# (p.867) Appendix 9.2 Sources and Methods for Chapter 9

# (p.867) Appendix 9.2 Sources and Methods for Chapter 9

# I. Data Sources and List of Industries 1998

The following data was taken from the US Bureau of Economics, http://www.bea.gov/iTable/index_industry_io.cfm: industry-by-industry sixty-five-order total requirements input–output tables ${\mathbf{\text{B}}}^{\prime},$ after redefinitions designed to match commodity flows to industries; the vector of direct sectoral wage bills $\mathbf{\text{W}}$ constructed from the Employee Compensation portions of value-added flows in the use tables, after redefinitions; and the market values of industry gross outputs ${\mathbf{\text{X}}}^{\prime}$ from the same use tables. All data is available for 1997–2009, but here we use 1998 to illustrate the general patterns.

Industry List: 1 Farms; 2 Forestry, fishing, and related activities; 3 Oil and gas extraction; 4 Mining, except oil and gas; 5 Support activities for mining; 6 Utilities; 7 Construction; 8 Wood products; 9 Nonmetallic mineral products; 10 Primary metals; 11 Fabricated metal products; 12 Machinery; 13 Computer and electronic products; 14 Electrical equipment, appliances, and components; 15 Motor vehicles, bodies and trailers, and parts; 16 Other transportation equipment; 17 Furniture and related products; 18 Miscellaneous manufacturing; 19 Food and beverage and tobacco products; 20 Textile mills and textile product mills; 21 Apparel and leather and allied products; 22 Paper products; 23 Printing and related support activities; 24 Petroleum and coal products; 25 Chemical products; 26 Plastics and rubber products; 27 Wholesale trade; 28 Retail trade; 29 Air transportation; 30 Rail transportation; 31 Water transportation; 32 Truck transportation; 33 Transit and ground passenger transportation; 34 Pipeline transportation; 35 Other transportation and support activities; 36 Warehousing and storage; 37 Publishing industries (includes software); 38 Motion picture and sound recording industries; 39 Broadcasting and telecommunications; 40 Information and data processing services; 41 Federal Reserve banks, credit intermediation, and related activities; 42 Securities, commodity contracts, and investments; 43 Insurance carriers and related activities; 44 Funds, trusts, and other financial vehicles; 45 Real estate; 46 Rental and leasing services and lessors of intangible assets; 47 Legal services; 48 Computer systems design and related services; 49 Miscellaneous professional, scientific, and technical services; 50 Management of companies and enterprises; 51 Administrative and support services; 52 Waste management and remediation services; 53 Educational services; 54 Ambulatory health care services; 55 Hospitals and nursing and residential care facilities; 56 Social assistance; 57 Performing arts, spectator sports, museums, and related activities; 58 Amusements, gambling, and recreation industries; 59 Accommodation; 60 Food services and drinking places; 61 Other services, except government; 62 Federal general government; 63 Federal government enterprises; 64 State and local general government; 65 State and local government enterprises.

# (p.868) II. Correction for Owner-Occupied Housing (OOH) 1998

The input–output matrix ${\mathbf{\text{A}}}^{\prime}$ and the gross output vector ${\mathbf{\text{X}}}^{\prime}$ incorporate entries for a fictitious real estate sub-industry because the BEA treats private homeowners as “businesses” renting out their own homes to themselves (Mayerhauser and Reinsdorf 2007). The BEA’s addition of the imputed rental value of owner-occupied housing doubles the listed gross output of the real estate sector, just as its addition of the imputed maintenance and repair costs of owner-occupied housing raises the listed intermediate inputs of the real estate sector by 50%. On the other hand, no addition is made to employee compensation because homeowners are not considered to pay wages to themselves.^{1} These imputations raise total real estate market price and intermediate input but not the corresponding labor requirements, thereby greatly enhancing the deviation between this industry’s market price and its corresponding labor values and prices of production. Removing the imputations brings us back to a more representative picture of actual real estate transactions. Two corrections are necessary. First, we reduce real estate gross output by the imputed gross output of owner-occupied housing, which is equivalent to dividing the original input–output coefficients in the real estate sector column by the ratio (x) of non-imputed gross output to originally listed gross output. Second, in order to remove home maintenance and repair expenditures from this column we multiply its coefficients by the aggregate ratio (a) of non-imputed intermediate input total to the originally listed intermediate total. The aggregate ratio is used, since we have no information on the detailed distribution of these imputed expenditures. The combination of the two steps amounts to multiplying the whole real estate sector column by $\text{a}/\text{x}$.

Input–output tables and labor coefficients for 1947–1972 were taken from Shaikh (1998a) as compiled in (Ochoa 1984). These tables were rebalanced to exclude the real estate sector, the great bulk of which is from OOH (Ochoa 1984, 252).

# III. Calculations

The total requirements matrix published by the US BEA is ${\mathbf{\text{B}}}^{\prime}\equiv (\mathbf{\text{I}}-{\mathbf{\text{A}}}^{\prime}{)}^{-1},$ where $\mathbf{\text{I}}$ is a 65-order identity matrix, from which we can derive the direct requirements input–output matrix ${\mathbf{\text{A}}}^{\prime}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\mathbf{\text{I}}-{\left({\mathbf{\text{B}}}^{\prime}\right)}^{-1}$. The *j ^{th}* component of the wage bill vector $\mathbf{\text{W}}$ is ${\text{W}}_{\text{j}}\equiv {\text{w}}_{\text{j}}\cdot {\text{L}}_{\text{j}},$ where ${\text{w}}_{\text{j}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}$ the average wage in the

*j*sector and ${\text{L}}_{\text{j}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}$ the total employment in the

^{th}*j*sector. This is used to derive the

^{th}*j*component of the labor coefficients vectors ${\mathbf{\text{l}}}^{\prime}$ as ${{\text{l}}_{\text{j}}}^{\prime}\equiv \frac{\left(\frac{{\text{W}}_{\text{j}}}{\text{w}}\right)}{{{\text{X}}_{\text{j}}}^{\prime}}\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}\left(\frac{{\text{w}}_{\text{j}}}{\text{w}}\right)\left(\frac{{\text{L}}_{\text{j}}}{{{\text{X}}_{\text{j}}}^{\prime}}\right)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}{\mathbb{w}}_{\text{j}}\left(\frac{{\text{L}}_{\text{j}}}{{{\text{X}}_{\text{j}}}^{\prime}}\right)$, where $\left(\frac{{\text{L}}_{\text{j}}}{{{\text{X}}_{\text{j}}}^{\prime}}\right)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}$ employment per unit gross output, and ${\mathbb{w}}_{\text{j}}\equiv \left(\frac{{\text{w}}_{\text{j}}}{\text{w}}\right)\phantom{\rule{thinmathspace}{0ex}}=\phantom{\rule{thinmathspace}{0ex}}$ the j

^{th}^{th}sector wage rate relative to the economy-wide wage rate w. The variable ${\mathbb{w}}_{\text{j}}$ is treated as a rough index of relative skills, (p.869) so that ${{\text{l}}_{\text{j}}}^{\prime}$ may be considered the skill-adjusted labor coefficient of the j

^{th}sector. The economy-wide wage rate w for 1998 was derived from NIPA tables as the ratio of aggregate employee compensation (table 1.10, line 2) and aggregate employment, full- and part-time (table 6.4D).

^{2}

It is important to note that while theoretical matrices $\mathbf{\text{A}},\mathbf{\text{l}},\mathbf{\text{X}}$ are in terms of physical quantities, empirical matrices ${\mathbf{\text{A}}}^{\prime},{\mathbf{\text{l}}}^{\prime},{\mathbf{\text{X}}}^{\prime}$ involve market prices. Since ${\mathbf{\text{A}}}^{\prime}$ is a similarity transform of **A**, it has the same eigenvalues as **A**. If we designate ${\text{p}}_{{\text{m}}_{\text{j}}}$ as the market price of a unit of output of the j^{th} sector, then $\mathbf{\text{A}}\equiv \left[{\text{a}}_{\text{ij}}\right]\equiv \left[\frac{{\text{X}}_{\text{ij}}}{{\text{X}}_{\text{j}}}\right],\mathbf{\text{l}}\equiv \left[{\text{l}}_{\text{j}}\right]\equiv \left[\frac{{\text{L}}_{\text{j}}}{{\text{X}}_{\text{j}}}\right],\mathbf{\text{X}}=\left[{\text{X}}_{\text{j}}\right]$, whereas ${\mathbf{\text{A}}}^{\prime}\equiv \left[\frac{{\text{p}}_{{\text{m}}_{\text{i}}}\cdot {\text{a}}_{\text{ij}}}{{\text{p}}_{{\text{m}}_{\text{j}}}}\right]\equiv \left[\frac{{\text{p}}_{{\text{m}}_{\text{i}}}\cdot {\text{X}}_{\text{ij}}}{{\text{p}}_{{\text{m}}_{\text{j}}}\cdot {\text{X}}_{\text{j}}}\right],{\mathbf{\text{l}}}^{\prime}\equiv \left[{\text{l}}_{\text{j}}^{\text{i}}\right]\equiv \left[\frac{{\text{L}}_{\text{j}}}{{\text{p}}_{{\text{m}}_{\text{j}}}\cdot {\text{X}}_{\text{j}}}\right],{\mathbf{\text{X}}}^{\prime}=\left[{\text{p}}_{{\text{m}}_{\text{j}}}\cdot {\text{X}}_{\text{j}}\right]$. These two sets are easily related through the diagonal matrix of market prices $\u3008{\text{p}}_{\text{m}}\u3009$. Then we can show that the empirical equivalents of the theoretical variables are the ratios of these variables to unit market prices (Shaikh 1984b, Appendix B, 82–82).

As noted in appendix 9.1 in the discussion of equations (9.1.6) and (9.1.7), the maximum rate of profit R is the reciprocal of the dominant eigenvalue of $\mathbf{\text{KT}}$. VR_{j}, ${\text{R}}_{\text{a}}(\text{r}),{\text{w}}_{\text{a}}(\text{r})$ were calculated as in equations (9.1.8)–(9.1.11) in that same appendix. All of these being ratios of price terms, the market price elements in $\mathbf{\text{p}}(\text{r}{)}^{\prime},{\mathbf{\text{Y}}}^{\prime}$ cancel out.

Finally, profit is defined as the difference between value added (VA) and employee compensation (EC), indirect business taxes (IBT) and depreciation (D). In the pure circulating model the stock of capital is assumed to be equal to the flow of material costs and depreciation is assumed to be zero (see the discussion in appendix 9.1 of equation (9.1.6)).

# IV. Construction of Capital Stock and Depreciation Matrices

The consideration of fixed capital in the calculation of prices of production requires matrices of capital stock and depreciation flow. Since capital stock figures represent end-of-year figures, the beginning-of-year stocks needed in (say) the 1998 price equations would come from 1997. The BEA only publishes the total value of net capital stock and depreciation flows in any given year for each of sixty-five industries, which yields a sixty-five-element vector for each of these (p.870) variables in any given year. The only available data on the composition of fixed assets appears in the form of capital flow (gross investment) matrices in benchmark years, in which each column displays the different asset types which enter into an industry’s gross investment.

For capital stocks, each asset type in a given industry’s capital stock is assumed to grow at the same gross rate of growth (retirement and expansion) as the total net stock of the industry. This implies that for any industry the proportions of asset types in net stock are the same as those in gross investment, that is, the capital flows (gross investment) and capital stock columns for a given industry have the same proportions. We can therefore use the data on the former to derive the proportions of each asset type in the industry’s capital stock and multiply these by the industry’s total net stock to derive the industry column of the capital stock matrix. For depreciation, it is assumed that the depreciation rate of an asset type is the same regardless of the industry in which it is used. This implies that the rows of the depreciation matrix are proportional to the rows of the capital stock matrix. The capital stock procedure turns out to be the same as in Ochoa, but the depreciation procedure is not, since he assumes that depreciation columns are proportional to gross investment flow columns (Ochoa 1984, 234–235, 242). The theory and empirical methods are elaborated next.

## 1. Theoretical procedures for constructing capital stock and depreciation flows matrices

In industry *j*, ${\text{K}}_{\text{j}}=$ value of aggregate capital stock, ${\text{IG}}_{\text{j}}=$ value of gross investment, and ${\text{g}}_{\text{j}}=\frac{{\text{IG}}_{\text{j}}}{{\text{K}}_{\text{j}}}=$ the gross rate of growth of the industry (so that ${\text{K}}_{\text{j}}=\frac{{\text{IG}}_{\text{j}}}{{\text{g}}_{\text{j}}})$. The gross rate of growth is the retirement rate plus the net expansion rate, so if an industry is expanding at (say) 3% and has an average retirement rate of (say) 2%, its gross rate of growth is 5%. Individual machines in the industry’s capital stock may have gross rates of growth which are different from the industry average. If we can assume that these differences are small, we can estimate each component of industry *j*’s capital stock by dividing each corresponding gross investment component by the industry’s growth rate. Then for asset types h, i in the *j ^{th}* column of capital stock matrix, ${\text{K}}_{\text{hj}}=\frac{{\text{IG}}_{\text{hj}}}{{\text{g}}_{\text{j}}}$ and ${\text{K}}_{\text{ij}}=\frac{{\text{IG}}_{\text{ij}}}{{\text{g}}_{\text{j}}}$, so that $\frac{{\text{K}}_{\text{hj}}}{{\text{K}}_{\text{ij}}}=\frac{{\text{IG}}_{\text{hj}}}{{\text{IG}}_{\text{ij}}}$, that is, the

*j*column in the capital stock matrix has the same proportions as that in the capital flows matrix.

^{th}On the issue of depreciation, it seems plausible to assume that each type of capital asset depreciates at roughly the same rate regardless of which industry it is used in. Let ${\text{K}}_{\text{ij}},{\text{K}}_{\text{ik}}=$ the capital stocks of the *i ^{th}* asset type used in industry

*j*and

*k*, respectively, with corresponding depreciation flows $\mathcal{D}{\mathcal{F}}_{\text{ij}},\mathcal{D}{\mathcal{F}}_{\text{ik}}$ and common depreciation rate ${\mathfrak{d}}_{\text{i}}$. Then $\mathcal{D}{\mathcal{F}}_{\text{ij}}={\mathfrak{d}}_{\text{i}}\cdot {\text{K}}_{\text{ij}}$ and $\mathcal{D}{\mathcal{F}}_{\text{ik}}={\mathfrak{d}}_{\text{i}}\cdot {\text{K}}_{\text{ik}}$ so that $\frac{\mathcal{D}{\mathcal{F}}_{\text{ij}}}{\mathcal{D}{\mathcal{F}}_{\text{ik}}}=\frac{{\text{K}}_{\text{ij}}}{{\text{K}}_{\text{ik}}}$, that is, the row elements in the depreciation matrix have the same proportions as those in the previously calculated capital stock matrix. Finally, the capital and depreciation coefficient matrices are calculated by dividing the column elements by industry gross outputs.

## 2. Empirical procedures for constructing capital stock and depreciation flows matrices

Data for total capital stock and total depreciation for private industries in each given year from Fixed Asset Tables 3.1ES (Current-Cost Net Stock of Private Fixed Assets by Industry) and 3.4ES (Current-Cost Depreciation of Private Fixed Assets by Industry), and these were mapped
(p.871)
into the sixty-one industries appearing in BEA input–output tables. The remaining four industries in the latter tables are Federal and State/Local General Government, and Federal and State/Local Enterprises. Depreciation totals for these industries are available in NIPA Table 7.5 (Consumption of Fixed Capital by Legal Form of Organization and Type of Income), lines 23–24, 26–27, respectively. Current cost net capital stock industry totals are available in Fixed Asset Table 7.1B (Current-Cost Net Stock of Government Fixed Assets) for Federal, State/Local, General Government, and Government Enterprises in lines 18, 46, 63, and 66, respectively. In order to disaggregate these into the four government categories in the input–output tables, it is necessary to assume one relation among the four elements. I assume that the ratio of the net stock of Federal Government Enterprises to total Federal net stock is the same as the corresponding ratio of gross investment, the latter being available from NIPA Table 5.8.5A–B as the ratio of line 58 to line 2.^{3}

Private industry capital flows (gross investment) matrices are available in benchmark year such as 1997, with 180 commodity rows (plus a row of column sums) and 123 industry columns (government is not shown). The first step is to aggregate these into a matrix of $61\phantom{\rule{thinmathspace}{0ex}}\times \phantom{\rule{thinmathspace}{0ex}}61$ industries.^{4} This is then supplemented by adding four government rows (all zero since government is not a producer of capital goods) and four government columns of capital flows. Investment by asset type for Federal and State/Local government is available http://www.bea.gov/industry/xls/Annual_IOUse_Before_Redefinitions_1998-2010.xls in which total Federal is the sum of columns $\text{FO}6\text{I}$ and $\text{FO}7\text{I}$, and total State/Local is the sum of columns $\text{FO}8\text{I}$ and $\text{FO}9\text{I}$^{5}. The next step is to split each of these two columns into general government and government enterprises, which is done by using appropriate investment ratios derived from NIPA Tables 5.8.5A–B. Finally, the full $65\times 65$ industry by industry capital flows matrix is normalized by dividing all industry elements by their total so as to obtain capital flow proportions which are then utilized along with the total capital stock and depreciation vectors to derive corresponding sixty-five-order matrices in the manner outlined in section I.

# V. Construction of Real Wage Curves 1947–1998

As shown in section 9 of chapter 9, we can create real wage curves by multiplying each year’s actual wage share curve by a productivity index taken from the Penn World Tables $\text{PWT}71$, Real GDP per worker in PPP terms (rgdpwok), converted to an index $2005=1$, 1950–2010. For 1947, the 1950 ratio was multiplied by the 1947–1950 ratio of the NBER index of output/labor, taken from (BEA 1966, Series A163, 209).

# VI. Appendix 9.3: Data Tables

(p.872) Data for figures 9.1–9.9 and 9.21–9.32 is in the Excel file Appendix 9.3. Data Tables for Chapter 9 (available online at http://www.anwarshaikhecon.org/).

Data for figures 9.10–9.20 of chapter 9 is calculated in various MathCad files as detailed in appendix 9.2. This data set is too large to be assembled here, but the underlying tables and flows can be made available on request.

## Notes:

(^{1})
The gross output of the real estate industry is directly available in ${\mathbf{\text{X}}}^{\prime}$ while that of owner-occupied housing ($\text{OOH})$ is from NIPA Table 7.12, line 133. In 1998, the latter imputed figure of $681.10 billion was added to the $607.35 billion of the gross revenue of the actual real estate business sector. The total imputed intermediate inputs of the fictitious real estate sub-industry in NIPA Table 7.12, line 134 (imputed homeowner repair and maintenance expenditures) was $114.4 billion, in comparison to the total $342.23 billion intermediate input of the actual real estate sector. Nothing is added to employee compensation of the overall real estate sector, since homeowners are not assumed to pay themselves wages.

(^{2})
This procedure differs somewhat from Ochoa (1984, 225), who uses the lowest sectoral wage as the deflator.

(^{3})
Let ${\text{A}}_{1},{\text{A}}_{2},{\text{A}}_{3},{\text{A}}_{4}$ represent the listed totals in Fixed Asset Table 7.1B for current cost net stocks of Federal, State/Local, General Government, and Government Enterprises, respectively, and let ${\text{x}}_{1},{\text{x}}_{2},{\text{x}}_{3},{\text{x}}_{4}$ the desired variables representing net stocks of Federal General Government, State/Local General Government, Federal Enterprises, and State/Local Enterprises, respectively. Then ${\text{A}}_{1}={\text{x}}_{1}+{\text{x}}_{2}$, ${\text{A}}_{2}={\text{x}}_{3}+{\text{x}}_{4}$, ${\text{A}}_{3}={\text{x}}_{1}+{\text{x}}_{3}$ and ${\text{A}}_{4}={\text{x}}_{2}+{\text{x}}_{4}$. This has four variables but the four equations are not independent (the matrix of coefficients has a zero determinant), so it takes an additional assumption such as ${\text{x}}_{2}=\mathrm{\alpha}{\text{x}}_{1}$ to solve for ${\text{x}}_{1},\dots ,{\text{x}}_{4}$.

(^{4})
The row of non-comparable imports was discarded (entries are less than 0.075% of industry sums).

(^{5})
The four government capital flow columns can only be derived for 1998; whereas, the sixty-one private capital flow columns can only be derived in the benchmark year 1997. Our final capital flow matrix is therefore a hybrid, but since we only utilize the proportions of the columns, the error is not likely to be significant.