Technical note

Technical Note

Total Factor Productivity:  Total factor productivity measures are derived by dividing an index of real
industry output by an index of the combined inputs of labor, capital, and intermediate inputs. The total
factor productivity indexes do not measure the specific contributions of capital, labor, and intermediate
inputs. Rather, they reflect the joint influences on economic growth of a number of factors that are not 
specifically accounted for on the input side, including technological change, returns to scale, improved 
skills of the workforce, better management techniques, or other efficiency improvements.

Output:  Manufacturing industry output is measured as annual sectoral output, the total value, in real 
terms, of goods and services produced for sale outside the industry. Industry value of production is 
derived by adjusting industry shipments for changes in inventories and subtracting intra-industry 
transfers and resales. For most manufacturing industries, real output is measured by deflating nominal 
value of production, but for some industries physical quantities of output are measured. For air 
transportation and line-haul railroads, output is measured by aggregating passenger-miles and freight 
ton-miles with weights based on revenues or operating expenses.  

Output measures for manufacturing industries are constructed using data primarily from the economic 
censuses and annual surveys of the Bureau of the Census, U.S. Department of Commerce, together with 
information on price changes chiefly from the Bureau of Labor Statistics (BLS). Output measures for air 
transportation and line-haul railroads are constructed using data primarily from the Bureau of 
Transportation Statistics (BTS) and the Surface Transportation Board (STB), both in the U.S. Department 
of Transportation (DOT), together with information from the Association of American Railroads (AAR), 
AMTRAK, and several other sources.  

Combined Inputs:  The index of combined inputs is a Törnqvist index of separate quantity indexes of 
capital, labor, and intermediate inputs (including fuels, electricity, materials, and purchased 
services). The annual growth rates of the various inputs are aggregated using their relative cost shares
as weights. The labor weight is based on labor compensation, including fringe benefits. The weight for 
intermediate inputs is based on the total cost of materials, fuels, electricity, and purchased services. 
The capital weight is based on total capital cost, which is calculated as the value of sectoral 
production minus the costs of labor compensation and intermediate inputs.

Capital Input:  Capital input reflects the flow of services derived from the stock of physical assets. 
Capital services are estimated by calculating productive capital stocks and are assumed to be 
proportional to changes in these capital stocks for each asset. The capital index is a Törnqvist index 
of separate quantity indexes of equipment, structures, inventories, and land. 

For manufacturing industries, physical capital is comprised of 24 categories of equipment, 10 categories
of structures, 3 categories of inventories, and land. Measures of total capital services for each 
industry are estimated by aggregating the capital stocks of individual asset types. Estimates of 
investment by asset type for each industry are derived using annual capital expenditures for detailed 
industries from the economic censuses and annual surveys of the Bureau of the Census. Additional annual 
investment data comes from the fixed asset accounts from the Bureau of Economic Analysis (BEA).

Annual investment data is supplemented with the 1997 benchmark capital flow table from the BEA as well 
as the 2008, 2012, and 2017 Annual Capital Expenditures Surveys from the Bureau of the Census. Price 
changes are removed from the annual investment data before calculating stocks. Price deflators for each
asset category are constructed by combining detailed price indexes (mostly BLS Producer Price Indexes) 
with weights that reflect each industry's use of individual asset commodities.

The capital stocks for the different assets are combined using weights based on estimated annual rental 
prices for each asset type, averaged between two time periods. Each rental price reflects the nominal 
rate of return to all assets within the industry and the rates of economic depreciation and revaluation
of the specific asset. Rental prices are adjusted for the effects of taxes. 

For air transportation, a weighted index combining 23 categories of airframes and 21 categories of 
engines is derived from annual carrier operating inventories reported to BTS. For assets other than 
airframes and engines, capital stocks are calculated similarly to manufacturing industries. For these 
assets, a more detailed breakdown of annual expenditures on equipment and structures from the BEA is 
used. Inventories of parts and supplies are also included; the current dollar series is deflated with 
a weighted cost index based on data from Airlines for America (A4A) and BTS. Indexes for aircraft and 
engines, non-aircraft assets, and parts and supplies inventories are aggregated using cost share weights
to derive an overall measure of capital input.

For line-haul railroads, current dollar investment for 10 categories of equipment and 13 categories of 
structures, obtained from STB and AMTRAK, are deflated with BLS PPIs and deflators based on BEA data. 
The capital stocks for each of the items are calculated similarly to manufacturing industries. 
Inventories of materials and supplies are also included. Estimates of investments in land from STB and 
AMTRAK are deflated with price indexes from BEA.  

Labor Hours:  For manufacturing industries, labor hours are measured as annual hours worked by all 
employed persons in an industry. Data on industry employment and hours come primarily from the BLS 
Current Employment Statistics (CES) survey and the Current Population Survey (CPS). CES data on the 
number of total and production worker jobs held by wage and salary workers in nonfarm establishments are
supplemented with CPS data on self-employed and unpaid family workers to estimate industry employment. 
Hours worked estimates are derived using CES and CPS employment, CES data on the average weekly hours 
paid to all employees, CPS data on hours of self-employed and unpaid family workers, and ratios of hours 
worked to hours paid based on data from both the CPS and the National Compensation Survey (NCS). For some
industries, employment and hours data are supplemented or further disaggregated using data from the BLS 
Quarterly Census of Employment and Wages (QCEW), the Census Bureau, or other sources. Hours worked are 
estimated separately for different types of workers and then are directly aggregated; no adjustments for
labor composition are made. 

For air transportation, annual labor input estimates are based on monthly employment data from BTS 
supplemented by employment and hours from the CES program and hours from the CPS. For line-haul railroads,
labor input measures are derived primarily from STB data and supplemented with data from AAR. For the 
railroad industry, the labor input measure includes an adjustment to remove capitalized labor hours in 
order to avoid double-counting because some capitalized labor costs are embedded in the railroad 
investment data.

Intermediate Inputs:  The index of intermediate inputs is a Törnqvist index of separate quantities of 
materials, purchased services, fuels, and electricity consumed by each industry. Except for electricity 
consumed by manufacturing industries, for which direct quantity data are available, quantities are derived
by deflating current dollar values with appropriate price deflators. 

For manufacturing industries, nominal values of materials, fuels and electricity, along with quantities of
electricity consumed by each industry are obtained from economic censuses and annual surveys of the Bureau
of the Census. To avoid double counting, an adjustment is made to the materials estimates to exclude the 
value of intra-industry commodity transfers. Purchased business services are estimated using annual 
industry data and benchmark input-output tables from BEA.  

Constant dollar materials consumed are derived by dividing annual current dollar industry purchases by a 
weighted price deflator for each industry. Aggregate materials deflators are constructed for each industry
by combining producer price indexes and import price indexes from BLS for detailed commodities.  The 
deflators are combined using weights based on detailed commodity data from the BEA benchmark input-output
tables. Aggregate price indexes to deflate purchased business services are constructed in a similar manner
using consumer price indexes (CPIs), PPIs, and deflators developed by BEA. The value of fuels consumed by 
each industry is deflated with a weighted price deflator based on PPIs for individual fuel categories; the
weights reflect fuel expenditures by industry from the Energy Information Administration (EIA), U.S. 
Department of Energy.

For air transportation, detailed cost of materials, services, fuels, and electricity from the BTS are 
deflated using cost indexes from A4A. For line-haul railroads, intermediate inputs data from STB are 
supplemented with data from other sources including AAR and AMTRAK. The nominal values are deflated with 
producer price indexes from BLS and implicit price deflators calculated from BEA investment data.

Labor Productivity:  Labor productivity describes the relationship between real output and the labor hours
involved in its production. These measures show the changes from period to period in the amount of goods 
and services produced per hour worked. Although the labor productivity measures relate output in an industry
to hours worked of all persons in that industry, they do not measure the specific contribution of labor to 
growth in output. Rather, they reflect the joint effects of many influences, including: changes in 
technology; capital investment; utilization of capacity, energy, and materials; the use of purchased 
services inputs, including contract employment services; the organization of production; the characteristics
and effort of the workforce; and managerial skill.

Contributions to Labor Productivity:

Contribution of Capital Intensity:  Capital intensity is the ratio of capital services to hours worked in 
the production process. Multiplying the change in capital intensity times capital's share of combined inputs
yields the contribution of capital intensity. 

Contribution of Intermediate Inputs Intensity: Intermediate inputs intensity is the ratio of intermediate 
inputs to hours worked in the production process. Multiplying the change in intermediate inputs intensity 
times intermediate inputs' share of combined inputs yields the contribution of intermediate inputs intensity.

When positive, both the contribution of capital intensity and the contribution of intermediate inputs 
intensity represent sources of labor productivity growth. These statistics represent factor substitution in 
the production process. In other words, positive change in the contribution of capital intensity indicates 
that labor productivity growth is being achieved in part through the substitution of capital for labor. 
Likewise, positive change in the contribution of intermediate inputs intensity indicates that labor 
productivity growth is being achieved in part through the substitution of intermediate inputs for labor.

Over a given time period, the average logarithmic growth rate of labor productivity will equal the sum of 
the average logarithmic growth rates of the contribution of capital intensity, the contribution of 
intermediate inputs intensity, and total factor productivity. However, because both output and input data 
are expressed annually, average annual (as opposed to logarithmic) rates of change are calculated. 
Therefore, the sum of growth rates of total factor productivity, the contribution of capital intensity, and 
the contribution of intermediate inputs intensity may not precisely equal the rate of change of labor 
productivity.

Annual Percent Change:  The annual percent change is the compound annual growth rate in an index series 
over a period of more than one year. The change of an index series varies from year to year. However, the 
annual percent change is the constant rate that can be applied to each year in a period, from the start to 
the end, that would give the same total result. It is calculated as 
(Ending Value/Starting Value)^(1/Number of Years)-1.