The deviation of production prices from labour values

Cambridge Journal of Economics 1987,11, 197-210

The deviation of production prices from labour values: some methodology and empirical evidence Pavle Petrovic*

•University of Belgrade. I am grateful to Dj. Suvakovic and anonymous referees for many comments and suggestions. I have also benefited from the comments by M. Landesmann, L. L. Pasinetti and B. Schefold. The usual disclaimer applies. 1 See various attempts to solve the so-called transformation problem. Pasinetti (1977) gives some references. 0309-166X/87/030197+14S03.00/0

C 1987 Academic Press Limited

Downloaded from cje.oxfordjournals.org at University of Cambridge on April 23, 2011

Introduction The empirical proposition that relative prices of commodities are predominantly determined by the relative quantities of labour time required for their production was put forward by Ricardo. He stated that a 1 % fall in profits would change relative production prices by only 1 % (Works, I, p. 36) and, as the change in profits does not usually exceed 6 or 7%, Ricardo ended up with the '93% labour theory of value' (Stigler, 1958). Being an empirical proposition, it calls for evidence which will support or refute it. There is hardly any empirical work dealing directly with the extent of discrepancies between relative production prices and relative labour values, the one exception being Shaikh (1984), while conclusions drawn, indirectly, from some other empirical work suggest huge deviations (Burmeister, 1984). Large deviations are also suggested by numerical examples derived from a two-commodity hypothetical model (Barkai, 1967). The aim of this paper is to provide evidence on deviations of production prices from labour values in an actual economy. Besides testing Ricardo's 1% rule, the results obtained can be also used to assess the empirical significance of Marx's proposition that total profit equals total surplus value, even though this is known to be not true in general.1 The paper consists of three parts. In the first part various linear price models are defined, and a number of relations which will be used in the empirical investigation are derived. The second part, the main one, deals with empirical results concerning deviations between relative prices and labour-value ratios. Ricardo's 1% rule is tested, employing first the value of total output as the unit of measurement and then the prices, one for each of 47 sectors that are considered. As the deviations obtained have varied when different numeraires were used, the question of which one to choose arises. Following Ricardian tradition one would look for an 'invariable standard of value' as the unit of measurement.

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Samuelson (1983) derived such a standard for the model we were using, so we calculated and employed it as a numeraire commodity. An alternative solution, and in our view the proper one, is to look at all possible exchange ratios i.e. 1081 in the 47 sectors model we have been considering. We then examine the generality of the results, which refer to the Yugoslav economy in 1976 and 1978. Alternative estimates of the deviations between prices and values based on procedures used by Shaikh (1984), are also reconsidered. Finally, the third part examines the effects of deviations between production prices and labour value ratios on some aggregates, particularly on the relation between total profit and surplus value. 1. The price models The model used for the empirical analysis is an input-output model with fixed capital, the framework that has often been employed for price calculations.1 The following expressions will be used (with the notation explained below): (1)

P° = P°(A + D + G) + tvL + rP°B

(2)

m

P» = P (A +D+G) 1

+ r^P-B

P = P\A +D + G) + tvL + P'Br

(3) (4)

1

P - = kL

(5)

We are primarily interested in the relation between labour-value ratios or value prices (1), that is prices proportional to labour embodied,2 and production prices (2). We shall also consider how the distance between the value prices and production prices changes when the profit rate increases from the actual level to the maximal one; thus, production prices with the profit rates above the actual one will be calculated, including the case in which the profit rate is equal to its maximum (3). Further, instead of a uniform profit rate one can use different rates of profit across sectors arising from different risks in production (Krelle, 1977) or from non-proportional growth (Pasinetti, 1981). For the price calculations, the latter approach is adopted, so that profit rates across sectors are proportional to sectoral long-run growth rates (4). Finally, the prices that are proportional to current labour are defined (5) for comparison with value prices, that is prices that are proportional to both current and past labour. The individual symbols have the following meaning: V = (v,), P° = (p°), P- = U>7), P1 = (/>,'), / * = (Pf) and Pa = (p°), i = 1,. . .,» are the row vectors of the following prices: value prices, production prices, prices with maximal profit rate, prices with different profit rates across sectors, prices proportional to current labour and actual prices; •£- = ('j)j i = l , . . -,n is the row vector of labour coefficients where /, denotes the input of labour in man-years required per unit of production in sector i;3 A = (atJ), i,j— 1,. • -Jt is ' See, for example, Brody, 1970; Kyn et al., 1970; Yefimov and Movshovitch, 1973; Carter, 1970 and Fink, 1981. 2 It can be shown that the value prices denned by (1) are proportional to labour values. 1 Reduction of heterogeneous labour to unskilled worker equivalents is performed by using weights which are ratios between the average net wage rate for a given qualification and the average net wage rate for unskilled workers.


V = V(A + D + G) + (1 + s) wL

Production prices and labour values

199

the square matrix of technical coefficients where atj denotes the amount of ith commodity used per unit of output in sector j ; D = {dtj), i,j= 1 , . . . , n is the square matrix of depreciation-output coefficients where dtj denotes the annual depreciation of the ith capital good in thej'th sector; G=gT%~l is the square matrix ofgovernment consumption, where column vector^ = (£,), i = 1 , . . .ji denotes the government consumption basket normalised such that Pa g= 1; T=(t[),i=l,.. .,n is the row vector where t, denotes the number ofgovernment consumption baskets paid, through taxes, by sector i, while JC=(xt), i= 1,...,« is the diagonal matrix where x, denotes the output of the ith sector; X=(xt), i = 1,. ..,«is the corresponding column vector; A = L(I — A — D — G)~ ' is the row vector A = (A,), i = 1 , . . .,« where kj is the labour value of commodity i; it is also known as the vertically integrated labour coefficient (see Pasinetti, 1973);

H = B(I—A — D — G)~i = (hij) i,j=\,.. .,n is the square matrix of integrated capital coefficients where ht] gives the amount of good i tied up directly and indirectly (in other sectors) per unit of output in sector./ (see Pasinetti, 1973); f = {rl), i = 1 , . . .,« is the diagonal matrix where r, is the profit rate of the ith sector. Profit rates across sectors are proportional to the sectoral long-run growth rates, i.e. r = n1v where fi2 is a scalar and v = (v(), i= 1 , . . .,n diagonal matrix with growth rates v, on the main diagonal; s, w, r, rnai and k are the following scalars: rate of surplus value, wage rate, maximum profit rate and mark-up. Solutions of equations (l)-(4) turn out to be positive characteristic vectors: V, P°, P", and Pl with characteristic roots: 1/1 +s, \\r, 1/r,^,, and l///2 respectively. Combining expressions (1) and (2) one can obtain the relation between relative prices of production and relative value prices.1

1-

P°J »i v

l

1 -

r P°h, w k{ rP0/:, TU

(6)

k.

where column vectors ht and hj are the vectors of integrated capital coefficients while scalars /., and ?.) are integrated labour coefficients for commodities i and j respectively. It follows from (6) that deviation between the relative prices considered is due to the difference in corresponding integrated capital-labour ratios, that is, the fact that 1

A similar result is obtained by Shaikh (1984).


B = (bi]),ixj= 1,. . .,nis the square matrix of capital coefficients where b(j gives the amount of the j'th commodity tied up per unit to output in sector^'. It includes both fixed and inventory capital i.e. the matrix B = K+In where K=(kij) and /M = (in,y), ij,= \,.. .ji are the square matrices of fixed capital coefficients and inventory capital coefficients respectively;

200

P. Petrovi* p°h,

p°hj

where the total tied up capital P°h, is aggregated at production prices.1 Accordingly, when direct capital labour ratios are uniform across sectors (and therefore integrated ones as well) there will be no discrepancies between value and production prices (see also Samuelson, 1983). An alternative invariance condition to secure V=P° = P" can be derived following the corresponding proof in the circulating capital model (Pasinetti, 1977, p. 79): P" = rMIP"B(/ - A - D - G)~' V = (1 + s) wL (I - A - D - G)~1 therefore

V = /"" implies +s)wL(I-A-D-Gy1B(I-A-D-G)-i

(1 + s) wL (I - A - D - G)~ • = !•„,„( 1

and multiplying by (/ — A — D — G) one gets: (7)

The condition obtained (7) differs from the one in circulating capital model i.e. PL= V; therefore small discrepancies between production and value prices, in the fixed capital model, do not imply that value prices are close to labour coefficients. The choice of numeraire commodity affects the size of the deviation of production prices from value prices so the distance between them will vary with various numeraire commodities. One way to deal with this problem is by finding an 'invariable standard of value', that is a composite commodity the value of which will remain unchanged with variations in profit rate, and use it as a numeraire. S raff a (1960) defined such a (standard) commodity for the circulating capital model andfixedcapital model with joint production, while Samuelson (1983) defined it for the durable capital model we are using. In the latter case, a standard commodity turns out to be the right-hand characteristic vector X* of the matrix H? rmHX* = X*

(8)

We can now proceed to Ricardo's 1% rule. In his example {Works, I, pp. 33-36) he calculates the change in price of commodity i (cloth) relative to that of commodity 7 (corn) caused by a fall of profit rate from 10% to 9% as: pflp9. 5995/5500 ' ' - 1= ' w lO n p ,lp j 6050/5500

1 = - 0009

that is, approximately 1 %. As we are going to compare value and production prices where corresponding profit rates are equal to zero and r% respectively, Ricardo's 1% rule corresponds to: - 1

(9)

' The result that differences in integrated capital—labour ratios is what matters has also been shown in the circulating capital model by Parys (1982). 2 Samuelson (1983) gives an alternative proof of this result and calls it the key theorem. 3 One should note that in our calculations depreciation coefficients dy are fixed, i.e. independent of r, so that the problem of existence of a standard commodity, put forward by Samuelson (1983), has been avoided.


L = r^AS.2


201

The expressions we have derived so far contain the coefficients given in physical terms, while the actual data are denominated in current prices. It is well known that when proceeding from actual input-output data one can only derive ratios of value prices and production prices to actual prices' e.g.

vt

vjVX

. = 1,. . .,n

and

ft = p^PX From (10) and (11) the deviation of production prices from value prices can be determined: _p°!P°X 1

'•••'"

(12)

v, The last expression will be used to test Ricardo's 1% rule (see 9) when the numeraire is total output. In the same manner one can obtain production price-value price deviations when the standard commodity X* is used as the numeraire:

P°IP°X* . _ vjVX* ' ' - 1 ' " - "

(13)

Since deviations of production prices from value prices are affected by the choice of unit of measurement, we need to obtain relative prices for different numeraires. This can be done by combining the results we have already obtained, that is (10) and (11): P°IP°X IP?IP°X =P ?/p° . = i vjVXl vjVX »>,'' '"""

(14)

Expression (14) gives these deviations taking the first commodity as numeraire; the same procedure can be applied when other commodities are taken as numeraires. As has been pointed out, it is possible to derive various price ratios (10-14), but not the relative prices themselves, for example V^VJVXOT p°=p IP°X, i = 1,.. .,n. A possible way to get rid of price ratios is to obtain seaors' outputs evaluated in terms of an appropriate price system. For production prices the procedure is as follows:

P°X = Ip,x, = 1.

(15)

In the same manner one can obtain t),x,,p™x( and so on, for all other types of prices. An alternative expression for (6) can now be derived (see also Shaikh, 1984). r ftov Pixi

P°X 1

P°htXl

ff*/ * + — ; — \

tO AtX

( V

r P°HX\ to AX )

Cf. Kyn ei al. 1970; Fink 1981. The result can be provided for our model as well.

(16)


p°

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P. Petrovifi

Setting P°X = 1 and VX= 1 it is possible to obtain a log-linear relation between output across sectors evaluated at production and value prices: lnp°x, = lnf.x, — lnz,

(17)

where z, =

r P°HX w AX The data used are from the Yugoslav economy in 1976 and 1978, and 47 sectors are considered.1 1+

2. Deviations of relative prices from labour—value ratios

Value of total output as a numeraire Deviations between relative prices and labour-value ratios measured by the RMS°oEs are given in Table 1. Table 1.

1976

RMS% E 1978 RMS% E

P>

pOilv

p02lv

p-/v

pLlv

r = 4-10% 6-08% r = 519%

r= 10-07 14-83 r=1007 1266

r=1503 22-24 r= 15-04 18-77

•„,, = 24-03 38-53 • „ , = 28-56 40-46

4610

6-62%

pl/v

r, = 0-694v, 7-47 r, = 0-873v, 4415 8-68

pllp°

3-69 4-45

The distance between production prices and value prices increases with increases in the profit rate: RMS°OE approximately doubles when the profit rate is doubled. Thus deviations increase considerably when the profit rate moves from its actual values (4-1 and 1

An explanation of the data used as well as the empirical results that are not reported are available to readers on request. 1 A suitable measure of distance between two vector of prices is root-mean-square-per-cent-error (RMS° 0 E). When production and value prices are considered, it is equal to:

where s denotes the standard of value. In order to calculate it one needs only price ratios [(p°/u),] and that is all that can be empirically determined. For definitions of various measures see Pindyck and Rubinfeld (1983, pp. 362-365).


The empirical results obtained enable us to assess the extent of discrepancies between various relative prices and labour-value ratios. To that end we use the root-mean-squarepercent-error (RMS%E)2 as well as Ricardo's l°/0 rule. As the deviations change with the numeraire, we shall first review empirical results when total output is taken as the unit of measurement [VX = P°X= 1], and afterwards those for others numeraires.


203

519%) to 10% and 15%, and particularly when it reaches maximum values (2403% and 28-56%). RMS%Es (38-53 and 40-46) are then about six times larger than those corresponding to production price-value price deviations (608 and 6-62%). Prices with different profit rates across sectors (P1) exhibit somewhat larger deviations from value prices than production prices do, and are quite close to production prices. Prices proportional to direct labour coefficients (P1) are calculated to see whether they are a good approximation for value prices and, via them, for other prices, especially actual ones. It turns out that these prices exhibit the largest deviations from value prices, so it would be misleading to use labour coefficients in order to estimate value or actual prices (see Table 2). Comparisons of the results between the two years show that the discrepancies from value prices, taking into account equal profit rates, are somewhat smaller in 1978 than in 1976. Deviations of various types of calculated prices from actual prices can be assessed by the values of RMS%E given in Table 2.

1976 1978

v\p°

P°If

PO1/P°

P02 If

P" If

f If

PLlf

11-84% 11 80

11• 1 1 11•01

14-46 12-87

19 67 16- 53

34 •80 36 •45

10•66 10 15

48-81 47-74

The smallest deviations occur for production prices calculated on the basis of profit rates across sectors varying with long-run growth rates, followed by production prices and value prices, although all three stand at almost the same distance from actual prices. On the other hand, prices proportional to labour coefficients deviate the most from actual prices, a result we have already commented on. Let us now examine more closely production price-value price deviations by applying Ricardo's 1% rule. This states that a 1% change in profit rate will cause a change in relative prices of at most 1%. In the same manner we could speak of a 2% rule when relative prices change by up to 2%. By applying equation (9) across sectors it is possible to discover the number of sectors in which the deviations of production prices from value prices is less than the profit rate (or twice the profit rate). Results are given in Table 3. Table 3. l%rule

2% rule

1976 Number of sectors ° 0 of total number °-0 share in total output

X= 1), the sum of profits differs from the total surplus value. This, of course, brings into question Marx's explanation of the origin of profit. By comparing these aggregates, we can see whether Marx's proposition survives at an empirical level.2 The calculated price ratios enable us to determine total profit and total surplus value evaluated at the corresponding prices, and the ratios between latter and the former are: 1976:1-036; 1978: 1055. As one can see, these deviations are quite small. The second well-known objection to Marx's value analysis refers to his determination of the average profit rate as the ratio between total surplus value and capital denominated in value prices, instead of the ratio between total profit and capital denominated in production prices (r). As can now be expected, the empirical deviations, given in Table 8, are very small:


209

1979, p. 335, Table 2). The ratios obtained are somewhat higher than ours but still indicate moderate discrepancies. By employing an indirect procedure in relation to the US economy Shaikh has estimated that the ratio between surplus value and actual profit is equal to 1064 and the ratio between Marx's profit rate and actual rate is 107 (Shaikh, 1984, pp. 57 and 58). The indirect procedure employed assumes, among other things, a circulating capital model. Shaikh suggested that the inclusion of fixed capital would reduce these deviations. The values of most of other aggregates are also hardly affected by the deviations of production prices from value prices. For example, the value of the consumption basket denominated in prices with maximal profit rate (P"c) and value prices (Vc), while VX=PmX= 1, will differ by 9-9% in 1976 and 6-4% in 1978. Conclusions Downloaded from cje.oxfordjournals.org at University of Cambridge on April 23, 2011

Contrary to what might be expected from indirect evidence and two-commodity hypothetical examples, the evidence we have provided supports Ricardo's empirical proposition that relative production prices are mainly determined by labour-value ratios. The results obtained indicate that a 1 % change in profit rate will cause most relative prices to change by less than 2%, while half of them will change by less than 1%. As actual profit rates are of the order of 5%, we end up with a '90% labour theory of value'. Let me stress that these results are independent of the choice of numeraire commodity, for we have considered the changes in all possible exchange ratios: 1081 in the 47-sector model. Certainly when various numeraire commodities are used the deviations observed have varied. They are somewhat smaller when total output and a standard commodity are used as numeraire commodites. These two composite commodites produce quite similar results concerning deviations. The evidence refers to an actual economy but the variations in integrated capital-labour ratios and in the profit rate—the factors that determine the deviations considered—turn out to be of the same order of magnitude as in other economies. The results obtained therefore have general significance. Our findings contradict those implied by other empirical evidence that stresses variations in capital-labour ratios. Large observed or expected variations in these ratios are put forward as an argument for expecting huge relative price-labour value deviations. But, as has been shown in this paper, widely differing capital-labour ratios do not significantly affect the deviations we are analysing. This is partly because integrated capitallabour ratios are what matter and these ratios are less variable than direct capital-labour ratios. We have also seen that an increase in the profit rate substantially increases the deviations considered between production prices and value prices so that, contrary to what stems from the procedure that employs correlation for the measure of deviations (Shaikh 1984, Wolf 1979), discrepancies between labour-value ratios and production prices with maximal profit rate are large. The aggregate effects of the deviations between production prices and labour—value ratios are small, so one can accept as empirically valid Marx's proposition that the sum of profits is equal to total surplus value. An important implication of the small deviation obtained between relative prices and labour values is that the price effect on the wage—profit curve will be insignificant, and the

210

P.Petrovifc

curve will be close to a straight line. Corresponding empirical evidence on the shape of wage-profit curve is, however, beyond the scope of this paper. Bibliography


Barkai, H. 1967. The Empirical Assumptions of Ricardo's 93 per cent Labour Theory of Value, Eamomica, NS vol. 34, no. 136, November Brody, A. 1970. Proportions, Prices and Planning, Budapest, Akademia Kiado Burmeister, E. 1984. Sraffa, labour theories of value, and die economics of real wage rate determination, Journal of Political Economy, 92, no. 3 Carter, A. 1970. Structural Change in the American Economy, Cambridge, Mass., Harvard University Press Fink, G. 1981. Price distortions in the Austrian and in the Hungarian economy, Zeitschrift fur Nationalokonomie, nos 1, 2 Krelle, W. 1977. Basic Facts in Capital Theory, Revue D'Economie Politique, no. 2 Kyn, O. et al. 1970. Price System Computable from Input-Output Coefficients, in Carter, A. and Brody, A. (eds), Input-Output Analysis, Amsterdam, North Holland Movshovitch, S. M. and Yefimov, M. N. 1973. Analiz sbalansirovanogo rasta v dinamicheskoj modeli narodnogo hozjajstva (An analysis of balanced growth in a dynamic model of the economy), Ekonomika i matematicheskie melody, no 1 Parys, W. 1982. The deviations of prices from labour values, American Economic Review, December Pasinetti, L. L. 1973. The notion of vertical integration in economic analysis, Metroeconomica, vol. 25 Pasinetti, L. L. 1977. Lectures on Theory of Production, London, Macmillan Pasinetti, L. L. 1981. Structural Change and Economic Growth, Cambridge, CUP Pindyck, R. and Rubinfeld, D. 1981. Econometric Models and Economic Forecast, New York, McGraw-Hill Company Ricardo,D. 1953. The Works and Correspondence of David Ricardo, vol. l,ed. P. Sraffa, Cambridge, CUP Samuelson, P. 1983. Durable capital inputs: conditions for price ratios to be invariant to profit-rate changes, Zeitschrift fur Nationalokonomie, vol. 43, no. 1 Shaikh, A. 1984. The Transformation from Marx to Sraffa, in E. Mandel and A. Freeman (eds), Ricardo, Marx, Sraffa, London, Verso Sraffa, P. 1960. Production of Commodities by Means of Commodities, Cambridge, CUP Stigler, G. 1958. Ricardo and the 93% Labour Theory of Value, American Economic Review, 48, June Wolf, E. 1979. The rate of surplus value, the organic composition, and the general rate of profit in the US economy, 1947-1967, American Economic Review, March