An official website of the United States government 
	  
	Here is how you know
 Department of Labor Logo  United States Department of Labor
 **The .gov means it's official.**
			 Federal government websites often end in .gov or .mil. Before sharing sensitive information,
			make sure you're on a federal government site.
 **The site is secure.**
			 The
			**https://** ensures that you are connecting to the official website and that any
			information you provide is encrypted and transmitted securely.


U.S. Bureau of Labor Statistics
 Follow Us
 
 Release Calendar Blog
 Search button
          Search:


Menu

  Search button
          Search:
	
	  Follow Us Follow Us on X, formerly referred to as Twitter| Release Calendar | Blog
    Home
        About BLS
		History	
		Our Leadership
		Budget and Performance		
		Questions and Answers	
		A-Z Index	
		Glossary	
		Careers at BLS	
		BLS Speakers Available
		Errata  				
		Contact BLSOverview of BLS Statistics
		Guide to Programs & Services
		Demographics
		Industries
		Business Costs
		Occupations
		GeographyResources For
		Business Leaders
		Consumers
		Developers
		Economists
		Investors
		Jobseekers
		Media
		Public Policymakers
		Researcher		
		Students & Teachers
		Survey Respondents
    Subjects
        Inflation & Prices
		Consumer Price Index
		Producer Price Indexes
		Import/Export Price Indexes
		Contract Escalation
		Price Index Research
		Pay & Benefits
		Employment Cost Index
		Employer Costs for Employee Compensation
						Wage Data by Occupation
		Earnings by Demographics
		Earnings by Industry
		County Wages  
		Benefits
		Modeled Wage Estimates		
		Compensation Research
		Strikes & Lockouts
		Wage Records from Unemployment Insurance
		Occupational RequirementsUnemployment
		National Unemployment Rate
		State & Local Unemployment Rates
		Unemployment Research
		Employment
		National Employment
		State & Local Employment
		County Employment 
		Worker Characteristics
		American Indian Report
		Employment Projections
		Job Openings & Labor Turnover Survey
		Business Response Survey
		Employment by Occupation
		Work Experience Over Time
		Business Employment Dynamics
		Foreign Direct Investment
		Employment Research
		Workplace InjuriesProductivity
		Labor and Total Factor Productivity
		Productivity Research
		Spending & Time Use
		Consumer Expenditures
		How Americans Spend Time			
		International
		International Technical Cooperation
		Import/Export Price Indexes
		Geographic Information
		Northeast (Boston-New York)
		Mid-Atlantic (Philadelphia)
		Southeast (Atlanta)
		Midwest (Chicago)
		Southwest (Dallas)
		Mountain-Plains (Kansas City)
		West (San Francisco)
    Data Tools
        Data Retrieval Tools
		BLS Popular Series
		Series Report
		Top Picks, One Screen, Multi-Screen, and Maps
		Data Finder
		Public Data APICharts and Applications
		Charts for Economic News Releases	
		CPI Inflation Calculator	
		Injury and Illness Calculator		
		Pay Measure Comparison
		Demographic Data Sources
		COVID-19 Economic Trends
		Industry Productivity Viewer
		Employment and Wages Data Viewer
		Industry Finder from the Quarterly Census of Employment and WagesCustomized Tables
		U.S. Economy at a Glance
		Regions, States & Areas at a Glance
		Industry at a Glance			
		Text Files
		News Release Tables
		
		More Sources of Data
		Restricted Data Access
		Discontinued Databases
		Questions & Answers
		Special Notices
    Publications
        Latest Publications
		The Economics Daily
		Monthly Labor Review
		Beyond the Numbers
		Spotlight on Statistics
		Reports
		Commissioner's Corner
		Videos
		Career OutlookOccupational Outlook Handbook
		Geographic Profile
		Handbook of Methods
		Research Papers
		Copyright Information
		Contact & Help
		Weekly Update
    Economic Releases
        Latest Releases
		Major Economic Indicators
		Schedules for news Releases
		By Month
		By News Release
		Current Year
		Prior Years
		Archived News ReleasesEmployment & Unemployment
		Monthly
		Quarterly, annual, and other
		Inflation & Prices
		Pay & Benefits & Workplace Injuries
		Productivity & Technology
		Employment Projections
		Regional News Releases
    Classroom
        K-12 Student & Teacher Resources		
        Games & Quizzes
		Student's Desk
		Teacher's Desk
		Questions & Answers
    Beta
        Redesigned News Releases


Overview
                     Concepts
                     Data sources
                     Design
                     CalculationOutputHours workedLabor productivityUnit labor costsCapital servicesIntermediate purchasesCombined InputsMultifactor productivityLimitations
                     Presentation
                     History
                     More Info
                 
                 
              
			                   	
                                  
		
		












	
	
	
	
	Methods Home
        About
		Programs
		Glossarysearch
	                 

				 				 				                                                                                                                                                                                                                                                                                                                                                                                                                 					
				                
	             Handbook of Methods > 
					                      Industry Productivity Measures

                 Handbook of Methods Industry Productivity Measures Estimation
                     Industry Productivity Measures: Estimation
				  
				   				   
			  
			   				
				
				
				
			  				
				
				
				
			  				
				
				
				
			  				
				
				
				
			  				
				
				
				
			  				
				
				
				
			  				
				
				
				
			  				
				
				
				
			  				
				
				
				
			   



Productivity statistics describe how efficiently an industry produces goods and services. This is done is by separately calculating industry inputs and outputs. Change in the quantity of output that is not explained by an equivalent change in inputs is defined as growth (or decline) in productivity. The following section will describe the mathematical principles and techniques by which industry output, inputs, and productivity series are measured.
Output
Industry output is usually measured as a weighted index of the changes in the various products or services (in real terms) provided for sale outside the industry. Real output is most often derived by deflating nominal sales or values of production using BLS price indexes, but for a few industries it is measured by physical quantities of output. Each of these methods is explained below.
*Deflated value output indexes* are derived from data on the value of industry output adjusted for price change. The adjustment for price change is accomplished by dividing the nominal value of output by one or more price indexes. Nominal values of industry output are adjusted where possible to reflect the total value of industry output from both employer and nonemployer firms and to remove sales between establishments in the same industry, referred to as intraindustry transfers. Value of industry output is further adjusted by removing resales and accounting for changes in inventory. Revenues are deflated at the detailed product-class level to take into account changes in the mix of products over time. The deflated revenues for each output category are then aggregated together using the Törnqvist formula. The resulting real output indexes are conceptually equivalent to indexes that are developed using data derived from physical quantities of output.
Where possible, *physical quantity output indexes*are Törnqvist aggregations of the quantities of individual products. The basic data on quantities generally reflect primary products for each industry, at the most detailed level possible. Physical quantity output indexes are used for some industries within the mining, utilities, and transportation sectors, and at community hospitals, commercial banking, and postal services.
For an industry producing a single uniform product or service, the output index is simply the ratio of the number of units produced in the current year divided by the number of units produced in the base year. More typically, industries produce a number of different products or perform a number of different services. For these industries, output is calculated with a Törnqvist formula:
QtQt-1=exp∑i=1nwi,tlnqi,tqi,t-1whereQ=the aggregate quantity of industry output,expressed as an annual index,n=number of products,q=quantity of productiin yeart;upon whichis computed the natural logarithm of the ratioof quantity in the current year to that of the previous year, andw=the average value share weight for productiin timet.The average value share weight for productiis computed as:wi,t=si,t+si,t-1÷2wheres=the yearly share weight for productiin yeart,which in turn is computed as:si,t=pi,tqi,t÷∑i=1npi,tqi,twherep=price of productiat timet.formula
The Törnqvist formula yields the ratio of output in a given year to that in the previous year. The ratios arrived at in this manner then must be chained together to form a time series. If *t* = 3 and the base year is denoted by *o*, then
QtQ0=Q3Q0=Q3Q2Q2Q1Q1Q0.The resulting chained output index,QtQ0,is used in the productivity formula.formula
Hours worked
Total annual hours worked for each industry are estimated separately for three categories: paid employees, self-employed, and unpaid family workers. Average weekly hours of paid employees are collected by the BLS Current Employment Statistics (CES) on an hours-paid basis. This includes time when employees are on leave. Therefore, ratios of hours worked to hours paid from the National Compensation Survey are applied to the CES data. This results in the average weekly hours worked, which are multiplied by employment to calculate total annual hours worked. The annual hours of the three categories of workers are then aggregated to derive total industry hours worked. Hours worked of all persons at the industry level are treated as homogeneous, with no distinction made between the quality of hours worked by employees with different skill levels and education.
Labor productivity
To calculate a**labor productivity index, an index of industry output is divided by an index of hours worked:
QtQ0÷LtL0whereQ=quantity of aggregate output, expressed as an annual index,L=total labor hours worked, expressed as an annual index,t=the current year, and0=the base year.formula
Unit labor costs
Indexes of unit labor costs are computed by dividing an index of industry labor compensation by an index of real industry output:
CtC0÷QtQ0whereC=total labor compensation, expressed as an annual index,Q=quantity of aggregate output, expressed as an annual index,t=the current year, and0=the base year.formula
Capital services
Capital services are estimated by calculating productive capital stocks; capital services are assumed to be proportional to changes in the quantity of capital stocks for each asset. The capital index is a Törnqvist index of separate quantity indexes of equipment, structures, inventories, and land.
Capital stocks are composed of numerous different assets purchased at different times, including categories of equipment, structures, inventories, and land. The measure of capital stock for each year includes that year’s investment in an asset plus the remaining productive stock from all previous years’ investments. Capital stocks of equipment and structures for each industry are calculated using the perpetual inventory method, which takes into account the continual additions to and subtractions from the stock of capital as new investment and retirement of old capital occur. Real (constant dollar) investments in various assets are estimated by deflating current dollar investments with appropriate price deflators. The perpetual inventory method measures real stocks at the end of a year equal to a weighted sum of all past investments, where the weights are the asset’s efficiency relative to a new asset. A hyperbolic age efficiency function is used to calculate the relative efficiency of an asset at different ages.
The hyperbolic age-efficiency function can be expressed as
St=L-t/L-βtwhereSt=the relative efficiency of at-yearold asset,L=the service life of the asset,t=the age of the asset, andβ=the parameter of efficiency decline.formula
The service life of the asset for each cohort of each type of equipment and structure is assumed to be normally distributed around an average service life for that asset type. For most assets, these service lives are the same across all industries. The parameter of efficiency decline is assumed to be 0.5 for equipment and 0.75 for structures.⁠1
These parameters yield a function in which assets lose efficiency more slowly at first, and then lose efficiency more rapidly later in their service lives.
Stocks are calculated for three separate categories of inventories: finished goods, work in process, and materials and supplies. Inventory stocks for each year are calculated as the average of the end-of-year stocks in years *t* and *t-1* to represent the average on hand during the year as a whole. Current-dollar values of finished goods and work in process inventories are deflated with an implicit price deflator of each industry’s output. Current-dollar values of materials and supplies inventories are deflated with a weighted aggregation of PPIs matched to various amounts of materials consumed, specific to each industry.
Land stocks are estimated as a function of the change in constant-dollar structures stocks for each industry.
The various equipment, structure, inventory, and land stock series in constant dollars are aggregated into one capital input measure using a Törnqvist formula. Capital stocks multiplied by implicit rental prices yield cost share weights. Rental prices are calculated for each asset as
RP=P*R+P*D-∆p1-uz-k/1-uwhereRP=the rental price,P=the deflator for the asset,R=the internal rate of return,D=the rate of depreciation for the asset, and∆p=the capital gain term representing the price change of the asset over the prior three years(1-uz-k)/(1-u)reflects the effects of taxation whereu=the corporate tax rate,z=the present value of $1 of deprecation deductions, andk=the effective investment tax credit rate.formula
Rental prices are expressed in rates per constant dollar of productive capital stocks. Each rental price is multiplied by its constant-dollar capital stock to obtain asset-specific capital costs, the shares of which are used for Törnqvist aggregation.
Intermediate purchases
The materials, purchased services, fuels, and electricity consumed by each industry are called intermediate purchases.****Except for electric­ity consumed by manufacturing industries, for which direct quantity data are available, quantities arederived by deflating current-dollar values with appropriateprice deflators.
Constant-dollar materials consumed are calculated by dividing annual current-dollar industry purchases by a weighted price deflator for each industry. 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 Bureau of Economic Analysis (BEA) benchmark input-output tables. Price indexes to deflate purchased business services are constructed in a similar manner using CPIs, PPIs, or 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.
The separate indexes of real materials, services, fuels, and electricity are aggregated into a total intermediate purchases index using the Törnqvist formula. The weights for each component are derived by dividing the current dollar cost of each by the total combined cost of intermediate purchases, and averaging these weights at times *t* and *t*-1.
Combined Inputs
The annual growth rates of hours worked, capital services, and intermediate purchases are aggregated using their relative cost shares as weights. The cost of hours worked equals labor compensation. The weight for intermediate purchases is the sum total cost of materials, fuels, electricity, and purchased services. The cost of capital is derived as a residual, which is calculated as the value of sectoral production minus the costs of labor compensation and intermediate purchases.
The quantity indexes of hours worked, capital services, and intermediate purchases are combined using the Törnqvist formula to create an index of combined inputs.
Multifactor productivity
Multifactor productivity measures are derived by dividing an index of real industry output by an index of combined inputs of hours worked, capital services, and intermediate purchases.
lnAtAt-1=lnQtQt-1-wklnKtKt-1+wllnLtLt-1+wiplnIPtIPt-1whereln=the natural logarithm of the variable,A=multifactor productivity,Q=output,K=capital services,L=labor input (i.e. hours worked),IP=intermediate purchases input, andwkwlwip=cost share weights.formula
The input cost share weights are 2-year averages of the cost shares for each input (capital services, hours worked, and intermediate purchases), in years t and t-1, where
wi=pixi∑ipixit+pixi∑ipixit-12formula
pi=price of inputicapital, hours worked, or intermediate purchasesin periodtformula
xi=quantity of inputiin periodtformula
Since the growth rates are represented by differences in logarithms, the antilogs of the differences must be chained to form the index of multifactor productivity. A chain index is an index number in which the value of any given period is related to the value of its immediately preceding period; this is distinct from a fixed-base index, where the value of every period in a time series is directly related to the same value of one fixed base period.
Labor productivity is related to multifactor productivity in the manner given by the following formula:
lnQtQt-1-lnLtLt-1=lnAtAt-1+wklnKtKt-1-lnLtLt-1+wiplnIPtIPt-1-lnLtLt-1formula.
The equation above shows that the rate of change in labor productivity (on the left side of the equation) is equal to the change in multifactor productivity plus the effects of factor substitution; that is, the combined effects of changes in the weighted capital–labor ratio and the weighted intermediate purchases–labor ratio (which are represented on the right side of the equation). Conversely, the equation also shows that the change in multifactor productivity equals the change in labor productivity, adjusted to remove the weighted changes in capital and intermediate purchases relative to labor.
Limitations
Labor productivity and multifactor productivity measures are subject to certain limitations. First, existing techniques may not fully take into account changes in the quality of goods and services produced, which could result in reduced measures of output. Second, although efforts have been made to maintain consistency of coverage between the output and input estimates, some statistical differences may remain. For example, establishment lists might differ between surveys of industry revenue and surveys of hours. Third, estimates of outputs and inputs for detailed industries are subject to more volatility and error than estimates for more aggregate sectors. Finally, year-to-year changes in productivity are volatile and, therefore, are not necessarily indicative of changes in long-term trends. Conversely, long-term trends are not necessarily applicable to any one year or to any period in the future. Because of these and other statistical limitations, these indexes cannot be considered precise measures; instead, they should be interpreted as general indicators of movements of productivity.
Notes
⁠1See “The Measurement of Capital by Charles R. Hulten, available at http://www.nber.org/chapters/c5974, page 25.

				    				   
				   				   
                 Last Modified Date: August 08, 2017 Facebook
 Twitter
 LinkedIn


Home
   Subjects
   Data Tools
   Publications
   Economic Releases
   Classroom
   Beta


**U.S. Bureau of Labor Statistics**
Division of Information and Marketing Services 
PSB Suite 2850 
2 Massachusetts Avenue NE  
Washington, DC 20212-0001 Telephone:1-202-691-5200 Telecommunications Relay Service:7-1-1 www.bls.gov/OPUB Contact Us
 resources  Inspector General (OIG)
								Budget and Performance	
								Department of Labor Grants									
								No Fear Act
								USA.gov
								Vote.gov
 about the site  Sitemap						
								Freedom of Information Act
								Privacy and Security Statement
								Disclaimers
								Linking and Copyright Info	
								Important Website Notices
								Help and Tutorials
								Babel Notice: accessing language services


Twitter Youtube Email


- Publications
- Handbook of Methods