NOAA Technical Report NMFS 67

Index Numbers and

Productivity Measurement

in Multispecies Fisheries:

An Application to the

Pacific Coast Trawl Fleet

Dale Squires

July 1988

U.S. DEPARTMENT OF COMMERCE

National Oceanic and Atmospheric Administration

National Marine Fisheries Service
NOAA TECHNICAL REPORT NMFS _

The major responsibilities of the National Marine Fisheries Service (NMFS) are to monitor and assess the abundance and geographic distribution of fishery resources, to

understand and predict fluctuations in the quantity and distribution of these resources, and to establish levels for their optimum use. NMFS is also charged with the development

and implementation of policies for managing national fishing grounds, development and enforcement of domestic fisheries regulations, surveillance of foreign fishing off United

States coastal waters, and the development and enforcement of international fishery agreements and policies. NMFS also assists the fishing industry through marketing service

and economic analysis programs, and mortgage insurance and vessel construction subsidies. It collects, analyzes, and publishes statistics on various phases of the industry.

The NOAA Technical Report NMFS series was established in 1983 to replace two subcategories of the Technical Reports series: "Special Scientific Report-Fisheries" and

··Circular." The series contains the following types of reports: Scientific investigations that document long-term continuing programs of NMFS; intensive scientific reports on

studies of restricted 'cope; papers on applied fishery problems; technical reports of general interest intended to aid conservation and management; reports that review in considerable detail and at a high technical level certain broad areas of research; and technical papers originating in economics studies and from management investigations. Since

this is a formal ~eries. all submitted papers receive peer review and those accepted receive professional editing before publication.

Cupie~ of NOAA Technical Reports NMFS are available free in limited numbers to governmental agencies, both Federal and State. They are also available in exchange for

other scientific and technical publications in the marine sciences. Individual copies may be obtained from: U.S. Department of Commerce, National Technical Information Service,

5285 Pon Royal Road, Springfield, VA 22161. Although the contents have not been copyrighted and may be reprinted entirely, reference to source is appreciated.

34. Additions to a revision of the shark genus Carcharhinus: Synonymy of AprionoJon and Hypoprion, and description of a new species of Carcharhinus (Carcharhinidae),

by J. A. F. Garrick. November 1985,26 p.

35. Synoptic review of the literature on the Southern oyster drill Thais haemaslOma

tloridana, by Philip A. Butler. November 1985, 9 p.

36. An egg production method for estimating spawning biomass of pelagic fish:

".pplication to the northern anchovy, Engraulis mordax, by Reuben Lasker (editor).

December 1985, 99 p.

37. A histopathologic evaluation of gross lesions excised from commercially important North AtlantiC marine fishes, by Robert A. Murehelano, Linda DespresPatanJo, and John Ziskowski. March 1986, 14 p.

38. Fishery allas of the northwestern Hawaiian Islands, by Richard N. Uchida and

James H. Uchiyama (editors). September 1986. 142 p.

39. Survey of fish protective facilities at water withdrawal sites on the Snake and

Columbia Rivers, by George A. Swan, Tommy G. Withrow, and Donn L. Park. April

1986, 34 p.

40. Potential impact of ocean thermal energy conversion (OTEC) on fisheries, by

Edward P. Myers, Donald E. Hoss, Walter M. Matsumoto, David S. Petas. Michael

P. Seki, Richard N. Uchida, John D. Ditmars, and Robert A. Paddock. June 1986

33 p.

41. A stationary visual census technique for quantitatively assessing community structure of coral reef fishes, by James A. Bohnsack and Scott P. Bannerol. July 1986, 15 p.

42. Effects of temperature on the biology ofthe northern shrimp, Pandalus borealis.

in the Gulf of Maine, by Spencer Apollonio, David K. Stevenson, and Earl E. Duntun,

Jr. September 1986, 22 p.

43. Environment and resources of seamounts in the North Pacific, by Richard N.

Uchida, Sigeiti Hayasi, and George W. Boehlert (editors). September 1986, 105 p.

44. Synopsis of biological data on the porgies, Calamus arcrifrons and C. proridens

(Pisces: Sparidae), by George H. Darcy. September 1986, 19 p.

45. Meristic variation in Sebasres (Scorpaenidae), with an analysis of character

association and bilateral pattern and their significance in species separation, by Lochai Chen. September 1986, 17 p.

46. Distribution and relative abundance of pelagic nonsalmomd nekton off Oregon

and Washington 1979-84, by Richard D. Brodeur and William G. Pearcy. December

1986, 85 p.

47. Reproduction, maturation, and seed production of cultured species: Proceedings

of the twelfth U.S.-Japan meeting on aquaculture, Baton Rouge, Louisiana, October

25-29, 1983, by Carl J. Sindermann (editor). February 1987, 73 p.

48. Widow rockfish: Proceedings of a workshop, Tiburon, California, December

11-12, 1980, by William H. Lenarz and Donald R. Gunderson (editors). January 1987,

57 p.

49. Reproduction, movements, and population dynamics of the southern kingfish,

Menricirrhus americanus, in the northwestern Gulfof Mexico, by Stephen M. Harding

and Mark E. Chittenden, Jr. March 1987,21 p.

50. Preparation of acetate peels of valves from the ocean quahog. Arc/ica islandica,

for age determinations, by John W. Ropes. March 1987, 5 p.

51. Status, biology, and ecology of fur seals: Proceedings of an international

workshop, Cambridge, England, 23-27 April 1984, by John P. Croxall and Roger

L. Gentry (editors). June 1987, 212 p.

52. Limited access alternatives for the Pacific groundfish fishery, by Daniel D.

Huppert (editor). May 1987, 45 p.

53. Ecology of east Florida sea turtles: Proceedings of the Cape Canaveral, Florida,

sea turtle workshop, Miami, Florida, February 26-27, 1985. by Wayne N. Witzell

(convener and editor). May 1987, 80 p.

54. Proximate and fatty acid composition of 40 southeastern US finfish species,

by Janet A. Gooch, Malcolm B. Hale, Thomas Brown, Jr., James C Bonnet, Cheryl

G. Brand, and Lloyd W. Reiger. June 1987, 23 p.

55. Proximate composition, energy, fatty acid, sodium, and cholesterol content of

finfish, shellfish, and their products, by Judith Krzynowek and Jenny Murphy. July

1987, 53 p.

56. Some aspects of the ecology of the leatherback turtle Dermochelys coriacea at

Laguna Jolova, Costa Rica, by Harold F. Hirth and Larry H. Ogren. July 1987, 14 p.

57. Food habits and dietary variability of pelagic nekton off Oregon and Washington,

1979-1984, by Richard D. Brodeur, Harriet V. Lorz. and William G. Pearcy. July

1987, 32 p.

58. Stock assessment ofthe Gulf menhaden, Brevoonia pa/ronus, fishery, by Douglas

S. Vaughan. September 1987, 18 p.

59. Allantic menhaden, Brevoonia tyrannus, purse seine fishery, 1972-84, with a

brief discussion of age and size composition of the landings, by Joseph W. Smith,

William R. Nicholson, Douglas S. Vaughan, Donnie L. Dudley, and Ethel A. Hall.

September 1987, 23 p.

60. Gulf menhaden, Brevooniapatronus, purse seine fishery, 1974-85, with a brief

discussion of age and size composition of the landings, by Joseph W. Smith, Eldon

J. Levi, Douglas S. Vaughan, and Ethen A. Hall. December 1987, 8 p.

61. Manual for starch gel electrophoresis: A method for the detection of genetic varialion, by Paul B. Aebersold, Gary A. Winans, David J. Teel, George B. Milner, and

Fred M. Utter. December 1987, 19 p.

62. Fishery publication index, 1980-85; Technical memoradum index, 1972-85, by

Cynthia S. Martin, Shelley E. Arenas, Jacki A. Guffey, and Joni M. Packard.

December 1987, 149 p.

63. Stock assessment of the Atlantic menhaden, Brevoonia tyrannus, fishery, by

Douglas S. Vaughan and Joseph W. Smith. January 1988, 18 p.

64. Dlustrated key to penaeoid shrimps of commerce in the Americas, by Isabel Perez

Farfante. April 1988, 32 p.

65. History of whaling in and near North Carolina, by RandalJ R. Reeves and Edward

Mitchell. March 1988,28 p.

66. Atlas and zoogeography of common fishes in the Bering Sea and northeastern

Pacific, by M. James Allen and Gary B. Smith. April 1988, 151 p.
NOAA Technical Report NMFS 67

Index Numbers and

Productivity Measurement

in Multispecies Fisheries:

An Application to the

Pacific Coast Trawl Fleet

Dale Squires

July 1988

u.s. DEPARTMENT OF COMMERCE

C. William Verity, Jr., Secretary

National Oceanic and Atmospheric Administration

William E. Evans, Under Secretary for Oceans and Atmosphere

National Marine Fisheries Service

James W. Brennan, Assistant Administrator for Fisheries
The National Marine Fisheries Service (NMFS) does not approve, recommend

or endorse any proprietary product or proprietary material mentioned in this

publication. No reference shall be made to NMFS, or to this publication furnished by NMFS, in any advertising or sales promotion which would indicate

or imply that NMFS approves, recommends or endorses any proprietary

product or proprietary material mentioned herein, or which has as its purpose

an intent to cause directly or indirectly the advertised product to be used or

purchased because of this NMFS publication.

ii
CONTENTS

PART 1

Introduction 1

Purpose of study 1

Applications of productivity measures 1

The Pacific trawl fleet 2

Methodological background 3

Productivity defined 3

Growth-accounting framework 3

Economic index numbers 4

Index-number formulae 4

Laspeyres and Paasche indices 5

Tornqvist and Fisher Ideal indices 5

Chain and fixed-base indices 5

Bilateral and multilateral indices 6

Economic performance index 6

Data and index construction 6

Introduction 6

Output indices 6

Input indices 7

Labor 7

Fuel 7

Capitai 8

Empirical Results 9

Introduction 9

Pacific trawl fleet 9

Economic performance 10

Methodological evaluation 10

Multilateral vs. bilateral indices 10

Chain vs. fixed-base indices 11

Laspeyres, Paasche, Fisher Ideal,

and Tornqvist indices 11

Recommended index-number procedure 11

Data evaluation 11

Concluding remarks 12

III

PART 2

Additional topics in productivity measurement 12

Growth-accounting and structural frameworks 12

Assumptions of the growth-accounting

framework 12

Intertemporal and interspatial productivity 13

Duality-based measures of productivity 13

Economic theory of index numbers 14

Introduction 14

Aggregator functions 14

Fisher's weak factor-reversal test 14

Laspeyres and Paasche (mean-of-order-r) indices 14

Geometric indices 15

Quadratic mean-of-order-r indices 15

Quadratic lemma 16

Superlative indices 16

Quadratic mean-of-order-r 16

Fisher's Ideal 16

Tornqvist 17

Choice among superlative indices 17

Divisia indices 17

Chain and fixed-base indices 17

Consistency in aggregation 19

Homogeneity and homotheticity 19

Multilateral and bilateral indices 19

Productivity measurement and stock effects in marine

fisheries 21

Acknowledgments 21

Citations 21

Tables 24

APPENDICES

1. Summary of different index formulae 31

2. Quadratic mean-of-order-r indices and

Fisher's tests 31

3. Superlative implicit indices 32

4. Aggregation of outputs 33

5. Quadratic approximation lemma 33

6. Constructing index numbers on electronic spreadsheets with personal computers 34

Index Numbers and

Productivity Measurement

in Multispecies Fisheries:

An Application to the

Pacific Coast Trawl Fleet

DALE SQUIRES

Southwest Fisheries Center

National Marine Fisheries Service, NOAA

P.O. Box 271

La Jolla, California 92038

ABSTRACT

This study is concerned with the measurement of total factor prodnctivity in the

marine fIShing industries in general and in the Pacific coast trawl fIShery in particular. The study is divided into two parts. Part I contains suitable empirical

and introductory theoretical material for the examination of productivity in the

Pacific coast trawl Deet. It is self-contained, and contains the basic formulae, empirical results, and discussion. Because the economic theory of index numbers

and productivity is constantly evolving and is widely scattered throughout the

economics literature, Part D draws together the theoretical literature into one

place to allow ready access for readers interested in more details.

The major methodological focus of the study is upon the type of economic

index number that is most appropriate for use by economists with the National

Marine Fisheries Service. This study recommends that the following types of

economic index numbers be used: chain rather than fIxed base; bilateral rather

than multilateral; one of the class of superlative indices, such as the Tornqvist

or Fisher Ideal.

PART 1

Introduction _

Purpose of study

Measurement of industry productivity is important to planning,

public regulation, and monitoring industry performance over time.

Yet, with the exceptions of Bell and Kinoshita (1973), Norton et al.

(1985), Kirkley (1984), and Duncan (undated), little attention has

been given to measuring productivity in marine fishing industries.

Moreover, these initial studies can be extended in scope and

methodology to draw upon recent advances in the economic theory

of index numbers and productivity measurement.

This study addresses these issues and has five explicit purposes.

First, the study applies recent advances in the theory of index

numbers to the Pacific coast trawl fleet. Second, an informal evaluation is made of the data sources available for productivity measurement ofthis fleet. Third, many different index-number procedures

are available, and this study evaluates the most important and widely

used procedures. Fourth, the interpretation ofproductivity is unclear

for marine fishing industries, and this study clarifies this issue. Fifth,

the economic theory of index numbers is constantly evolving and

is widely scattered throughout the literature. This study draws this

literature together to make it more accessible to other applied

economists.

The productivity indices are developed for U.S. vessels of the

Pacific coast trawl fleet homeported in Washington, Oregon, and

northern and central California over the 1981-85 period. These

vessels employ bottom, shrimp, and midwater trawl gear to harvest

numerous species of groundfish and pink shrimp which are sold

to domestic shoreside processors or to foreign processors participating in joint venture operations.

Applications of productivity measures

Productivity measures, used as indicators of relative economic

performance in fisheries, portray fishery trends and problems.

Productivity measures can be an effective means to monitor the

economic performance of a fishery. Only by accurate knowledge

of a fleet's performance over time can effective policies be designed.

For example, declines in fleet productivity could signal a need for

government assistance or regulation. Alternatively, productivity

declines after years of government assistance could imply that public

resources committed to the fishery have been dissipated through

overcapitalization. Government support might be either scaled back

or refocused to foster efficiency without encouraging increases in

input usage. Ifpublic regulation offisheries is to be concerned with

net economic and social benefits, regulators should be aware of

changes in productivity and in resource stock levels.

Small or negative productivity gains in fishing industries can be

associated with lagging profits, returns to labor, and employment,

because fishermen must compete with foreign fishery imports and

other protein substitutes, such as meat and poultry, where productivity is a main component of competitive advantage. Rising

productivity in the fishery sector can also help mitigate inflationary

pressures in fish products, because products can be supplied at

declining costs over time. In tum, this helps to maintain the competitive capability of fisheries in relationship with its close substitutes

such as meat and poultry (Bell and Kinoshita 1973).
International comparisons of fishing fleet productivity can help

clarify differences in international competitiveness of fishing

industries. Decline in a fishery's productivity coupled with rising

imports of the species being harvested could suggest a need for

corrective action by industry bodies or government. Declines in

productivity coupled with a structural shift in consumption patterns

toward more fish could signal an increase in imports or a decrease

in exports in the future.

Regional differences in productivity can be linked to the geographical distribution of productive resources within the fishery.

For example, the empirical results presented later in this report show

that in 1983 trawl fleet capital declined in Washington while it increased in northern and central California. Moreover, Washington

and Oregon experienced positive growth rates in fleet total-factor

productivity over the time period 1981-85, while northem and central California experienced declines. Public regulation might become

more effective by adopting a more explicitly regional-based

approach.

The Pacific trawl fleet _

The Pacific trawl industry off California, Oregon, and Washington

is composed ofseveral different commercial gear and vessel types

harvesting a wide array of species. The most important harvested

species include pink shrimp, Dungeness crab, flatfish (Dover,

English, petrale, rock, and rex sole), roundfish (sablefish, Pacific

cod, ling cod, and Pacific whiting), the Sebastes complex (yellowtail, canary, widow, boccacio, chillipepper, and shortbelly rockfish),

and thornyheads.

The contributions of each species and species assemblage to total

revenue are provided in Table 16. This table shows the proportions or shares of total regional exvessel revenue received for each

species assemblage in each year. Pink shrimp, rockfish, and Dover

sole consistently provide the highest proportion or share of total

revenue. The contributions of each region to total revenue, or

revenue shares, are reported in Table 17. Oregon generally provides about 45% of the total revenue, Washington around 22%,

northern California around 20%, and central California the balance.

The species' and regions' revenue shares are not static, but change

over time.

Three separate trawl fisheries exist for pink shrimp, groundfish,

and midwater species. Gear switching occurs, most notably between

otter and shrimp trawls and between otter and midwater trawls.

Huppert and Korson (1987) note that smaller inshore trawlers in

the Crescent City, California, area also shift among target specie~

by harvesting Dungeness crab in the winter and otter trawling for

soles and rockfish during the fall and spring. The midwater trawl

fishery harvests Pacific whiting in joint ventures and widow rockfish

for shoreside landings. For a brief period in 1982, midwater trawlers

caught shortbelly rockfish. Many vessels that participate in the pink

shrimp and groundfish fisheries have insufficient horsepower to

operate the midwater trawls.

Increased profits and the availability of vessel financing led to

a significant expansion of the shrimp fishery in the late 1970s

(Dewees 1986). Since the early 1980s, however, the shrimp fishery

steadily declined in importance, dropping from 19,923 short tons

worth $20 million in 1981 to 4,814 short tons worth $3.81 million

in 1984. A resurgence began in 1985, with landings rising to 12,779

short tons worth $7.61 million (all dollars are $1981 values)

(Table 1). The contribution ofshrimp to regional exvessel revenue

declined from 36% and 46% of total 1981 revenue in Washington

2

and Oregon, respectively, to 13% and 9% in 1984, before rebounding to 29% and 22% in 1985 (Table 16).

During the early 1980s, the groundfish fishery continued the expansion begun following the Magnuson Fisheries Conservation and

Management Act of 1976. Table 2 indicates the number of coastwide

otter and shrimp trawl vessels in the fishery between 1981 and 1985.

Some of the otter trawl vessels were formerly pink shrimp trawlers

that switched to otter trawling after the decline in pink shrimp

catches and abundance. The decline in the number of otter trawl

vessels during 1983 and 1984 can be attributed, in part, to declining rockfish stocks (particularly widow rockfish), continual decline

in real groundfish prices, and the effects of high interest loans taken

out to finance the fleet's expansion. Some of the larger vessels

transferred their operations to Alaska, while other vessels have sunk,

burned, been repossessed, transferred to other fisheries, or simply

tied up due to financial difficulties.

Otter-trawl groundfish landings in Washington, Oregon, and

northern and central California during the 19808 have declined from

a 1982 peak of 113,492 short tons valued at $44.33 million (1981

constant dollars) to 79,938 short tons worth $32.26 million in 1984,

before increasing slightly in 1985 to 82,988 short tons worth $34.86

million (Table 1). Although the number of vessels, tonnage landed,

and total revenue all peaked in 1982, the total frequency of landings (Le., the number offish receipts recorded) reached a high one

year later, with 15,436 landings (Table 4).

The midwater trawl fishery developed in the late 1970s as joint

ventures (JV) with foreign fishing companies. With the exception

oflimited fishing for shortbelly rockfish in 1982, the joint-venture

fishery targets Pacific whiting almost exclusively. This fishery

occurs primarily in the summer months. Additional sources of

revenue for these vessels include widow rockfish and JV operations in Alaska. Annual landings and revenue by Pacific whiting

N vessels were stable during 1982-85 at around 80,000 short tons

valued at around $10 million ($1981 dollars) before declining in

1985 to 34,934 short tons worth $3.2 million (Table 1).

The Pacific trawl fleet has undergone a rapid modernization since

1976. Dewees (1986) examines the rates of adoption of eight

technological innovations during this period. The innovations

ostensibly contributing most to increases in the technical capabilities

(or' 'fishing power") oftrawlers have been the development of midwater trawling, chromoscopes, sonar, and track plotters. Midwater

trawling represents a major change in the harvesting process. Midwater trawlers have shown a greater reliance on these electronics

since otter trawlers can still function satisfactorily without them.

With other factors affecting productivity held constant, these

technological innovations should increase productivity in the Pacific

trawl industry. Benefits from technological progress can be realized,

however, only if productivity gains are not dissipated through overcapitalization in the fishery. Moreover, the rate oftechnical progress

should be comparatively high in open-access fisheries because of

the intense competition among fishermen to harvest limited fish

stocks.

In recent years, most of the commercial groundfish stocks have

been harvested at or near the maximum sustainable yields (MSY)

estimated by the Pacific Fishery Management Council's Groundfish Management Team. Dover sole, Petrale sole, other flatfish,

Canary, yellowtail, and widow rockfish, ling cod, Pacific ocean

perch, and sab1efish in particular are all harvested at levels close

to, or surpassing, MSY (PFMC 1985; Huppert and Korson 1987).

The Pacific whiting catch remains well below its estimated MSY.

The pink shrimp resource is somewhat ephemeral and receives

intensive exploitation (Korson 1984).
y(t)

Y 1(t) = A1(t)f [K(t),L(t)]

Changes in resource abundance and composition will affect productivity in marine harvesting industries. With the fishery stocks

being harvested at levels close to or at their MSYs in the Pacific

coast trawl fisheries, economic index measures of fleet productivity

are likely to remain unchanged or exhibit downward movements

over 1981-85, reflecting constant or declining levels of stock

abundance.

Methodological background _

This section provides an introduction to the methodology of productivity measurement and the theory of economic index numbers.

Productivity is first defined, then the concept of economic index

numbers is discussed, followed by discussions of different indexnumber formulae, chain and fixed-base indices, and bilateral and

multilateral indices. Readers interested in additional methodological

issues can refer to Part 2 and its Appendices.

y"

y'

y

x X' X(t)

Growth-accounting framework

Productivity defined

The standard framework for estimating productivity change is

derived from the theory of production. Consider the following one

output-two input production function:

where yet) denotes total landings at time t, K(t) denotes the flow

of capital services at time t, L(t) is the flow oflabar services used

at time t, and A(t) is an efficiency parameter allowing for shifts

in the production function. The production function defines the maximum output achievable with the given quantity ofinputs, L(t) and

K(t), and is determined by the state of technical knowledge and

resource abundance, A(t). Total landings can grow from several

sources: (1) as existing firms expand their input usage, (2) as new

firms enter the industry, and (3) as technology advances and resource

abundance increases, causing shifts in the aggregate production

function.

(2)

[8InY(t)/8InK(t)] [dlnK(t)/dt]

+ [8InY(t)/8InA(t)] [dlnA(t)/dt]

+ [8InY(t)/8InL(t)] [dlnL(t)/dt]

dlnY(t)/dt = (dY/dt)(lIY)

Figure 1

Production function representation of total-factor productivity.

Intuition into the meaning of productivity is provided by Figure 1.

Two different levels of the production function in equation (1) are

presented: Yo(t) = Ao(t}f[K(t),L(t)] and Yt(t) = AI(t)f[K(t),L(t»),

where Y1(t) > Yo(t). The vertical axis represents different catch or

output levels, where y' > Y. The horizontal axis represents different levels of an index of aggregate input, X, where X' >X. When

the state of technical knowledge and resource abundance both remain constant but a larger quantity of inputs is used to harvest fish,

X' > X, and firms move along the existing production function,

Yo(t) = Ao(t)f[K(t),L(t»), from point B to point C. Firms harvest

more fish by using more capital and labor, and total catch increases

from yet) to Y' (t). Total catch can also increase when technological

innovations are adopted by the fleet, even if the same amount of

inputs is used and resource abundance remains constant. In this case,

the state-of-technology index increases from Ao to AI' and the production function shifts from Yo(t) = Ao(t)f[K(t),L(t)] to YI(t) =

A1(t)f[K(t),L(t)]. At a constant input bundle X, the shift upwards

of the production function is denoted by moving from point B to

point D and catch increases from Y to y'. Fishing firms can now

harvest a larger catch, y', with the same level of inputs, X. Economists say that fishing firms are now more productive.

The basic problem of productivity analysis is to use data on the

prices and quantities of inputs and outputs to allocate the growth

of yet) among the growth rates of K(t), L(t), and A(t). The growthaccounting framework used in this study proceeds nonparametrically

by first taking logarthims of equation (I) and then logarithmically

differentiating equation (1) with respect to time. The logarithmic

differential of (I) can be written as:

yet) = A(t)f[K(t),L(t»), (1)

Productivity is traditionally used to explain the physical output per

unit ofinput. Higher productivity means that more can be produced

with the same bundle of inputs or, conversely, that the same output bundle can be produced from fewer inputs.

Historically, productivity measurement focused upon one factor,

such as output per unit of capital or output per man-hour (Bell and

Kinoshita 1973). These partial productivity measures may provide

misleading results, since output increases may arise from the inc

creased use of other inputs or changes in capacity utilization. This

limitation to partilil productivity has led to emphasis upon totalfactor productivity.

Dividing the level of production (total output) by an index of all

inputs creates an index of total-factor productivity. Properly constructed, the total-factor productivity index accounts for all changes

in the quantities of inputs. Variation in the total-factor productivity

index tracks the productivity residual which is not accounted for

by changes in the volume of economic inputs. With this introduction, the concept of total-factor productivity is now rigorously

developed.

3
where alnY(t)/alnA(t) is set equal to unity because it is a

technology-shift parameter, alnY(t)/alnK(t) = [ay/aK] [K/Y]

is the output elasticity for capital (the percentage increase in

output with a I percent increase in capital), denoted EK, and

alnY(t)/alnL(t) = [a YI aLl [LlY] is the output elasticity for labor

(the percentage increase in Qutput with a I percent increase in labor),

denoted EL .

The logarithmic derivatives are interpreted as rates of growth so

that the rate of output growth becomes:

Y(*)/Y = EKK(*)/K + ELL(*)/L + A(*)/A, (3)

where the asterisk * denotes time derivatives. The rate of output

growth is thus allocated among growth in capital and labor, technical

progress, and changes in resource abundance.

Because EK and EL are unobservable, equation (3) cannot be

used for empirical analysis. One further step is required. Assuming that inputs are paid their value of marginal product:

aY(t)/aK(t) = PK(t)/P(t), aY(t)/aL(t) = PL(t)/P(t), (4)

where pet), PK(t), and PL(t) are the prices of output, capital

services (rental price), and labor services (wage rate), respectiveIy. Substituting (4) into (3) gives:

Y(*)/Y = SKK(*)/K + SLL(*)/L + A(*)/A. (5)

Because in competitive open-access equilibrium, industry profits

are dissipated (Gordon 1954) and firms display locally constant

returns to scale (Baumol et al. 1982), total costs equal total revenue,

and any input's cost share equals its revenue (or income) share.

Therefore, SK = [PKK]/[PY] and SL = [PLL]/[PY] , that i~, the

cost (equals revenue) shares of capital and labor. Given (3), these

shares are equal to the production elasticities EK and EL• These

shares provide weights to the growth of capital and labor over time.

The [mal step is to rearrange equation (5) to give:

A(*)/A = Y(*)/Y - SKK(*)/K - SLL(*)/L. (6)

Productivity in fisheries, that is, technical progress and change in

resource abundance, is therefore measured as the residual of output growth after accounting for the growth of inputs. Intuitively,

output grows over time as inputs increase and is reflected in movement along the frontier of the industry production function, while

technical progress and changes in resource abundance cause shifts

in the production function. The residual (6) thus is a measure of

production function shift, and is called the productivity residual.

Tracking the total-factor productivity index for a fishing industry

therefore provides information on technical progress and changes

in resource abundance of exploited fish stocks. Because the productivity index is measured as a residual in equation (6), changes

in productivity might also include changes in the economic efficiency of the individual fishing firms, altered fishing regulations,

variations in economic capacity utilization, or variations in exogenous conditions like weather.

The effects of changing resource abundance can be disentangled

from the productivity residual. This topic is the subject of current

research, &nd will be discussed in a future report.

After defining and clarifying the issue of productivity in marine

fishing industries, attention must be turned to measuring productivity in some way. Economic index numbers have been developed

by economists for tasks such as productivity measurement. The next

4

section provides a brief introduction and survey of this important

topic, while readers interested in additional details can refer to

Part 2.

Economic index numbers

Productivity, production, and input use are more effectively

measured by economic index numbers than by physical measures.

Physical measures (e.g., total catch per hour towed) fail to distinguish changes or differences in composition or quality over time

and space. or between fishing firms. Simply lumping together total

tonnage of catch in one time period and comparing with total tonnage from a subsequent time period neglects the change in catch

composition. Different products are then compared, and the assumption is implicitly maintained that a ton of Pacific whiting, for instance, is perfectly substitutable by a ton of pink shrimp.

Economic index numbers deal with situations in which industry

outputs and inputs are too diverse to measure simply by weighing

or counting. Economic index numbers provide weighted measures

of the different kinds of outputs (species) or inputs (capital, labor,

fuel). Shares or proportions of total revenue (revenue shares), for

example, can be employed to combine the different outputs into

a weighted measure of total output, and shares of total costs (cost

shares) can similarly be employed to aggregate different inputs into

a weighted measure of total input.

The different outputs (inputs) are combined into weighted

measures of total output (input) by functions. These functions are

called aggregator functions since they aggregate the individual components (e.g., outputs) into the composite (e.g., total output).

Different formulae for the aggregator functions have different implications for the properties of the index numbers formed. The

economic theory of index numbers is concerned with these relationships between the properties ofindex numbers and the properties of the underlying aggregator functions they represent.

Both individual quantities and individual prices can be aggregated

into a composite quantity or price. Quantity aggregator functions

aggregate quantities ofindividual outputs or inputs into composite

measures oftotal-output quantity or total-input quantity, while price

aggregator functions aggregate prices of individual outputs or inputs into composite measures of total output or input prices.

Index-number formulae

A number of different types of economic indices exist. Each type

of index offers an approximate scalar measure of a multidimensional change over time in prices, quantities, or productivity. The

different indices approximate these intertemporal changes in different ways, according to their theoretical properties (manifested by

their formulae). Differences in indices can be viewed as differences

in their abilities to provide approximations to the intertemporal

changes in prices, quantities, or productivity.

Consider a concrete example of the way in which the different

indices provide different approximate scalar measures ofintertemporal changes. Suppose the problem is to measure intertemporal

changes in an aggregate output bundle, which in fisheries is the

change in total catch over time. One of the most important issues

in constructing an economic index number for cases ~uch as these

is to account for intertemporal changes in the relative composition

of this bundle, that is, the changes in species mix. When output

(species) prices change relative to one another, fishermen alter the
individual species (output) composition of their catch (aggregate

output). An index number that does not properly incorporate these

intertemporal changes in output composition into the aggregate

measure becomes increasingly biased over time, that is, the errors

in approximation increase.

The different index numbers approximate changes such as these

in different ways, and thus have different degrees of accuracy. When

intertemporal changes (in productivity or output and input prices

and quantities) are relatively small, the different indices all provide reasonably accurate and similar approximate scalar measures

of these changes. Typically, the larger the changes over time, the

more the measures from different indices diverge from one another.

This departure occurs because the indices provide approximations

in differing ways.

Four economic index numbers are commonly applied: Laspeyres,

Paasche, Tornqvist, and Fisher Ideal. These indices correspond to

different methods of approximation (reflected in the formulae of

their aggregator functions) with correspondingly different properties. The Laspeyres and Paasche indices have tradionally been widely applied, but the Tornqvist and Fisher Ideal are increasingly used.

Laspeyres and Paasche indices-The Laspeyres and Paasche

indices are the most widely used. In forming aggregates, these

indices weight individual outputs or inputs with prices or quantities.

The Laspeyres index for quantities of inputs or outputs may be

written:

where P/ and X/ represent the price and quantity of good i in time

t. Since prices are held fixed at their base time-period levels, the

Laspeyres index indicates how much of the change in value of total

quantity resulted from pure quantity changes. A Laspeyres price

index can similarly be specified in which quantities, used as weights,

are held fixed at their base time-period levels.

The Paasche quantity index for inputs or outputs may be written:

In contrast to the Laspeyres quantity index, prices (rather than quantities) are held fixed at their new levels. The Paasche price index

holds quantities, used as weights, fixed at their current levels.

In summary, the Laspeyres quantity index weights the individual

quantities to be aggregated with base time-period prices; the

Laspeyres price index weights with base time-period quantities; the

Paasche quantity index weights with current prices; and the Paasche

price index weights with current quantities.

The Laspeyres and Paasche indices provide approximations to

intertemporal changes which capture only the two most extreme

classes of changes in the composition of the aggregate: either perfect

or no substitution among the individual elements ofthe aggregate.

(This is because the indices correspond to linear or fixed-coefficient/

Leontief aggregator functions.) If, over time, substitution among

inputs or outputs occurs, the indices can provide biased measures

of the true aggregate either because substitution is not allowed or

perfect substitution occurs.

Tornqvist and Fisher Ideal indices-The Tornqvist and Ftsher

Ideal indices provide more accurate approximations to changes than

the Laspeyres or Paasche indices because intermediate :;ubstitution

5

possibilities are incorporated. The individual components (e.g.,

species) ofthe aggregate (e.g., total catch) do not have to be either

perfect or nonsubstitutes; instead, intermediate substitution possibilities are allow·ed. This occurs because the prices or quantities from

both time periods under comparison enter the index to account for

the possible changes in the mix ofthe inputs or outputs ofthe index.

The Tornqvist quantity index may be specified:

where X/ is the value of the ith price or quantity in time k, S/ is

the share of total revenue (cost) in time k of output (input) i, In

denotes natural logarithm, and IIi is the product operator. Revenue

or cost shares are used as weights.

The Fisher Ideal index is the geometric mean of the Laspeyres

and Paasche indices, and may be written as:

In order to empirically assess the differences among the Laspeyres, Paasche, Tornqvist, and Fisher Ideal index formulae, this

study computes all four types of output indices in each ofthe four

regions and for all regions combined. Part 2 provides additional

theoretical discussion on this general topic.

Chain and fixed-base indices

Indices may be formed by either the chain or fixed-base methods.

The fixed-base procedure directly compares all changes in prices,

quantities, or total-factor productivity to some initial base period.

The base period may remain constant or may be changed after some

period of time.

Fixed-base indices can be expressed in general form as:

where POt represents the fixed-base index comparing price in time

t with that of base time 0, Pt is price in time t, Po is price in the

initial time 0, and the prices are calculated by some index-number

formula (e.g., Laspeyres or Tornqvist).

Chain indices directly compare adjacent observations in a sequence ofindex numbers. Nonadjacent observations are compared

indirectly by using the intervening observations as intermediaries.

This practice results in transitive comparisons. The general form

of the chain index can be written:

where each individual term, P,}, is computed by the index-number

formula used, and represents the change from time period i to time

period j [i <j]. POtch thus compares output in time t with output

in time 0, the base time period. This formula reflects the basic

relationship:

Since all values are represented in terms of the reference period °

(in this study, 1981), comparisons between adjacent time periods,

say i and i-I, are achieved with the following formula:
where a< i-I < i < t.

The values used to weight the individual quantities or prices (aggregated into a composite quantity or price) are kept up-to-date in

the chain index, while the fixed-base index compares time periods

for which the weights can be very different. As producers change

their production patterns in response to changes in relative price,

fixed-base indices maintain weights which may have changed and

are no longer representative of current output or input mixes. Chain

indices are generally preferred on a priori grounds for these reasons.

This study empirically assesses the fixed-base and chain indices

for total output in each region and the entire fleet. Additional

methodological discussion is provided in Part 2 for interested

readers.

Bilateral and multilateral indices

Two basic types of indices can be used, bilateral and multilateral.

Bilateral indices provide intertemporal comparisons of total-factor

productivity (TFP) for any region or interspatial comparisons

between regions for any given time period. Because of the large

number of possible binary combinations which are not necessarily

transitive, bilateral indices are inappropriate for comparisors that

are not binary (e.g., TFP in region i in time t with TFP of region

j in time t+ I).

The Tornqvist bilateral index of total-factor productivity can be

written:

where k and l are adjacent time periods (or regions), the Yij are

output indices for output i of time j, the Xij are input indices for

input i of time j, the Rij are product revenue shares, and the Wjj

are input cost shares.

Multilateral indices have been proposed by Caves et al. (1982a)

to provide transitive comparisons in a multilateral setting. Transitive comparisons are achieved by making all possible binary comparisons in terms of the geometric mean of all observations. For

example, any two regions in different time periods are compared

with each other by comparing both with the geometric mean.

Multilateral indices directly compare adjacent and nonadjacent

observations but only by destroying the fixity of historical comparisons. As additional observations are added over time, thereby

expanding the set of comparisons, the multilateral index changes

because the geometric mean of the observations changes. In contrast, bilateral indices do not directly compare nonadjacent observations and the historical comparisons remain intact.

The Tornqvist multilateral index for total-factor productivity

(TFP) may be written (Caves et al. 1982a):

InTFPk - InTFP1 = Lj 0.5 (Rjk + Rj*)[1nYjk - InY;']

- L; 0.5 (Ri/ + Rj*)[lnYi/ - InY;']

- L; 0.5 (W;k + W;*)[IoXjk - loX;']

- L; 0.5 (Wi/ + Wj*)[loXi/ - loX;'],

where an asterisk associated with a variable indicates the arithmetic

mean, and an an apostrophe indicates the geometric mean.

6

This study empirically evaluates the bilateral and multilateral procedures for all of the indices computed. The Tornqvist formula is

used because it is the only one for which the theoretical properties

of both the bilateral and multilateral indices have been examined.

Additional discussion is provided in Part 2.

Economic performance index

Norton et aI. (1985) present an index of industry economic performance which attempts to measure economic performance over time

or space. The index recognizes that economic performance could

change due to changes in prices as well as productivity. The index

incorporates the effects upon industry of economic performance

of real prices for aggregate output, aggregate input, and total-factor

productivity .

The economic performance index in general form may be written:

EPkl = TFP (Pd / pJ),

where TFP again refers to total-factor productivity, P Y refers to

an aggregate-output price index, px refers to an aggregate-input

price index, and k and l refer to time periods or regions.

This study provides economic performance indices using the

Tornqvist direct-chain index ofTFP and the Tornqvist implicit realprice chain indices of agregate output and aggregate input. Both

bilateral and multilateral indices are formed.

The economic performance index is developed through an ad hoc

procedure and the theoretical properties of the index are unknown.

For example, when productivity is increasing (decreasing) and the

price ratio is widening (narrowing), the results are unambiguous:

industry economi; performance is improving (declining). However,

when productivity is increasing (decreasing) and the price ratio is

narrowing (widening), the overall effect is not clear. Instead, the

systematic properties of the index require further analysis, and

results should therefore be treated as preliminary.

Data and index construction _

Introduction

This section provides a description of the sources and methods used

in the construction ofthe panel (pooled cross section and time series)

data set used for analysis of the Pacific trawl fleet.

Output indices

The output indices are developed for U.S. vessels with landings

in Washington, Oregon, and northern and cental California which

harvest in the fishery conservation zone for years beginning in 1981.

The Washington landings exclude fish harvested in Puget Sound

but do include vessels harvesting in the fishery conservation zone

and landing in Puget Sound ports. The northern California or Eureka

region includes landings in ports in the Crescent City, Eureka, and

Fort Bragg areas. The central California or Monterey region includes landings in ports in the Bodega Bay, San Franci~co, and

Monterey areas. Nine species assemblages are specified: Dover sole,

Petrale sole, other flatfish, rockfish, Pacific cod and ling cod,

sablefish, pink shrimp, and Pacific whiting.
The output indices require total dollar value and total pounds of

landings for each geographical region and each year for each species

of concern. The revenue and catch data are from the PacFIN

Management Database. All species and market categories within

each ofthe nine species groups are linearly aggregated. The Pacific

whiting species category includes fish harvested by both domestic

and joint-venture vessels. Joint-venture whiting revenue and landings data are apportioned between Washington and Oregon in the

Columbia area according to the home ports of the joint-venture

whiting vessels. Revenue shares by region are reported in Table 17.

Input indices

Three major categories of inputs are distinguished: labor, capital,

and fuel. The share or proportion of each input in total costs by

region is reported in Table 33.

Labor index-The labor input indices are constructed from three

categories oflabor: ordinary crewmember, engineer, and captain.

Total crew size is currently unavailable for the fleet, but since most

vessels have a total crew of three (captain, engineer, and ordinary

seaman), all vessels are assigned this crew size. Greater refinement

will be possible as more information becomes available. As such,

the labor indices presented in this study are strictly preliminary and

may be subject to revision after refinement of the database.

Crew sizes are stock values and alone do not provide a satisfactory measure of the annual flow of labor services. Crew sizes are

converted into annual flows of labor services by multiplying the

number of people in each labor category in each region and time

period by the corresponding measure of fishing time used in the

study, the number of landings (discussed below). The flow oflabor

services is thus in man-landings per year.

Quality adjustments of effective annual flows of labor services

are not possible with the level and extent of data available. For example, Jorgenson and Griliches (1967) adjust for changes in the

quality of labor due to changes in the educational composition of

the labor force. In fishing industries, years of fishing experience

would be desirable. Ideally, the flow of labor services could also

be adjusted for changes in labor efficiency that accompany changes

in intensity of effort or time per person (Denison 1962).

Each labor category is valued at its opportunity cost. 1 This provides an exogenous representation of both remuneration to labor

and food costs. The data sources include:

(1) County Business Patterns for California, Oregon, and Washington, Bureau of the Census, U.S. Department of Commerce,

Wash., D.C.

(2) Area Wage Survey ofGrays Harbor and Pacific Counties 1984,

Labor Market and Economic Analysis Branch, Washington

State Employment Security Department, Olympia, WA 98504.

(3) Oregon Occupations, 1984-85, Oregon Career Information

System, University of Oregon, Eugene, OR 97403.

(4) Area Wage Survey, Bureau of Labor Statisitics, U.S. Department of Labor, Wash., D.C.

'In economics, the tenn "opportunity cost" describes the cost of undertaking a particular activity in tenns ofthe foregone benefits of the next-best alternative. Thus the

opportunity cost to capital owners is usually specified to be the interest they could

receive if they placed this money in a financial institution. Similarly, the opportunity

cost to a fishennan is the wage foregone (or opportunity lost) to this person by not

working in the next-most-rewarding alternative.

7

(5) Area Planning 1nformation, Employment Development Department, State of California, Sacramento, CA 95814.

All mean annual incomes used for opportunity costs in all labor

categories are in 1981 dollars after deflation by the GNP implicit

price index.

The opportunity cost of crew labor is an economic measure based

upon reported earnings of workers in the counties in which the

trawlers are homeported. Data on mean annual income for ordinary

crewmembers are from source (1) above, where the opportunity

cost is assumed to be the mean wage earned in manufacturing,

transportation, and retail trade sectors. Captains are assumed to have

managerial and entrepreneurial skills which imply a higher opportunity cost than for ordinary crewmembers. For lack of any specific

alternatives, captains are given an opportunity cost 20 percent higher

than ordinary crewmembers. The same data source is used as for

ordinary crewmembers.

Vessel engineers are assumed to have an annual opportunity cost

equivalent to the wages of an auto mechanic in their home ports

(complete data are not available for the preferred category of diesel

mechanic). Hourly wage rates for individual California ports are

obtained from source (5). Since these wage rates for experienced

journeymen auto mechanics are given only in ranges, the midpoint

of each year's range is selected. Because data prior to 1984 are

usually absent, these rates are assumed to change year-to-year at

the same proportional rate of change as in San Francisco, where

more timely data are obtained from source (4). Hourly wage rates

for Oregon coastal auto mechanics are obtained for 1984-85 from

source (3), and are assumed to change over time at the same rate

as Portland auto mechanics, from source (4). Hourly coastal

Washington auto mechanic rates are obtained for 1984 from source

(2), and are assumed to change at the same rate as Seattle wages,

from source (4) for previous years. Auto mechanics are assumed

to work 40 hours per week, 50 weeks per year.

The home port of each vessel in every year is not known because

home ports are obtained from the PacFIN Research Database, whose

timeliness lags the annual vessel inventory used to compile annual

numbers of vessels by region. Moss Landing is selected as the

representative port for central California, Crescent City for northern California, Newport for Oregon, and Westport (Grays Harbor)

for Washington.

In order to compute a single index of real labor input services

for each region, a Tornqvist multilateral index is employed:

InLk - InLl = IL 3 0.5 (Wik + W;*)[lnLik - InL/]

- IL 3 0.5 (Wj{ + Wi*)[lnLj{ - InL/].

where Lik is the quantity of labor services in the ith labor category

for the kth (time-differentiated) region, Lk is the aggregate index

of labor input for the kth region, W;k is the compensation share

for category i in region k, W;* is the arithmetic mean over all

regions and time periods of compensation shares for category i,

and L;' is the geometric mean of the number of labor services in

category i over all regions and time periods. A Tornqvist bilateral

labor index is also constructed for use in the Tornqvist bilateral

TFP and aggregate input indices. Tornqvist multilateral and bilateral

chain aggregate labor indices are reported in Table 29 and Table

30, respectively.

Fuel index-Fuel consumption rates are estimated following an

economic-engineering procedure. Annual fuel cost data for 120
vessel-years covering the 1981-83 period are divided by port-specific

mean annual prices for cash purchases of NO.2 marine diesel fuel

for 400-gallons. The estimate of annual fuel consumption for each

vessel is then divided by its respective number of landings to give

its mean fuel consumption per landing by region. Average fuel consumption per landing is tl:ien calculated for all vessels in each region.

Each region's total annual fuel consumption is then derived by

multiplying the regional mean fuel consumption per landing by the

total number of landings in each year for that region. All vessels

are assumed to use diesel fuel rather than gasoline. All prices are

deflated by the GNP implicit price index to provide constant 1981

dollars. Fuel cost data are from confidential federal financial

statements.

Annual diesel fuel cash prices are for 400 gallons of No.2 marine

diesel fuel. The 1981-83 port prices were obtained in the following manner. First, 1985 prices from marine fuel docks in each sample port were obtained by telephone interviews in February, May,

and November with operators of marine fuel docks from 31 ports

in Washington, Oregon, and California. These 1985 prices are

averaged and deflated to 1981 levels by the GNP implicit price index, and are assumed to vary over time at the same rate as diesel

fuel prices at petroleum terminals in San Francisco, Portland, or

Seattle reported in Platt's Oilgram Price Report: An International

Daily Oil/Gas Price and Marketing Letter (McGraw-Hill). It is

reasonable to assume that individual port prices follow prices at

major oil terminals, since marine-fuel dock port prices are essentially established on a formula basis from the terminal prices.

Regional Tornqvist bilateral and multilateral chain indices fOT fuel

are reported in Tables 29 and 30, respectively.

The total number of landings for Washington vessels of U. S.

ownership fishing in the fishery conservation zone and landing

anywhere in Washington is from the Washington Department of

Fisheries, Olympia, WA 98504, while the number of landings for

the other three regions is from the PacFIN Management Database.

Joint-venture vessels' fishing time is calculated in weeks offishing,

where the beginning and ending dates of each vessel's fishing season

are obtained from logbooks for the years 1981-84. The total number

of days for each vessel is then divided by 7. When more complete

information becomes available, the number of actual days fished

can be taken from the logbooks rather than the beginning and ending

dates of a season.

Capital index-The quantity of capital actually used in production

is not the stock of capital (e.g., the number of vessels) but the flow

of productive services from this capital stock. Thus more services

for production are available from vessels actually fishing than from

the same vessels tied up in ports.

The price of these capital services is a rental price for capital

services on organized markets (e.g., tool rental). When capital

services are not exchanged on markets, costs are imputed to firms

to reflect the opportunity cost to capital owners of their money tied

up in the capital stock and the depreciation of the capital equipment (Jorgenson 1974).

The capital services price per vessel for any given year (t) and

size class (j) is given by:

where PAj is the mean vessel acquisition price per vessel in size

class j, r, is the opportunity cost of capital in time t, and d, is the

depreciation rate. Depreciation measures the present value of all

future declines in productive capability.

8

This capital services price is an imputed price, and provides an

accurate measure of the economic value of capital services to capital

owners when the stock of capital is in full static equilibrium in any

given time period. That is, the actual capital services are equal to

the optimum flow ofservices, and firms are capable of making the

required adjustments to their stock of capital in order to attain the

optimum amount in each year. When, for example, the existing

capital stock is inadequate relative to demand, firms face a relative

shortage of capital and have incentives to invest. An additional unit

of capital then has an economic valuation greater than that measured

by the capital services price. Alternatively, when the stock of capital

is greater than that required for full equilibrium in any year, firms

have a relative surplus of capital stock and incentives to disinvest

or even leave the industry. In this case, the economic valuation of

capital services is lower than the measured imputed price. This study

assumes that capital is in full static equilibrium and that the imputed price of capital services accurately measures the economic

value of these services. 2 The value of all capital services is also

assumed equal to the sum of the values of the individual capital

services (Christensen and Jorgenson 1969, 1970). Aggregation from

the firm to the industry is therefore assumed possible.

PAj is expressed in 1981 prices, and is the mean vessel acquisition price per vessel from vessels purchased 1976-82. Of the total

106 vessels used to calculate these vessel acquisition prices, 15 are

class I (1-49 registered feet), 81 are class II (50-74 registered feet),

and 10 are class III (75+ registered feet). The vessel acquisition

prices are from confidential financial statements. Stable and consistent functioning of capital markets and industry expectations are

assumed after the Magnuson Fishery Conservation and Management Act was announced, but no changes are assumed after this

time. Limited data require the assumption of no capital gains or

losses. The relatively limited number of years for vessel acquisition prices mitigates the effects of capital vintage. Property taxes

are not applied to fishing vessels on the U.S. west coast, and are

therefore not included in the capital services price.

The opportunity cost of capital in any year, r" is assumed equal

to the annual corporate bond rate on seasoned issues rated BAA

by Moody's, reported in the Federal Reserve Bulletin of San Francisco. The annual depreciation rate is set at 7 percent, as suggested

by the Southwest Regional Office of the National Marine Fisheries

Service, NOAA, Terminal Island, California, and roughly (but not

exactly) corresponds to straight-line depreciation with a 15-year

economic life and zero scrap value of the vessel and gear.

The price of capital services is used in constructing the index

of capital services for the fishery. Constructing this index requires

weights that reflect the annual capital cost. The annual capital cost

of vessels in each region and year for each length class is estimated

as the product ofthe number of vessels and the annual price of capital

services per vessel. The capital services price is assumed constant

across regions due to the general mobility of vessels.

The index of capital services also requires annual quantity flows

of capital services. The first step in measuring the flow of capital

services is to collect annual vessel counts. These are compiled by

the National Marine Fisheries Service at Terminal Island, California (Korson 1981-85). Northern and central California vessels since

1985 are assumed to have the same home ports on a percentage

'Current research is in the process of incorporating variations in capacity utilization

of capital following the procedure of Hullen (1986) and Berndt and Fuss (1986). This

procedure adjusts the price of capital services rather than the stock of capital. The

same research is accounting for changes in population abundance ofthe resource stock

and variations in its catchability.
basis as those of the 1984 vessels in the PacFIN Research Database

in La Jolla, CA. Three vessel size-classes based on Coast Guard

registered length are distinguished: 1-49 ft; 50-74 ft; 75+ ft. The

assignment of home states for a few vessels in the annual vessel

inventories is inconsistent with the home port classification of the

PacFIN research database, in which case the PACFIN assignment

is followed.

The annual vessel counts are stocks of capital potentially available

for productive purposes in any given year. This study assumes that

all vessels fully utilize the potential productive capacity available

in each year, that is, the firms are at full static equilibrium in capital.

Capital services are therefore assumed to be proportional to capital

stocks. Ongoing research will relax this assumption.

The index of real capital services is aggregated over the three

vessel length-classes by the Tornqvist multilateral index formula:

InKk - InK[ = ,23 0.5 (Wik + Wj*)[lnKik - InK/]

,23 0.5 (Wit + Wj*)[lnKit - InK/].

where Kik is the number of vessels in size class i for the kth (timedifferentiated) region, Kk is the aggregate index of annual capital

services for the kth region, K;' is the geometric mean of the

number of vessels in category i (over all regions and time periods

in i), Wik is the kth region's share of total annual vessel capital

cost attributed to vessels of type i, and W;* is the arithmetic mean

of annual capital costs of vessels in class i (over all regions and

time periods in i). Tornqvist bilateral indices are also constructed

for use with the Tornqvist bilateral aggregate input and TFP index

numbers. Tornqvist multilateral and bilateral capital chain indices

are reported in Tables 29 and 30, respectively:

Empirical results _

Introduction

This section has three objectives: (I) Review the empirical results

and relate them to industry events; (2) evaluate empirically the different types of index procedures; and (3) assess the sources of

available data.

The empirical results are reported in Tables 6 through 36. These

tables include annual indices by region and fleet of total-factor productivity (TFP), TFP growth rates, aggregate output, each individual

output, aggregate input, each individual input, implicit prices, and

an index of industry economic performance in the spirit of Norton

et al. (1985). Revenue and cost shares are also reported. Tornqvist

multilateral and bilateral chain indices are reported for all categories

in order to evaluate the multilateral and bilateral indexing approaches. Laspeyres, Paasche, Fisher Ideal, and Tornqvist fixedbase indices for aggregate outputs are reported in Tables 21-24 in

order to empirically evaluate fixed-base versus chain indices and

to evaluate the four different types of index formulae.

The growth-accounting results presented here assume constant

returns to scale, full capacity utilization, technical efficiency, and

marginal cost pricing. The empirical results are strictly preliminary

and may be subject to revision because crew sizes and detailed vessel

counts require additional refinement.

9

The Pacific trawl fleet

The bilater~ total-factor productivity (TFP) indices are reported

in Table 7. The empirical results indicate a substantial decline in

productivity during 1982 and 1983 compared with 1981. Total-factor

productivity then grew in 1984 and 1985. Fleet TFP grew at an

average growth rate of nearly 0.38% for the industry over 1981-85

with particularly strong growth for 1984-85 (Table 9).

The 1982-83 multifactor productivity decline may largely be due

to the harvesting of widow rockfish at levels beyond estimates of

maximum sustainable yield and the decline ofthe pink shrimp fishery

(Table 13). Rockfish and pink shrimp are the two most important

species groups by share of total revenue (Table 16). Fleet aggregate

output as a whole declined by 1.3% in 1982 and 0.47% in 1983

(Table 15). General increases in aggregate input also contributed

to the 1982-83 decline in TFP (Table 26). Industry aggregate input

usage grew by 8.73% in 1982 and 1.28% in 1983 (Table 28).

The TFP rise in 1984 and 1985 (Tables 7 and 9) can be attributed

to an important decline in aggregate input usage from the 1983 level

(Table 28). Industry aggregate output declined by 0.46% in 1984

and grew by 2.65 % in 1985 (Table 15). No individual species appears to dominate the general rise in aggregate output, although

pink shrimp landings were up (Table 13), and in 1985 pink shrimp

constituted 29 % and 22 % of total revenue in Washington and

Oregon, respectively (Table 16), and 16% of fleet revenues (Table

18). Fleet aggregate input declined by 11 % in 1984 and by 1.5%

in 1985 (Table 28).

Capital is an important component of aggregate input in terms

of cost share (Table 33). Table 30 reports individual capital indices.

Capital began to decline in Washington in 1983, while northern

and central California experienced increases in capital in 1983 before

a decline in 1984. Vessels may have transferred from Washington

and Oregon to California in 1983. A number of vessels may also

have left the active fleet due to sinkings, burnings, or fmancial difficulties. Other vessels, particularly larger ones, are known to have

transferred fishing activities to Alaska.

Additional factors may have contributed to the 1984-85 rise in

TFP. It is likely that the vessels leaving the fishery were relatively

inefficient harvesters, in which case TFP would increase if many

ofthe remaining vessels were operated by skippers and crews with

fishing skills superior to those vessels that left the active fleet. The

larger vessels that transferred to Alaska may have been more inefficient than medium and smaller vessels under the reduced level

of resource abundance. This could be due to decreasing overall

returns to scale as stocks, particularly widow rockfish, declined.

As those larger vessels left the fleet, productivity measures of the

remaining vessels should have increased. Productivity may also have

increased as fishermen became more skilled with the technological

innovations previously introduced in the production process (an in-·

crease in technical efficiency).

Regional differentials in total-factor productivity growth are

demonstrated by the bilateral total-factor productivity indices

reported in Table 7. By 1985, total-factor productivity for the entire fleet, and in Washington and Oregon surpassed 1981 levels,

but not in northern and central California. Tables 14 and 26 indicate that the 1985 aggregate output level is below the 1981 level

in Oregon and northern California regions and for the fleet, but

that important regional variations exist in input usage. While input

usage in Washington and Oregon remains well below 1981 levels,

and in fact continued to decline through 1985, input usage in California is still above the 1981 level.
Fleet economic performance depends upon the real prices of outputs and inputs in addition to total-factor productivity. Tornqvist

bilateral implicit chain indices for constant-dollar aggregate output prices and aggregate input prices are reported in Tables 20 and

32, respectively. 3 By 1985, aggregate output real prices increased

above 1981 levels for the entire fleet and northern and central

California while remaining below 1981 levels for Washington and

Oregon. By 1985, aggregate input real prices for the fleet had remained below 1981 levels, while the 1985 aggregate input prices

rose above the 1981 level for Washington. The general increase

in aggregate output price and a general decline in aggregate input

price by 1985 suggest an improvement in the ratio of product price

to input price for the Pacific trawl fleet which reinforced the recent gains in TFP. On the whole, the economic performance of

the fleet should have returned to 1981 levels although economic

conditions for individual vessels may differ.

Economic performance

The economic performance index of Norton et al. (1985) attempts

to combine indices ofprices and productivity into a single measure

of fleet economic performance. Tornqvist bilateral chain economic

performance indices are reported in Table 36. As discussed above,

the results should be interpreted with caution due to the uncertain

theoretical basis of these indices.

The indices indicate that overall fleet economic performance in

1985 is above that of the initial time period, 1981. Reinforcing recent productivity gains are the general increase in aggregate output price and general decline in aggregate input price as noted above.

Methodological evaluation

Multilateral vs. bilateral indices-Comparisons of mutlilateral and

bilateral indices are made for the Tornqvist chain indices. These

indices are computed for total factor productivity (TFP) in Tables

6 and 7, aggregate outputs in Tables 11 and 14, individual outputs

in Tables 10 and 13, aggregate inputs in Tables 25 and 26, and

individual inputs in Tables 29 and 30.

All of the multilateral indices are normalized in terms of 1981

Washington by dividing all values by the value of 1981 Washington.

Setting 1981 Washington equal to 1.00 provides a more convenient

basis for making comparisons. All relative relationships are preserved by normalization. Multilateral indices for the fleet are similarly normalized in terms of the 1981 value.

The computed Tornqvist multilateral and bilateral chain indices

generally differ little from one another in tracking turning points

(i.e., increase to decrease or decrease to increase) and trends. For

example, both sets of TFP indices (Tables 6 and 7) indicate productivity declines in 1982 or 1983 and agree with the occurrences

of all increases and decreases. This coincidence of turning points

occurs because the growth rates of multilateral and bilateral indices

generally differ little in magnitude.

Similarity in growth rates can be demonstrated for the most important index, the total-factor productivity index, by regressing the

1982-85 multilateral growth rates upon the 1982-85 bilateral growth

'hnplicit prices are calculated through Fisher's weak-factor reversal test. This relationship states that the product of the price index and the quantity index should equal

the expenditure ratio between the two time periods. Part 2 and Appendix 3 provide

additional discussion.

10

rates with an intercept term. Using this result, it is possible to (1) test

to determine whether the intercept term is significantly different

from zero and thus whether one set of growth rates over- or understates growth by a fixed percentage relative to the other; (2) test

to determine whether the slope coefficient is significantly different

from unity as a check on the proportionality of one set of growth

rates to another; and (3) examine the R2 as a measure of the linear

proximity ofthe two sets of growth rates. Regressing the multilateral

TFP growth rates upon the bilateral TFP growth rates produces

the expected result: the intercept is not significantly different than

zero, the slope is not significantly different from unity, and the R2

is very high. (Intercept coefficient = 0.005, S.E. 0.005; slope coefficient = 1.054, S.E. 0.031; R2 = 0.984, and F-statistic for

overall regression = 1123.004.)

Comparability between the multilateral and bilateral indices is

diminished at the highest levels of aggregation, because relatively

minor differences for individual outputs and/or inputs begin to

accumulate at higher levels of aggregation. For example, proportionately greater differences are exhibited among TFP indices than

among individual outputs or inputs. Thus [he regional individual

species indices (Tables 10 and 13) track individual outputs by region

in a similar manner, but the 1983-84 indices for the fleet (Tables

11 and 14) differ (the multilateral index indicates a small increase,

whereas the bilateral index indicates a slight decrease).

Although the multilateral and bilateral indices nearly always agree

on turning points and usually agree on trends and growth rates,

the magnitudes of the indices relative to the initial year (1981) can

nonetheless differ in important ways.

To formally compare the magnitudes of multilateral and bilateral

TFP indices (rather than growth rates), the multilateral TFP index

is regressed upon the bilateral TFP index for the years 1982-85

with an intercept term. The estimated intercept coefficient is 0.319

(S.E. 0.250), the estimated slope coefficient is 0.577 (S.E. 0.266),

the Rl is 0.208, and the overall F is 4.707. These results suggest

that a fixed displacement between indices does not exist but that

the multilateral TFP measures tend to be about 40 percent lower

than the bilateral TFP measures. The R2 of 0.21 suggests that an

additional 79 percent of the variation in the multilateral index

estimates exists after the 40 percent proportionality difference has

been accounted for (Hazilla and Kopp 1984a,b).

The principal reason for the difference in magnitude relative to

the initial year lies in interpretation of the index. The multilateral

index for the initial year represents deviations from the geometric

mean (of all regions and years), while the bilateral index for the

initial year is the constant value 1. That is, the multilateral and

bilateral initial values can differ considerably because of the different initial-period magnitudes. This difference is accentuated if

the initial year differs markedly from the geometric mean of all

years and regions. Both base year and chain multilateral indices

are affected by the difference in interpretation of the initial time

period.

The computed multilateral and bilateral indices also differ markedly in the years 1982-85 when an intertemporal change occurs which

is substantially different from the geometric mean and the value

of the preceding year. Consider the Pacific whiting multilateral and

bilateral output indices. The Pacific whiting catch from joint-venture

vessels is relatively large in total tonnage and can vary considerably

from year-to-year and region-to-region.

The computed multilateral and bilateral indices can also differ

in an important way when very small intertemporal chariges occur.

The bilateral index can track the small change but the multilateral

index can fail to pick up a change in trend from increasing to
decreasing or vice versa, because all ofits comparisons are in tenns

of the geometric mean. Thus the multilateral procedure tracks

turning-points most effectively when the values are closer to the

geometric mean, but may experience difficulties when all the values

are substantially different from the geometric mean.

In summary, the multilateral index has superior theoretical properties to the bilateral index, but can demonstrate empiricallirnitations in relatively extreme situations. Fundamentally comparable

results are demonstrated with the two indices. However, for official

reports likely to receive widespread distribution, the bilateral procedure should be used since fixity of historical comparisons remains

intact (explanations are not required when different numbers occur

in subsequent years) and interpretations are easier.

Chain vs. fIXed-base indices-As noted above, fixed-base indices

compare all changes to some initial base period, while chain indices make comparisons by a process of chaining binary (periodto-period) comparisons back to the original time period (in this case,

1981).

Table 14 reports Tornqvist bilateral chain indices oftotal output

by region, and Table 24 reports Tornqvist bilateral fixed-base indices

of total output by region. Comparison between the two tables

indicates that very different results and types of information are

provided by the two approaches. For example, the fixed-base indices

indicate that 1985 total output is less than 1981 in all regions, while

the chain indices indicate that 1985 total output is greater for

Washington and central California. This difference may be due to

the changing species composition of catch over time (see Tables

16 and 18) with which the fixed-base index procedure has difficulty in dealing. Although the choice between the two procedures

is somewhat dependent upon the type ofinformation to be presented,

the chain procedure is generally preferred on theoretical grounds.

Laspeyres, Paasche, Fisher Ideal, and Tornqvist indices-As

discussed above, a number of different index-number procedures

exist, each index corresponding to different functional fonns ofthe

aggregator function and consequently each index number having

different theoretical properties. The Laspeyres and Paasche indices

implicitly assume either no or perfect substitution between individual

commodities, while the class ofsuperlative indices does not require

commodities to be perfect or zero substitutes. Changes in the composition of an aggregate are therefore correctly captured.

To facilitate an empirical evaluation of these indices, fIXed-base

aggregate-output bilateral indices are developed for each region.

Laspeyres indices are presented in Table 21, Paasche indices in

Table 22, Fisher Ideal indices in Table 23, and Tornqvist indices

in Table 24. An examination ofTables 21-24 indicates that different

results are possible, depending upon the choice of index number.

Consider, for example, the years 1983-85 for northern California:

The Laspeyres and Fisher Ideal indices report a decline in landings

from 1983 to 1984, with an increase from 1984 to 1985. In contrast, the Paasche and Tornqvist indices report continual increases.

The Fisher Ideal index lies between the values.of the Paasche

and Laspeyres indices as expected, since the Fisher Ideal index is

the geometric mean of the Paasche and Laspeyres indices. Values

of the Fisher Ideal and Tornqvist indices do differ.

As the degree of data disaggregation increases (so that a quantity may become zero), the Fisher Ideal index number formula

remains well defined while the Tornqvist does not (because of the

log transformation which is undefined when the untransformed

11

variable is zero). The consequence of this property is clearly

demonstrated in Tables 10 and 13 for Pacific whiting landings in

Washington. In 1981, no landings of Pacific whiting were made

in Washington by either the coastal fleet or joint-venture vessels.

However, large landings in 1984 were reported by joint-venture

vessels homeported in Washington (with landings consequently

assigned to Washington). To allow a Tornqvist index number to

be calculated for 1984, a small value is given to 1981 (an ad hoc

procedure), but an enormous value is calculated for the Tornqvist

index. This result carries over to the 1984 fleet value, where the

Fisher Ideal (Table 23) indicates a decrease from 1983 to 1984 and

an increase from 1984 to 1985. In contrast, the Tornqvist (Table

24) reports just the opposite result.

Recommended index-number procedure-The recommended

index-number procedure for analysis with a widespread and

disparate audience is either the Tornqvist or Fisher Ideal bilaterllJ

chain index. The major advantage of the Fisher Ideal is that it is

well defmed when the data are so highly disaggregated that a zero

output or input occurs, while the Tornqvist is not. Chain indices

are preferred to fixed-base indices. Multilateral indices have greater

versatility than bilateral indices, but do not have fixity of historical

comparisons. The Tornqvist multilateral chain index can be quite

suitable for technically sophisticated audiences.

Data evaluation

The data currently available are sufficient for potentially satisfactory construction of productivity, quantity, and price indices for

outputs and inputs. Nevertheless, certain limitations exist, many

of which can be corrected or at least mitigated with time.

The PacFIN Management Database provides timely data for

revenues and quantities of outputs. Fuel prices and interest rates

are readily obtainable on a consistent and timely basis through

telephone surveys and various issues ofthe Federal Reserve Bulletin,

respectively. Wage and income data used to construct the opportunity cost of labor are available, but there is a 1-2 year lag in the

most recent data of the Bureau of the Census. Nevertheless, this

data lag does not do great harm to the analysis. Crew-size information for all vessels is currently incomplete, and scope for updating and refmement certainly exists.

Cost data from the NMFS confidential financial-cost database are

adequate for the task at hand. The cost sample is not comprehensive for all vessels, nor is it systematically derived on the basis of

sampling theory. Yet the rather large sample does provide a

reasonable degree of confidence in its adequacy. This confidence

should improve as the cost data are updated on a continual basis.

The Pacific whiting joint-venture data are among the least satisfactory of all the data currently available. The fishing-time information needs to be updated from logbooks and from confidential

sources.

Refinement ofthe concept of a fisherman's opportunity-cost and

better information about the most likely alternatives would improve

the analysis. Ifthe information was available, capital gains or losses,

Le., the revaluation of assets, could be important in computing the

real cost of capital services (even though most capital gains are not

realized). Little is known about the actual rate of economic depreciation of capital assets.

The use of an engineering approach to measuring fuel consumption may also be subject to limitations since the effects of changing
economic conditions might not be fully incorporated into this procedure. Fuel consumption and more accurate fishing-time data may

become available from logbooks. The quality of the analysis might

improve in this case, and ~e engineering approach to measuring

fuel consumption can be replaced by direct measurement. Finally,

the absence of prices and the spotty and incomplete data on

materials, supplies, ice, and other trip costs preclude their use in

the analysis. In tum, this may lead to some form of omitted variable

bias.

In summary, sufficient data of acceptable quality are available

to construct output, input, and productivity indices. These data

require additional refinement, upgrading, and updating to improve

the quality of the analysis, particularly for crew sizes and vessel

homeports. As such, the empirical results presented in this study

are strictly preliminary, and may be subject to revision after final

refinement of the database is completed.

Concluding remarks _

This study is developed to address five issues of productivity

measurement in the Pacific trawl fleet: (I) Apply the most recent

advances in productivity measurement and the economic theory of

index numbers; (2) utilize more extensive data sources than those

previously used in productivity and performance studies of marine

fishing industries; (3) evaluate the many different index-number

procedures that are currently available; (4) clarify the meaning of

productivity in marine fishing industries; and (5) collect into one

accessible place the recent advances in productivity measurement

and the economic theory of index numbers. Part 2 and its Appendices address task (5).

In conclusion, this study recommends the use of chain indices

rather than fixed-base indices, either the Fisher Ideal or Tornqvist

index, and bilateral indices for recurring publications. The data currently available are adequate for satisfactory construction of productivity, quantity, and price indices. Certain limitations do exist

but are potentially correctable if existing efforts at the Southwest

Fisheries Center (National Marine Fisheries Service, La Jolla, CA)

aimed at expanding and updating economic bases are continued

along their current lines. These current efforts include expanding

and developing the PacFIN Research Database, logbook data, the

cost database, and input prices.

12

PART 2

Additional topics in productivity measurement _

This section considers several additional aspects of productivity

measurement to supplement the general framework developed in

Part I. This section proceeds by first considering measurement of

productivity by the growth-accounting framework using economic

index numbers and by the structural framework using econometric

methods. Next, the assumptions behind the growth-accounting

framework are reviewed, followed by a discussion of interspatial

and intertemporal productivity, and a review of duality-based

measures of productivity.

Growth-accounting and structural frameworks

Total-factor productivity can be interpreted and measured by either

the growth-accounting framework with economic index numbers

or by the structural framework using econometric methods. The

structural approach measures productivity change as the rate of

technological progress when measured by a production, cost,

revenue, or profit function. It also allows detailed examination of

the structure of production, including measurement of economies

of scale, tests for full static equilibrium (Kulatilaka 1985, Schankerman and Nadiri 1986), technical inefficiency (Nishimizu and Page

1982), and corrections for deviations from competitive markets and

marginal cost pricing (Denny et al. 1981). This detailed information requires estimation of an econometric model.

The structural approach is parametric and global, since it can yield

information about the full range of the estimated aggregate production or cost function and requires parametric specification of

this function. The growth-accounting framework is nonparametric

since it is based on Divisia indices ofmultifactor input and multifactor productivity. It is also local, since the only information about

the nature of the production technology is embodied in the marginal

productivity conditions. These conditions allow calculation of the

slope of the aggregate production function using only relative prices,

but only along the observed surface of the aggregate production

function. The results are thus local to this observed range (Hulten

1986).

Assumptions of the growth-accounting framework

The growth-accounting framework for measuring total-factor productivity makes several assumptions: Constant returns to scale,

technical efficiency, perfect competition in input and output markets,

and full static equilibrium of all inputs. When these assumptions

are not satisfied, conventional indices of total-factor productivity

growth include not only the effect of technical change, but may

also include some or all of the effects from nonconstant returns to

scale, technical inefficiency, market imperfections, and departures

from full static equilibrium.

Consider first the assumption of perfect competition: This assumption is in part acceptable in marine fishing industries. Most importantly, fishermen are generally pricetakers in the product and factor

markets, but entry and exit may be difficult. Readers interested in

the concept are referred to Kendrick (1973) and Denny et al. (1981).

Consider next the returns to scale: Parametric representation of the
Duality-based measures of productivity

where K(t) and L(t) are the cost-minimizing quantities of capital

and labor, respectively. Under constant returns to scale, this can'

be written:

The growth-accounting model developed in Part 1 measures

multifactor productivity as the residual output not accounted for

by the share-weighted growth rates of the factor inputs. As Ohta

(1975) notes, total-factor productivity can also be equivalently

measured by the residual diminution in average cost not accounted

for by the input prices (and changes in scale economies if constant

returns to scale are not assumed). Although this result is not ofimportance in this study, its general outlines are developed along the

lines of Hulten's (1986) discussion for the sake of completeness

and because so much of the recent productivity literature is

developed in terms of costs.

Under certain regularity conditions (McFadden 1978, Lau 1978),

the existence of a cost function dual to the production function Y

= f(K,L) is implied:

the aggregate-production function, while the multifactor productivity residual is associated with shifts in the aggregate production

function. This interpretation is generally attributed to Solow (1957),

Jorgenson and Griliches (1967), and Hulten (1975). Denison (1962)

and Star (1974) instead stress a diversity of factors which might

be captured in the residual total-factor productivity measure.

Changes in total-factor productivity in marine fishing industries

might also be due, in part, to changes in resource availability and

species composition, since higher levels of resource abundance

should allow any given input-bundle to harvest more outputs.

Changes in resource availability might also impact upon capacity

utilization of a quasi-fixed factor such as capital and upon the rate

of fuel utilization.

Interspatial total-factor productivity has a somewhat different

interpretation than intertemporal total-factor productivity. Interspatial total-factor productivity can be defmed as the proportional

differences in an index of outputs between different production

regions (or firms) relative to the proportional differences in an index

of inputs. Interspatial productivity differences arise not from the

dynamic process of technological change, but rather from static differences in technology across producing regions (or firms). Interspatial productivity differences in marine fishing industries are also

likely to directly reflect differences in resource availability and composition (and indirectly through economies of scale).

C[pK(t), PL(t), Y(i),t] (1)

= pK(t)K(t) + PL(t)L(t),

C(t)

production technology is necessary to identify and estimate the

separate effects due to scale economies and technical change (Diamond et al. 1978, Chan and Mountain 1983). Once a measure of

nonconstant returns to scale is available, the TFP index can be adjusted by dividing by the measure ofscale economies (Caves et al.

1982b). Alternatively, James Kirkley (Va. Inst. Mar. Sci.,

Gloucester Point, VA 23062, pers. commun. Feb. 1987) suggests

that the nonparametric method of Diewert and Parkan (1985) can

be used to examine the effects of scale economies and technical

change (although only the lower bounds can be examined).

Nishimizu and Page (1982) note that a distinction should be drawn

between technological change and changes in the efficiency with

which known technology is applied to production. Given a level

of technology, explicit resource allocation may be required to reach

the "best-practice" level of technical efficiency over time. After

the adoption of a new technology (e.g., stem trawling), productivity gains are possible as firms master the new technology over

time. A parametric approach is required to address this issue. The

growth-accounting framework adopted in this study implicitly

assumes that all firms are technically efficient, that is, that production is on the frontier of the industry-production function.

Fourth, consider the effects of vessels not in full static equilibrium

in all of their inputs: If producers are assumed to be in long-run

equilibrium when in fact they may be in short-run temporary

equilibrium, the productivity residual may be systematically

underestimated. (Berndt and Fuss 1986, Hlllten 1986, Winston

1974). The traditional method for TFP measurement is appropriate

only if the firm's output is always produced ai the long-run

equilibrium point, i.e., the point of tangency between the shortrun unit or average total cost curve and the long-run unit cost curve.

Instead, if there are divergences from static equilibrium, the firm

is not operating along the long-run average cost curve, and the conventional measure of TFP includes variations in capacity utilization of the fixed inputs. For example, if a decline in overall resource

availability or a change in its composition (species mix) causes a

temporary equilibrium due entirely to underutilization of harvesting

capacity, then a perfectly competitive fishing firm would not be

in long-run equilibrium and measurement ofTFP might be biased.

The problem of disequilibrium is corrected by one of two

methods. Most frequently, the quantity of the quasi-fixed factor

is adjusted to reflect the degree of capacity utilization (Jorgenson

and Griliches 1967). Alternatively, the price of the fixed factor is

adjusted to reflect its true shadow value, that is, the contributions

of quasi-fixed inputs are valued at their shadow prices rather than

market prices (Berndt and Fuss 1986). Morrison (1985a,b, 1986)

provides further discussion on economic measures of capacity

utilization obtained by econometric means, while Hulten (1986) and

Berndt and Fuss (1986) discuss this in a growth-accounting framework. Current empirical research is addressing this issue.

Intertemporal and interspatial productivity C(t) = B(t)C[PK(t), PL(t)]Y(t), (2)

Intertemporal total-factor productivity can be interpreted as a rate

of shift over time in a production function4 • As discussed in Part

1, the mechanism generating these rates of change is usually assumed

to be technological progress, so that measurement oftechnological

change is equivalent to the measurement of a change in intertemporal TFP. The input effect is associated with movements along

where c(t) is termed the unit cost function, since C(t)/Y(t) is the

average cost of producing yet) under cost minimization.

The sources of growth implications of equation (1) are derived

from Shephard's Lemma, which implies that 8C(t)/8pK(t)

K(t) and 8 C(t)/ 8pL(t) = L(t). This implies that:

C(*)/C - Y(*)/Y = B(*)/B + SKCPK(*)/PK) (3)

'Either full or partial equilibrium econometric models of production can be used.

13
This expression states that the growth rate of average cost equals

the growth rate ofthe shift parameter, B(*)/B, plus a Divisia index

of input prices. Under constant returns to scale, PY = pKK + pLL

= C, implying that A(*)/A = - B(*)/B. In other words, real

average cost decreases at a rate equal to the growth rate ofthe Hicksian efficiency parameter.

This result means that the total-factor productivity residual can

be measured as the residual growth rate of output not explained

by the Divisia index of inputs, or as the residual diminution rate

of average cost not explained by the Divisia index of input prices.

Economic theory of index numberss _

is a unit-revenue function for prices and a factor-requirements function for quantities (Diewert 1974).

An index number, such as a quantity index, is exact for a particular functional fonn F of the aggregator function if the ratios

of the outputs (the values ofF) between any two periods or regions

are identically equal to the index of outputs: QI(PO,PI,XO,XI) =

F(X')/F(Xo) (Diewert 1976).

Diewert (1976a) provides a strong argument for considering only

flexible aggregator functions, that is, those aggregator functions

which can provide a second-order approximation to an arbitrary

aggregator function. Diewert (1976a) tenns that class of index

numbers that are exactly represented by flexible aggregator functions as superlative.

Introduction

Fisher's weak factor-reversal test

The economic theory of index numbers is concerned with the relationships between the properties of index numbers and the properties of the underlying aggregator functions they represent. The

demonstration in recent years that numerous index-number fonnulae

can be explicitly derived from particular aggregator functions implies that rather than starting the selection process with a number

of index-number fonnulae, an aggregator function with desirable

economic properties can be specified and the corresponding indexnumber procedure derived.

To be more concrete, consider price and quantity data for N commodities for two periods (or economic entities), pO = (Plo,

P2

0 ,., .,PNo), pI = (P1l,P20, •.. ,PN'), XO = (XI

O,X20, .. . ,XNO),

and X' = (X,I,X2 ', .. .XN'). A price index PI(p°,P',Xo,X') is

defined to be a function of prices and quantities, while a quantity

index QI(po,P' ,XO,XI) is defined to be another function of observable prices and quantities for the two periods.

PI and QI are generally assumed to satisfy Fisher's (1922) weak

factor reversal test:

(4)

Fisher's weak factor-reversal test (4) states that the product of the

price index multiplied by the quantity index should equal the expenditure ratio between the two periods. PI is to be interpreted as

the ratio of the price level in period 1 to the price level in period

0, while QI is the ratio of the quantity levels ofthe two time-periods

(or economic entities). Given either a price index or quantity index,

the other function can be defined implicitly by Fisher's weak

factor-reversal test (4).

Aggregator functions Laspeyres and Paasche (rnean-of-order-r) indices

The Laspeyres and Paasche indices have traditionally been the most

widely used of all index numbers. They belong to the general class

of mean-of-order-r indices.

Defme the mean-of-order-r quantity index using period-l shares

as (Allen and Diewert 1981):

An aggregator function f is a particular fonnula or procedure for

aggregating the price and/or quantity data into some price or quantity

index. Thus an aggregator function for the quantity index QI would

specify some particular functional fonn for QI(PO,Pl,Xo,X'). The

aggregator function for an input-quantity index is essentially a production function, while the aggregator function for an input-price

index is a unit-cost function. The aggregator function for outputs for r *

0 (5)

for r = 0

= LSO(X'/XO) I l I ,

(6)

PL = LP/Xjo/ LPPXjo (7)

= LSO(P.1/Xo) J I I'

where Sl = pIXI/iLNPj'X' J I I I'

Some well-known indices are special cases ofthe mean-of-order-r

index. The Laspeyres quantity index may be written:

QL = LPPX;'/ LPPX;o

where Sp = PPX;o/ LPjOXr Since prices are held fixed at their

base period (or economic entity) levels, the Laspeyres index indicates how much of the change in value of total quantity resulted

from pure quantity changes. Similarly, the Laspeyres price index

may be written:

'Frisch (1936) distinguishes three approaches to index number theory: (I) "statistical"

approaches, (2) the test approach, and (3), the functional approach, now called the

economic theory of index numbers by Samuelson and Swamy (1974). The statistical

approach assumes that all prices are affected proportionately (except for random errors) by the expansion ofthe money supply. Therefore it does not matter which price

index is used to measure the common factor of proportionality, as long as the index

number contains a sufficient number of statistically independent price ratios. Theil

(1960) provides a "neostatistical" approach. The test or axiomatic approach initiated

by Fisher (1911, 1922) assumes that the price and quantity indices are functions of

the price and quantity vectors pertaining to two periods. Tests are a priori reasonable

properties that the price and quantity vectors should possess. However, not all a priori

reasonable properties are consistent with each other, that is, there are various impossibility theorems. Moreover, the family of index-number formulae resulting from

a consistent restricted set oftests is often not uniquely determined (Diewert and Parkan

1985).

The economic and test approaches to index-number theory can be partially resolved.

After assuming explicit functional forms for the underlying aggregator function and

cost-minimizing behavior by the producer, certain functional forms for the aggregator

function can be associated with certain functional forms for index-number formulae.

Many of the index-number formulae, such as Fisher's Ideal formula, have been suggested as desirable in the literature on the test approach to index-number theory

(Diewert 1981).

14
The Paasche quantity index is also a specific form of the meanof-order-r index, and may be written as:

(14)

(15)

where the logarithmic mean function L introduced by Vartia (1974)

and Sato (1976) is defmed by L(a,b) = (a - b)/(lna - lnb) for

a*, b and L(a,a) = a. Thus for equation (14), L(P/X;' ,P;OX;O) =

(P;'X;' - PpXp)/[ln(p;'X;I) - In(PpXp)] and L('i.p;'X;I, LPPXP)

= (LP;'X;I - LPPXp)/[ln(P/X;') - In(LPpX;O)].

The Vartia I quantity index Qv(p°,P',XO,Xl) is defined by:

since the geometric index satisfies the circular test, there is no difference between the chain and direct (fixed-base) indices (discussed

further below).

The Vartia I (1974, 1976b) index provides another example of

one in which the aggregator function is Cobb-Douglas. 7 The Vartia

I price index Pv(p°,P',XO,Xl) may be written:

(9)

(8)

I(X) = min;[X;la;: i = 1.2, .. .,N], (10)

where a, is a fixed constant. The Laspeyres and Paasche indices

are also exact for a linear aggregator function 1: 6

In contrast to the Laspeyres quantity index, prices are held fixed

at their new levels. The Paasche price index may be specified as:

The Laspeyres quantity index weights with base prices; the

Laspeyres price index weights with base quantities; the Paasche

quantity index weights with current prices; and the Paasche price

index weights with current quantities.

The Laspeyres and Paasche indices are exact for a Leontief or

fixed-coefficients aggregator function I:

I(X) = La;X;i. (11)

The Laspeyres and Paasche indices are thus exactly equal to their

corresponding true indices ifthere is either no substitution between

commodities or if there is perfect substitution between commodities.

The Laspeyres and Paasche indices always offer a first-order approximation to the true index. Therefore, if substitution between

commodities lies exactly or close to either no or perfect substitution. the Laspeyres and Paasche indices provide acceptable per-.

formances. However, the larger the time period or the greater the

difference between'the base and comparison levels, the greater the

likelihood that substantial price changes may occur (leading to an

intermediate substitution case) and the greater the misrepresentation ofthese indices. Kirkley (Va. Inst. Mar. Sci., Gloucester Point,

VA 23062, pers. commun. Feb. 1987) also notes that if the production function from which the Laspeyres and Paasche indices is

derived is convex, then the Laspeyres index overstates and the

Paasche index understates, while ifthe production function is concave, then the Laspeyres index understates and the Paasche index

overstates.

Geometric indices

The class of geometric indices is also used with some regularity

in empirical work. The geometric indices are defined by:

i.e., the price and quantity indices have the same functional form

except that the roles of prices and quantities are interchanged. The

Vartia I price and quantity indices satisfy the Fisher factor-reversal

test (4) and have the property (defined below) of consistency in

aggregation (Diewert 1978).

Diewert (1978) further notes that the Vartia I index approximates

to the second order any superlative index. Thus the Vartia I (and

geometric mean) index Zv(pO,pl,XO,XI) will be close to any

superlative index Zs(po, pI, XO, X') provided that po is close to P'

and XO is close to XI. Diewert calls this type of index pseudosuperlative. Moreover, this property holds without the assumption

of optimizing behavior on the part of economic agents, since this

property is founded lIpon theorems in numerical analysis rather than

economics.

Diewert (1978) further suggests that the Vartia I price and quantity indices have serious defects which preclude their empirical

applications. The Vartia I quantity index has the property such that

rescaling the prices in either period will generally change the index,

(i.e., in general QV(TPO,pl,XO,XI)*'TQv(p°,P',XO,X') for T

*'1), while the Vartia I price index has the property such that rescaling the comparison period (or economic entity) price does not in

general change the value of the price index by the same scale factor (i.e., in general, Pv(po, TP' ,Xo,X')

*'TPv(PO,P' ,XO,XI) for

T*'1).

where ;LNA; = 1: The geometric mean index is one example of

(12) with an aggregator function like (13). Frisch (1936) notes that

where Sp = P;OX;O/;L N PPX;o and S/ = P/X/I;LN Pr'X;'. These

indices are exact for a Cobb-Douglas aggregator function defined by:

(12)

(13)

Quadratic mean-of-order-r (superlative) indices

Quadratic-mean-of-order-r indices are increasingly used in applied

economics. These indices improve upon the mean-of-order (Laspeyres, Paasche) and geometric indices without extending the

information required. The quadratic mean-of-order-r indices are

superlative, because they are exactly represented by flexible aggregator functions. Fundamental to this approach is the quadratic

lemma.

"Thus more than one index-number formula can be exact for the same aggregator

function, and one index-number formula can be exact for quite different aggregator

functions (Diewert 1981). 'Sato (1976) shows that the Vartia II indices are exact for aCES aggregator function.

15
Quadratic lemma-Diewert (l976a) shows that a superlative index

can be expressed in terms of only first-order derivatives. This is

the basic property that allows construction of quadratic mean-oforder-r indices with the same information as mean-of-order-r

indices. Diewert provides aquadratic approximation lemma which

uses the following homogeneous quadratic functional form:

(16)

proportional to their new prices. Therefore, the prices from both

periods or economic entities enter the superlative index to represent the marginal productivities in both periods or economic entities.

Superlative indices also offer a solution to errors of aggregation

(which occur because there are changes in the mix ofthe components

making up the aggregate). Use of a superlative index-number procedure on the components of an aggregate will further capture correctly any changes or differences in the quality of the components

over time or between economic entities (Christensen 1975).

where Ao, Ai' and Ai} are constants for all i and j, X is an Ndimensional vector, and T represents the transpose operator.

Diewert's quadratic approximation lemma states that if the homogeneous quadratic aggregator function f is defined by (16), then:

Quadratic mean-of-order-r-The general class of quadratic meanof-order-r indices are exact for the quadratic mean-of-order-r

aggregator function. For r'" 0, the quadratic mean-of-order-r quantity index can be written:

(17) (18)

Fisher's ideal-Fisher's Ideal index can be written as:

[ L SO(X1 /XO)r/2]l/r [L SI(X1 /xO)-rl2]-l/r I NI I I J N) ) 'j ,

where SiO = PPXP/iLNPPXiOand S/ = P/~l/iLNP/X/ The

quadratic mean-of-order-r price index can similarly be written:

Pr(PO,Pl,XO,X (19) I

)

[ L SO(NipO)rl2]lfr [L SI(P1 lpO)-rI2]-llr I Nl I I J N] } ) •

'The quadratic mean-of-order-r aggregator function Ir is defined by /,.{Z) = (k~N

/~NAklZ/2Z,'/2)lfr. The aggregator function for Fisher's Ideal index number is

thereforefi{Z) = {liz k~N/~NAk/ZkZ/)lf2, Le., it is the square root of a homogeneous

quadratic function.

where S/j and Ski are value share weights for the two economic

entities or time periods being compared (k, I). and the Zi are the

corresponding prices or quantities. This index is simply the

geometric mean of the Laspeyres and Paasche indices. The Fisher

Ideal index is the exact index for the linearly homogeneous quadratic

mean-of-order two-aggregator function. 9 Diewert (1974b) shows

that this aggregator function is flexible, implying that the Fisher

Ideal index is superlative.

The Fisher Ideal index possesses several nice properties. (Diewert

1976a, 1981) The Fisher Ideal price and quantity indices can be

obtained by simply interchanging the quantities and prices in the

same general formula. They are also consistent with both a linear

aggregator function (perfect substitution) and a Leontief aggregator

function (no substitution); no other superlative index-number formula has this rather nice property. The Fisher ideal index numbers

are the only pair among the quadratic mean-of-order-r numbers

which satisfies the Fisher weak factor-reversal test. As a consequence, the implicit indices equal the direct indices, and no difficulty

arises in choosing whether to use an implicit price or quantity index

according to the relative variation in price and quantity data going

A multiplicity of superlative price and quantity indices exist,

depending upon the value of r. Two superlative index numbers are

widely used. The Fisher Ideal index is defined for r = 2, and the

Tornqvist index is a limiting case as r tends to O.

'Denny and Fuss (1983) generalize the quadratic lenuna to the case of discrete variables.

They demonstrate that the quadratic lenuna can still be applied without alteration when

a subset of X is a vector of noncontinuous variables. In effect, the discreteness of

the variables is ignored. Denny and Fuss (1983) note that the quadratic lemma can

be interpreted as resulting from a differencing of two linear approximations. They

further note that the quadratic lemma is exactfor linear and quadratic functions urness

the laller's second-order parameters are not independent of the point of approximation. Appendix 5 discusses this laller topic in greater detail.

Superlative indices-An index is superlative when it is exact for

an aggregator function which provides a second-order approximation to a linear homogeneous function. The superlative indices (and

the quadratic mean-of-order-r aggregator function) do not require

commodities to be either perfect or zero substitutes. Ifthe relative

price of a commodity increases, the economic agent decreases its

use (substituting other inputs) until all marginal productivities are

where Df(xr) represents the gradient vector off evaluated at xr,

i.e., the matrix of second-order derivatives. Diewert (1981) notcs

that this lemma follows simply by differentiatingfand substituting

the partial derivatives into (17). Intuitively, the quadratic lemma

states that the difference between the values of a quadratic function evaluated at two points is equal to the average of the gradient

(first-order information) evaluated at both points multiplied b~1 the

difference between the points.

All first-order approximations satisfy the quadratic lemma. All

second-order approximations of the form given in equation (16)

also satisfy the quadratic lemma. Moreover, even quadratic functions in which the zero-order and first-order parameters are specific

to a data point satisfy the quadratic lemma (the second-order te-rms

are constant across all data points). However, should the secondorder parameters be specific to a data point, then the quadratic

lemma is not satisfied. Denny and Fuss (1983) provide further

discussion on this point.

Contrasting Diewert's lemma with the usual Taylor's series

expansion for a quadratic function indicates that knowledge of

D2f(Xo), i.e., second-order terms, is not required to construct

superlative indices. 8 Moreover, it is not necessary to econometrically

estimate the (generally unknown) coefficients which occur in the

matrix of coefficients; only the observable price and quantity vectors are required.

Due to the fundamental importance of the quadratic approximation lemma. Appendix 5 provides additional discussion. Particular

attention is given to providing intuition into the lemma.

16
(21)

(23)

from one observation to another. (See Appendices 2 and 3 for

additional details.) Moreover, as the degree of data disaggregation

increases (so that a quantity may become zero), the Fisher Ideal

index formula remains well defined. Kirkley (1984) notes that the

Fisher Ideal may be biased unless the biases inherent in the Paasche

and Laspeyres exactly counterbalance one another.

Tornqvist-The Tornqvist index in its logarithmic form can be written as (Tornqvist 1936):

InTlkl = 2 V2(S/i + SkJln(Zk;lZ/i),

or, without the logarithmic transformation:

(22)

The Tornqvist index is superlative, since it is exact for the linearly

homogeneous translog aggregator function. 10 The Skj (revenue or

cost shares) are the values of the logarithmic derivatives Df(Xl)

and Df(XO) of the quadratic lemma when logarithms of Xl and XO

(output or input quantities or prices) are used. Intuitively, the use

of these shares as weights incorporates any factor substitution or

product transformation which may have occurred. This index

requires the assumption of constant share derivatives across comparisons k and l (Denny and Fuss 1983).

Choice among superlative indices-The choice among the various

possible superlative indices, i.e., the choice of r in equation (18),

for empirical applications may not be important, provided that the

variation in prices and quantities is not too great going from period

(or economic entity) 0 to I (Diewert 1981). This occurs because

all superlative indices differentially approximate one other to the

second order, provided prices and quantities are the same for the

two periods or economic entities. Moreover, the assumption of optimizing consumer or producer behavior is not required to achieve

these results. Appendix 3 provides some related discussion in this

area.

Maddala (1979) suggests that differences in functional form of

the aggregator function (and therefore the choice of index number)

produce negligible differences in measures of TFP. Intuitively, the

different functional forms suggested in the literature differ in their

elasticities ofsubstitution (which depend on the second derivatives

of the production function), whereas from the quadratic approximation lemma only the first derivatives matter. Maddala suggests

that for productivity measurement, other matters such as disequilibrium, measurement errors, aggregation problems, and economies

ofscale are more important than functional forms ofthe aggregator

function. The choice of functional form may then be advocated for

other reasons.

Denny and Fuss (1983) and Hazilla and Kopp (l984b) provide

evidence that suggests caution when using the growth-accounting

framework to quantify changes in productivity, outputs, and inputs,

since such procedures calculated with the modern theory of index

numbers ignore second-order price effects. This topic receives

additional attention in Appendix 5.

IC'The linearly homogeneous translog aggregator function/is defined as In/(Z) = Ao + ,L.A; InZ; + Ih;LNjLNAijlnZ;lnZj , where ;LNA; = I, ;LNA;j = ;LNAij = ,LN jLNA;j = O. If input prices are being aggregated then this represents a rranslbg unitcost function, while if quantities are being aggregated, this represents a transJog production function.

17

Divisia indices

An alternative to approximation is the construction of Divisia indices

(Deaton and Muellbauer 1980, Hulten 1973). Divisia indices analyze

the continuous effects of price, quantity, or TFP changes instead

of comparing two discrete price, quantity, or TFP situations. The

Divisia index for period t is defined by:

Z(t) = Z(O)exp[ I tj 2 N Sj(t)(dlogZj/dt)dt], o

where Sj(t) is the value share of the ith commodity defined as S;(t)

= Pj(t)Xj(t)/jLNPj(t)Xj(t) and Z(O) is an arbitrary base-period (or

economic entity) price, quantity, or TFP level. The Divisia index

comparing, say, ZO and Zl can be written as:

Zl

10g(ZI/Zo) = I j2N Sj dlogZj. (24) zO

The Divisia index is a line integral, defmed with respect to

infinitesimal changes in Zj(t), so that discrete approximations to

the Divisia involve approximations to the continuous rate of change

of components of the index and to the value share in some infinitesimal interval shares. These approximation errors could accumulate over time causing the index to drift over time. Discrete

approximations to the Divisia converge to the Divisia as the discrete

units of prices and quantities become small enough and if relative

value shares are constant over time (or economic entities). Ifshares

are not constant, the discrete approximation involves an error that

depends on the variability of the relative shares and the length of

the time period (Trivedi 1981).

Chain and fIXed-base indices

Indices may be constructed using either the chain principle or the

fixed-base method. The fixed-base method directly compares all

changes in prices, quantities, or total-factor productivity to some

initial base period or economic entity level. The base period may

remain constant or may be changed after some period oftime. Two

overlapping series of binary comparisons using base periods can

also be spliced. The chain method directly compares adjacent observations, while nonadjacent observations are only indirectly compared, using the intervening observations as intermediaries. This

practice results in transitive comparisons. Frisch (1936) notes that

any chain index satisfies the factor-reversal test and the circular

test. Appendix 2 provides further details.

Chain indices make use of all the data from the initial year

cumulated to the current year. The concept of using cumulated data

leads to the Divisia index in theory and the chain index as its practical realization. In contrast, a base index provides a sequence qf

direct binary comparisons between the current year and the base

year and no refercnce to the course of prices and quantities in

between. Allen (1975) notes that from a statistical angle, fixed-base

indices are inefficient in that they do not make full use of all the

data as they unfold over time. Fixed-base indices also imply that

a price (quantity) index in year t is not influenced by prices (quantities) before year t as well as those achieved in year t.

The difference between chain and fixed-base indices can be

intuitively presented. The functional form of the true aggregator

function is unknown, so that different index-number formulae provide first- or second-order approximations to the true underlying,

but unknown, aggregator function. Approximation errors which

arise with these indices are smaller, as are the changes in prices
and quantities from one period to the next. These changes, and

therefore approximation errors, are typically (although not always)

smaller when the time periods are adjacent to one another than when

separated by wide intervals.

To illustrate, suppose the surface of the true aggregator function

leX) is concave to the origin and smooth. Intuitively, over time

a chain index "creeps along" the surface of this true aggregator

function, providing in effect a piecewise approximation to this surface over the relevant range. The errors of approximation should

then be relatively small. In contrast, the fixed-base index compares

increasingly divergent points on the true function over time, so that

the approximation errors are often increasing over time and are

generally larger than with chain indices. Thus the degree of approximation should usually be closer if the chain principle rather

than the fixeC ·base principle is used to construct index numbers.

Additional intuition can be developed in terms of the quadratic approximation lemma by reference to Appendix 5, and Appendix

Figure 5-1. In this case, the aproximation error with chain indices

should generally be lower than with fixed-base indices beca~se the

two linear functions being averaged lie next to one another in adja·

cent time periods.

Errors also arise for the Paasche and Laspeyres fixed-base indices,

because the base-period quantities or prices (used as weights) reflect

a bundle of inputs or outputs whose composition is increasingly

likely to change over time. Not a great deal of meaning can be

attached to base-period indices which compare distant periods for

which the relative quantities or weights may be very different. The

reason why Laspeyres and Paasche index numbers (and their

derivatives, the Marshall-Edgeworth and the Fisher Ideal iLdices)

do not meet the circular test is because the weights in these indices

depend on the period for which the comparisons are being made

(Karmel and Polasek 1970).

Norton et al. (1985) use a fixed-base index in which the weights

are not changed from period-to-period. Consequently, less information needs to be collected in order to calculate it. However, fixing the weights implies that they are increasingly out-of-date as time

passes (Le., the base-period relative prices at which outputs and

inputs are being valued cease to be relevent).

Longer-term comparisons are made with chain indices by a

process of chaining direct binary comparisons (also called price

relatives). Such an index is called a chain index, and the formula is:

where the separate links in the chain are binary comparisons between

adjacent periods (two-period base indices where the base is updated

each period) made according to some index-number fonnula. The

formula reflects the basic relationship:

For example, let the prices in period-Q be $1.00, in period-1 $1.10,

in period-2 $1.32, and in period-3 $1.19. Then roo = 1.00, POI

= 1.10, P 12 = 1.12, P 23 = 0.90, P 03 = 1.19, and P 03 = 1.00

x 1.10 x 1.20 x 0.90 = 1.19. (Allen (1975), Frisch (1936),

Karmel and Polasek (1970), Kirkley, Va. Inst. Mar. Sci., Gloucester

Point, VA 23062, pers. commun. Feb. 1987).

Although the precise meaning of a chain index POtch is not

simple in character, because it is based on a changing collection

of items, nevertheless there is a sense in which weights are kept

up-to-date in the chained index; the weights are unlikely to change

radically between adjacent periods. The value of a chaili index

18

P O/ h will not be the same as that of a base-period index POt where

a direct point-to-point comparison is involved (Karmel and Polasek

1970).

To compare prices in period 2 relative to period 1 with chain

indices, either a direct binary comparison can be made (calculating

the price relative to P12 ) or a figure obtained by dividing P02 by

PUl' This example is actually a specific case of the general problem of changing the base of a series. Suppose that there exists an

index for a number of periods with a certain base and it is desired

to change the period used as a basis for comparison. The usual practice is to divide through the whole series by the original index

number for the new base. In the case of a chained index or an

aggregate index with fixed weights, this correctly accomplishes the

change in base. For chained indices:

(27)

In general, for two time periods s and t, where s <t, then the chain

index between the two points pstch can be defined as above, but

this simply reduces to:

Ps,c" = P,.•..-I X Fs-t-l.s+2 X ... X Pr-I,r' (28)

Strictly speaking, such a procedure is not valid where indices with

changing weights are being used, since a change in the period of

reference then requires a change in weights. In practice, this is usually ignored (Allen 1975; Frisch 1936; Karmel and Polasek 1970).

The divergency which exists between a chain index and the corresponding direct or fixed-base index (when the latter does not

satisfy the circular test) will often take the form of a systematic

drifting (Frisch 1936). This means that with increasing time t, the

ratio Ps/h/Psr (t >s), where ps/h denotes the chain index between

periods s and t, increasingly departs from unity. The Laspeyres

index tends to drift upwards, the Paasche index tends to drift

downwards, and the Fischer ideal index tends to drift downwards.

Geometric indices should not drift over time, because there are no

differences between chain and fixed-base geometric indices. FIisch

notes that drifting must not be taken to mean that the fixed base

index is right and the chain index wrong. Frisch (1936), Allen

(1975), and Karmel and Polasek (1970) provide additional discussion and methods of measuring the amount of drifting.

Diewert (1978, 1981) generally recommends the use of chained

rather than fixed..base indices. All superlative, pseudosuperlative,

Paasche, and Laspeyres index numbers should coincide quite closely

if they are constructed using the chain principle. The chained

Paasche, Laspeyres, or any superlative index number can also be

regarded as discrete approximations to the continuous-line integral

Divisia index, which has some useful optimality properties from

the standpoint of economic theory. These discrek approximations

will be closer to the Divisia index if the chain principle is used.

Moreover, the IJse of chained indices avoids problems of discontinuities which arise when the base year in the fixed-base indices

is changed. The use of chained indices avoids the discontinuities

introduced by period changes in the base year.
Consistency in aggregation

Indices of prices and quantities (input and output) and TFP might

be constructed from data at the level of the individual firm or consumer (or even region or nation) or from previously constructed

subindices (Deaton and Muellbauer 1980; Diewert 1978, 1981).

In the first case, an index is constructed in a single step. In the

second case, there are two or more stages of construction. This

raises the issue of consistency in aggregation. Vartia (1974) defines

an index-number formula to be consistent in aggregation if the

numerical value of the index constructed in two or more stages

necessarily coincides with the value of the index calculated in a

single stage. Thus, for example, a discrete Divisia index of discrete

Divisia indices would be the discrete Divisia inoex of the components. Vartia (1976a) notes that the Paasche, Laspeyres, and the

geometric indices, including the Vartia I, are consistent in aggregation. Unfortunately, the superlative indices are not consistent in

aggregation.

Consistent aggregation providing a perfectly satisfactory overall

index that can be applied to individual periods in an intertemporal

context, to individual economic entities, or to subgroups of commodities requires homothetic weak separability of the underlying

aggregator function. II Thus to justify the two-stage method of

calculating index numbers for any partition of variables requires

an aggregator function, such as the Cobb-Douglas, which is

homothetically separable in the same partition that corresponds to

the two stages. The Paasche and Laspeyres indices are consistent

in aggregation since the underlying aggregator function is either

linear or Leontief, the Vartia I's underlying aggregator function

is the Cobb-Douglas, and the Vartia II's underlying aggregator function is the CES. If the underlying aggregator function is not

separable, any attempt to construct an overall or group ,quantity

index by using subgroup indices will result in the group-quantity

index varying with variations in quantities of commodities outside

of that group. An implicitly separable underlying aggregator function for an index also allows consistent aggregation. Blackorby et

al. (1978) (Deaton and Muellbauer 1978; Diewert 1978, 1981;

Blackorby et al. 1978), provide further details.

Although superlative indices are not consistent in aggregation

when constructing overall indices out of individual subindices,

Diewert (1978) shows that they are approximately (second-order

differentially) consistent in aggregation. Thus a practical objection

to the use of superlative index-number formulae loses its force.

Moreover, the degree of approximation will become closer if, for

the time-series data, indices are constructed by chaining observations in successive periods rather than by the fixed-base method.

To summarize, constructing aggregate indices by aggregating two

(or more) stages will give approximately the same answer that a

one-stage index would, provided that either a superlative index or

the VaItia I index is used. Further, given the otherwise superior

properties of the superlative index formulae, this procedure is

preferred.

Homogeneity and homotheticity

The discussion of superlative index numbers has been developed

in the context of aggregator functions which are linearly homogeneous. Moreover, the commodities to be aggregated are implicitly

assumed to be separable. Graphically, the set of all isoquants for

input quantities to be aggregated into a composite input quantity

lie on a straight line from the origin and are equally spaced. The

marginal rates of transfonnation between these inputs are therefore

fixed and independent of the particular combinations of other inputs

and outputs.

Should the true aggregator function (which the superlative indexnumber formulae approximate) be homothetic but not linearly

homogeneous, then no serious difficulties arise. In the homothetic

case, all isoquants (or isoproduct curves) are the same shape and,

for any factor ratio, lie on a straight line emanating from the origin

(the expansion path). Therefore, the distance between any pair of

isoquants is the same on any ray from the origin, and bundles of

inputs can be compared directly (Christensen 1975).

If the true aggregator function is nonhomothetic, then isoquants

can have different shapes and do not lie on a straight line from the

origin. Marginal rates ofsubstitution (and transfonnation) and factor

(and product) shares vary with the aggregate's level. Comparison

of input (or output) bundles can be made only by reference to an

isoquant corresponding to a particular level ofthe aggregate. Even

ifthe true aggregator function is nonhomothetic, however, economic

theory provides some appealing justification for the use of flexible

index numbers. For example, Diewert (l976a) has shown that the

Tornqvist index is exact for the nonhomothetic translog aggregator

function when the isoquant for the geometric mean output (of the

base and comparison-period input bundles) is the basis for comparison. (This is particularly appealing for multilateral indices.)

Diewert (1976a) also notes that the Fisher Ideal index will indicate

correctly the direction of change in the aggregate, even if the true

aggregator function is nonhomothetic. Diewert (1976a, 1981) further shows that the Tornqvist and Fisher Ideal superlative indexnumber fonnulae are exact for nonhomothetic aggregator functions.

In particular, he shows that any quadratic mean-of-order-r index

can approximate an arbitrary nonhomogeneous function to the

second order (Diewert 1976a,b, 1981; Christensen 1975; Swamy

and Bingswanger 1980).

Bilateral and multilateral indices _

Two basic types of output, input, and total-factor productivity (TFP)

indices can be developed: bilateral and multilateral. Bilateral indices

provide intertemporal comparisons of, say, TFP for any given

economic entity or interspatial comparisons among economic entities

for any given time period. The Tornqvist bilateral index of TFP

may be written as:

"Separabilily is the relevant property of aggregator functions which allows aggregation. Weak separability requires that the marginal rates oftechnical substitution (MRTS)

between all pairs of variables (e.g., prices) in a particular group of commodities be

independent of changes in the levels of variables not in that group. While weak

separability of the aggregator function for some group of commodities is necessary

and sufficient for the existence of an aggregate, homotheticity is a necessary and sufficient condition for the validity of the multistage aggregation procedure. Homotheticity

insures expansion paths for the subgroup of commodities comprising the aggregate

which are straight lines emanating from the origin. Homothetic separability for the

aggregate exists if the aggregator function is weakly separable in the individual components of the aggregate and the aggregator function is linearly homogeneous.

Homotheticity is a characteristic of production technology which restricts expansion paths to be straight lines from the origin. Marginal rates of substitution or transformation and factor or product shares are therefore constant along any expansion

path. In contrast to homogeneity, homotheticity does not place any restrictions on the

spacing of isoquants, and returns-to-scale are a function of the initial level of inputs

or outputs.

19

lnTFPk - InTFP, Lj 0.5 (Rik + Ri/)[lnYjk - InYi/] (29)

Li 0.5 (Wik + Wi/)[lnXik - InXi/],
where k and l are adjacent time periods (or economic entities), the

Y;j are output indices for output i of economic entity j, tht Xij are

input indices, the R jj are output revenue shares, and the W;j are

input cost shares. Diewert (1976a) shows that (29) can be derived

from a homogeneous translog product· transformation function that

is separable in inputs and outputs and exhibits neutral differences

in technology. Caves and Christensen (1980) show that separability and Hicks neutral technological change are not required to derive

(29) from a homogeneous translog product-transformation function.

The direct use of the bilateral index of TFP is limited for comparisons that are not binary, e.g., TFP of one economic entity in

somt' time period with TFP of another economic entity of a different time period. For example, interpreting k and l as time periods

or firms, the total number of possible binary comparisons of the

kl time-differentiated firm observations is given by the formula for

combinations. There is no guarantee of transitivity in such comparisons. As Caves et al. (1981, 1983) note, in a given year firm

k might be found to be more productive than firm I and less productive than firm m; yet a direct comparison of land m might

indicate that m is less productive than I. This possible lack of transitivity occurs because weights Rij and W,j specific to the firms in

question are used. The traditional solution to this problem i~ to use

weights that are not specific to the individual observation. The dis·

advantage of this solution is that the comparisons lose what i~: called

characteristicity, that is, they are no longer based on economic conditions specific to the two entities being compared (Caves et al.

1981, 1983).

Although transitivity of comparisons and complete characteristicity cannot be simultaneously achieved. transitive n:sults can

be achieved in a multilateral selting by the following compromise

formula (Caves et al. 1982a; Caves et a!. 1981, 1983):

+ L j 0.5 (Wil + W;*)[lnXii - InX;'],

where an asterisk associated with a variable indicates the arithmetic

mean and an apostrophe indicates the geometric mean. 12 The use

of this Tornqvist multilateral index for binary comparisons results

in transitive multilateral comparisons that retain a high degree of

characteristicity (revenue or cost-share weights specific to the entities

and time periods). The weights used to compute the productivity

comparisons reflect the economic conditions faced by all economic

entities (through Ri

* and W/), but at the same time more than half

"Caves et aI. (1981) note that the equation for Tornqvist multilateral TFP comparisons

can be derived directly from a translog transformation structure by taking the difference between each firm's transformation function and the function resulting from

averaging arithmetically the transformation functions across all observations.

Moreover, the use of revenue shares as weights implies that the structure of production exhibits constant returns-to-scale and that the prices of the outputs are proportional to their marginal costs. Relaxation of these assumptons would require extensive econometric estimation. Finally, Caves et aI. (l982a,b) note that multilateral

comparisons can be obtained from Fisher Ideal bilateral indices in exactly the same

way that translog multilateral indices are obtained from translog bilateral indices.

However, it is not known whether or not the resulting index can be directly derived

from a flexible transformation function that is nonseparable in inputs and outputs and

permits non-neutral differences in productivity among economic entities. For these

reasons, Tornqvist bilateral and multilateral indices are used in this study rather than

the Fisher Ideal index procedure.

20

of each weight is specific to k or l. In effect, each economic entity

is compared with all others by the multilateral index via a hypothetical entity having the geometric average characteristics of all

entities. Transitive comparisons are achieved by using this representative firm as the basis for making all possible binary comparisons,

i. e., any two firms are compared with each other by comparing

both with the representative firm (the geometric mean).

The issue of bilateral vs. multilateral indices also applies to output and input indices. Consider outputs first. The Tornqvist bilateral

output index may be written as:

where there are say i = 1, ... ,M outputs. The Tornqvist multilateral

output index may be written as:

InYk - InY; = L; 0.5 (Rik + i*)[lnY;k - InY/] (32)

- L; 0.5 (Ril + Rji*)[lnY;l - InY/].

where the definitions are as before. Tornqvist bilateral and multilateral input indices can be derived in exactly the same manner,

substituting X for Yand W for R.

Comparisons ofthe output and input index formulae with the TFP

formula readily confirm that productivity comparisons can be

interpreted as comparions of outputs to inputs.

Caves et al. (1982b) discuss the application of multilateral indices

in the time-series context. Multilateral indices are applicable to

cross-section data, combinations of cross-section and time-series

data (panel data), and time-series data. Multilateral methods are

attractive for cross-section data because there is generally no natural

ordering of the data points, In contrast, time-series data have a

natural chronological order. For this reason, adjacent observations

in time-series data are usually directly compared, while nonadjacent observations are only indirectly compared, using the intervening

observations as intermediaries. This procedure is called chainlinking, and results in transitive time-series comparisons.

Time-series comparisons using superlative bilateral chain-linked

indices have the undesirable property that nonadjacent observations

are only indirectly compared. Superlative multilateral indices directI~' compare adjacent and nonadjacent observations, but only by

destroyLTJ.g the fixity of historical comparisons. As additional observations are added with time. expanding the set of comparisons, the

chain-linked bilateral approach leaves the historical comparisons

intact, but the multilateral procedure results in new comparisons

for the entire time series. This occurs because the multilateral approach compares one observation with another via a hypothetical

entity having the average characteristics (geometric mean) of all

entities, and as observations are added over time, the hypothetical

average entity changes (Caves et al. 1982a).

The choice between the bilateral and multilateral approaches

dep~nds in large part upon the importance attached to the conflicting traits of symmetry of treatment and fixity of historical comparisons. The issue ofsymmetry becomes important with panel data.

The set of time-series comparisons could be linked together through

any single cross section, but the results would differ from those

obtained by choosing any other cross section. An equally unattractive alternative would construct all the cross-section comparisons

and combine them by chain-linking the results through an arbitrarily

chosen economic entity (finn, region). The results would then differ

from those obtained by choosing any other country. The multilateral

approach to panel-data comparisons treats all economic entities
(finns, geographical areas) and time periods symmetrically (Caves

et al. 1982a).

In conclusion, Caves et al. (1982a,b) state that the superlative

multilateral indices are very attractive for cross-section and paneldata comparisons, but that they are not necessarily preferable to

chain-linked bilateral indices for time-series comparisons. This

follows because chronology provides a natural ordering of timeseries data and historical fixity of constructed indices that is lacking for cross-section or panel data.

Productivity measurement and stock effects

in marine fisheries _

Intertemporal productivity measurement in marine fishing industries

faces a measurement problem peculiar to all natural resource industries: variations in the composition, quantity, and quality of the

natural resource being exploited. This problem exists in both

renewable and nonrenewable resource industries.

In marine fishing industries, variations in the quantity and species

composition of the resource stock over time affect the costs and

revenue of harvesting for any given quantity of fish landed. These

effects on production might in tum obfuscate the efficiency with

which inputs would otherwise be converted into outputs, and could

lead to biased measures of productivity. For example, fishennen

might adopt fish fmders which should allow them to locate and target

desired species and quantities of fish while reducing search time,

fuel costs, risk, spoilage, and other opportunity costs of harvesting.

For any given level and species composition of fish stocks, fishermen's productivity should increase. However, if overall resource

abundance is declining and lower-valued species are simultaneously

increasing as a proportion, fishennen might actually have to increase

their search time, fuel costs, and trip length to attain a constant

revenue or catch level.

Changes in resource abundance and composition thus shift the

fishery-product transfonnation fr~ntier: increased resource abundance likely causes an expansion outwards, away from the origin,

while decreased resource abundance likely causes a contraction of

the frontier toward the origin. These shifts can either reinforce or

counteract the productivity trend that would otherwise occur with

a constant and homogeneous resource. The residual TFP measure

might then capture resource changes along with productivity

changes.

Resource changes can further and indirectly affect productivity

measurement by affecting (multiproduct) economies-of-scale and

capacity utilization (the latter by temporary, short-run disequilibrium). Changes in resource composition are less likely to affect

TFP measurement, since induced changes in catch or effort composition should be captured by Divisia indices.

Accounting for the effects of the resource stock upon productivity measurement depends upon the approach taken toward

productivity measurement. If the structural approach is adopted,

a production, cost; or profit function can be econometrically

estimated incorporating measures of resource abundance. Resource

abundance should be interpreted as a technological constraint, since

it is beyond the control of any individual firm but nevertheless affects

the environment within which fishing finns operate. Changes in

resource abundance may then be viewed as shifts in the technology

that relate the generation of outputs to inputs. The finn's producttransfonnation frontier may then be written as T(Y,XIA)., with

feasibility written as T(Y,XIA) ~O, where Yrefers to a vector of

outputs, X refers to a vector of inputs, and A refers to an index

21

oftechnology. Resource abundance can then enter the transfonnation frontier as either a dummy variable or as a parameter (McFadden 1978, Daughety et. al. 1986)13 Allowing this measure to interact with the other inputs further allows estimation of the effects

of resource abundance upon economic capacity utilization and scale

economies.

Current research is focusing upon incorporating the effects of

resource abundance into the growth-accounting or index-number

procedure. Until this research is completed, the resulting TFP

measure is interpreted as not strictly the rate oftechnological change,

but as a measure of both the technical efficiency with which inputs

are converted into outputs and the effects of resource availability

and composition. The TFP measure may also include changes in

technical efficiency, scale effects and effects from changes in capacity utilization (which may change with changes in resource abundance and composition), and effects of public regulation possibly

restricting productivity in the short run (but presumably increasing TFP over the long run as resource stocks rebuild to desirable

levels).

Acknowledgments _

A study of this nature could not be possible without considerable

assistance from a number of people. The comments of Dan Huppert,

Sam Herrick, Jim Kirkley, Jeff Lee, and an anonymous reviewer

are gratefully acknowledged. Chuck Korson and Wes Silverthorne

collected and processed a large portion of the cost data. Ed Ueber

provided many of the crew sizes. The very helpful cooperation of

Will Daspit, Frank Henry, Jim Golding, Jack Taggert, Dale Ward,

and Joe Terry was essential for this project. Any errors remain the

responsibility of the author.

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23
Table 1

Trawl Landings and Revenue:

Washington, Oregon, northern and central California

Pink shrimp JV Pacific whiting Groundfsh

------

Year Tons Revenue Tons Revenue Tons Revenue

1981 19923 20.01 48012 6.17 100414 38.06

1982 13913 13.37 74367 9.64 113492 44.33

1983 6613 8.78 79476 9.28 89538 36.71

1984 4818 3.81 86959 10.29 79938 32.26

1985 12779 7.61 34934 3.2 82988 34.86 ._------------ --~

Note: All values millions of dollars in 1981 dollars

All weights are short tons

Source: PacFIN Management database

Table 2

Number of coastwide otter and shrimp trawl vessels by region

and year

Year Wash. Oregon N. Calif. C. Calif. Total --_.__ ._._-

1981 87 148 99 n 406

1982 93 169 97 69 4~~

1983 88 170 98 96 452

1984 97 174 80 63 414

1985 92 146 73 59 370

Note: Vessels employing both otter and shrimp trawl gear arc counted .mly

once.

Source: Annual vessel inventory and PacFIN Research database

Table 3

Total number of groundflSh and shrimp landings and jointventure weeks fished by region and year

Year Wash. Oregon N. Calif. C. Calif. Total ---_._._----_..- --

1981 2488 7410 2687 2529 15114

1982 2107 7097 4181 4470 17855

1983 2302 7723 3565 3984 17574

1984 1976 6071 3441 3544 15032

1985 1908 5686 3959 4857 16410

Source: PacFIN Management database and joint-venture logbooks

Note: Excludes Puget Sound vessels

Table 4

Number of groundfish landings by region and year

Year Wash. Oregon N. Calif. C. Calif. Total ._----

1981 1474 4188 2062 2497 10221

1982 1570 4767 3479 4347 14163

1983 1491 6552 3500 3893 15436

1984 1615 5377 3260 3452 13704

1985 1391 4735 3761 4846 14733

Source: PacFIN Management database and Washington Dept. of Fisheries

Note: Excludes Puget Sound vessels

24

Table 5

Number of pink shrimp landings by region and year

Year Wash. Oregon N. Calif. C. Calif. Total

------

1981 1014 3108 625 31 4778

1982 53, 2082 702 75 3396

1983 8li 1102 43 48 2004

1984 322 ~O8 118 49 997

1985 488 951 171 4 1614 ._--_._-_.-

Source: PacFIN Management database

Note: Excludes Puget Sound vessels

Table 6

Multilateral total-factor productivity

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.00000o 0.666324 0.988394 0.878428 1.00000o

1982 0.969277 0.657095 0.784490 0.780530 0.920491

1983 0.996574 0.619725 0.781322 0.672099 0.883903

1984 L080838 0.714634 0.869972 0.808189 1.006902

1985 1.186149 0.768584 0.871066 0.737701 1.040016 .- -------,-------- Note: Tornqvist multilateral cham indices.

Normalized on either 1981 Washington or 1981 fleet.

Table 7

Bilateral total-factor productivity - -_.- _._._-_._--_.

Year Wash. Oregon N. Calif. C. Calif. Fleet - _.._.--- _.._-_._._-

1981 1.00000o 1.000000 1.00000o 1.00000o 1.00000o

1982 0.979454 0.984080 0.812086 0.905919 0.928293

1983 0.985601 0.905697 0.811335 0.762710 0.874973

1984 1.086189 0.993388 0.889217 0.919418 0.973892

1985 1.174060 1.097958 0.893955 0.836522 1.015422

Note: Tornqvist bilateral chain indices with 1981 base
Note: Computed following conventional productivity practice as In (T+ I) -

In (T).

Percentages are obtained by multiplying by 100.

1981

1982 -0.02075 -0.01604 -0.20814 -0.09880 -0.07440

1983 0.006256 -0.08300 -0.00092 -0.17207 -0.05915

1984 0.097178 0.092416 0.091659 0.186863 0.107107

1985 0.077792 0.100086 0.005314 -0.09448 0.041759

Avg. 0.040117 0.023363 -0.02802 -0.04462 0.003826

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 0.385915 -0.02006 0.374241 0.256295 0.232110

1982 -0.03120 -0.01394 -0.23104 -0.11816 -0.08284

1983 0.027773 -0.05855 -0.00404 -0.14956 -0.04056

1984 0.081168 0.142494 0.107473 0.184391 0.130286

1985 0.092975 0.072778 0.001256 -0.09125 0.032357

Avg. 0.042677 0.035693 -0.03159 -0.04364 0.009809

Note: Averages computed for 1982-85.

Computed following conventional productivity practice as 1n(T+ I) -

In(T).

Percentages are obtained by multiplying by 100.

Table 9

Tornqvist bilateral TFP growth rates

Table 10

Tornqvist multilateral-output chain indices for individual

species

Year Wash. Oregon N. Calif. C. Calif.

Dover sole

1981 1.00000o 1.134581 1.235892 0.969079

1982 1.031920 1.213503 1.249156 1.014966

1983 1.043814 1.231094 1.165989 1.029052

1984 1.060182 1.165946 1.178834 1.058754

1985 1.033139 1.155265 1.243946 1.149278

Petrale sole

1981 1.00000o 1.039037 0.998366 1.007475

1982 0.995872 1.077271 0.993645 1.009050

1983 1.017418 1.058683 0.983387 0.983778

1984 1.010374 1.029645 0.990106 0.976597

1985 1.003475 1.020638 1.004848 1.005584

Other flatftsh

1981 1.00000o 1.063230 1.021449 1.019490

1982 1.060928 1.080922 1.006804 1.016159

1983 1.039346 1.066653 0.997134 0.978593

1984 1.056870 1.038617 0.986737 0.966470

1985 1.072833 1.049883 1.006727 1.000616

RoclUtsh

1981 1.00000o 1.135445 0.845552 0.799196

1982 0.987301 1.068244 0.870419 0.960247

1983 0.921153 0.985263 0.791875 0.739806

1984 0.787059 0.940486 0.738121 0.706176

1985 0.746628 0.950586 0.780170 0.706032

Pacific cod and lingcod

1981 1.00000o 0.961548 0.944272 0.950172

1982 0.984703 0.974115 0.946402 0.949718

1983 0.988245 0.981781 0.936098 0.928333

1984 1.015783 0.965144 0.930080 0.928432

1985 1.014086 0.963693 1.013049 0.916561

Sableftsh

1981 1.00000o 1.028505 1.065513 1.006452

1982 1.039216 1.061494 1.113609 1.019816

1983 1.026009 1.060395 1.069461 0.997292

1984 1.055970 1.061871 1.058095 0.998186

1985 1.001191 1.069056 1.065822 1.017845

Pacific whiting

1981 1.00000o 1.980067 3.186918 1.266354

1982 3.155684 2.200408 1.465981

1983 1.228060 3.726936 1.827943 1.891140

1984 1.829728 4.331311 1.937834 1.333818

1985 0.569838 2.575733 1.718608 1.083178

Mise. groundftsh

1981 1.00000o 0.928832 0.930768 0.925377

1982 0.930179 0.921754 0.927003 0.92393~

1983 0.923390 0.924609 0.926919 0.922923

1984 0.941897 0.920984 0.922016 0.921810

1985 0.931889 0.915865 0.926429 0.921569

Pink shrimp

1981 1.00000o 1.455978 0.685070 0.594482

1982 0.776709 1.039321 0.720047 0.562585

1983 0.837948 0.784123 0.537546 0.585361

1984 0.704627 0.723615 0.623711 0.466517

1985 0.918707 0.953571 0.635173 0.376092

Note: Each species normalized on 1981 Washington

Oregon N. Calif. C. Calif. Fleet

Table 8

Tornqvist multilateral TFP growth rates

Year Wash.

25
Table 11 Table 13

Tornqvist multilateral total-output chain index Tornqvist bilateral-output chain indices for individual species

Year Wash. Oregon N. Calif. C. Calif. Fleet Year Wash. Oregon N. Calif. C. Calif.

-------

1981 1.00000o 1.291362 1.021075 0.760186 I.OC{)()()() Dover sole

1982 0.949664 1.316348 0.989674 0.843147 1.006640 1981 1.00000o 1.00000o 1.00000o 1.00000o

1983 0.948678 1.301456 0.916246 0.777224 0.979096 1982 1.030778 1.046032 1.005249 1.044329

1984 0.985571 1.337039 0.912741 0.763521 0.997392 1983 1.039376 1.052024 0.919681 1.068231

1985 1.042698 1.306008 0.948681 0.789267 1.012015 1984 1.056653 1.007226 0.929270 1.116155

Normalized on either 1981 Washington or 1981 fleet

1985 1.032961 0.998675 0.992297 1.234016

Note:

Petrale sole

1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.997656 1.025436 0.996988 1.001759

1983 1.019186 1.007527 0.991650 0.972668

Table 12 1984 1.011397 0.984124 0.999488 0.966593

1985 1.003844 0.976959 1015331 1.007015

Tor,lqvist multilateral total-output growth rates

Other DatflSb

Year Wash. Oregon N. Calif. C. Calif. Fleet 1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 1.050860 1.012182 0.986576 0.993858

1981 ,0.209308 0.465006 0.230164 -0.06488 0.29870~ 1983 1.035842 1.000413 0.982697 0.945585

1982 -0.05164 0.019163 -0.03123 0.103577 0.006613 1984 1.048375 0.980072 0.967930 0.933938

1983 -0.00103 -0.01137 -0.07709 -0.08141 -0.02774 1985 1.058343 0.988099 0.988802 0.979600

1984 0.038151 0.026973 -0.00383 -0.01778 0.018514

1985 0.056346 -0.02348 0.038620 0.033162 0.014554 RockflSb

Avg. 0.010453 0.002819 -001838 0.009385 0.002985 1981 1.00000o 1.00000o 1.00000o 1.00000o ._---,-- 1982 0.969789 0.963165 1.025003 1.244426 Note: Average computed over 1982-85.

Computed following conventional practice as In(1'+ I) - In (1'). 1983 0.906027 0.903112 0.942332 0.909872

1984 0.772757 0.864293 0.868502 0.881363

1985 0.738265 0.868268 0.919292 0.855615

Pacific cod and lingcod

1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.981546 1.008163 1.001947 0.998303

1983 0.985724 1.012166 0.994721 0.972500

1984 1.007928 0.999340 0.988133 0.976240

1985 1.001678 0.998151 1.065126 0.958621

SableflSb

1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 1.038586 1.018711 1.035889 1.011607

1983 1.025752 1.016554 0.982902 0.991006

1984 1.054688 1.016632 0.972060 0.995523

1985 0.996965 1.018245 0.979190 1.022964

Pacific whiting

1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 1.238506 0.836257 1.006033

1983 1.000293 1.288067 0.745557 1.187542

1984 1.847661 1.280507 0.768798 0.919833

1985 0.578058 1.075752 0.729935 0.918422

Misc. grouodflSb

1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.952279 0.998376 0.997840 0.998862

1983 0.946974 0.999049 0.997561 0.998054

1984 0.962312 0.998225 0.994299 0.997104

1985 0.957000 0.997644 0.996707 0.996811

Pink sbrimp

1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.817113 0.884780 1.032789 0.961276

1983 0.843731 0.713472 0.829293 1.002105

1984 0.754689 0.688080 0.892004 0.858257

1985 0.927733 0.817190 0.899803 0.855178

26
Table 14 Table 16

Tornqvist bilateral total-output chain index Revenue shares for individual species

Year Wash. Oregon N. Calif. C. Calif. Fleet Year Wash. Oregon N. Calif. C. Calif.

1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o Dover sole

1982 0.951994 1.017866 0.989733 1.128879 1.012927 1981 0.062378 0.090002 0.239311 0.136368

1983 0.938590 0.981250 0.919959 1.010007 0.967034 1982 0.109959 0.116553 0.262092 0.145698

1984 0.983368 0.957565 0.906041 1.003958 0.962535 1983 0.107973 0.139658 0.286786 0.188439

1985 1.024315 0.966535 0.947134 1.035043 0.988344 1984 0.137110 0.127130 0.308158 0.263875

1985 0.133598 0.133288 0.319552 0.299553

Petrale sole

1981 0.027498 0.036147 0.024773 0.072473

1982 0.035259 0.057619 0.026134 0.067279

Table 15 1983 0.057294 0.056145 0.041042 0.054172

Tornqvist bilateral total-output growth rates 1984 0.061184 0.043535 0.047721 0.059309

1985 0.058398 0.039441 0.054731 0.090266

Year Wash. Oregon N. Calif. C. Calif. Fleet

Other flatfISh

1981 1981 0.035638 0.056740 0.073943 0.175925

1982 -0.04919 0.017708 -0.01031 0.121225 0.012844 1982 0.085919 0.062413 0.065854 0.140396

1983 -0.01417 -0.03663 -0.07310 -0.11126 -0.04636 1983 0.052500 0.064428 0.103705 0.110846

1984 0.046604 -0.02443 -0.01524 -0.00600 -0.00466 1984 0.069220 0.055251 0.083194 0.116338

1985 0.040795 0.009323 0.044356 0.030493 0.026460 1985 0.094835 0.060986 0.090399 0.141190

Avg. 0.006006 -0.00850 -0.01357 0.008610 -0.00293

RockfISh

Note: Computed following conventional pr~ciice as In (T+1) - 1n(T). 1981 0.345909 0.258226 0.199817 0.449818

Percentages are obtained by multiplying by 100. 1982 0.430116 0.219123 0.254779 0.532013

1983 0.372185 0.219397 0.356498 0.407205

1984 0.273139 0.254203 0.295879 0.486615

1985 0.286877 0.294933 0.314285 0.391329

Pacific cod and lingcod

1981 0.060677 0.016345 0.016549 0.050374

1982 0.059709 0.021286 0.019564 0.039840

1983 0.060207 0.030539 0.022901 0.019670

1984 0.087306 0.022967 0.016025 0.028949

1985 0.093287 0.023374 0.044540 0.009347

SableflSh

1981 0.016350 0.015484 0.057081 0.037595

1982 0.052120 0.029913 0.096263 0.038811

1983 0.036808 0.035748 0.095665 0.026525

1984 0.064435 0.040548 0.084996 0.037598

1985 0.039281 0.058915 0.093637 0.064697

Pacific whiting

1981 0.00000o 0.064884 0.288935 0.000367

1982 0.00000o 0.232024 0.138482 0.003749

1983 0.000143 0.298068 0.072521 0.099129

1984 0.160048 0.370051 0.099579 0.000731

1985 0.000161 0.167617 0.032845 0.000012

Misc. groundflSh

1981 0.089734 0.001264 0.003812 0.004731

1982 0.009507 0.000691 0.002876 0.004788

1983 0.001591 0.001229 0.005071 0.004764

1984 0.014890 0.000605 0.001598 0.004165

1985 0.005931 0.000252 0.003622 0.003538

Pink shrimp

1981 0.361813 0.460904 0.095773 0.072343

1982 0.217406 0.260374 0.133951 0.027422

1983 0.311294 0.154783 0.015806 0.089246

1984 0.132663 0.085705 0.062846 0.002416

1985 0.287628 0.221189 0.046384 0.000063

Note: Columns sum to one for each year

27
Table 17 Table 21

Revenue shares for aggregate output by region Laspeyres bilateral total-output fIXed-base output index

Year Wash. Oregon N. Calif. C. Calif. Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 0.214827 0440626 0.241314 0.103231 1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.168605 0.500048 0.211973 0.119373 1982 0.821997 1.153909 0.912882 1.273951 1.061186

1983 0.223064 0.500821 0.153800 0.122313 1983 0.772182 0.910060 0.539013 1.021493 0.835867

1984 0.223811 u.474046 0.183672 0.118469 1984 0.727997 0.760679 0.513517 0.797237 0.712299

1985 0.204127 0.433124 0.218770 0.143977 1985 0.814834 0.786996 0.719087 0.965476 0.803519

Note: Rows sum to one for each year

Table 18

Revenue shares by species

Pacific Misc. Pink

Year Flatfish Rockfish Sablefish whiting groundfish shrimp

1981 0.228841 0.282746 0.027990 0.098352 0.050674 0.311393

1982 0.274514 0.299606 0.048784 0.145825 0.032744 0.198523

1983 0.288509 0.297536 0.044072 0.172589 0.036986 0.160305

1984 0.299684 0.293630 0.053708 0.229618 0.041208 0.082149

1985 0.337891 0.311401 0.063336 0.079819 0.042878 0.1646n

Note: Rows sum to one for each year

Table 19

Tornqvist multilateral-chain implicit aggregate real-output

price indices

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.00000o 1.588297 1.100108 0.632123 1.00000o

1982 0.875465 0.913271 0.940230 1.452849 0.784505

1983 1.145988 0.632382 0.650732 1.083332 0.646321

1984 0.870856 0.605587 1.119241 1.085154 0.655304

1985 0.881346 0.704910 1.265058 1.551498 0.791198

Note: Formed by Fisher weak factor-reversal test (see text for explanation).

Normalized on 1981 Washington or 1981 fleet.

Table 20

Tornqvist bilateral-ehain implicit aggregate real-output price

indices -------------------------

Year Wash. Oregon N. Calif. C. Calif. Fleet -_._---..-

1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.864217 1.168767 0.930372 1.073803 1.055549

1983 1.146229 0.830000 0.641348 0.824957 0.870908

1984 0.863707 0.836760 1.115762 0.816667 0.891656

1985 0.887810 0.942562 1.253913 1.170751 1.032354

Note: Formed by Fisher weak factor-reversal test (see text for explanation).

28

Table 22

Paasche bilateral total-output fixed-base index -----

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.847758 1.152493 0.909703 1.255725 1.061971

1983 0.785450 0.800268 0.537646 0.929191 0.772340

1984 0.810597 0.768892 0.559243 0.775773 0.740534

1985 0.707631 0.762588 0.695146 0.931342 0.760912

Table 23

Fisher Ideal bilateral fIXed-base total-output index

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.000000 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.834778 1.153201 0.911291 1.264805 1.061557

1983 0.778787 0.853400 0.538329 0.974249 0.803080

1984 0.768188 0.764775 u.535893 0.786432 0.726065

1985 0.759343 0.774696 0.707016 0.948255 0.781744

Table 24

Tornqvist bilateral fIXed-base total-output index

Year Wash. Oregon N. Calif. C. Calif. Fleet -----_.

1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.831325 1.174962 0.906550 1.262224 1.070544

1983 0.751070 0.865508 0.513394 1.150764 0.820717

1984 2.840610 0.763753 0.544107 0.744312 1.185931

1985 0.792929 0.784339 0.666361 0.793430 0.761591
Table 25 Table 29

Tornqvist multilateral-aggregate input chain index Tornqvist multilateral-input chain indices for individual

inputs

Year Wash. Oregon N. Calif. C. Calif. Fleet

Year Wash. Oregon N. Calif. C. Calif.

1981 1.00000o 1.938041 1.033066 0.865394 1.00000o

1982 0.979766 2.003286 1.261553 1.080224 1.093590 Capital

1983 0.951940 2.100056 1.172688 1.156414 1.107696 1981 1.00000o 1.493297 1.016260 0.712498

1984 0.911859 1.870943 1.049162 0.944732 0.990555 1982 1.164689 1.725777 1.009586 0.680118

1985 0.879063 1.699240 1.089104 1.069901 0.973076 1983 0.965152 1.731570 1.013056 0.926326

Note: Nonnalized on either 1981 WaShington or 1981 fieet 1984 1.048221 1.771869 0.806910 0.634371

1985 0.999909 1.467446 0.747832 0.594396

Labor

1981 1.00000o 2.978295 1.079983 1.016479

1982 0.846864 2.852492 1.680466 1.796623

Table 26 1983 0.925241 3.104099 1.432877 1.601286

1984 0.794212 2.440112 1.383038 1.424437

Tornqvist bilateral-aggregate input chain index 1985 0.766881 2.338022 1.591238 1.952170

Year Wash. Oregon N. Calif. C. Calif. Fleet Fuel

1981 1.00000o 1.314194 0.924861 0.914602

1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o 1982 0.968086 1.261695 1.007578 1.022959

1982 0.971964 1.034332 1.218754 1.246114 1.091170 1983 0.984833 1.316068 0.974642 0.996861

1983 0.952302 1.083419 1.133882 1.324235 1.105216 1984 0.952958 1.198347 0.964424 0.969880

1984 0.905337 0.963938 1.018920 1.091949 0.988339 1985 0.949028 1.168710 0.994190 1.045427

1985 0.872455 0.880302 1.059488 1.237316 0.973333

Note: Nonnalized on 1981 Washington

Table 27

Tornqvist multilateral-aggregate input growth rate Table 30

Tornqvist bilateral-input chain indices for individual inputs

Year Wash. Oregon N. Calif. C. Calif. Fleet

Year Wash. Oregon N. Calif. C. Calif.

1981 -0.17660 0.485069 -0.14407 -0.32117 0.066592

1982 -0.02044 0.033111 0.199812 0.221738 0.089465 Capital

1983 -0.02881 0.047175 -0.07304 0.068155 0.012816 1981 1.00000o 1.00000o 1.00000o 1.00000o

1984 -0.04301 -0.11552 -0.11130 -0.20217 -0.11177 1982 1.141725 1.145958 0.992528 0.955572

1985 -0.03662 -0.09626 0.037363 0.124420 -0.01780 1983 0.847359 1.003822 1.004162 1.336298

Avg. -0.03222 -0.03287 0.1>13206 0.053034 -0.00682 1984 1.064866 1.023683 0.803897 0.700222

1985 0.953517 0.837494 0.931737 0.938812 Note: Average computed for 1982-85.

Computed following conventional practice as In(T+I) - In(T). Labor

Percentages obtained by multiplying by 100. 1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.846864 0.957759 1.556010 1.767497

1983 0.925241 1.042240 1.326758 1.575326

1984 0.794212 0.819298 1.280610 1.401344

1985 0.766881 0.785020 1.473390 1.920521

Table 28 Fuel

Tornqvist bilateral-aggregate input growth rate 1981 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.968045 0.981431 1.078369 1.108289

Year Wash. Oregon N. Calif. C. Calif. Fleet 1983 0.983202 1.018130 1.047477 1.082092

1984 0.956085 0.916900 1.040827 1.056834

1981 1985 0.950329 0.901658 1.070286 1.138541

1982 -0.02843 0.033756 0.197829 0.220030 0.087251

1983 -0.02043 0.046365 -0.07218 0.060804 0.012789

1984 -0.05057 -0.11684 -0.10690 -0.19287 -0.11177

1985 -0.03699 -0.09076 0.039042 0.124980 -0.01529

Avg. -0.03411 -0.03187 0.014446 0.053236 -0.00675

Note: Computed following conventional practice as In(T+ I) - In(T).

Percentages obtained by multiplying by 100.

29
Table 31

Tornqvist multilateral-aggregate input implicit real-price

chain indices

Table 34

Share of all inputs in total costs by region

Note: Sum by row

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.00000o 0.908036 0.951392 0.852814 1.00000o

1982 1.203309 0.603971 0.958440 1.123094 0.968105

1983 1.004202 0.512719 0.814385 1.144682 0.860481

1984 1.270734 0.561590 0.898666 0.802676 0.889275

1985 1.161675 0.539658 0.930371 1.013478 0.910484

Note: Formed by Fisher's factor-reversal test (see text for explanation).

Normalized on either 1981 Washington or 1981 fleet.

Year

1981

1982

1983

1984

1985

Wash.

0.223180

0.218652

0.196351

0.229767

0.238080

Oregon

0.392755

0.394892

0.399427

0.423828

0.394362

N. Calif.

0.219352

0.220400

0.197730

0.188271

0.193572

C. Calif.

0.164711

0.166055

0.206490

0.158132

0.173984

Table 32

Tornqvist bilateral-aggregate input implicit real-price chain

indices

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o

1982 1.098898 1.059757 0.898798 0.882023 1.003326

1983 0.909419 0.900371 0.763048 0.905610 0.876076

1984 1.159524 0.987504 0.838317 0.629150 0.942273

1985 1.060399 0.943734 0.866437 0.793935 0.930485

Note: Formed by Fisher's factor-reversal test (see text for explanation)

Table 33

Share of inputs in total costs -----_.-

Year Wash. Oregon N. Calif. C. Calif.

Labor

1981 0.433897 0.401721 0.435379 0.413456

1982 0.464813 0.412169 0.427663 0.~75625

1983 0.466212 0.454291 0.439758 0.450216

1984 0.480218 0.465550 0.445895 0.431485

1985 0.492314 0.470206 0.443233 0.411349

Fuel

1981 0.180059 0.225103 0.117663 0.153649

1982 0.133997 0.184159 0.157210 0.232437

1983 0.147428 0.178804 0.132978 0.148997

1984 0.109134 0.135108 0.140781 0.180984

1985 0.106073 0.145023 0.164822 0.236210

Capital

1981 0.386043 0.373175 0.446956 0.432893

1982 0.401188 0.403670 0.415125 0.391936

1983 0.386358 0.366904 0.427262 0.400786

1984 0.410646 0.399341 0.413322 0.387529

1985 0.401612 0.384769 0.391943 0.352440

Note: Sum by column

30

Table 35

Tornqvist multilateral economic-performance chain index

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.00000o 1.165505 1.142894 0.651108 1.00000o

1982 0.705195 0.993601 0.769585 1.009704 0.745921

1983 1.137283 0.764362 0.624313 0.636077 0.663914

1984 0.740717 0.770621 1.083505 1.092607 0.741983

1985 0.899914 1.003936 1.184420 1.293200 0.903759

Table 36

Tornqvist bilateral economic-performance chain index

Year Wash. Oregon N. Calif. C. Calif. Fleet

1981 1.00000o 1.00000o 1.00000o 1.00000o 1.00000o

1982 0.770281 1.085306 0.840614 1.102895 0.976610

1983 1.242249 0.834910 0.681934 0.694784 0.869811

1984 0.809081 0.841746 1.183507 1.193450 0.921575

1985 0.982972 1.096595 1.293736 1.233551 1.126590
APPENDIX 1

Summary of different

index formulae _

This appendix summarizes the most widely applied indices in order

to provide an easily accessible reference of the different formulae.

All indices are expressed for quantities, but extension to price indices

is straightforward. Deaton and Muellbauer (1980) and Lau (1979)

provide fine references. The indices compare two time periods or

economic entities (firms, regions) and aggregate over i = 1, .. .,N

commodities.

Paasche

Qp = (X'/XO) = LP;!Xi

l

/ LPi'X?

Laspeyres

QL = (Xl/XO) = LpoX'/LpoXo = LSO(X1/XO) l I I I I I I

Tornqvist

QT = (X'/XO) = Il;£X//X?]'h(Sii + SiO)

APPENDIX 2

Quadratic mean-of-order-r indices

and Fisher's tests _

Diewert (l976a, 1981) discusses which oflrving Fisher's tests are

satisfied by the quadratic mean-of-order-r quantity and price indices.

The quadratic mean-of-order-r quantity index for r *

0, XO »

ON, Xl » ON' pO> ON' and pI > ON may be defined as:

(A2.1)

where SjO = PjOX?/iLNPjOXjO and S/ = P/X//jLNP/X{ For r =

2, equation (A2.1) becomes Fisher's Ideal quantity index, while

when r tends to 0, the quadratic mean-of-order-r index becomes

the Tornqvist index as a special case. ~or r*

0, the (homogeneous)

quadratic mean-of-order-r aggregator function may be written:

(A2.2)

Fisher Ideal

QF = (Xl/XO) = [LS;'(Xjl/XjO)/LS?(Xjl/X?)]'h

Geometric

QG = (Xl/XO) = Il;£Xi'/X?]si

Divisia

where Ai} = Ajj for i*j.

Diewert (l976a, 1981) states that Qr satisfies the (1) commodity reversal test, Le., the value of the index number does not change

ifthe ordering ofthe commodities is changed; (2) identity test, i.e.,

Qr(p°,Po,Xo,Xo) = 1 [also Qr(Po,Pl,Xo,Xo) = 1] so that the

quantity index = 1 if all quantities remain unchanged; (3) commensurability test, i.e., the quantity index remains invariant to

changes in units of measurement; (4) determinateness test, i.e., Qr

does not become zero, infinite, or indeterminate if an individual

price becomes zero for any r *

0 and Qr does not become zero,

infinite, or indeterminate if any individual quantity becomes zero

if 0 < r 2 (thus the quantity indices Qn for 0 < r 2, are

somewhat more satisfactory than the Tornqvist); (5) proportionality

test, Le., Qr(p°,PI,Xo,TXo) = Tfor every T> 0; (6) time or

point reversal test, Le., Qr(p°,pl,XO,Xl)Qr(P',pO,Xl,XO) = 1.

Define the quadratic mean-of-order-r price index Pr for pO »

ON' P' » ON' XO > ON' Xl > ON' for r *0, as:

Pr(P°,pl,XO,X')

= [jLN(Pil/pjOyI2SjOj 'h CLN(P//P/)-rl2S/]-lIr

(A2.3)

31

where SiO and S/. are defined as before. Pr also satisfies Fisher's

tests (1) to (6).

The only Fisher tests not satisfied by the quadratic mean-of-order-r

price and quantity indices, Pr and Qn are: (7) the circularity test,

Le., Pr(p°,P',XO,Xl)Pr(PI,P2,X',X2) * Pr(PO,P2,XO,X2), so

that transitivity is not satisfied by binary comparisonst; and (8) the

factor-reversal test, Le., Pr(P°,pl,XO,X')Qr(p°,pl,XO,X')

*

LP;!Xil/LP?X?, except that P2and Q2' the Fisher Ideal price and

quantity indices, satisfy the factor-reversal test (so that implicit

indices equal direct indices).

tIf this test is not passed. the implication is that the price index Pen over the time

period 0 to 2 does not depend upon how prices develop over time in the intermediate

year(s), Po to P2 via Pl'
APPENDIX 3

Superlative implicit indices _

This appendix examines in greater detail the relationship between

superlative index numbers and Fisher's factor-reversal test. It borrows heavily from Diewert (1976a, 1981) and Allen and Diewert

(1981), and demonstrates that if either a price or quantity direct

index is defined, then a corresponding quantity or price index can

be defined implicitly by using the weak factor-reversal test [equation

(13) of the text].

Define the implicit quadratic mean-of-order-r price index P,* as:

Pr*(po, , , pi XO XI) = LPIXI/(LPOXO)(Q I I I I r(po , , , pi XO XI»

(A3.1)

and the implicit quadratic mean-of-order-r quantity index Q,* as:

Thus implicit quadratic mean-of-order-r indices defined in this

manner will satisfy the weak factor-reversal test of Fisher, equation (13). Then (P,*,Q,) and (P"Q,*) are both superlative pairs

of index-number formula. However, this is not necessarily so for

the pair of direct superlative indices (P"Q,).

The Fisher Ideal index, where r = 2, does satisfy the weak factorreversal test. Therefore, PZ(p°,pl ,Xo,X1)QZ(P°,pl ,XO,X!) =

LPj'Xjl(LP?X?, (Pz,Qz) is a pair of superlative indices, and

(Pz,Qz*) = (Pz*,Qz) = (Pz,Qz)·

The Tornqvist index, where r tends to 0, does not satisfy the weakfactor reversal test in general, i.e., PO(p°,pl,XO,XI)QO(pO,PI,

XO,XI)*LpjIXjIILP?X? This occurs because the quantity index

Qo is cons~stent with a homogeneous translog aggregator function,

while the price index Po is consistent with an aggregator function

which is dual to the translog unit-cost function, and the two aggregator functions do not in general coincide. They instead correspond

to different (aggregation) technologies, i.e., they are not self-dual.

Thus, given Qo, the corresponding price index, which satisfies

the weak factor-reversal test, is the implicit index defined by

Po*(PO,PI,XO,XI) = LP/Xjl/(LPjoXjO)[Qo(p0, pi ,XO,XI)].

Alternatively, given Po, the corresponding quantity satisfying the

weak factor-reversal test is the implicit index defined by Qo* (p0,pl,XO,XI) = LPjIXjl/(LP?X?HPo(P°,pl,XO,XI)]. The

price-quantity indices Po, Qo* correspond to a translog unit-cost

(or revenue) function, while Po*, Qo correspond to a homogeneous

translog aggregator function.

The price-quantity index pair (Po, Qo*) is advocated by Kloek

(1967) over the pair (Po*, Qo). He argues that as data and the level

of study are increasingly disaggregated, the individual consumer

or producer will utilize positive amounts offewer and fewer goods

(i.e., as N grows, components of the vectors XO and Xl will tend

to become zero), but the prices which the producer or consumer

face are generally positive irrespective ofthe degree of disaggregation. Since the logarithm of zero is not finite, Qo will tend to be

indeterminate as the degree of disaggregation increases, but Po will

still be well defined (provided that all prices are positive).

The choice between (P" Q,*) and (P,*, Q,) can alternatively be

made by comparing the variation in the N-quantity ratios (XjI/X?)

to the variation in the N-price ratios (NIP?). If there is less variation in the ~'rice ratios than in the quantity ratios, i.e., prices are

more highly proportional than quantities, then the various types of

direct superlative price indices P, are essentially share-weighted

32

averages of the price ratios (PjIIP?), and will tend to be in closer

agreement with each other than the implicit price ratios P,*. In this

case, the aggregates generated by direct-price indices should all

be numerically close (and they can be approximately justified using

Hick's Aggregation Theorem). Thus in this situation, use of a superlative direct-price index and the corresponding implicit-quantity

index, (P" Q,*), is preferred for some r (Allen and Diewert

1981).

If there is less variation in the quantity ratios than in the price

ratios, then the direct quantity indices Q, are essentially shareweighted averages of the quantity ratios and will tend to be more

stable than the implicit-quantity indices Q,*. Use of a direct

superlative-quantity index Q, and corresponding implicit-price

index P, may be preferable for some r, and the aggregates

generated by these indices should all be numerically close (and can

be justified by Leontiefs Aggregation Theorem) (Allen and Diewert

1981).

Allen and Diewert (1981) present a simple procedure by which

to empirically determine whether prices are more highly proportional than quantities. They suggest individually regressing

In(P/IP?) and In(X/IXjO) on constants. The sum of squared

residuals of the regressions (SSR) will then be measures of nonproportionality of the vectors po and P I and XO and Xl, respectively. Prices are then less proportional than quantities if the SSR

from the price-ratio regression is greater than the SSR from the

quantity-ratio regression. In this case, the use of the superlative

index-number pair (P,*, Q,) for some r is recommended in order

to aggregate the data. If the converse holds, then the superlative

index-number pair (P" Q,*) for some r is recommended.

Ifthe proportionality criterion cannot distinguish whether prices

are more proportional than quantities, Fisher's Ideal index may be

preferred since Pz = Pz* and Qz = Qz* (and thus the formula is

approximately consistent with both Hick's and Leontiefs aggregation theorems). Moreover, the Fisher Ideal index also lies between

the Paasche and Laspeyres indices, since it is the geometric mean

of the Paasche and Laspeyres indices.

Tornqvist indices are preferred over Fisher Ideal indices ifmultilateral indices are employed, since to date it is not known whether

or not Fisher Ideal multilateral indices can be directly derived from

a flexible transformation function that is nonseparable in inputs and

outputs and permits non-neutral differences in productivity among

economic entities, while this has been demonstrated for the

Tornqvist multilateral index by Caves et al. 1981. Moreover,

because the data for this study are available in quantity and revenue

values, the Tornqvist implict index for prices and the Tornqvist

direct index for quantities, (Po*, Qo), is used.
APPENDIX 4 APPENDIX 5

Aggregation of outputs _ Quadratic approximation lemma _

which in log form becomes:

(AS. 1)

Y = Yr + LJ/[Xj - Xn + (\12!) LLi)i/[Xi - Xn

[~- ~r] + ... + (11m!) iJ, ... ,m=I, ... ,Nlij ...mr

Denny and Fuss (1983) provide intuition into the quadratic approximation lemma given in equation (17). In particular, they show that

the quadratic lemma of Diewert (1976a) can be interpreted as

resulting from a differencing of two linear approximations. The

discussion in this appendix follows their discussion quite closely.

Consider the true aggregator function Y = I(X), where X = (Xj ,X2 , ... ,XN~' with mth-order continuous partial derivatives

with respect to the Xi' Y = I(X) can then be approximated by an

mth-order Taylor-series expansion around the point xr:

where Rm+t' is the (m+ I)-order remainder term and the superscript r denotes evaluation at r.

Replace Xj by Xr Then the right-hand side of the Taylor-series

expansion is just equal to ys. If the roles rand s are reversed (i.e.,

evaluate the derivatives at s and replace X/ by X/ and Xi by Xn,

an expression is obtained for yr. Subtracting the two expressions

and dividing by 2 provides an exact representation ofthe difference

ys - yr. This difference, with suitable interpretation, is the

general growth-accounting equation.

Consider now a linear approximation to the true aggregator function Y = I(X) and that two data points are observed, {Xr} and

{xs}. These data points and the corresponding true values of the

true aggregator function, yr and ys, are labelled A and B in

Appendix Figure AS.l (again, adapted from Denny and Fuss 1983).

(M.l)

(A4.2)

(M.3)

where jLMAj = 1, jLMAij = jLMAij = jLMjLMAij = 0, and the

producer is producing M outputs, Y = (Yt,Y2,· .. ,YM).

The Tornqvist output quantity index may then be specified as:

Aggregation ofindividual outputs into a composite output is accomplished by a factor-requirements function for quantities and a unitrevenue function for prices. A factor-requirements function relates

the minimum amount of aggregate input required to produce the

vector of outputs, while the unit-revenue function provides the maximum amount of total revenue for a given level of an aggregate

input. Kirkley (1986) provides additional theoretical background

on revenue functions.

The quadratic mean-of-order-r functional form can be used to

provide superlative price and quantity indices by aggregating

individual prices and quantities, respectively. For example, the

linearly homogeneous translog functional form provides a secondorder approximation to an arbitrary twice-differentiable factorrequirements function:

where Sjl = Pjlyit/LP;!yil and SiD = PioljoILPPY?,

Similarly, if the unit-revenue function R(P) is translog over the

relevent range of data and the producer is maximizing revenue, the

Tornqvist product-price index can be defined as:

(M.4) y

Diewert (1974a, 1976a) provides additional details, including the

dual theoretical relationship between revenue functions and factor

requirement functions.

A. Y ------------------

y

A

Y

y=f(x)

L.....---~X7r-----------I.X-=.---X

Figure AS.l

Quadratic approximation lemma.

33
It is assumed that the true function f(X) is unknown and hence ys

and yr and the difference ys - yr are also unknown.

To approximate ys, take a linear approximation off(Y) around

A and evaluate it at xs. This point is called C and the corresponding value on the vertical axis, Y*-'o The approximation can be

written as:

APPENDIX 6

Constructing index numbers on

electronic spreadsheets with personal computers _

Utilizing the last set of equalities and solving for ys - yr provides:

= (yr - YS) + 2dft + HHX/ - X/J.

This appendix suggests ways of constructing Tornqvist chain index numbers on electronic spreadsheets using personal computers.

After the data is placed on a spreadsheet (e.g., LOTUS), create

a table composed of only the share-weighted binary logarithmic

changes from time-period to time-period. Unfortunately, the relative

copy command of LOTUS cannot be used to make copies of a

general formula. These numbers are binary period-to-period

changes. Next, create a second table composed of the exponents

ofthe logarithmic changes. The copy command of LOTUS can be

used to create this table. This table represents the Tornqvist binary

index numbers for period-to-period changes, that is, as if the base

period is updated with each time period.

To create chain indices, create a third table in the electronic

spreadsheet. Index the intial time period 0, the second time period

I, the third time period 2, and so forth. Then the value ofthe chain

index in time T is simply POTchain = Po x POI X P I2 X ... x

PT-I.T' Thus the value of the chain index in period 1 is simply the

value of the first period's binary index POI (from the second of the

created tables) multiplied by the initial base-period value Po. The

value of the chain index in period 2 is then the value ofthe second

period's binary index PI2 (from the second of the tables created)

multiplied by the product Po X POI' which is the value of period

I 's chain index and which has already been created in the third table

(the one currently being worked in). Once a single column of chain

indices has been created, the relative copy command can then be

used for other economic entities ifmore than one is being analyzed.

The initial base-period value Po differs, depending upon whether

or not a bilateral or multilateral index is constructed. The baseperiod value for a bilateral index is simply 1.0. The base-period

value for a multilateral index is the logarithmic change of the base

period not from a preceeding period as created in the first of the

three tables, but from the geometric mean. This value for an output index of a single species is simply O.S(Rjo + Rj*)(lnYio -

InY;'); for a single input it becomes O.S(Ujo + Uj*)(1~o -

InX;'); and for total-factor productivity it is simply the difference

between these two, where R indicates a revenue share, W indicates

a cost share, * indicates the arithmetic mean, , indicates the geometric mean, Y indicates output, X indicates input, and In indicates

natural logarithm. These logarithmic changes are placed in the rust

table created (which is composed of binary, share-weighted periodto-period changes). Because ofthe interpretation of the first period

as deviations from the geometric mean, the initial period's revenue

ratio used in Fisher's factor-reversal test used to construct implicit

indices is the initial period's revenue to the arithmetic mean of all

revenues.

(AS.2)

(AS.4)

(AS.3)

(AS.3)

ys _ yr = 0 s2.rfr + fS][X.s - xr] . I Vi Ji t t

y*S = ys _ R2s = yr + 2J{ [X/ - X{].

y*S _ yr = (ys _ yr) + (R2S - R{)

Similarly, to approximate yr, take the linear expansion of f(X)

around B and evaluate it at xr, denoting the point D and the relevent value y*r. This approximation can be written as:

By ignoring the error of approximation O.S[R{ - R2s], this last

equation becomes the quadratic approximation lemma.

All first-order approximations satisfy the quadratic lemma. All

second-order approximations of the form given in equation (16)

also satisfy the quadratic lemma. Moreover, even quadratic functions in which the zero-order and first-order parameters are specific

to a data point (e.g., s) satisfy the quadratic lemma. However,

should the second-order parameters be specific to a data point, then

the quadratic lemma is not satisfied.

The unknown difference ys - yr can be approximated by the

approximate difference y*s - y*r, where:

34