Paul Cockshott

I have been working on producing a table of capital stocks for the UK by industry and type of asset. The aim is to have the data required to set out hypothetical national economic plans for the UK. These would enable exploring things like how to restructure the economy to shift to low CO 2 emissions or how to eliminate the trade deficit. But having done the initial preparation of a capital matrix it becomes possible to use it for other purposes. One is to test price of production theory.
A substantial body of literature now exists on empirical tests of Marxian price theories. Researchers have performed comparisons between theoretical price structures derived from Marx’s Capital[8] and observed market prices[109112124515616]…. Many of these studies only use flows of constant capital shown in the IO table, rather than actual capital stocks for different industries. Where they do use capital stocks, these are just taken from money totals for capital rather than evaluating the constant capital itself in terms of labour values or prices of production. Given that one of the main points about the transformation problem debate has been whether Marx properly transformed capital inputs, this is somewhat of a weakness.
In the context of this literature the recent work of Han Cheng and Minqi Li[1] is a particularly careful and detailed contribution, in that it covers multiple time periods and attempts to use a very detailed modeling of prices of production. One of the interesting results that they report is that according to their methodology prices of production are significantly better at predicting actual prices in China than labour values are. This is an interesting result since most other studies had shown that prices of production and labour values were not much different in accounting for market prices.
It is therefore worth looking in detail at how their calculations are performed. Their equation for prices of production is
m=m( w β α +A+D+T ) + π m[ K+( w β α +A ) Γ ]
Their notational conventions are that m is the row vector of Marxian price of production coefficients for each economic sector, α is the row vector of direct labor input for each Yuan of gross output value, w is the average wage for one unit of simple labor, β is the column vector of household consumption distribution coefficients, T is the diagonal matrix of net producer tax coefficients, π is the Marxian uniform profit rate, K is the square matrix of fixed capital stock, and Γ is the diagonal matrix of turnover time.
Their work is unusual in that it attempts both to take into account taxes and to use a square matrix of capital stock. However there are some limitations to their treatment of these issues.


  1. Whilst they include data on taxes and subsidies when modelling prices of production, they leave these out in their labour value model. Whilst leaving taxes out of labour value calculations is not unusual, if one is to compare two value theories, the inclusion of tax data in second of the models and not the first, then that gives the second model a predictive advantage. Since neither Marx’s account of prices of production nor labour values include taxes, including taxes in your model of the former and not the latter, vitiates the procedure as a test of Marx’s theories.
  2. Whilst the inclusion of both depreciation via D and flows via A in the model are theoretically sound, it is questionable whether they are operationally so. The A matrix has to be derived, in practical studies, from the input output table. This table shows total commodity flows between sectors. It will therefore include some flows of the replacement machinery that compensates for depreciation. If one includes an additional imputed depreciation matrix there is a danger of introducing double counting.
  3. The construction of the K matrix is based on performing a vector outer product between the vector of gross fixed capital formation distribution with the vector of fixed capital stock coefficients. Whilst better than nothing, this is a relatively crude estimate. If there are n sectors this procedure produces a matrix with only 2n numbers of actual information, unlike the n 2 numbers that a full matrix would have. Since it gives all production processes the same fixed capital composition, it is only marginally better than simply using a fixed capital vector as other studies have done.
The last problem arises largely from the limited data available on the Chinese capital stock. It is doubtful that anything better could have been done given what was available to the authors. It would, however, be of interest to see what the result would be if one had a more detailed capital stock matrix available.

UK data

The UK Office of National Statistics(ONS) publish capital stock data annually that give net capital stocks for distinct combinations of 12 asset types and some 90 sectors. Some sectors are given in both aggregate and disaggregated form so the total number of independently specified sectors is slightly less. Whilst the sector names are not identical to those used in the IO table, industrial sector codes are provided so it is relatively easy to translate to the IO table sectors. The data for 2015 was used as this corresponds to the most recent year that a British input output table has been published.
The capital data from ONS are in relational rather than matrix form so an expanded stock matrix was created such that for each column in the original stock matrix for which several sub industries exist in the io table the capital values in the original are spread among the new multiple columns in proportion to their share in the final output of this group of industries. The final output matrix has column names in the same order as the input output table. The resulting intermediate stock matrix has rows with the asset types:
  • Dwellings
  • Other buildings and structures
  • Transport equipment
  • Computer hardware
  • Telecommunications equipment
  • ICT equipment
  • Cultivated biological resources
  • Research & development
  • Computer software and databases
  • Intellectual property products
  • Machinery, equipment and weapons systems
  • Other machinery, equipment and weapons systems
It is again relatively easy to identify the input output table industries producing these categories of goods. Using this and the intermediate capital stock matrix produced by the previous step, software was used to produce an expanded stock matrix with the same layout as the iotable such that


  • For each row in the intermediate stock matrix for which several source industries exist in the source index file these are mapped to IO table rows using Table 1.
  • Capital values in the original matrix are spread among the new multiple rows in proportion to the flows shown in the corresponding columns in the iotable.
  • The underlying assumption for this approach is that the flows shown in the io table are replacement for depreciation and will be proportional to the corresponding capital stocks.

Handling of tax and imports

The IO table contains rows for taxes on products and production and for the import content of each column listed. Since we are concerned to do an unbiased comparison between prices of production and since Marx’s Vol I and Vol III price theories ignore the effect of taxes we do not include them in the calculation. It should however be born in mind that in this and other studies of correspondence between labour values and market prices, the differential impact of taxes of industries will constitute a source of unaccounted for noise in the market prices.
Imports are dealt with by computing the labour content of £1 of exports and imputing the same labour content to each £1 of imports used by an industry.


Labour values and prices of production were computed via a Jacobi iterative procedure with 12 iterations. Two temporary vectors v,p are used. The vector v holds the labour content of each £ of output of the corresponding industry, p holds the production price per £ of output. Both vectors are initialised to zero.
On each iteration for each industry i the total labour content L i is computed by adding the direct labour λ i to U i T .v , that is to the total obtained by converting the £ costs in the i th column of the use matrix U into labour using v . Then v is updated by setting v i = L i F i where F is the final output vector in £.
An analogous procedure is used to update p .
In this case the total production cost P i for each industry is computed as
P i = λ i + U i T .p+r( K i T .p )
where K is the capital stock matrix with the same shape as U and r the rate of profit for the whole economy obtained by dividing the Gross Operating Surplus of the economy as a whole by the total capital stock of the economy.
At the end of each iteration P,L are re-normalised to ensure that their totals are equal to the total in £ of final output F.

Circulating capital

Circulating capital is ignored because it can readily seen that it will produce a very small effect on the final result. Consider the circulating capital supposedly ‘advanced’ as wages. In many sectors the turnover time of this will actually be negative – a worker in a restaurant for example is not paid till the end of the month, but the employer pockets the value added by their labour on the day that the work is done. In other industries like air transport the airline gets paid even before the work is done. If we turn to industries like food processing, they have very short turnaround time of material. Coffee beans arriving at an instant coffee factory leave in trucks to supermarkets as jars of instant the next day. If we assume they are delivered against a 30 day invoice, the average time between coffee workers being paid and their wages returning in sales will be only half a month. Some industries like construction and shipbuilding do have long circulation times. Suppose we assume an average time of one month. Since the rate of profit in the UK in 2015 was just over 10%, the net effect of including circulating capital in the price of production figures will be only of the order of 10 12 %, which is negligible compared to the typical 10% to 15% MAD between price of production and market values shown in such studies.


The results are presented graphically in Figure 1. It is evident to the eye that the labour values are clustered more closely to the diagonal than the prices of production. This is born out by the respective R 2 for the two trend lines. Figure 1 plots the logs of money values of sector outputs against logs of labour values and logs of prices of production. This is done to spread out the plot. Correlations of the raw data are:


labour value with total output for market sectors
price of production with total output for market sectors
Figure 1: The relationship between output prices, and two sorts of Marxian prices for the UK industrial sectors 2015.


Clearly, for the UK in 2015 market prices were better predicted by labour values than by prices of production. The MAD for labour values is only slightly greater than the average value given for the China years in[1] of 0.168 and well within the range of variations (0.103 to 0.219). The MAD for price of production is much greater than the average given for China (0.086) and outside the range of variation (0.063 to 0.120).
A contributory factor to the poor performance of the prices of production must certainly be the very dispersed rate of profit shown in Figure 2. Regressing profit rate against organic composition reveals no negative slope unlike the data for the USA in [3]. Indeed using correlation weighted by capital stock you get a slight positive correlation (0.14). However all industries with an organic composition above 25 had below average profit rates and all industries with above average profit rates had lower organic compositions.
It is unclear whether the difference between this result and that obtained by Han Cheng and Minqi Li for China reflects real differences between the two economies or differences in methodology. This could only be determined by repeating their work with a calculation procedure that excluded taxes from their measure of price of production.
Figure 2: Spread of profit rates as a function of organic composition.



1 Cheng, Han and Li, Minqi, “Do Labor Values Explain Chinese Prices? Evidence from China’s Input-Output Tables, 1990–2012“, Review of Radical Political Economics (2019), pp. 0486613419849674.
2 Cockshott, W Paul and Cottrell, Allin, “The Scientific Status of the Labour Theory of Value“, IWGVT conference at the Eastern Economic Association meeting, in April (1997).
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