CONTENTS

 Introduction

 Description of Multifactor Productivity

 Defi ning Productivity

 Sources of Multifactor Productivity

 Benefi ts of Multifactor Productivity

 Increases

 Description of Benefi ts

 Application to Air Transportation

 Uses of Multifactor Productivity Data

 Qualifi cations to MFP Estimates: External Costs

 References

 Appendix A. Basic Growth—Accounting Methodology

 Appendix B.—Methodology with the Tornqvist Index

A Primer on Multifactor Productivity: Description, Benefi ts, 

and Uses

ANTHONY D. APOSTOLIDES, Ph.D.

April 2008

U.S. Department of Transportation

Research and Innovative Technology Administration

Bureau of Transportation Statistics

Offi ce of Advanced Studies

anthony.apostolides@dot.gov
2 A PRIMER ON MULTIFACTOR PRODUCTIVITY: DESCRIPTION, BENEFITS, AND USES

INTRODUCTION

This primer presents a description of multifactor 

productivity (MFP) and its calculation. Productivity is an important measure of the state of the 

economy at various levels: fi rm, industry, sectoral, 

and the macroeconomic. The method described is 

the one used by government agencies, such as the 

Bureau of Labor Statistics (BLS) and the Bureau 

of Transportation Statistics (BTS), as well as the 

Economic Research Service (U.S. Department of 

Agriculture). The method is also used outside of 

government, including academia and other private researchers. The estimation of multifactor 

productivity includes the measurement of output 

(of an industry, etc.) and the inputs used to create 

the output. It also includes a weighting of the inputs; the weights indicate the relative importance 

of the inputs in industry output. 

 Besides the description and measurement of 

multifactor productivity, the primer includes a discussion of the benefi ts of increasing MFP. These 

relate to increases in real income per capita in the 

country, which affects increases in the standard of 

living of the population. A rising standard of living has been a key goal of U.S. government policy 

as well as of other governments.

 Lastly, the primer discusses the uses that can be 

made of MFP estimates. They are indications of 

production effi ciency and can be used as proxies 

to indicate rates of return to capital and other resources used in production, in an industry—such 

as the various transportation industries. The MFP 

numbers provide knowledge that can help policymakers decide on the allocation of public funds 

(resources) to various sectors of the U.S. economy, 

including the transportation sector.

DESCRIPTION OF MULTIFACTOR 

PRODUCTIVITY

Defi ning Productivity 

Productivity refers to the effi ciency with which 

output—of a factory, company, industry, etc.—is 

produced with the available inputs used to produce the output—e.g., labor, capital, etc. Output, 

for example, can refer to the number of cars produced by an automobile factory in a year or the 

number of passengers transported by an airline 

company in a year (or some other time period). 

The inputs used in the making of a company’s or 

industry’s output typically are: labor (the workers employed by the company or industry), capital (e.g., machines used in production), land (on 

which production takes place—for a factory or 

airline terminal), and intermediate inputs (such as 

materials and energy purchased). 

 Productivity can be defi ned in terms of a single 

factor input, such as labor productivity, or in terms 

of many or all factor inputs, which is MFP. It can 

be estimated at various levels: industry, economic 

sector, or the economy. 

 In the basic growth-accounting methodology, 

used by R. Solow (1957), W. Kendrick (1973), and 

E. Dennison (1974), multifactor productivity (or 

“total factor productivity”) is typically estimated 

as a growth rate. In the second main approach, 

the Tornqvist methodology, MFP is calculated 

as an index number (level), which is obtained by 

dividing the output index by a combined input index (BLS, 1983). Growth rates can be calculated 

from the index numbers. Appendixes A and B 

present the computational frameworks of the two 

approaches described above. 

Labor productivity is defi ned as output per 

unit of labor, and is calculated by dividing output 

produced by a measure of the labor input used 

to produce the output (number of employees or 

labor hours). For example, suppose that a factory 

that produces automobiles makes 1,000 cars in a 

year and 50 people work in that factory. In that 

case, labor productivity is 20 cars per employee—

obtained by 1,000/50. In the making of cars (the 

output), workers also use capital, in the form of 

machinery, and other inputs (e.g., metal parts). In 

a transportation example, the output of the airline 

industry is the delivery of a service—that is, the 

transportation of passengers (and freight). If the 

industry transports 100,000 passengers per year 

and employs 100 employees, labor productivity 

would be 1,000 passengers per employee.
APOSTOLIDES 3

Labor productivity can increase in the car factory or in the airline industry over time. Increases 

in labor productivity refl ect the effects of two basic 

factors: 

1. increased use of capital (e.g., machines) in 

production—which increases the amount of 

capital per worker, and 

2. “technological progress” which can include a 

number of factors, such as improvements in 

labor or capital. Technological progress affects 

labor productivity and multifactor productivity, and it will be described at a later point. 

Multifactor productivity refers to the productivity of all the inputs used in the production process. 

These include: labor, capital, land, and intermediate 

inputs (e.g., energy inputs and purchased services). 

Consequently, MFP measures output per unit of 

combined inputs, and indicates the overall production effi ciency of an industry (sector or economy). 

In illustrating the two productivity measurements: 

labor productivity can be represented as 1,000 

cars/50 workers; and MFP can be represented as 

1,000/(Labor + Capital + Land + Intermediate 

Inputs). Thus, multifactor productivity is a more 

comprehensive measure of productivity than labor 

productivity or other single-factor productivity 

measures. 

For estimating MFP at the industry level, there 

is need for data on the output and the inputs used 

to make that output. The output measure used is 

total output (rather than value added). The inputs 

used are: labor, capital, and intermediate inputs. 

Output and inputs are measured in quantity 

terms or in constant dollars. The output measurement in constant dollars is obtained by defl ating a 

current-dollar output measure by a price index (or 

indexes). The constant-dollar measure is equivalent to the quantity measure of output. Thusly, a 

percentage change (for a given time period) in the 

quantity measure should be similar to the percentage change in constant-dollar output.

For the transportation sector, output in quantity 

terms refers to measures such as passenger-miles 

(in air transportation) or ton-miles (in rail transportation or truck transportation). Output in constant dollars—e.g., for trucking—is derived by fi rst 

obtaining data on output in current dollars. That 

relates to industry revenue, which is the product 

of the quantity of freight services sold times the 

price per service. Subsequently, the current-dollar 

output measure is defl ated by a price index (which 

is meant to capture the increase in costs and prices 

over time). An index for measuring output in real 

terms is the Fisher Chain Quantity Index, used by 

the Bureau of Economic Analysis (BEA).

Output of an industry may change over time 

with regard to quality. That dimension should 

be incorporated into the output measure. A fair 

amount of writing has been devoted to this topic. 

In this primer, one notes that in the case of the 

constant-dollar output measure, the impact of 

quality changes can be incorporated in the price 

indexes used for defl ation. In the quantity measure 

of output (e.g., ton miles) that measure (index) can 

distinguish freight of higher charge—of higher value freight though not necessarily of higher weight 

freight—by assigning weights to the various segments of freight carried by a transportation industry. These weights can relate to revenue shares of 

each product’s segment in total industry revenue. 

 The inputs used in productivity analysis are 

also measured in real terms: in quantity terms or 

in constant dollars. Quantity measures relate to 

the measurement of the labor input by the number of employees or work hours. For capital and 

intermediate inputs, a measure in constant dollars 

involves a measure in current dollars defl ated by 

an appropriate price index (or price indexes).

The labor input, as noted, is typically measured 

in either number of workers or number of work 

hours. If labor hours are obtained by multiplying 

the number of employees by an expected number 

of hours worked in a year, the two measures will 

be essentially the same. One also needs to address 

the measurement of part-time employees. In rail 

transportation, the labor input is measured by 

the Bureau of Labor Statistics in labor hours. In 

trucking, the Bureau of Economic Analysis data 

measures labor by the number of employees. 
4 A PRIMER ON MULTIFACTOR PRODUCTIVITY: DESCRIPTION, BENEFITS, AND USES

The quality of the labor input can be incorporated in the labor input index—by using numbers 

of workers in various skill categories weighted by 

the wages of each skill category. To the extent that 

that is not done in productivity measurement, the 

impact of quality changes will be included in the 

number for the MFP (residual). The estimation of 

multifactor productivity at the industry level by 

BLS and BTS does not make a distinction for different types of labor; that is, for hours of different types of employees in the measurement of the 

labor input (Bureau of Labor Statistics, 1983).

The capital input includes structures, equipment, inventories, and land (in a broad defi nition 

of capital). Intermediate inputs include purchased 

electricity, fuels, materials, and services. One can 

regard reproducible tangible capital—structures 

and equipment—and land as two distinct factors 

of production. For one, structures and equipment 

are man-made; they are the output of a production process. Land, on the other hand, is not manmade; it is a natural resource. Moreover, structures 

and equipment depreciate over time as they are 

used in production; land does not depreciate—at 

least for practical purposes. 

The capital input is typically measured in 

constant dollars; that is a measurement of current dollars is defl ated by a price index (or price 

indexes). In this measurement, improvements in 

the quality of the capital input could be measured 

by the price indexes used to defl ate current-dollar 

capital data. However, such a measurement has 

been only partial in the case of air transportation 

(per communication with staff of the Bureau of 

Labor Statistics). Consequently, the impact of 

improvements in capital would be included in the 

number for MFP. 

 In estimating multifactor productivity, the inputs are weighted. The weight of each input is the 

share of the input in the total cost of production 

of the industry. These weights indicate the relative 

importance of the inputs in production and are 

used to estimate the contribution of each input to 

changes or increases in output. 

The framework to examine multifactor productivity can be illustrated as follows:

 Output = f (Labor, Capital, Land, Intermediate

 Inputs)

This equation states that output depends on (is 

a function of) the inputs of labor, capital, land, 

and intermediate inputs. That is, in order to transport passengers (the output), an airline company 

(or airline industry) needs: 

1. Labor: pilots, cabin attendants, maintenance 

mechanics, ticket sales people, etc.; 

2. Capital: airplanes, air terminals, etc.; 

3. Land: on which to build the air terminals, runways, and maintenance facilities; and 

4. Intermediate inputs: airplane fuel, etc.

Given the above production relationship, the 

output of the air transportation industry (number of passengers carried in a year) can increase 

through two basic ways: 1) By increasing the quantity of the inputs: that is, by using more airplanes 

and pilots, or 2) By increasing the productivity of 

the inputs. This would include the use of bigger 

airplanes (better-quality capital) that can carry 

more passengers (more output); the use of faster 

airplanes (better-quality capital) that can reach 

their destination faster (hence, more output); and 

the reduction in the number of pilots from three to 

two—made possible by technological advances of 

the airplane piloting system. This would increase 

the number of passengers (output) per pilot. Increases in the productivity of the factor inputs, 

such as those described above (under #2), would 

result in higher multifactor productivity.

 Thus, a change or increase in the output of, for 

example, the airline industry is composed of two 

parts: the part that is affected by increases in the 

quantity of the inputs and the part that is affected 

by increases in the productivity of the inputs 

(MFP). This is the analytical framework used in 

the estimation of MFP. 

 Labor productivity is the most widely used 

measure of single-factor productivity. For this reason it seemed useful to compare estimates of la-
APOSTOLIDES 5

bor productivity and multifactor productivity for 

transportation industries and the U.S. economy 

(the private business sector). Relevant data are presented in table 1. It can there be observed that for 

rail, truck transportation, and the U.S. economy, 

the overall increase of labor productivity, over the 

periods for which data are available, exceeds the 

increase of multifactor productivity. In air transportation, however, the increase of MFP exceeds 

that of labor productivity. The second thing to 

observe is that the increase in labor productivity, 

over the period, is rather close to the increase in 

MFP. The exception to this is rail transportation, 

where the increase in labor productivity exceeded 

substantially the increase in MFP. 

 MFP and labor productivity levels can be higher for one measure than for the other at various 

times. For example a chart of the two variables 

for the private domestic economy, over 1947 to 

1967, shows that until 1960, MFP levels exceeded 

labor productivity; while from 1960, the situation 

reversed and labor productivity exceeded MFP 

(National Research Council, 1979, p. 40).

 Factors to explain the degree to which one 

productivity number may exceed the other are not 

available. Such an explanation would relate to the 

mechanics of calculating the two numbers. When 

labor productivity exceeds MFP “substantially,” 

it indicates that the effi ciency of all the factors of 

production is increasing substantially slower than 

labor productivity.

Sources of Multifactor Productivity

At a given point in time, MFP is affected by the 

technology used by the fi rm or industry in making 

TABLE 1 Data on Labor and Multifactor Productivity

Air transportation

NAICS 481

Rail transportation

SIC 4011

Truck transportation

NAICS 484 U.S. economy

Labor 

productivity 

(1987=100)

Multifactor 

productivity 

(1987=100)

Labor 

productivity 

(1987=100)

Multifactor 

productivity 

(1987=100)

Labor 

productivity 

(1987=100)

Multifactor 

productivity 

(1987=100)

Labor 

productivity 

(1987=100)

Multifactor 

productivity 

(1987=100)

1987 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

1988 100.6 100.2 108.4 105.8 100.8 107.7 101.6 100.7

1989 97.1 97.9 114.7 109.8 98.8 101.1 101.0 100.3

1990 97.9 100.8 118.5 113.7 104.6 100.8 102.2 100.7

1991 100.9 99.9 127.8 117.5 103.9 104.2 101.7 99.3

1992 104.1 103.6 139.6 125.0 105.5 103.8 104.3 102.7

1993 104.1 97.0 145.4 129.0 98.0 100.8 100.5 100.2

1994 107.2 106.5 150.3 131.8 102.5 100.5 101.0 100.8

1995 105.0 104.0 156.2 139.6 97.5 97.7 100.2 99.7

1996 103.7 103.8 167.0 144.8 99.5 100.1 102.9 101.7

1997 101.2 101.1 169.7 144.9 104.9 99.8 101.9 100.9

1998 97.6 99.0 173.3 143.4 99.4 99.5 102.8 101.3

1999 100.6 101.8 182.5 147.9 99.7 98.7 103.1 101.3

2000 100.0 103.0 102.8 99.4 102.9 101.3

2001 93.6 95.9 101.2 97.5 102.8 100.2

2002 103.7 101.4 104.2 101.7

2003 103.5 100.7 103.9 102.7

Sources: For data on labor productivity: 

 1) Air transportation: ftp://ftp.bls.gov/pub/special.requests/opt/dipts/indmfp3.txt.

 2) Rail transportation: ftp://ftp.bls.gov/pub/special.requests/opt/dipts/indmfp.txt.

 3) Truck transportation: http://data.bls.gov/cgi-bin/dsrv

 4) U.S. Economy: http://www.bls.gov/mfp/mprdload.htm

 For data on multifactor productivity:

 1) Air transportation: ftp://ftp.bls.gov/pub/special.requests/opt/dipts/indmfp3.txt.

 2) Rail transportartation: ftp://ftp.bls.gov/pub/special.requests/opt/dipts/indmfp3.txt.

 3) Truck transportation: http://www.bts.gov/programs/economics_and_fi nance.

 4) U.S. Economy: http://www.bls.gov/mfp/mprdload.htm

Note: For the U.S. economy, labor productivity and multifactor productivity are measured by the “Private Business Sector” (excluding government enterprises).
6 A PRIMER ON MULTIFACTOR PRODUCTIVITY: DESCRIPTION, BENEFITS, AND USES

the output (product or service). Over time, changes—hopefully, increases—in MFP are affected by 

improvements in technology. These concepts are 

described below.

Technology and Advances in Technology

Technology is the “recipe,” the “know-how,” 

that is used by producers in different industries 

in order to produce a product or deliver a service. 

The technology utilized should be the best available technology, in order to produce a product or 

service at the greatest possible level (and quality), 

given the available inputs or resources and their 

costs. The production of a product or service at the 

maximum level (given resources) also implies that 

it is produced at the lowest possible cost (cost per 

unit). If a fi rm or industry uses the various factor 

inputs—labor, capital, intermediate inputs, land—

as effi ciently as possible, the output produced 

would be the largest amount of output. In producing this output, there would be costs incurred (by 

the fi rm or industry) for using the factor inputs 

(wages, supplies, etc.), which would sum to the total costs of producing the output. In this case, since 

output is the greatest possible, the cost per unit of 

output (obtained by Total Costs/Output) would be 

the lowest possible. 

 Over time, changes (increases) in multifactor 

productivity can be affected by a number of factors, 

and these are typically classifi ed under the term 

“technical progress” or “advances in technology.” 

This category includes factors other than increases 

in the quantity of the inputs that affect increases in 

output. These factors include improvements in the 

quality of the inputs—labor, capital, etc. In the case 

of capital, quality improvements refer to the use 

of more effi cient machines, such as computers of 

more recent vintage, with higher levels of technology incorporated in them. It can also include more 

effi cient structures or buildings. In this regard, a 

study points out the technological progress in construction technology that has enabled the building 

of much higher buildings over time. The height of 

commercial buildings increased from 10 fl oors in 

1885 (The Home Insurance Building in Chicago) 

to 110 stories in 1974 (Sears Tower in Chicago). 

In addition, technological progress has resulted in 

a reduction of the amount of space necessary to 

provide the expected level of comfort in buildings 

(air conditioning, etc.). This increases the value of 

the building because more fl oor space will be available to rent (Gort, Greenwood, and Rupert, 1999). 

Therefore, improvements in technology have led 

to quality improvements of the structures, which 

resulted in higher levels of output per structure 

and, thus, higher value per structure. 

In air transportation, technical progress includes 

the use of advanced or improved equipment used 

in the inspection of cracks in airplane frames. This 

signifi cantly reduced the amount of time needed to 

carry out airplane inspections and, consequently, 

the amount of an airplane’s out-of-service time. 

This, in turn, leads to increases in its output—the 

number of passengers transported—which increases MFP. Factors, such as this, which affected 

increases in technical progress in air transportation are examined in a recent study (Apostolides, 

2006-b). 

Factors affecting “technical progress” also include changes or improvements in the arrangement 

of the production process so that more output is 

produced with the same inputs. This may include 

a rearrangement of workstations on a factory fl oor 

so that work fl ow proceeds more effi ciently, with a 

corresponding increase in output. 

Factors affecting MFP can also include changes 

in industry structure, brought about by mergers 

or acquisitions and bankruptcies, that result in 

increasing production effi ciency of the industry. 

In the case of acquisitions, industry effi ciency can 

increase when more profi table fi rms—presumably 

because they have higher effi ciency or productivity—acquire less profi table, less effi cient, fi rms. In 

the typical case after acquisition, there is a combining of certain central administrative functions 

of the two fi rms, such as personnel and payroll. 

This can lead to higher levels of output—of the 

two combined fi rms—but with the same number 

of labor (and perhaps capital or machines) input 
APOSTOLIDES 7

as was used by the acquiring fi rm. This leads to 

increases in fi rm and industry productivity. 

 In cases of bankruptcies, the fi rms going 

bankrupt and leaving the industry are typically 

unprofi table fi rms. They would generally be characterized by relatively low levels of effi ciency or 

productivity. Consequently, when they leave the 

industry, there is expected to be an increase in the 

overall level of industry effi ciency.

BENEFITS OF MULTIFACTOR PRODUCTIVITY 

INCREASES

Description of Benefi ts 

Increases in multifactor productivity have crucial 

benefi ts for the economy and society. Productivity 

increases result in increases in output, on one side, 

and incomes of various economic groups, on the 

other side. The increase in output is a direct contribution to the country’s economic growth. The 

increase in (real) incomes contributes to a rising 

living standard in the country. Real income of an 

individual refers to the purchasing power of his or 

her money income, that is, the amount of goods 

and services that a person’s money income will 

buy. Consequently, real income can increase when 

there is an increase in a person’s money income or 

when there is a decrease in the prices of goods or 

services. Economic growth and rising living standards have been basic and important economic 

goals of U.S. economic policy.

 According to economic theory, increases in 

productivity in an industry (or fi rm) can affect 

profi ts, prices, and labor compensation. The basic benefi t of increased productivity is that more 

output can be produced with the same quantity 

of inputs—while some inputs will be of improved 

quality or there can be changes in production arrangements. Alternatively, the same output can be 

produced with fewer resources. Therefore, other 

things being equal, increases in productivity result 

in a bigger difference between total revenues and 

total costs, and thus higher profi ts for the affected 

industry. This would be the initial and basic effect. 

The existence of higher profi ts can subsequently 

be followed by three effects:

1. The fi rms in the industry can keep a portion 

of the increased profi ts for internal use (additional investment, etc.), or distribute a part 

of it to their owners or shareholders;

2. The fi rms can decrease the prices of their service 

(or products) to the consumers, or—perhaps 

more likely—they may increase prices by less 

than they would have in the absence of productivity increases; and

3. The fi rms can provide higher compensation to 

their employees—in the form of higher wages 

and/or fringe benefi ts.

On the other hand, a decline in productivity 

would result in opposite effects of a productivity increase—and could lead to declines in labor 

wages and, in extreme cases, to bankruptcies of 

companies, accompanied by job losses.

As noted, the productivity increase benefi ts 

directly the affected industry, by increasing profits. Subsequently, the increase in profi ts can be 

followed by lower prices of the industry—particularly when there is competition among the producers of the industry. Competition is affected by 

the number of producers in the industry, among 

other things. The higher the number of producers 

in an industry, the more the expected competition 

in that industry.

All three impacts of a productivity increase can 

result in higher real incomes in the economy. In the 

case of the business enterprise, where productivity 

increases take place, there is a resultant increase 

in its profi t or income. A portion of the higher 

profi t can be kept by the company in the form of 

retained earnings—with which to fi nance future 

investment that can be lead to higher levels of 

productivity. If part of the increased profi t goes to 

the stockholders of the fi rm in the form of higher 

dividends, their incomes would also increase. In 

the case of labor, there can be an increase in labor 

compensation of the workers working in the affected industry; this directly results in increased 
8 A PRIMER ON MULTIFACTOR PRODUCTIVITY: DESCRIPTION, BENEFITS, AND USES

incomes. In the case of consumers or users of 

the industry’s service (or product), there can be a 

decrease in the price of that service. Such a price 

decrease results in an increase of real incomes 

(the amount of goods and services which can be 

purchased with a given level of money or nominal 

income).

Productivity increases result in increases in output (industry to national) and rising incomes in 

the society—resulting in higher living standards. 

The rise in particular types of income depends on 

what companies do with the increased profi ts from 

productivity increases—as described above. Rising 

output and incomes contribute to, and are part of, 

the country’s economic growth, which has been 

a most important objective of national economic 

policy. These are the basic benefi ts of productivity increases, and the reasons why productivity 

increases are desirable from the perspective of the 

company, industry, and the economy or society. 

Productivity increases can also have secondround effects, as when labor uses its higher income 

to increase its consumption of various goods and 

services. This increased consumption stimulates 

sales of various products or services and subsequent production in other industries, with possible 

increases in employment and incomes there. Thus, 

the benefi ts of an initial productivity increase can 

have a ripple effect from the affected industry 

and infl uence positively other industries and the 

economy.

Application to Air Transportation

The economic framework described above has 

been used to analyze the impacts of productivity 

changes or increases in air transportation (Apostolides, 2006-a). The objective of that study was 

to assess the impacts of changes or increases in 

labor productivity and multifactor productivity of the air transportation subsector—in three 

economic areas: 1) industry profi ts, 2) consumers, 

and 3) airline employees.

The analysis resulted in several fi ndings. First, 

there was a marked association between productivity changes (increases or decreases) in air transportation and industry profi ts. Second, the benefi ts 

of productivity increases in air transportation did 

not seem to transfer to consumers of air passenger 

services in the form of lower prices. On the other 

hand, users (shippers) of scheduled airfreight services benefi ted from lower rates of price increases. 

For these shippers, lower price rises restrained 

increases in their (distribution) costs, contributing 

to higher profi t or income.

In addition, a portion of the benefi t of productivity increases went to industry employees in the 

form of relatively high levels and increases of labor compensation. This resulted in an increase of 

their real incomes. Such increases in real incomes, 

to shippers and labor—as well as industry profits—are the important contributions of greater 

productivity. Increases in real incomes result in 

higher standards of living—a major objective of 

U.S. economic policy.

USES OF MULTIFACTOR PRODUCTIVITY 

DATA 

Estimates of MFP can be calculated for the economy, for different economic sectors—e.g., manufacturing, transportation—and for individual 

industries such as air transportation and truck 

transportation. 

Estimates of MFP can be used in several ways, 

described below:

 MFP is an indication of the productivity, or effi ciency, with which all factors of production are 

used in the creation of the output (of products 

or services)—of a fi rm, industry, etc. An increase 

in the MFP number indicates an increase in the 

productivity of utilizing the factor inputs. Such 

an increase in production effi ciency means that 

the fi rm or industry is getting more benefi t—in 

terms of output—from using the available inputs 

(resources). Thus, more output can be produced. 

And more people can, hopefully, share the benefi ts 

from higher levels of output.

The productivity of an industry affects the overall productivity of the U.S. economy. Thus, if MFP 

in a transportation industry grows faster than 
APOSTOLIDES 9

MFP in the U.S. economy, that industry would 

contribute positively to the growth of productivity 

of the U.S. economy and to increases in the standard of living in the country. As an illustration, 

and with regard to the contribution of trucking to 

the economy’s multifactor productivity, data show 

that over 1978-2000, MFP in truck transportation 

increased at a higher annual rate than MFP in the 

U.S. business sector—with particularly high rates 

during 1987-1995 (Apostolides, 2007). Consequently, during this period of time, trucking MFP 

contributed positively to the economy MFP. 

Productivity increases in the U.S. economy over 

time have contributed signifi cantly to economic 

growth and to improvements in the standard 

of living in the country. The goals of economic 

growth and rises in living standards—increases in 

real income per capita in the country—have been 

basic and important objectives of U.S. government 

policy. Consequently, measuring and assessing the 

productivity of the transportation sector is, or 

should be, a key priority. In addition, the enhancement of transportation productivity would be a 

relevant economic goal. 

Multifactor productivity measures can be used 

as an indicator of the rate of return to all the 

resources used in an industry. This would be an 

indicator similar to the rate of interest or rate of 

return earned in the use of capital. For example, 

the rate of interest earned on Certifi cates of Deposit is an indicator of the rate of return earned by 

capital (fi nancial capital, in this case). MFP numbers would indicate the benefi t—increase in the 

output—that an industry, sector or an economy 

gets for investing its resources—labor, capital, 

land, and intermediate inputs, land—in a particular pattern. That pattern would relate to the types 

of inputs and the combination of those inputs 

used in production—such as the capital-intensity 

of techniques of production, the types of physical 

capital used, the labor-intensity of production, the 

types of labor used, etc.

If MFP fi gures are relatively high in transportation industries, compared to the economy (which 

includes other industries), these fi gures suggest that 

increased investment for transportation-related 

infrastructure and other support for transportation industries have a higher rate of return than 

for the economy as a whole. 

QUALIFICATIONS TO MFP ESTIMATES: 

EXTERNAL COSTS

The price of a fi rm’s or industry’s service, or product, is expected to be suffi cient to cover all costs of 

production (including a profi t). Stated somewhat 

differently, all the costs of production are expected 

to be covered by the price of the service or product (total revenue = total costs, or price per unit 

= cost per unit). That is if the price of oranges is 

$2.00 per pound, that is expected to cover all the 

cost of producing and transporting that product 

to market (including, for example, $0.50 a pound 

for transportation costs). However, often the 

stated market price does not cover all the costs 

of production and distribution. This is due to the 

existence of costs that are incurred by the fi rm but 

are not included in the fi rm’s price of the product 

or service. These costs are instead passed on from 

the fi rm to outside the fi rm or society; hence, they 

are called “external costs.” These external costs 

typically refer to environmental pollution, such as 

air and water pollution (as discussed, for example, 

in Greene, et. al., 1997).

 From the economic perspective, such effects 

typically result in lower stated production costs 

for the producer (of a service or product); however, they entail higher costs to the society. This 

happens when, for example, air, train, or truck 

transportation (or the manufacturing of soap) result in increased pollution of the air, and this pollution results in adverse effects on people’s health, 

in both the short-run and the long-run. 

Typically, the effects of external costs have not 

been measured in studies of productivity, which 

may relate to diffi culties in their measurement. 

However, in a comprehensive measurement, they 

should be included in the analysis.
10 A PRIMER ON MULTIFACTOR PRODUCTIVITY: DESCRIPTION, BENEFITS, AND USES

REFERENCES

Apostolides, A. 2006-a. “Impacts of Productivity 

Changes in Air Transportation on Profi ts, Prices, 

and Labor Compensation: 1990-2001”, Journal 

of Transportation and Statistics, Volume 9, Number 1.

Apostolides, A. 2006-b. “An Analysis of Labor 

and Multifactor Productivity in Air Transportation: 1990-2001”, on the BTS website: www.bts.

gov/programs/economics_and_fi nance.

Apostolides, A. 2007. “Estimating Multifactor 

Productivity in Truck Transportation, 1987-

2003”, on the BTS website: www.bts.gov/programs/economics_and_fi nance.

Dennison, E. F. 1974. Accounting for United 

States Economic Growth 1929-1969. Washington, D.C.: The Brookings Institution. 

Gort, M., Greenwood, J., and Rupert, P. 1999. 

“How Much of Economic Growth Is Fueled by 

Investment-Specifi c Technological Progress?”, 

Economic Commentary, Federal Reserve Bank of 

Cleveland; available at: http://ideas.repec.org/a/

fi p/fedcec. 

Greene, D.L. (ed.), Jones, D.W. (ed.), and Delucchi, M.A. (ed.). 1997. The Full Costs and Benefi ts of Transportation: Contributions to Theory, 

Method and Measurement. New York: Springer.

Kendrick, J.W. (assisted by M.R. Pech). 1973. 

Postwar Productivity Trends in the United States, 

1948-1969. New York: Columbia University 

Press.

National Research Council. 1979. Measurement 

and Interpretation of Productivity. Washington, 

D.C.: National Academy of Sciences.

Solow, Robert. 1957. “Technical Change and the 

Aggregate Production Function”, Review of Economics and Statistics.

U.S. Department of Labor, Bureau of Labor Statistics. 1983. Trends in Multifactor Productivity

(Bulletin 2178). Washington, D.C.: U.S. Government Printing Offi ce.
APOSTOLIDES 11

APPENDIX A. BASIC GROWTH - ACCOUNTING METHODOLOGY

ΔT = ΔQ – [(α∗ ΔLabor) + (β∗ ΔCapital) + ( γ ∗ ΔIntermediate Inputs )]

 T Q Labor Capital Intermediate Inputs 

Where:

ΔT = Growth of MFP

 T

ΔQ = Growth of gross output

 Q

ΔLabor = Growth of labor

 Labor

ΔCapital = Growth of capital

 Capital

ΔIntermediate Inputs = Growth of intermediate inputs

 Intermediate Inputs

α = Share of labor cost in output

β = Share of capital cost in output

γ = Share of intermediate inputs cost in output.

The empirical relationship used to estimate growth of multifactor productivity by the basic 

growth-accounting methodology is shown below:
12 A PRIMER ON MULTIFACTOR PRODUCTIVITY: DESCRIPTION, BENEFITS, AND USES

APPENDIX B. METHODOLOGY WITH THE TORNQVIST INDEX 

Multifactor productivity is the ratio of the output index to a weighted average of the input indexes. A 

Tornqvist formula expresses the change in multifactor productivity as the difference between the rate of 

change in output and the weighted average of the rates of change in the inputs. Let:

Ln = the natural logarithm of a variable

A = multifactor productivity

Q = output

I = combined input

K = capital input

L = labor input

M = intermediate input

Wk = the average share of capital cost in total cost in two adjacent periods

Wl = the average share of labor cost in total cost in two adjacent periods

 Wm = the average share of intermediate input cost in total cost in two adjacent periods,

The change in the multifactor productivity is then:

 (1)

Or

 (2) 

A multifactor productivity index can be further developed by calculating the antilogs of ΔLnA, 

chaining up the resulting annual rates of change, and expressing the resulting series as a percentage of 

a selected base year. Equivalently, the change in the multifactor productivity can be directly expressed 

as At/At-1 = (Qt/Qt-1) / (It/It-1). Again, At/At-1 can be chained over time and converted into 

an index number.

 All variables, except for cost shares, are in the form of a constant dollar quantity index. The 

output quantity index is usually derived by defl ating the industry output in current dollars by an 

appropriate price index when the industry output is a single measure. When an industry produces 

multiple products and the output measure of each individual product is available, such individual 

∆LnA= Ln( )= Ln( )- [Wk(Ln )+Wl(Ln )+Wm(Ln )] At

At-1 Qt-1

Qt Kt

Kt-1

Lt

Lt-1

Mt

Mt-1

∆LnA= Ln( )= Ln( )- Ln( ). At

At-1 Qt-1

Qt It

It-1
APOSTOLIDES 13

outputs may be defl ated separately by more detailed price indexes. In that case, the total output 

quantity index can be derived through a Tornqvist aggregation such as: 

 (3) ∑ Δ nwi LnQi 1 ,

 where Qi is the output of the ith product, and

 wi is the average share of the ith product in the total output.