist market economies with socialist centrally-managed economies, takes into account only differences between levels of consumer (retail) and producer (wholesale) prices and disregards differences in relative prices within each group. Jerzy Osiatinski 1) has shown that this short-cut method was quite frequently used in reaching conclusions about the price bias generated by the ”two-level price systems” in the East European countries.

He is perfectly right in maintaining that the differences in relative prices can change the picture considerably, but the empirical calculations to which he refers do not fully support his argument, and his theoretical model is not quite satisfactory.

Actually the situation is not as hopeless as he seems to believe. The artificial prices, computable from input-output models, can provide quite a solid ground for the comparison of otherwise incomparable economic indicators. Specifically, they can help to assess the influence of alternative rules for price setting.

In spite of all the difficulties involved, it can be demonstrated that the two-level price system is likely to produce a one-sided bias in certain macroeconomic indicators. It leads particularly to the undervaluation of material costs and of the share of investment in national income as well as to the overvaluation of the share of consumption. Osiatynski claims that such a bias is unidentifiable or even that it does not exist. But this is not true. It is, however, true that the bias may not influence all indicators equally and that it may be negligible in some of them. In a few, rather rare cases, a very special coincidence of technological production, and consumption structures may result in a converse bias.

It will be shown in this paper that the differences generated by deviations in relative prices can also have a consistent and empirically identifiable impact on macroeconomic indicators. The direction of the bias generated by these differences depends on properties of the particular technological matrices, and cannot be assessed purely on the grounds of the theoretical model.

Of course, under special conditions it can happen that both of the above-mentioned influences assume opposite directions for some indicators so that they partly or fully compensate each other. This is probably what happened with the share of investment in the quoted empirical studies; the undervaluation resulting from the two-level price system became partially compensated by the overvaluation resulting from “distorted” relative prices. In this particular case, it was true that the transition from the actual two-level prices to the calculated one-level “prices of production” would not lead to any radical change in the share of investment in the national income. This conclusion is, however, conditional. It cannot be used as an automatic generalization for all countries, all stages of technological development, and all other macroeconomic indicators.

The empirical calculations used by Osiatynski also show a considerable price bias in other important macroeconomic indicators - particularly in the share of total material costs in gross social product and in the share of wages in national income.

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) l_{C(I
- A) -1} of the matrix C(I - A)^-1.The rate of surplus value m is then determined by:

(7) m = (1/ l_{C(I
- A) -1} ) - 1

from which it follows that m > 0 if and only if 0 < l_{C(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) l 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

where
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 > 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).

Table 1 Basic Price Models Calculated for Czechoslovakia

(Indices of Calculated Prices to Wholesale Prices of 1966)

		Labor- Value Prices	Cost Prices	Production Prices	Capital- Value Prices	Material- Value Prices
	Sector	m=1.711	n=.286	r=.228	*r=.42**	*n=535**
1.	Electric Power	144.8	149.3	203.0	222.9	145.1
2.	Coal and Oil	176.7	152.3	173.3	158.4	125.3
3.	Metallurgy	152.1	175.7	166.4	167.7	198.6
4.	Chemicals	167.7	194.4	172.1	169.0	221.6
5.	Engineering	176.2	174.8	157.6	141.9	176.4
6.	Consumer Goods	176.0	172.0	145.7	124.9	167.9
7.	Building Materials	186.6	164.5	169.4	143.0	141.6
8.	Food	122.9	144.0	149.3	176.4	169.8
9.	Other Ind.Prod.	109.1	119.9	226.8	280.8	126.2
10.	Construction	185.8	155.1	127.0	86.4	125.3
11.	Agriculture	98.6	89.8	137.1	165.1	81.3
12.	Services	170.9	122.0	123.2	88.8	76.5
13.	Foreign Trade	170.0	205.8	149.6	137.2	243.45

Source: B. Sekerka, O. Kyn, L.Hejl (12.), p.196.

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.

Table 2 Belkin's Calculations of Production Prices for the USSR and the USA

Indices of Calculated Prices to Real Producer Prices)

	Sector	USSR 1966	USA 1947	USA 1958
1.	Metallurgy	152	119.6	109.8
2.	Coal	191	104.9	113.3
3.	Oil Mining	—	125.8	122.0
4.	Oil Processing	163	126.9	116.8
5.	Electric Power	160	140.8	128.0
6.	Engineering	117	102.7	104.1
7.	Chemicals	127	108.9	102.0
8.	Lumber and Wood Products	125	l0l.5	105.7
9.	Building Materials	134	108.3	104.1
10.	Light Industry	77	96.1	103.7
11.	Food	120	l00.2	95.9
12.	Construction	-	95.4	94.4
13.	Agriculture		l00.6	93.1
14.	Transport	127	151.1	118.6
15.	Servces	283	—	—
16.	Trade & Misc.	-	90.8	93.8

Source: Belkin (2) pp.113,127.

3. Extensions of a Price Model

It follows from the preceding section that comparability of East European statistical data with Western data can be improved if they are recalculated into artificially computed “production prices”. Of course, a total elimination of the price bias cannot be expected, but even a small improvement would be valuable.

The reevaluation of East European statistics by prices comparable to market prices is not the only feasible way. It is also possible to recalculate Western statistics into prices comparable with the East European price systems, or to recalculate both sets of statistics into artificial prices. In all of those cases, international comparability can be improved and some very interesting information can be gained.

Let us first increase the number of price models by two. These two models are totally artificial and inapplicable to any real economy, nevertheless, they are theoretically interesting and useful. They correspond, in a certain way, to “labor-value prices”, but reflect the dependence of prices on capital and material inputs respectively, rather than on labor inputs. We can, therefore, label them “capital-value prices” and “material—value prices” 6).

“Labor-value prices” (la) cover material costs and distribute value-added in proportion to direct wage costs. Let us, therefore, construct “capital-value prices” as prices covering material costs and distributing value-added in proportion to capital stock tied in the production of each commodity.

“Capital-Value Prices”

(14a) p’ = p'A + r* p'B

(14b) p'b = 1.

The subsistence wage equation p'c = w is redundant in this case, because wage costs do not enter the price equations. The numeraire equation (14b), as before, serves only to fix price level and has no effect on relative prices. If the system is productive, (i.e. if the dominant characteristic root of the matrix A is less than one) , as originally assumed, then (14a) can be rearranged to read:

(15a) p' = r* p'B(I - A)^-1

The matrix B(I - A) ^-1 is semipositive, therefore, there exists a semipositive vector p, associated with the dominant characteristic root l _{B(I - A) -1} of the matrix B(I - A) ^-1 The parameter r* is determined by:

(15b) r* = 1/ l _{B(I - A) -1}

If the matrix B (I-A) -1 is indecomposable, then uniqueness and strict positivity of p is guaranteed. In a similar way, we can construct “material-value prices” by distributing value-added proportionally to material costs.

“Material—Value Prices”

(16a) p' = p'A + n* p'A,

(16b) p'b = 1

No subsistence wage equation is needed in this case either, and the numeraire (16b) has no impact on relative prices.

A simple rearrangement of (16a) gives:

(17) p' = (1 + n*) p'A,

We can immediately see from (17) that p' is semipositive (strictly positive if A is indecomposable) and

(18) n* = 1/ l_A

where l_A is the dominant characteristic root of the matrix A. Because l_A < 1 by assumption, n* is always positive.

What is the purpose of introducing these two models of purely artificial prices ? First of all, it can be demonstrated that “production prices” are an “intermediate case” between “labor-value prices” and “capital-value prices”. Certainly it is clear from (10a) and (11) that

(a) if real wage is raised enough to make the dominant characteristic root of the matrix A + C equal to 1, the rate of profit will fall to 0, and “production prices” will become “labor-value prices”;

(b) If, on the contrary, real wage is pressed down to 0 the rate of profit will reach r* and “production prices” will be transformed into “capital-value prices”.

Reasoning along these same lines shows that “cost-prices” (2a) are a certain intermediate case between “labor—value prices” and “material-value prices”.

Still more interesting is the property of artificial prices to measure various macroeconomic indicators without bias.

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 >= 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* = l_A /(1 - l_A) = l_{A(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 = l_A /(1 - l_A) = l_{A(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 l 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) -1}x

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_{B(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.

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 + dp'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/l_C(I-A)-1

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

(47) d_max = 1/l_C(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 l_C(I-A)-1)

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

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

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

(53) g* = 1/ (2 - l_C(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* = l_B(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* = l_A(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 + n) (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/l_C(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 = l_A(I-A)-1

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

(64c) d = d_max = (1/ l_C(I-A)-1) - 1 implies a* = a** = l_A(I-A)-1 /(2 -l_C(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 l_C(I-A)-1 = 1 implies a** = a. An increase in d from its lower limit d = 0 to its upper limit d_max= (1/ l_C(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 = l_B(I-A)-1

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

(65c) d = d_max = (1/ l_C(I-A)-1) -1 implies b*=b** = l_B_(I-A)-1 /(2 -l_C(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 = l_C(I-A)-1

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

(66c) d = d_max = (1/ l_C(I-A)-1) -1 implies g* = g** = 1 /(2 -l_C(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 p'B

and if: d < (1/ l_C(I-A)-1 ) - 1

then:

(69) p' = rp'B[ I - 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/l_C(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.

6. The Impact of Changes in Relative Prices

The changes in relative prices can be studied apart from the “two-levelness” of prices, if we introduce two new price systems which imply the same relative prices as two-level “cost prices” or two-level “production prices”, but do not contain turnover tax.

Such systems are known as “two-channel prices”8). Let us define them:

“M - Two Channel Prices”

(70a) p' = (1 + n) p'A + (1 + m)w l',

(70b) p'c = w

(70c) p'b = 1.

“C - Two-Channel Prices”

(71a) p' = p’A + (1 + m )w1' + r p'B,

(71b) p'c = w

(71c) p'b = 1.

As before, we can proceed to:

(72) p' = (1 + n)p'[A + (1 + m)C],

and:

(73) p' = r p'B[I - A - (1 + m)C ]-1

A comparison of (72) and (73) with (62) and (69) shows that relative “M-two channel prices” are equal to relative two level “cost prices” and that relative “C-two channel prices” are equal to relative two-level “production prices”, provided m = d.

If we assume:

(74) n > 0, m > 0, r > 0

then, the following upper limits can be derived for n, m and r:

(75) n < = 1/l_{A+C - 1} , m < = 1/l_C(I-A)-1- 1 , r < = 1/l_B(I-A)-1.

When m = 0 “M-two channel prices” become “cost prices”, and when n = 0 they become “labor-value prices”. The gradual increase in m from 0 to its upper limit (75) will result in a gradual decline in n from its upper limit to 0. Relative prices will be also gradually changed. The direction and the degree of changes in relative prices will depend on the particular coincidence of matrices A and C. If these matrices are known, the whole spectrum of “M-two channel prices” can be empirically computed by solving the equation (72) with various values of the parameter m. It is important to stress that the gradual increase in m would not cause any random shifts in the price vector; they would rather result in very systematic and predictable changes.

The same can be said about “C-two-channel prices” (73). The gradual increase in m from 0 to its upper limit, will cause a gradual decline in r from its upper limit to 0, and also smooth and predictable changes in relative prices from “production prices” to “labor-value prices”. The direction and the degree of these changes depend on the particular coincidence of matrices A, C and B.

Clearly, if the “physical content” of a certain macroeconomic indicator is given, then the changes in its nominal value are explainable from the changes in parameters m, n, and r.

The concept of “two-channel prices” may be extended to include “capital value prices” and “material value prices”. In order to do that, it is sufficient to move the lower limit of m from 0 to -1:

(76) n >= 0, m >=-1, r >= 0

which implies the new upper limits for v and r

(77) b <= (1/l_A) - 1 , m <= (1/l_C(I-A)-1) - 1 , r <= 1/l_B(I-A)-1

The pushing of m down from 0 to -1, will cause a further increase in n towards its new upper limit, and, “M-two-channel prices” will be pushed beyond “cost prices” towards “material value prices”.

A similar effect is achieved in “C-two channel prices” by depressing m down to -1.

Diagrams 1 and 2 show the effect of changing parameters m, n and r on both types of two-channel prices. It is based on Czechoslovak price calculations for 1966.9)

Diagram 1.

Dependence of Relative Prices on m

(M-Two Channel Prices)

Source : B. Sekerka, O. Kyn, L. Hejl (12), p. 93, 94

Note: Nos 1 to 13 denote sectors as given in Table 1.

am 2.

Dependence of Relative Prices on r

(C-Two Channel Prices)

Source : B. Sekerka, O. Kyn, L. Hejl (12), p. 93, 94

Note: Nos 1 to 13 denote sectors as given in Table 1.

It is obvious that the movement along the spectrum of two-channel prices brings about systematic and predictable changes in relative prices. It is not surprising, therefore, that such a movement results also in systematic and clearly explainable changes in the ratios: a , b, and g.

Table 3 shows the values of the following macroeconomic indicators for “C-two-channel prices”.

1, 2	parameters of the price model		m n
3	Gross Social Product	p'x	GSP
4	Material costs	p'Ax	MC
5	National Income	p'(I-A)x	NI
6	Total Wages	p'Cx	WAG
7	Total Profits (surplus)	p'(I-A-C)x	PROF
8	Total Capital Stock	p'Bx	CAP
9	Cost-Income ratio	p'Ax/ p'(I-A)x	a
10	Capital-Output ratio	p'Bx/ p'(I-A)x	b
11	Share of Wages in NI	p'Cx/p'(I-A)x	g
12	Average rate of profit	p'(I-A-C)x/ p'Bx	r

Table 3 Basic Macroeconomic Indicators

(Measured at “C-two-channel prices”)

1	2	3	4	5	6	7	8	9	10	11	12
m	n	GSP	MC	NI	WAG	PROF	CAP	a	b	g	r
0	1.71	585.8	332.1	253.7	93.6	160.1	743.1	1.309	2.929	.369	.215
.05	1.28	583.3	334.6	248.7	93.6	155.1	712.3	1.345	2.864	.376	.218
.l0	.88	580.4	336.5	243.9	93.6	150.3	681.5	1.380	2.794	.384	.220
.15	.51	576.9	337.7	239.2	93.6	145.6	652.0	1.412	2.726	.391	.223
.20	.18	572.9	338.3	234.6	93.6	141.0	623.2	1.442	2.656	.399	.226
.23	.00	570.5	338.4	232.1	93.6	138.5	607.5	1.458	2.618	.403	.228
.25	-.13	568.4	338.3	230.1	93.6	136.5	595.5	1.470	2.588	.407	.229
.30	-.41	563.3	337.6	225.7	93.6	132.1	568.9	1.496	2.521	.415	.232
.35	-.67	557.6	336.3	221.3	93.6	127.7	543.0	1.520	2.454	.423	.235
.40	-.90	551.4	334.4	217.0	93.6	123.4	517.9	1.541	2.387	.431	.238
.42	-.98	548.7	333.4	215.3	93.6	121.7	507.9	1.549	2.359	.435	.240
.43	- 1.00	548.1	333.2	214.9	93.6	121.3	505.6	1.550	2.353	.436	.241

Let us now return to the two - level price systems. The previous exposition laid the groundwork for a decomposition of the price bias into two parts:
(i) an effect of changed relative prices and
(ii) an effect of differences in the levels of wholesale and retail prices.

Suppose, for example, that we want to estimate the impact of turnover tax with the rate levied on “production prices”.

We shall use the following notations

a^o, b^o, g^o - the respective ratios measured at one-level “production prices”

a*,b*, g* - the ratios measured at one-level “C-two-channel prices” with m = d*

a**, b**, g** - the ratios measured at two-level “production prices”.

The imposition of turnover tax d* on “production prices" does change relative prices because the uniform rate of profit in the two - level system, is smaller than r in the one-level system. The rates r and r* are determined in the following manner:

(78) r = 1/ l _B(I-A-C)-1 r* = 1/ l_{B[I-A-(1+d*)C]-1}

The formula (73) can be used to calculate “C-two-channel prices” corresponding to the parameters r = r* and m = r* . Relative prices of such a price system are the same as if the tax d* were imposed, however, there is no difference between the levels of wholesale and retail prices. The differences between a* , b* and g* measured at above-described prices and a^o , b^o and g^o measured at original “production prices”, represent the price bias caused purely by changes in relative prices.

Once the bias resulting from changed relative prices is determined, the rest of the bias is very easy to find. From (41), (42) and (43), it follows directly that:

(79a) a** = a* /(1 + d*g* )

(79b) b** = b* /(1 + d*g* )

(79c) g** = (1 + d*) g* /(1 + d*g* )

The differences between just defined a**, b**, g** and the previously obtained parameters a*, b*, g*, represent the second part of the price bias. This part has its origin purely in the “two-levelness” after elimination of the “relative price” effect.

If d* > 0 then a** - a* < 0, b** - b* < 0 and g** - g* > 0. It is, therefore, obvious that the “two-levelness” alone causes always undervaluation of a and b and overvaluation of g. This effect resulting from the “two-levelness” of prices can be either increased or diminished by the “relative-price” effect.

This will depend on the values and signs of differences a* - a^o , b* - b^o , and g* - g^o .

Let us illustrate the decomposition of the price bias using the data from the Table 3. The values of parameters a , b and g measured at one level “production prices” (m = 0) are:

a^o = 1.458 , b^o = 2.618 , g^o = .403.

Suppose that the turnover tax with the rate d* =.511 is levied on consumer goods. The “relative-price" effect of such turnover tax can be read directly from the Table 3. We find a* , b* and g* as the value of parameters a , b and g measured at one-level “C-two-channel prices” with the parameter m equal to .511:

a* = 1.412 , b* = 2.726 , g* = .391

The “relative-price” effect leads to slight under valuation of a and b and to overvaluation of g . The values a** , b** and g** corresponding to two-level “production prices” with turnover tax d* = .511, can be now calculated from (79a,b,c):

a** = 1.177 , b** = 2.272 , g** = .492

The pure “two-level” effect is much stronger than the “relative-price” effect and causes considerable under valuation of a and b and overvaluation of g.

Table 4 summarizes the decomposition of the price bias for all three parameters.

Table 4.

		a		b		g
values of parameters	at one level production prices	a^o	1.458	b^o	2.618	g^o	.403
	at one level C-two-channel prices	a^*	1.412	b^*	2.726	g^*	.391
	at two level production prices	a^**	1.177	b^**	2.272	g^**	.492
price bias	Relative-price effect	*a^- a^o**	-.046	*b^- b^o**	.108	*g^- g^o**	-.012
	pure two-level effect	a^*- a^	-.235	b^*- b^	-.454	g^*- g^	.101
	total bias	a^- a**^o	-.281	b^- b**^o	-.346	g^- g**^o	.089

7. Conclusions

The main purpose of this paper was to disprove the assertion of Jerzy Osiatynski that it cannot be theoretically proven that the two-level system of prices used in East European countries, creates a systematic price bias in the macroeconomic indicators. Osiatynski correctly pointed out that the problem is much more complicated than it is usually believed. The main complication results from the fact that the imposition of turnover tax influences not only the level of prices, but also relative prices. Nevertheless, he was wrong in claiming that the change in relative prices is absolutely unpredictable and that there is an equal chance for each indicator to be biased upward or downward - except in the case of “labor-value prices”.

This paper has shown that:

(i) the price bias can be easily detected and measured not only at “labor-value prices”, but also at "material- value prices” and at “capital-value prices”;

(ii) the likelihood of the price bias can also be established for “production prices” and for “cost prices”, if measured on specially standardized output vectors;

(iii) the extent and direction of price bias can be empirically measured by decomposing the bias into two parts: the part which is due to the changes in relative prices, and the part which is due to the differences in the levels of retail and wholesale prices.

In section (4) of this paper, it was also shown that certain macroeconomic indicators can be made internationally comparable without recalculating statistics of the involved countries into the same price system. It is sufficient to measure the indicators at their own artificially computed price vectors or on the specially standardized output vectors. It was also suggested that the dominant characteristic roots of certain matrices may serve as the best unbiased measures of basic macroeconomic indicators.

	References
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3.	Brody, A.: Three types of Price Systems, economics of Planning 5, No. 3, 1965.
4.	Brody, A: Proportions, Prices and Planning, North-Holland Publishing Co., Amsterdam 1970.
5.	Carter,A.P. Brody, A.(eds): Contributions to Input-Output Analysis North-Holland Publishing Co. 1969.
6.	Costa, A.M.: The Structure of Development in the Soviet Union (manuscript) , 1973.
7.	Csikos-Nagy,B. Gancer, S., Racz, J. , Az elsö termerkrend szerü armodell, Kozgazdasagi Szemle,No.1, 1964 (pp. 17-35).
8.	Dmitriev,V.K.: Ekonomicheskiye ocherki, V.Richter, Moscow, 1904.
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FOOTNOTES

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Jerzy Osiatyñski: On the Price-Bias in Comparative Analysis of Planned and Market Economies. Forschungsberichte Nr. 13 Wiener Institut für Internationale Wirtschaftsvergleiche;

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See the selected bibliography at the end of this paper.

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In this paper it is assumed that only one homogenous type of labor exists. In the more general case, it is possible to operate with the matrix of labor-output coefficients. L, the elements lrj of which represent the quantity of the r-th type of labor needed per unit of j-th output. If “reduction coefficients” hr representing the degree of complexity of r-th type of labor are given, then the vector l (l' = h'L)can be regarded as the vector of “homogenized” labor-output coefficients. The attempts to correctly establish coefficients hr may create serious difficulties. Probably, the easiest solution would be to determine vector h as:

h = (1/ ws)w

where w is the vector of wage rates and ws is the “basic wage rate” or the wage rate for simple labor.

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The coefficients cij of the matrix C, represent the quantity of consumer good i, which must be supplied to workers in the j-th industry per unit of product j. In definition (5), it is implicitly assumed that both the level and the structure of consumption in all industries are identical. This restrictive assumption can be relaxed without impairing on the validity of price models.

Let D be a matrix with elements dir representing consumption of consumer good i per unit of the r-th type of labor. Then the matrix C can be redefined as:

C = DL

where L is the matrix of labor-output coefficients which was defined in the Footnote (3).

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Throughout this paper the dominant characteristic roots are denoted by the symbol lx with the subscript x standing here for the particular matrix.

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In Hejl, Kyn, Sekerka (11) and (12), these prices are called “F-income prices” and “N-income prices”.

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Sraffa (22) calls it “standard commodity”.

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By a “M-Two-Channel Price”, we mean a price which keeps one channel proportionate to material costs and the other proportionate to wage costs. By a “C-Two-Channel Price”, we mean a price which keeps one channel proportionate to wage costs and the other proportionate to capital (11) p.110.

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See Sekerka, Kyn, Hejl (12)