In This Chapter Chapter 10. Productivity Measures: Business Sector and Major Subsectors Calculation Procedures Labor productivityLabor productivity, or output per hour, is computed as: Labor productivity = (Output index) / (Hours of labor input) or P = O / H The computation of labor compensation per hour parallels the computation of output per hour. Unit labor costs (ULC) are computed as labor compensation (C) per unit of output, but are often represented as: ULC = (C / H) / (O / H) This form highlights the relationships between unit labor costs, hourly compensation, and labor productivity. Real compensation per hour (RC) is computed as hourly compensation deflated by the seasonally adjusted Consumer Price Index for All Urban Consumers (CPI-U): RC = (C / H) / CPI-U Unit nonlabor payments (UNLP) include all nonlabor components of gross product originating in a given sector — depreciation, rent, interest, and indirect business taxes as well as profits and profit-type income — whereas unit nonlabor cost (UNLC) excludes profit. These measures are computed as: UNLP = (CU - C) / O and UNLC = (CU - C - PR) / O where: CU is current-dollar output C is current-dollar compensation O is the output index PR is current-dollar profits. Labor's share in current dollar output in a given sector is simply the ratio of labor compensation paid in that sector to current dollar output: LS = C / CU and, analogously, the nonlabor or capital share is defined as: NLS = (CU - C) / CU = 1 - LS. Most of the measures noted above are presented quarterly in index form. Indexes are computed from basic data or analytic ratios by dividing the series by its own base year annual value (presently 1992) and multiplying by 100. In addition, quarterly percent changes at a compound annual rate and percent changes from the same quarter in the previous year are computed:20 Qt = 100 (Vt / Vt-1) 4 - 100 Yt = 100 (Vt / Vt-4) - 100, where: t is a time subscript denoting the quarter, V is a series described above, Qt is the quarterly percentage change in series V from quarter t-1 to quarter t, measured at a compound annual rate, Yt is the percentage change in series V from quarter t-4 (the same quarter 1 year before) to quarter t. Indexes and percent changes are published to one decimal point. In order not to lose precision, all computations are made from the underlying measures themselves rather than from the published indexes. Multifactor productivity BLS aggregates inputs for its multifactor productivity measures using a Tornqvist chain index. Some of the basic properties of this index are: It is calculated as a weighted average of growth rates of the components; the weights are allowed to vary for each time period; and the weights are defined as the mean of the relative compensation shares of the components in two adjacent years. Hence, the growth rate of the index (I) for major sectors is the proportional change over time (the triangle (delta) refers to discrete change with respect to time), such that: % Δ I = exp(Δ ln I) = exp {1/2 * [sk(t) + sk(t-1)] Δ ln K + 1/2 * [sl(t) + sl(t-1)] Δ ln L} where sl(t) = labor costs(t) / total costs(t) and sk(t) = capital costs(t) / total costs(t) Similarly, both capital, K, and labor, L are Tornqvist indexes. Each is a weighted average of the growth rates of detailed types of capital, ki, and labor inputs, li, respectively. % Δ K = exp ( Δ ln K) = exp{Si 1/2 * [ski(t) + ski(t-1)] Δ ln ki} where ski(t) = cki(t) * ki(t)/ total capital costs and where cki(t) is the rental price for capital asset ki. % Δ L = exp ( Δ ln L) = exp{Si 1/2 * [sli(t) + sli(t-1)] Δ ln li} where sli(t) = wli(t) * li(t)/ total capital costs and wli(t) is the hourly compensation for worker group li. Changes in the index of labor composition, LC, are defined as the difference between changes in the aggregate labor input index, L, and the simple sum of the hours of all persons, H. % Δ LC = exp (Δ ln LC) = exp (Δ ln L - Δ ln H) The Tornqvist index for major sector multifactor productivity growth, A, is: % Δ A = exp (Δ ln A) = exp(Δ ln Q - Δ ln I) where Q is the Fisher-Ideal index of sector output as measured by BLS. For manufacturing and the 20 industries which comprise manufacturing, aggregate input has a conceptually similar definition except that there are 5 inputs rather than just the 2 used in the major sector measures. % Δ I = exp (Δ ln I) = exp{1/2 * [sk(t) + sk(t-1)) Δ ln K + 1/2 * [sl(t) + sl(t-1)] Δ ln L + 1/2 * [se(t) + se(t-1)] Δ ln E + 1/2 * [sm(t) + sm(t-1)] Δ ln M + 1/2 * [ss(t) + ss(t-1)] Δ ln S} where L = total hours at work sl(t) = labor costs(t)/total costs(t) sk(t) = capital costs(t) / total costs(t) se(t) = energy costs(t)/total costs(t) sm(t) = materials costs(t) / total costs(t) ss(t) = purchased business services costs(t) / total costs(t) and total costs are the current dollar value of shipments adjusted for inventory change. Using this definition for aggregate input, multifactor productivity for manufacturing or any of the 20 industries which comprise manufacturing is identically defined as above. % Δ A = exp (Δ ln A) = exp(Δ ln Q - Δ ln I) where Q is a Tornqvist output index developed by BLS. Footnotes20 The estimation of quarterly (or subannual) changes at compound annual rates as the differences between movements in the underlying series involves approximations. For changes in the neighborhood of 1 or 2 percent, these approximations are good; however, the inexactness of these approximations is amplified by relatively large changes in the economic measures such as those experienced during periods of inflation, sharp recession, and rapid recovery. Since most of the productivity and costs measures are reported as percentages to one decimal place, e.g., 2.6 percent, questions sometimes arise because the greater precision carried in the automated computation results in differences in related measures in the final decimal place.