Appendix 4.A: Sources of data and estimating methods

The input—output table of 2018 is, as of this writing, the last available in the site of BEA as a direct requirement input—output matrix, that is, the Leontief inverse. The initial data include 72 industries and we brought down to 65 industries to make it suitable to 65x65 capital flows (investment) matrix of the year 1997.8 Some of the eliminated industries have zero rows and the entries in their columns are trivially small; thus, we removed these industries because their aggregation leads to more distortions. After all, aggregation of industries would require the initial input—output table (not Leontief’s inverse), which is not available unless we constructed it through the Use and Make matrices as is suggested in the BEA site. The changes from the elimination of the seven industries are only marginal, and our estimates are not affected in any empirically significant way.

Data on depreciation and fixed capital stock matrices for our 2018 come from BEA’s industry data for both the private sector and the government, federal and local. We allocated the total capital stock and depreciation of government enterprises, following information provided in past practices. In particular, we allocated 70% of the total capital stock and depreciation to local government enterprises and the remainder to the federal government enterprises.

For the estimation of the matrix of capital stock coefficients for the year 2018, we proceeded as follows: we formed weights derived from the capital flows matrix, which is post multiplied by the diagonal matrix of depreciation per unit of output. In so doing, we derive estimates of the matrix of depreciation coefficients per unit of output, D, while the post-multiplication of the capital flows matrix of weights by the diagonal matrix of capital stock per unit of output gives us the matrix of capital stock coefficients, K. The idea is that despite the old data of the matrix of capital (or investment) flows, one does not expect wild changes in its structure over the years, since the investment goods industries produce intermediate and capital goods. By contrast, the consumption goods industries and services do not produce investment goods. Consequently, only the investment goods industries will have their rows filled with data, while the remaining have many zero elements. In general, the resulting matrices of depreciation and capital stock coefficients are of the sparse kind: that is, they contain by far more zeros or near zero elements compared with the matrix of input—output coefficients.

The last investment matrix for the USA regrettably is available for the year 1997, and this is what we have used for our estimates of PRP trajectories and WRP curves for the year 2018. It is important to note the other available alternative was the OECD investment matrix for the year 1990, at the 34 input—output industry detail, which could not be used for our 65-industry structure of 2018. However, we used the 34x34 capital flow (investment) matrix of the USA of the year 1990 in Chapter 7, where we explicate our methods of estimations on the basis of 5 sectors input—output table of the year 2014.

For the estimation of the matrix of capital stock coefficients for the year 2014 and the other years, we proceed as follows: The vector of investment expenditures for the 54 (actually 49) industries for the period 2000—2014 is provided in the world input—output database (WIOD) whose link is: http:// and is accompanied by the necessary documentation (Tim- mer et al. 2015). The vector of capital stock in current prices is obtained from WIOD the Socio-Economic Accounts (SEA) whose link is: http://www. and it has been deflated by the investment deflator (with base year 2010). The real capital stock of each industry is divided by the respective industry’s real output; in so doing, we get the row vector of real capital stock per unit of output. The matrix of fixed capital stock coefficients is derived from the product of the column vector of investment shares of each industry times the row vector of capital stock per unit of output (see also Montibeler and Sanchez 2014; Tsoulfidis and Paitaridis 2017; Tsoulfidis and Tsaliki 2019; Cheng and Li 2020). Hence, it is important to note for the accuracy of our estimations that the column sums of the resulting square matrix are the same as those that we would have derived had we utilized the more accurate capital flow tables. The rank of the resulting matrix is one as the product of multiplication of two vectors and because of the presence of linear dependence, the maximum eigenvalue of the resulting matrix К [I — A] 1 R_l is equal to one with zero for all the subdominant eigenvalues. The resulting new matrix of capital stock coefficients, K, possesses the properties of the usual capital stock matrices derived and employed in the hitherto empirical studies (see Tsoulfidis and Paitaridis 2017; Mariolis and Tsoulfidis 2016a, and the literature cited therein). The idea is that the investment matrices contain many rows with zero elements (consumer goods and service industries do not produce investment goods) and so the subdominant eigenvalues will be substantially lower (indistinguishable from zero) than the dominant; this is another way to say that the equilibrium prices are determined almost exclusively by the dominant eigenvalue.

In similar fashion, the matrix of depreciation coefficient, D, was estimated as the product of the column vector of investment shares of each industry times the row vector of depreciation per unit of output. Data for depreciation by industry is not available in the world input—output database, so we used data from other sources, such as the database of Structural Analysis of the OECD (STAN) STAN08BIS. To minimize the effects of any possible methodological differences between databases, we estimated the ratio of depreciation to gross value added by industry from the OECD data sets and then we multiplied it by the corresponding gross value added data that is available in the WIOD.

The total wages are also derived from the industry data available in the BEA site. Each industry’s total wages for full-time equivalent are divided by the economy-wide average wage estimate at 57,000 USD. The employment coefficients are derived by dividing the industry employment by its respective real output available also in the same database.

The vector of consumption expenditures of workers is derived by dividing each industry’s personal consumption expenditures by the total personal consumption expenditures. The derived vector of relative weights is multiplied by the economy-wide average real wage.

The estimates of A, 1, b and К for the year 2014 are all in constant prices of 2010. So, the deviations of PP and DP from MP are expected to be somewhat higher because of the probable mistakes in the construction of price indexes.


  • 1 In the 1970s and early 1980s, there were many Sraffians arguing about the redundancy of labor theory of value although over the years the objections faded away. The often-cited book on this issue is Steedman’s (1977).
  • 2 The available official data do not include industry turnover times and one must estimate them from limited information. For example, Ochoa (1984, 1989) defined as turnover time the ratio of inventories to output for each of the 71 industries of his study. The trouble with this estimation is that the data are not readily available for all years and industries and he was forced to use the turnover time of 1972 to the other benchmark years as 1947, 1953, 1958, 1963 and 1967.
  • 3 This approach draws on Tsoulfidis and Mariolis (2007), for other similarly motivated approaches, see Parys (1982) and Caravale and Tosato (1980, pp. 85—87).
  • 4 The standard ratio is not too different from the economy-wide average capital- output ratio. The characterization of the effects is somewhat unfair to Sraffa, whose contributions inspired the distinction of the above two effects.
  • 5 The dimensions of 2018 BEA input—output table are 71x71 industries but we reduce it to 65x65, so as to make it compatible with the capital flows matrix. For details, see Appendix 4.A.
  • 6 The fixed capital model is typically studied within the joint production analytical framework, where fixed capital is considered along with the produced output. For the treatment of fixed capital in the Sraffian literature and its critique, see Semmler (1983, ch. 6), Shaikh (2016, pp. 804—807) and Cogliano et al. (2018, chs. 10 and 11).
  • 7 Details for the data and methods of derivation see Tsoulfidis and Tsaliki (2019 and the literature therein).
  • 8 Capital flow tables for the USA are no longer published by the BEA. The last capital flows table refers to the year 1997. The capital flows table for the year 2002 was not completed due to the lack of funding (Meade 2010).
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