Title Page
 Table of Contents
 List of Tables
 List of Figures
 Theoretical foundations
 Empirical model
 Empirical results
 Summary and conclusion
 Biographical sketch

Group Title: Bio-econometric analysis of the Gulf of Mexico commercial reef fish fishery
Title: A bio-econometric analysis of the Gulf of Mexico commercial reef fish fishery
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00084171/00001
 Material Information
Title: A bio-econometric analysis of the Gulf of Mexico commercial reef fish fishery
Alternate Title: Reef fish fishery
Physical Description: xii, 176 leaves : ill. ; 28 cm.
Language: English
Creator: Taylor, Timothy G
Publication Date: 1980
Subject: Fisheries -- Economic aspects -- Mexico, Gulf of   ( lcsh )
Fisheries -- Economic aspects -- Gulf States   ( lcsh )
Food and Resource Economics thesis Ph. D
Dissertations, Academic -- Food and Resource Economics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis (Ph. D.)--University of Florida, 1980.
Bibliography: Bibliography: leaves 172-175.
Statement of Responsibility: by Timothy Gordon Taylor.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00084171
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000014532
oclc - 07353188
notis - AAB7759

Table of Contents
    Title Page
        Page i
        Page ii
    Table of Contents
        Page iii
        Page iv
        Page v
    List of Tables
        Page vi
        Page vii
        Page viii
    List of Figures
        Page ix
        Page x
        Page xi
        Page 1
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    Theoretical foundations
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    Empirical model
        Page 51
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    Empirical results
        Page 77
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    Summary and conclusion
        Page 123
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    Biographical sketch
        Page 176
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Full Text








To acknowledge all who have assisted in my graduate career would be

an impossible task. I wish to express my appreciation and thanks to my

many unrecognized colleagues.

My greatest debt of gratitude is to my wife, Keri, whose patience,

support and love carried me through many trying times.

Fred Prochaska served not only as chairman of the supervisory

committee, but also as a close friend. His insight has enhanced my

ability as an economist. Jim Cato gave freely of his time in providing

an excellent critique of this dissertation which greatly improved the

final draft. I also wish to thank the other members of my supervisory

committee, Tom Spreen, John Reynolds and Bill Seaman, for their many

contributions to this study. To many of my fellow students who provided

help and friendship during my tenure as a graduate student, I wish to

say thanks.

I also wish to thank Leo Polopolus, Chairman of the Food and

Resource Economics Department of the University of Florida, and George

Maddala, Director of the Center for Econometrics and Decision Sciences,

for providing financial assistance during my graduate career.

I am also greatly indebted to Leigh Parsons and Janet Eldred for

their help in the seemingly impossible task of typing and putting this

dissertation into its present form.



ACKNOWLEDGMENTS .. . . . . ...

LIST OF TABLES . . . . . .

LIST OF FIGURES .. .......... . .

ABSTRACT ...........


I INTRODUCTION .. . ...........

Objectives . . . . .
Scope . . . .


Biological Theory .........

Sustainable Yield. .. .......
Stock Production Models .. ......

Basic Bioeconomic Models of the Fishery .

Static Bioeconomic Fishery Models .. ...
Dynamic Bioeconomic Fishery Models ...

Theoretical Extensions .. . . ..

Fishing Effort and Equilibrium Yield .
Variable Product Price ....
A Multi-Sector Fishery ....


Introduction . . . . .
Catch Equation Specification and Estimation .

Specification of Fishing Effort .. ....
Within Region Specification Considerations
Stochastic Approximation of Resource Stock
Effects . . . . .






















Cross-Sectional Specification
Considerations .... .. ..... ... .. 59
Choice of Estimator for the Catch
Equations . . . . .. 63

Price Equation Specification and Estimation ...... 68

Aggregation Across Species .... ..... .69
Choice of Estimator for the Price Equation ... 74

IV EMPIRICAL RESULTS .................. .. 77

Analysis of Production in the Gulf of Mexico Reef
Fish Fishery .. ... ................ 77

Fishing Power .................... 81
Catch Equations .............. 88
Derived Equilibrium Catch Equations ......... 89
Productive Interdependence and Average
Productivity . . . . 91

Analysis of the Price Equations ............ 94
Gulf of Mexico Reef Fish Optimization Model ...... 100

Total Revenue Equations .............. 101
Cost Equations ............. .... 102
Catch Equation Constraints ............ 106

Maximum Economic Yield .. .. ........... 107

Exogenous Changes in Fishing Power ......... 114
Comparison with Previous Studies ......... 119

V SUMMARY AND CONCLUSION ................ 123

Theoretical Conclusions ... ....... .125
Estimated Parameters . ........... 127
Condition and Management of the Fishery ......... 134








SUSTAINABLE REVENUE CURVE ...... ......... .157


Bartlett Test .. .. .......... 159
Max x2 Test .... . .. ..... 160
Akaike's Final Prediction Error (FPE) Test ..... 160
Durbin-Watson Test ................. .161


EFFORT . . . . . . 166


BIBLIOGRAPHY .... ... ................ ...... 172

BIOGRAPHICAL SKETCH .................. .. 176


Table Page

1 Ordinary least squares with dummy variables parameter
estimates for the Gulf of Mexico Reef Fish Fishery
catch equations .................. ... 79

2 Four state Aitken's parameter estimates for the Gulf
of Mexico Reef Fish Fishery catch equations ....... 80

3 Estimated relative fishing power indices by state,
1957-75 ..... . ..... .. .. ... ... 84

4 Estimated number of standardized reef fish vessels
and actual number of reef fish vessels in the Gulf
of Mexico Reef Fish Fishery by state, 1957-1975 ...... 85

5 Estimated differences in intercepts for the Gulf of
Mexico Reef Fish Fishery state catch equations ..... 93

6 Ordinary least squares parameter estimates for the
Gulf of Mexico Reef Fish Fishery price equations ..... 96

7 Two stage Aitken's parameter estimates for the Gulf
of Mexico Reef Fish Fishery price equations ....... 97

8 Estimated within and across state price flexibilities
for states participating in the Gulf of Mexico Reef
Fish Fishery ........ ......... .. 99

9 Estimated annual operating and maintenance costs for
reef fish vessels by state, 1979 .... ... .. 105

10 Adjusted and unadjusted intercepts for the estimated
Gulf of Mexico Reef Fish Fishery catch equations by
state ..... ......... . .... .108

11 Estimated catch, profits and effort levels cor-
responding to maximum economic yield in the Gulf of
Mexico Reef Fish Fishery ... .. ..... .. 109

12 Number of reef fish vessels in 1975 and the
economically optimum number of vessels by state ...... 110


LIST OF TABLES (Continued)


13 Estimated species composition of MEY catch of reef
fish . . . . . .

14 Fishing power components for proportional increases
along rays defined by constant vessel size-crew
size ratios . . . . .

A-1 Species in the management unit ....

A-2 Species included in the fishery but not in the
management unit . . . .

B-1 Red Snapper catch by countries, 1970-1973 .. .

B-2 Grouper catch by countries, 1970-1973 .. ...

B-3 Estimated catch and effort in Gulf of Mexico
recreational reef fishery for selected years .

B-4 Estimated number and weight of reef fish caught by
recreational fishermen in the Gulf of Mexico, 1975

B-5 Estimated catch of reef fish per handline vessel
in the Gulf of Mexico, 1957-1975 .. ......

B-6 Catch of reef fish by handline vessels in the Gulf
of Mexico Reef Fish Fishery, 1957-1975 .. ....

B-7 Average crew size of reef fish vessels by state,
1957-1975 . . . . . .

B-8 Average size of reef fish vessels by state, 1957-
1975 . . . . . .

B-9 Number of reef fish vessels by state, 1957-1975 .

F-1 Domestic marketing of grouper and snapper by Gulf
of Mexico commerical fish dealers, 1977 .. ..

F-2 Two stage Aitken's parameter estimates for Red
Snapper and Grouper price equations .. ......

H-1 Maximum economic yield in the reef fishery given
a 10 percent increase in average fishing power
per vessel . . . . .


. 115

S. 116

S. .. .140

. 141

S. 144

. ... 145

. 146

. .. 147

. 148

S. .. 149

S.... 150

S. ... 151

. 152

. 163

. 164

. 170


LIST OF TABLES (Continued)

Table Page

H-2 Maximum economic yield in the reef fishery given
a 15 percent increase in average fishing power
per vessel ................ .... .. 170

H-3 Maximum economic yield in the reef fishery given
a 20 percent increase in average fishing power
per vessel ................... ..... 171

H-4 Maximum economic yield in the reef fishery given
a 25 percent increase in average fishing power
per vessel .............. ...... .. 171

* 0










1 Relationship between population size and mature
progeny . . . .. . .

2 Quadratic sustainable yield function .........

3 Equilibrium yield relationships for various
values of m . . . . .. .

4 Allocation of fishing effort between fishing
grounds of different productivity or location .. ...

5 Cost and revenue in an unregulated fishery with
constant product price .. . ........

6 Phase diagram for equilibria between vessels and
resource stock in an open access fishery ...

7 Open access equilibrium and maximum economic yield
in a fishery with constant product price ...

8 Open access equilibria and maximum economic yield
in a fishery with variable product price ....

9 Equilibrium in a multi-sector fishery with
variable price and pecuniary externalities ..

10 Principal fishing grounds in the Gulf of Mexico
Reef Fish Fishery . . . . .

11 Estimated relative fishing power for proportionate
increases in average crew size and vessel size ...

12 Iso-fishing power contours for selected levels of
relative fishing power .......

13 Derived equilibrium catch equations for the Gulf
of Mexico Reef Fish Fishery .. . . ...

14 Optimum number of vessels corresponding to maximum
economic yield for increasing levels of average
fishing power . . . . . .

S. 30

S. 38



. 62

. 82

.. 87

. 92

S. 117

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philsosphy



Timothy Gordon Taylor

December, 1980

Chairman: Frederick J. Prochaska
Major Department: Food and Resource Economics

Commercial reef fish landings (primarily grouper and red snapper)

from the Gulf of Mexico have declined fairly consistently since the mid-

sixties while the number of reef fish vessels has increased in all

coastal states except Louisiana. Declining catch per unit of effort has

caused concern in the industry. The main objective of this dissertation

was to construct an aggregate econometric model of the commercial sector

of the Gulf of Mexico Reef Fish Fishery and estimate maximum economic

yield. The basic theoretical model developed was a multi-sector model

with variable production prices and pecuniary externalities. Each

state participating in the fishery constituted a single sector. An

alternative methodology for obtaining equilibrium catch functions was

developed and utilized.

Stochastic processes were identified and incorporated into the

residual components of the estimated catch equations to account for the

unobservable resource stock effects. Derived equilibrium catch func-

tions were obtained by taking the limit of the catch equations over

time, with fishing effort held constant. A non-linear optimization

model for the Gulf of Mexico Reef Fish Fishery was constructed through


incorporation of the derived equilibrium catch functions with a system

of estimated price and cost equations.

Maximum economic yield of reef fish was estimated to be 11.5

million pounds. The economically efficient number of nominal vessels

corresponding to this level of catch was estimated to be 180. This

result was conditioned by exogenously fixed average fishing power per

vessel at 1975 reported levels. Fishing power was systematically

changed to determine the effects of such changes on maximum economic

yield and the corresponding optimum number of vessels.

The results of the analysis implied that the reef fish fishery in

1975 was overfished biologically and economically. To support this

implication, a Schaefer type sustainable yield function was estimated

for the domestic Gulf of Mexico Reef Fish Fishery. Maximum sustainable

yield was estimated to be 13.7 million pounds which is consistent with

the implication of overfishing.



The Gulf of Mexico Reef Fish Fishery is one of the oldest

(Carpenter, 1965) and most important of the Gulf fisheries in terms of

both quantity landed and total dockside value. The fishery encompasses

a wide variety of fishes, including 15 species of snappers, 15 species

of groupers and 3 species of sea basses. Although the above species

constitute the management unit as defined by the Gulf of Mexico Fishery

Management Council (GMFMC, 1979), several additional species of fish are

taken incidentally. These incidental species include several species

each of tilefishes, jacks, tiggerfishes, wrasses, grunts, porgies and

sand perches (Appendix A). In spite of the sizable number and variety

of species taken, three species, red snapper (Lutjanus campechanus), red

grouper (Epinephelus morio) and black grouper (Mycteroperca bonaci), are

the most desired species and hence the most abundant in commercial

catches (Moe, 1963). All of the Gulf Coastal States participate in the

reef fishery with fishing activity widely dispersed throughout the Gulf

of Mexico.

The Gulf of Mexico Reef Fish Fishery (GMRFF) is the primary domes-

tic producer of reef fish, accounting annually for an average of 93

percent of total domestic catch. Total landings (U.S. NMFS, 1979) in

1The Gulf Coastal States include Florida West Coast, Alabama,
Mississippi, Louisiana and Texas.

1979 were reported to be 10.3 million pounds with a total dockside value

of $10.4 million. Red snapper landings were 4.2 million pounds with a

dockside value of $5.6 million, while grouper landings were reported to

be 6.0 million pounds at $4.7 million. The importance of red snapper

and grouper to the GMRFF can be seen from these figures. In 1976, red

snapper and grouper accounted for 82 percent of all reef fish landings

by weight and 86 percent of the total revenue generated in the commer-

cial Gulf of Mexico Reef Fish Fishery.

Three nations, Cuba, Mexico and the United States, are responsible

for the bulk of the world supply of reef fish (grouper and red snapper).

The U.S. is the leading producer of red snapper, accounting for approxi-

mately 52 percent of the world catch in 1973 (Klima, 1976). In that

same year the United States ranked third in grouper production, behind

Mexico and Cuba which accounted for 52 and 29 percent of the world

catch, respectively (Appendix B).

The Gulf of Mexico reef fish stocks also support a significant

recreational fishery. The recreational sector is composed of three

distinct segments: party or head boats, charter boats and privately

owned and operated boats.2 Although data pertaining to the recreational

fishery are somewhat scant, there is sufficient evidence to suggest

that it is larger, in terms of catch, than the commercial fishery.

Surveys conducted in 1960, 1965 and 1970 indicated recreational catches

of 122.6, 70.9 and 76.8 million pounds of reef fish, respectively

(Clark, 1963; Duel and Clark, 1968; Duel, 1973). Total recreational

2Party or head boats generally carry over 20 passengers while
private charter boats carry six passengers or less.

catch, however, dropped to 39.5 million pounds in 1975 (GMFMC, 1979).

The proportion of groupers and red snapper present in recreational reef

fish catches declined significantly over the 1960 to 1970 period. In

1960, approximately 69 percent of the total weight of the recreational

catch was comprised of these primary species. By 1970, this proportion

dropped to only 39 percent of the total. Preliminary data for 1975

indicated that this proportion rose to 64 percent of the 1970-1975

period, however (Appendix B).

Within the fishery the Florida West Coast is the dominant producer

with respect to both catch levels and industry size. In 1976, Florida's

West Coast accounted for 81 percent of total Gulf of Mexico reef fish

landings. In terms of the primary species, Florida's West Coast

accounted for 57 percent by weight and 67 percent by value of the total

red snapper catch and 96 percent by both weight and value of all grouper

landed in the Gulf of Mexico in that same year. Industry size, as

measured by the number of vessels reported by states, also illustrates

the dominant position of Florida. In 1976, 509 vessels were reported in

the reef fishery. Of these vessels, 449 fished out of Florida Gulf of

Mexico ports. During the 1970 to 1976 period, 82 percent of all GMRFF

vessels originated from Florida West Coast ports.

The Gulf of Mexico reef fishery is a hook and line fishery. The

fishing process mainly involves the location of high concentrations of

reef fish and the capture of these fish using hand or mechanically

operated fishing reels. Generally, each crewman on a vessel operates

only one reel. Fishing activity in the Gulf of Mexico occurs over a

wide geographic area ranging from the West Florida shelf to the Western

Gulf off Texas and as far south as the Campeche shelf near Mexico

(Appendix A). Given the ."search and capture" nature of the fishing

process, two of the most important "inputs" in the reef fish fishery are

the size of vessels and the crew sizes corresponding to fishing vessels.

Average vessel sizes are heterogeneous across states. Florida vessels

are the smallest, having an average size of 24.7 gross registered tons

in 1976. Vessels originating from Mississippi ports are the largest,

averaging 73 gross registered tons per vessel in the same year (GMFMC,

1979). Crew sizes also exhibit considerable variation across states

ranging from an average of three men per vessel for Florida vessels to

nine men per vessel aboard Mississippi vessels in 1976.

Very little information is available on costs and revenues of

commercial reef fish vessels in general. Some data, however, have been

accumulated for Florida vessels for the years 1974 and 1975 (Cato and

Prochaska, 1977). While these data cannot be assumed representative of

vessels originating from states other than Florida, they nevertheless

provide some indication as to the magnitude of costs and revenues cor-

responding to reef fishery vessels. During 1974 and 1975, the average

annual total revenue for Florida vessels operating in the reef fish

fishery was estimated to be $56,484. In general, $11,680 of the revenue

went to crew shares with the captain and/or owner receiving an average

of $22,752. The remaining revenue was used for payment of fixed and

variable vessel expenses. During this period average investment per

vessel ranged from $26,526 to $67,267.

Total commercial landings in the GMRFF have exhibited a fairly

consistent decline from a peak of 24.7 million pounds in 1965 to approxi-

mately 18.3 million pounds in 1972. Since 1972, reef fish landings have

shown no significant trend, fluctuating between 16.7 and 17.7 million

pounds. This appearance of constancy is somewhat misleading, however,

in that from 1972 to 1976 the total landings of the primary target

species, red snapper and grouper, continued to decline. Combined land-

ings of these primary species have decreased 37 percent from 22.5

million pounds in 1965 to 14.0 million pounds in 1976. Thus, the

apparent constancy of total reef fish landings is attributable to in-

creased landings of the less desired reef species. The behavior of

total landings since 1972 is especially interesting given the fact that

fishing effort as measured by the number of vessels operating in the

fishery has increased consistently during this time. Within the 1957

to 1976 period covered by current available data, three states--Florida,

Alabama and Mississippi--experienced their lowest average catch per

vessel in 1976 (Appendix B).

In spite of the trends in total catch and catch per vessel, only

partial conclusions can be offered with respect to the biological status

of the reef fish stocks and the extent to which economic efficiency

exists in the fishery. The Gulf of Mexico reef fish stocks are typical

of biological populations in that any given level of stock size

(measured by either numbers or weight) is capable of producing a sus-

tainable yield. That is, a given proportion of the population may be

harvested in any given time period while leaving the underlying stock

size unchanged. Biological theory has maintained that, in general,

sustainable yields can range from zero to some unique maximum level,

termed maximum sustainable yield (MSY) (Gulland, 1965). Furthermore,

this body of theory when used in conjunction with economic theory has

suggested that the common property nature of the fishery in combination

with the interdependency of producing units would, under a competitive


regime, lead to economically inefficient levels of catch and fishing

effort (Gordon, 1954). Generally, a fishery is said to be biologically

overfished if fishing effort being expended is greater than that

required to capture MSY, while the fishery is considered to be economi-

cally overfished if aggregate fishing effort exceeds the point where the
marginal cost of effort equals marginal revenue.

Production in the GMRFF has followed a competitive regime through-

out most of its long history.4 Whether or not his competition has led

to a situation of economic and/or biological overfishing remains largely

unanswered. To date, only limited aggregate economic analysis has been

conducted on this fishery. The most notable exception has been the pre-

liminary management plan constructed by the Gulf of Mexico Fishery

Management Council (GMFMC, 1979). Although some basic MSY calculations

presented in the plan suggest that the fishery is currently operating

near MSY, the bulk of the study is descriptive in nature. The basic

questions of economic efficiency, price structure and possible conse-

quences of instituting various management strategies on catch and effort

levels remain unanswered.


The passage of the Fishery Conservation and Management Act of 1976

(PL94-265) has made it necessary to develop management plans for all

3The yield from a fishery that results from a level of fishing
effort such that marginal cost equal marginal revenue is called maximum
economic yield (MEY).

4Minimum legal size limits for certain species of reef fish have
been instituted in some states.

domestic commercial fisheries (U.S. Department of Commerce, 1976).

These management plans are to be constructed from the best available

scientific data and directed toward achieving an "optimum yield" from

all fisheries. While the precise economic and biological definition of

optimum yield is still unclear, the necessity to develop empirical

fishery models describing the interrelationships between relevant

economic and biological agents is clear. The primary objective of this

study is to provide an econometric model of the Gulf of Mexico Reef Fish

Fishery to serve as an analytical framework within which a wide variety

of management questions may be analyzed. Specific objectives include:

1. Development of a conceptual and empirical fishing power
function for vessels operating in the Gulf of Mexico
Reef Fish Fishery.

2. Specification and estimation of aggregate catch equations
for each participating state in the GMRFF. Both produc-
tive interdependence between states and the unobserva-
bility of the fish stock will be explicitly considered by
means of stochastic specifications.

3. Identification of the price structure of the reef fishery
and estimation of a system of dockside price equations.

4. Development of a framework within which the concept of
fishing power and the estimated catch and price equations
can be integrated to analyze economic efficiency in the
Gulf of Mexico Reef Fish Fishery.

The results of this analysis will provide valuable information to these

individuals charged with the responsibility to make management decisions

in the Gulf of Mexico Reef Fish Fishery. Furthermore, it is hoped that

the results of this study will further the state of the art in the

statistical specification and estimation of empirical models describing

production in an ocean fishery.


The scope of the analysis is confined solely to the commercial

fishing sectors for Florida West Coast, Alabama, Mississippi, Louisiana

and Texas. In spite of the size of the recreational fishery in terms of

catch, severe data limitations preclude any detailed analysis of this

sector. This, however, should have only minimal effects on the analysis

of the commercial sector. The GMFMC has established that the commercial

and recreational sectors are, to a large extent, geographically distinct

in regard to the location of fishing activity.

The data utilized in this study consist primarily of secondary

data reported annually by the National Marine Fisheries Service. While

these data constitute the "best available" data, its highly aggregate

nature has forced considerable simplification in building the empirical

models. Within the context of limited data, it is anticipated that some

of the theoretical and statistical formulations presented will prove

useful to others engaged in the empirical analysis of the fisheries

under similar data confines.

Chapter II presents a review of current bioeconomic theory and

develops a theoretical model of a multi-sector fishery with variable

output prices. Chapter III presents the empirical specification of the

catch and price equations for each state participating in the GMRFF.

Also included is a discussion of the estimators utilized in estimating

these equations. The resulting estimated equations are discussed and

used to estimate maximum economic yield in Chapter IV. Chapter V con-

tains a summary of this study and all conclusions which have been

rendered. Also included are suggestions for further research.


The analysis of fishery production has led to the development of a

conceptual framework known as the bioeconomic theory of the fishery.

The term bioeconomics as used here refers to the theoretical integration

of biological and economic theory. Under this general definition, there

are several variations of theoretical models describing fishery

production. The main thrust of bioeconomics has been to create a

conceptual framework that enables the determination of economically

efficient input levels at the firm and/or industry level while simul-

taneously maintaining the underlying resource at some fixed level.

The first section of this chapter discusses the biological bases of

bioeconomic models. Specifically, the notion of sustainable yield will

be developed and some of the main stock production models will be

presented. Section two builds upon the technical biological relation-

ships by introducing prices and concepts dealing with economic effi-

ciency in production. Included in this section is a review of several

specific bioeconomic models of fishery production. The final section of

this chapter extends these basic bioeconomic models by presenting a

theoretical bioeconomic model of a multi-sector fishery with variable

product price and pecuniary externalities.

Industry here refers to the aggregate of all vessels operating in
a given fishery.


Biological Theory

The need to consider the biological characteristics of fish popula-

tions in economic analyses of fisheries becomes immediately evident when

the nature of the resource (fish population) is considered. Fish popu-

lations can be placed in the category of use-dependent flow resources

with a critical zone (Schaefer, 1957). A critical zone as used here is

defined to be a rate of decrease in flow which cannot be reversed

economically or technologically. Thus, in terms of fish population, a

critical zone would correspond to that level of population which has an

insufficient reproductive potential to remain viable.

Sustainable Yield

The main aspect of biological theory that is relevant to production

analysis of a fishery is that of population dynamics. More precisely,

the notion of sustainable yield (SY) is one of the cornerstones of bio-

economic theory.

The size of a non-exploited fish population (biomass) can, in

general, be assumed to be a function of three factors: growth, recruit-

ment and natural mortality (Gulland, 1965). Each of these variables, in

turn, is a function of the biomass. Individual growth is generally

assumed to be at a maximum at low levels of population and to decrease

as the population size increases. Natural mortality acts in the oppo-

site manner, being low at low levels of population. As, population size

increases, natural mortality increases due to increased competition for

food and other such factors. Recruitment, the rate at which individuals

enter into the fishable population, is generally assumed to be low at


both high and low populations, reaching a maximum at some intermediate

biomass. These three factors can be combined to yield a general func-

tional relationship between the population in time period t and mature

progeny in time period t + 1. This relationship is described by the

h(P) function shown in Figure 1. The h(P) function corresponds to the

actual production of mature progeny. For example, if the population at

time t is equal to P1, the mature progeny entering in the fishery will

be equal to MP,. The r(P) function is the replacement line representing

the production of progeny necessary to maintain the population at its

present level.



450 I I I

p p* p p Population (t)
1 max N

Figure 1. Relationship between population size and mature

A population of P1 need only produce MPr progeny to maintain

itself. The difference between MP1 and MPr (A B in Figure 1) cor-

responds to a yield of fish which can be harvested while maintaining


the population at a level of P1. This is the basis from which the
notion of sustainable yield derives. Before proceeding to a discussion

of sustainable yield, several aspects of the model in Figure 1 merit

comment. Under a given set of environmental conditions, a given popula-

tion will approach some natural equilibrium size. This equilibrium

occurs at the population size corresponding to the intersection of the

r(P) and h(P) functions. This population size is given by PN in the

diagram. The h(P) function is assumed to possess a unique maximum pro-

duction of mature progeny, corresponding to the underlying population,

P max As will be shown presently, the population size corresponding to
the maximum production of mature progeny is not the same as that cor-

responding to maximum sustainable yield.

Sustainable yield represents, for any given population level, the

surplus production of mature progeny over that needed to just maintain

the population at a fixed level. In terms of Figure 1, sustainable

yield for any given population is then simply the difference between the

h(P) function and the r(P) function. Mathematically, this can be

represented by

SY(Pi) = h(Pi) r(Pi) 0 < P< P (1)

where SY(Pi) refers to sustainable yield corresponding to population Pi"

Equation (1) defines a single valued function relating sustainable yield

to population size. Figure 2 illustrates one possible shape of the

sustainable yield function.2 This function can be seen to rise from a

20ther possible shapes of the sustainable yield curve are discussed




1 SY

SP P P2 Population
0 0

Figure 2. Quadratic sustainable yield function

level of zero at zero population to a unique maximum, and then back to

zero at the natural equilibrium level.

The maximum point on the curve, maximum sustainable yield (MSY),

1 2

occurs at population P This is precisely the same P as shown in

Figure 1. Thus, it can be seen that maximum sustainable yield occurs at

a population level which is smaller than that which produces the maximum

number of mature progeny, max in Figure 1.

The form of the sustainable yield function in Figure 2, the

inverted U-shape, corresponds to any population that obeys a logistic

growth process. While other shapes are possible, there is no loss of

generality in considering the above curve. Before proceeding to a

mathematical discussion of biological stock production models, two

aspects of the above sustainable yield curve need mention. First,

inspection of Figure 2 reveals that the same sustainable yield can cor-

respond to two different population levels. For example, populations

PO and P2 both produce a sustainable yield of SYO. The importance of


this fact will become obvious when costs and revenues are incorporated

into the theoretical models. More precisely, it is this property of the

sustainable yield function which leads to many conclusions rendered by

economists with respect to the workings of perfect competition.

Secondly, any given sustainable yield function is defined for a given

set of environmental and ecological parameters. Any change in these

parameters will bring about a shift in the sustainable yield function

(Schaefer, 1957).

Stock Production Models

The mathematical models describing the sustainable yield concepts

have mainly been in the form of biological stock production models. Two

of the more prominent models in fisheries theory are the Schaefer model

(Schaefer, 1957) and the Generalized Stock Production Model (Pella and

Tomlinson, 1969). These models are, in general, empirically oriented.

This orientation has resulted primarily from the lack of time series

data on population sizes. Thus, these models generally invoke several

assumptions which make it possible to express sustainable yields as a

function of fishing effort. While the term fishing effort is more

specifically dealt with in the following sections, it should suffice

here to define effort simply as some measure of fishing activity

directed toward the resource stock.

The Schaefer model is actually a special case of the Generalized

Stock Production Model. The following discussion will deal with the

most general model while in the process pointing out its relation to the

Schaefer formulation.


The Generalized Stock Production Model (GSPM) as developed by Pella

and Tomlinson (1969) is composed of two functions. These are a popula-

tion growth function and a catch function. These functions are combined

in such a manner as to create a function relating sustainable yield to

fishing effort. The rate of change in any given fish population over

time can be expressed as a function of the population size by

P(t) = HPm(t) KP(t) (2)

where H, K, m are constant parameters and P(t) is the time derivative of

population or biomass, P(t). Equation (2) is a general functional rep-

resentation of the sustainable yield function shown in Figure 2. For

populations to have an absolute maximum rate of growth or maximum sus-

tainable yield, the above equation must satisfy certain conditions on
the parameters. These conditions are H, K < 0 if m > 1 and H, K > 0

if m < 1.

Fishing effort is introduced into the GSPM by using the equation

C(t) = qE(t)P(t) (3)

where q is a constant, E(t) is fishing effort expended in time period t

and C(t) is the time derivative of catch. This relationship is hypothe-

sized under the assumption that effort units operate independently.

Equation (3) can be seen to represent the production function for the

fishery under non-equilibrium conditions. Equilibrium conditions are

3The m parameter in equation (2) measures the skewness of the popu-
lation growth function. A value of m = 2 leads to a symmetric function.
As will be shown presently, a value of m = 1 is not permissable.


defined to be those levels of effort and population that yield a catch

equal to sustainable yield.

The introduction of fishing into the GSPM necessitates that equa-

tion (2) be modified to

P(t) = HPm(t) KP(t) qE(t)P(t) (4)

where all terms have been previously defined. Equation (4) implies the
rate of increase of any given population over time is decreased pre-

cisely by the rate at which fish are caught through fishing activity.

The imposition of the equilibrium conditions described above can be

accomplished by constraining P(t) = 0 in equation (4). This constraint,

in effect, requires that catch always equal sustainable yield. By solv-

ing the equation

HPm(t) KP(t) E(t)P(t) = 0 (5)

for P(t) and substituting the result into equation (3), the equilibrium
effort yield function


C = qE(qE + m (6)

is obtained. This equation represents the Generalized Stock Production
Model relating equilibrium catch to fishing effort. Note that a value

of m = 1 would make equation (6) undefined.
The Schaefer model, which is a special case of the above model

named after its originator, M. B. Schaefer, was developed in 1954
(Schaefer, 1954). This model is based upon a logistic population growth


function. Using this function, the natural rate of increase as a func-

tion of population size can be expressed by

P(t) = K1 P(t) [M P(t)] (7)

where K and M are constants and all other terms are defined as above.

It can be easily shown that this equation corresponds to equation (2)

of the GSPM with m = 2, H = -K1, and K = -K1M. Thus, it becomes appar-

ent that the basis of the Schaefer model is merely a specific form of

the GSPM, with parameter m = 2. The Schaefer model also provides an

equation expressing equilibrium catch as a function of fishing effort.

Utilizing equation (3) once again as the non-equilibrium catch function,

the function

C = qE (M E) (8)

can be derived. As before, the appropriate redefinition of constants

(H = -K1, K = -KIM) illuminates the fact that equation (8) is merely a

specific form of equation (6) with m = 2.

The above has shown the widely used Schaefer model to be a special

case of the Generalized Stock Production Model. In part, one of the

primary goals of developing the GSPM was aimed at relaxing the con-

straint of the symmetric yield function generated by the Schaefer model
(Pella and Tomlinson, 1969). Figure 3 presents several possible equi-

librium yield functions that are possible utilizing the GSPM. It can be

seen that the shape of the equilibrium yield function takes a wide

variety of shapes as m varies.


50 m = .27

E 40

So 30

S 20 m = .80

10 \\ m = 4.94
m=11.1 2.0 m= 1.4

200 400 600 800 1000 1200 Effort

Figure 3. Equilibrium yield relationships for various values
of m

The foregoing has presented a brief introduction to the biological

bases underlying the bioeconomic models currently utilized in fisheries

management. The Generalized Stock Production Model provides a framework

which is more general than Schaefer's formulation, but still enables the

expression of equilibrium catch as a function of fishing effort. As

noted above, the type of result is significant in that the generally

unobservable population variable is eliminated in favor of those

variables (catch and effort) which are observable.

Basic Bioeconomic Models of the Fishery

The previous section of this chapter presented the basic biological

relationships that create the framework within which the economic

aspects of fisheries production may be analyzed. The biological rela-

tionships it will be recalled, led to the concept of sustainable yield,

and suggested that the harvest of any given fishery be restricted to be


equal to some sustainable yield in equilibrium. The economic models of

the fisheries have built upon this restriction and have attempted to

define the sustainable yield which is optimum in terms of economic effi-

ciency for both industry components (vessels) and society as a whole.

As would be expected, considerable debate has been generated over what

is the true socially optimum yield. This section presents a discussion

of several of the more popular theoretical bioeconomic models of the

fishery. The ensuing discussion of these models does not attempt to

derive the "true" optimum catch and effort levels of a fishery, but

rather provides a description of current theoretical models.

Static Bioeconomic Fishery Models

Common to all bioeconomic models is the recognition of the unique

aspects of the resource and its productive setting. Three aspects of

any given fishery, the use-dependence of the resource, the lack of

property rights (common property) and interdependence of producing

units, provide the motivation behind the development of bioeconomic

models of fishing. Using these aspects, bioeconomic models have almost

universally resulted in the conclusion that the workings of unregulated

competition in a fishery generally lead to a higher level of effort and

lower sustainable yield than that which is socially optimal.

The first attempt at constructing a bioeconomic model of a fishery

was done by H. Scott Gordon (1954). Gordon's analysis begins with the

assumption that a fishing ground can be treated in a manner similar to a

parcel of land in the traditional economic analysis of rents. Thus, the

conclusion reached is that the optimum degree of utilization of a fish-

ing ground occurs at the level of fishing effort which equates value of


the marginal product of effort (VMPE) with its marginal (average) cost,

r. To demonstrate how the common property nature of an unregulated

fishery encourages non-optimal levels of effort, Gordon (1954) analyzes

a fishery which is composed of two grounds of different productivity or

location and constant product price. Figure 4 depicts this situation.

$ a $ b

r \ r
SI Effort Efor
0 E1 E Effort 0 E2 E E2 Effort

Ground 1 Ground 2

Figure 4. Allocation of fishing effort between fishing
grounds of different productivity or location

The optimum degree of utilization of the fishery will, according

to Gordon, occur with OE1 units of effort being used on Ground 1 and OE2

units of effort on Ground 2. Under this allocation of effort, each

ground yields a rent corresponding to the shaded areas. This pattern of
fishing, however, does not represent a stable position for the fishery.

The reason for this instability relates to the lack of property rights

on any given fishing ground. Fishermen venturing from port are inter-

ested in grounds with the highest average productivity. Given the

constant marginal cost of effort, this is where the fisherman will


receive the highest return. Thus, in the case depicted by Figure 4,

fishermen will enter the fishery and allocate their effort such that the

value of average productivity (VAP) of the two grounds is equal to

average cost (r) and, hence, equal on all grounds. Thus, OE' units of

effort will fish on Ground 1 and OE2 units will produce on Ground 2.

Effort levels, El and E2, correspond to the effort levels that would

produce a catch equal to maximum sustainable yield. This makes it

apparent that the result of unregulated competition in this common

property resource industry leads to effort levels greater than that

necessary to harvest maximum sustainable yield. One final point of

note is that on both grounds, effort is employed past the point of nega-

tive marginal productivity. This represents Gordon's (1954) theory

explaining the results of production from a common property resource

under a competitive regime.

Gordon also proposed a bioeconomic model of the fishery at the

industry level. Schaefer (1957) presented essentially the same model.

Due to the wide use of the so-called Schaefer model in fishery theory

and its similarity to Gordon's formulations, the following analysis

follows Schaefer.

The Schaefer model begins with the definition of the long-run

equilibrium industry production function. This function, based on a

logistic population growth function, was shown in equation (8) to be a

special case of the GSPM. Equation (9) restates equilibrium catch

function as

C = aE(b E) (9)

2 K1M
where a and b in terms of the constants in equation (8), and
K1 q


E is defined as fishing effort. Further assumptions of the model are as

follows. The demand curve facing the industry is assumed to be infi-

nitely elastic, which implies a constant price, p, and industry costs

are proportional to effort. Thus, the cost function can be written as

K = rE


where r corresponds to average and marginal cost of effort. These

equations are shown in Figure 5. Given the constant production price,

the total revenue curve shown is simply the catch function in equation

(9) multiplied by the product price p. Recalling Gordon's (1954)

result that all profit in a common property fishery is dissipated

through entry of new firms, the equilibrium position of an unregulated

fishery will occur at the point where total revenue is equal to total



optm E

Figure 5. Cost and revenue in an unregulated fishery
with constant product price


The level of effort corresponding to this point can be derived from

the profit equation

n = paE(b E) rE (11)

Setting 0 = 0 and solving for E yields the open access equilibrium

effort level E= b r (see Figure 5). As shown in the above figure,
1e pa
the open access level of effort is greater than that needed to harvest

maximum sustainable yield (revenue), Em. The implications of this are

that a decrease in effort will not only free resources to be used in

other productive processes, but also an increase in equilibrium catch

will result as effort is decreased from the open access level of

E = b --, to the MSY level of effort E = (Appendix C). While
I pa 2
this is true in terms of Figure 5, this conclusion in fact depends upon

the average (marginal) cost of providing effort. It can be shown that

if average cost, r > bpa, any decrease in effort will result in a de-

crease in equilibrium yield. Further, if r = bpa- the effort levels

corresponding to MSY and open access equilibrium will coincide.

The economically optimum yield, termed maximum economic yield

(MEY), requires the maximization of the profit function shown in

equation (11). Differentiating and solving the first-order condition

for E yields the MEY effort level, Eopt = !(b ). A comparison of

Eopt and E1, the open access effort level, illustrates that it is always
necessary to restrict effort if returns to the resource are to be


The seminal paper by Gordon (1954) and the ensuing analysis by

Schaefer (1957) have provided the basis from which subsequent bio-

economic formulations have proliferated. These additional works have


both criticized and extended these basic models. In spite of these

extensions, the basic conclusions of Gordon's model have been maintained

in nearly all bioeconomic models. These conclusions are that the common

property nature of the resource creates a situation in which the open

access equilibrium in the fishery generates socially undesirable levels

of catch and effort and that some type of restrictions on the fishery

are necessary as a means of correction. Commensurate with this is

Gordon's creation of the management goal of attaining maximum economic

yield rather than the more traditional goal of maximum sustainable

yield generally proposed by the biological discipline.

Dynamic Bioeconomic Fishery Models

One of the first criticisms of the traditional model was made by

Scott (1955) and later extended by others, most notably Clark (1976).

This criticism was aimed at the concept of the static MEY. More pre-

cisely, it was argued that since catch was a function of population and

population was a function of catch, a dynamic concept of MEY was needed.

To attack this problem, Scott developed the concept of user cost.4

Basically, Scott argues that the feedback relationship between catch and

population size implies that "correct" regulation of a fishery requires

an examination of the discounted present value of returns in the

fishery. Thus, any increase in marginal current revenue (catch) must be

weighed against the cost of such an increase in terms of diminished

present value. Scott defines user cost to be the "effect of succeeding

4Scott's argument was couched in terms of the sufficiency of sole
ownership of the fishery with the attainment of MEY. The essence of his
argument, however, involves the optimality of static versus dynamic MEY.


units of current output on the present value of the enterprise" (Scott,

1955, p. 123). Dynamic MEY, therefore, occurs where marginal current

revenue is equal to marginal user cost. The determination of the catch

and effort levels necessary to achieve dynamic MEY is thus a function of

the discount rate. In general, as the discount rate rises, lower valua-

tion is put on landings in the future. Clark (1976) has shown that

dynamic MEY is bounded by static MEY levels of catch and effort and the

open access equilibrium position. More precisely, a discount rate of

zero would result in an equilibrium position identical with the open

access position while a discount rate of infinity would result in the

attainment of static MEY.

The bioeconomic models discussed above represent what can be termed

aggregate models. Aggregate as used here refers to the treatment of the

entire industry as the unit of analysis. One limitation of this type of
model is that the basic producing unit of any given fishery, the vessel,

is not explicitly included other than in its nebulous relationship with

the variable termed fishing effort. In answer to this problem, V. L.

Smith developed the first bioeconomic model which incorporated firm

behavior into the analysis (Smith, 1969). Others, most notably

Fullenbaum et al. (1971) and Anderson (1975) have all criticized and

sought to extend Smith's work. The following discussion is confined to

Smith's analysis.

Smith's formulation centers on dealing with three key aspects of a

fishery. These are the renewable nature of the resource stock, the

feedback relationship between industry catch and stock growth rate and

The term vessel and firm are used interchangeably.


the externalities of production. Three different types of production

externalities are said to exist. Stock externalities are assumed to

represent shifts in the firm cost function due to changes in the stock

size. Crowding externalities result from direct interdependence of pro-

duction (fishing) activities. Finally, Smith (1968) considers mesh

externalities which correspond to both the economic and biological

effects of changing mesh size. It should be noted that in any given

fishery, some or all of these externalities may or may not exist.

The general formulation of the model centers on the assumption of

V homogeneous vessels, each producing x units of output. Total industry

catch is thus equal to Vx. The sustainable yield function used is

defined in general function form to be f(X) where X corresponds to the

resource stock size. This function is assumed to possess the following

properties: f(X) = f(X) = 0 where X and X are the maximum and minimum

viable populations, respectively, f x0 = 0 for some X < X < X, an
interior maximum growth rate (MSY) exists, and finally 2 < 0 ruling
out any inflection point in the sustainable yield function. As noted

previously, the most common specific form of f(X) is the quadratic form

which corresponds to a logistic growth law.

To bring fishing activity into the model, Smith expresses f(X) X


X = f(X, m, Vx) (12)

where m = mesh size and other variables are defined as above. Equation

(12) thus states that sustainable yield, X, is a function of population

size, mesh size and total industry catch. It is further assumed that X

is an "inverted U" shape with f3 V < 0. By ruling out any type of


interaction between industry catch and the population growth rate,

equation (12) can be rewritten as

X = f(X, m) Vx (13)

This form of equation (13) can be interpreted to mean that the sus-

tainable yield produced by any given stock and fixed mesh size is

reduced by an amount precisely equal to industry catch.

In dealing with the individual firm, Smith (1969) chooses to define

behavior in terms of the firm's long-run cost function. It should be

emphasized that the firm's production function is implicity in the cost

function. The general form of the cost function is

c = 5(x, X, m, V) + ( (14)

where T is defined to be the firm's opportunity cost. Partial effects

are hypothesized to be, c = > 0, c2 0, c3 > 0 and
-1 a-x > 01c X = am
c4 E > 0. Of interest here is the fact that stock externality
4 aV -
effects, c2, and productive interdependency effects, c4, can be equal to


Industry revenue (R) is defined to be a function of industry catch

and mesh size. Mathematically, this relationship is expressed by

R = R(Vx, m) (15)

From this relationship the price of output received by individual firms

in the industry can be shown to be

P(m) R(Vx, m) (16)
P(m) = Vx


The use of the notation P(m) here implies that price is constant with

respect to variations in firm output, but does vary with changes in mesh

size due to the change in the size of fish caught.

Individual firm behavior is assumed to follow a profit maximization

goal with vessel catch rate, x, and mesh size, m, being the decision

variables. The profit function for the individual firm can thus be

expressed as

S= P(m)x ((x, X, m, V) v (17)

Maximization of this function yields the following first-order


P(m) = cl(x, X, m, V) (17a)

P'(m)x < c3(x, X, m, V) if < m = m (17b)

The inequality occurring in equation (17b) holds when the solution is

such that mesh size is below the point of technological feasibility.

Equation (17a) states the familiar profit maximization condition that

marginal cost equals price. Interpretation of equation (17b) is less

clear. In general, it states that the marginal revenue of varying mesh

size must equal the marginal cost of doing so. It may be, however, that

the mesh size which satisfies this condition is below that which is

technologically feasible. Hence, the inequality becomes effective in

this case and the optimal mesh size is assumed to be mrn.

The foregoing illustrates the determination of the optimal (profit

maximizing) levels of firm catch and mesh size. The rate of exit or

entry of firms operating at these levels is given by


61 Tr Tr > 0
S= { o (18)
62r ir < O
where 9 = dV
where 6, 62 are constants of proportionality and = firm

profit. Equation (18) illustrates that the entry and exit of firms is

asymmetrically proportional to profit. Generally, it is assumed that

firms leave the industry at a slower rate than firms enter.

Equations (17), (17a), (17b) and (18) provide a system of equations

in which the entire workings of the fishery can be analyzed.

Specifically, equations (17a) and (17b) provide unique values of catch

and mesh size for any given population size and industry size. Thus,

once the catch rate per vessel and mesh size is determined, changes in

industry output can be seen to be a function of changes in the industry

size and stock size. These effects are summarized by

X = F(X, V) (19a)

V = I(X, V) (19b)

Equation (19a) states that the change in the resource stock over time is

a function of both the stock size and the number of efficiently operat-

ing vessels in the fishery. When X = 0, a biological equilibrium occurs

in the sense that the industry harvest rate is equal to sustainable

yield. In equation (19b), the change in the number of participating

vessels is also seen to be a function of stock size and industry size.

The set of solutions represented by V = 0 correspond to those in which

investment in the fishery is in equilibrium in relation to alternative

Industry size here refers to the number of efficiently operating
vessels in the fishery.


productive uses. Thus, when equations (19a) and (19b) are simultaneously

zero, an open access bioeconomic equilibrium is said to exist.

An example of such a system of equations is pictured in Figure 6.

Points above the I(X, V) = 0 curve correspond to points where industry

profits are negative while the converse holds for points below the


I (X, V) = 0

( A

F(X, V) = 0

Resource stock (X)

Figure 6. Phase diagram for equilibria between vessels and
resource stock in an open access fishery

Similarly, points above the F(X, V) = 0 curve correspond to harvest

rates in excess of sustainable yield. The arrows in the diagram cor-

respond to the direction of change in vessels and resource stock.

Immediately obvious is the fact that there are three potential equi-

librium positions (points A, B and C). Only points A and C represent

stable solutions, however. The instability of point B can be seen by

examination. Any displacement from this point would result in a new

equilibrium being established at point A or C.


The above discussion has provided a review of Smith's (1969) steady

state representation of an unregulated commercial fishery. As with

previous writers, Smith's conclusion is that the unrestricted operation

of a fishery results in levels of catch and effort which exceed those

necessary to maximize returns to the resource. Furthermore, Smith

assumes if the fishery were managed by a sole owner, the appropriate

levels of catch and effort would result. These levels are harmonious

with previous writers in that they result in maximizing returns to the

resource; in other words, static MEY results.

The profit function for a sole owner can be written as

S = P(m)Vx VQ(x, X, m, V)


where all terms are defined as above. In contrast to the firm which

considers only m and x as decision variables, the sole owner must maxi-

mize equation (20) with respect to the arguments x, m, X, and V.

Furthermore, to insure that the stock remains in equilibrium, maximiza-

tion of equation (20) is constrained such that f(x, m, Vx) = 0.

Constrained maximization of the above profit function leads to the first-

order conditions

P(m) = c1 x f3

xp' (m) + 2 < c3
V_ 3 c

if < m = m


S= P(m) x- c = Vc f3x

F(X, m, Vx) = 0







where X represents the undetermined lagrange multiplier. Equation (20a)

states that the vessel catch rate be adjusted to the point where price

equals direct and user cost. Similarly, equation (20b) states that mesh

size should be adjusted to the point where marginal private and social

revenue is less than or equal to the cost of changing mesh size.

Condition equation (20c) states that the marginal profitability of total

industry catch equals the marginal social cost of adding a vessel to the

fishery. Using these criteria, the socially optimal levels of the

decision variables will be realized. Of course, this assumes that the

socially optimal position of a fishery is achieved by catching MEY.

As with previous writers, Smith (1969) concludes that in the

absence of sole ownership, an open access fishery must be regulated to

achieve economically efficient production. Smith proposes that an

extraction fee of -f3 on each pound of fish landed and a license fee of

Vc4 on each vessel would be sufficient to insure social and economic

efficiency (MEY) in an open-access fishery.

The main goal of bioeconomic theory can be seen to be that of

representing the productive activities of fishing within the bio-

technical constraints created by the growth pattern of the resource

stock. The works discussed above are by no means exhaustive. They

merely serve to illustrate the historical development of bioeconomic

theory and the general conclusions derived concerning economic effi-

ciency in fisheries production. Under the stated assumptions of con-

stant product price, a well-behaved growth law, and homogeneous units of

effort, be they vessels or some other economic entity, the above models

all arrive at the conclusion that in the absence of restrictions of

some type, suboptimal levels of catch and effort will result.


The economically "correct" degree of utilization of effort in the

fishery is shown to correspond to some form of maximum economic yield

(dynamic MEY or static MEY). Further, the failure to achieve MEY under

perfect competition was universally attributed to the common property

nature of the resource and externalities in production.

While the preceding analysis may seem to indicate that the "book

has been closed" on bioeconomic theory, the converse is true. As with

most theoretical constructs, when the assumptions change, so do the

conclusions. Thus, Anderson (1973) has shown that when price becomes

variable, the elasticity of demand becomes an important determinant in

defining the socially efficient production level for a fishery.

Furthermore, Bromley (1969) has eloquently questioned whether or not

externalities do, in fact, exist in fisheries production. By differ-

entiating between productive interdependence and externalities, Bromley

argues that perfect competition may not be as inefficient as the tradi-

tional writers above would lead one to believe. He also questioned the

social optimality of maximizing returns to the resource, suggesting that

maximizing net social benefits is perhaps more appropriate.

Theoretical Extensions

The foregoing has provided a review of the basic notions and

principles underlying the bioeconomic models used in analyzing fisheries

production. As suggested by the term bioeconomics, the models are char-

acterized by incorporating prices and costs into the biological surplus

stock production models. The resulting analysis then proceeded in the

neo-classical economic tradition to derive the results of the undesira-

bility of unregulated competition, the economic inefficiency of MSY


regulation and the "economically efficient" management goal of attaining

maximum economic yield. In all cases, these results were obtained from

models which treated the fishery as a single aggregate operating with a

constant product price. The purpose of this section is to first relax

the assumption of a constant product price and then extend the results

to consider the case of a multi-sector fishery where each sector cor-

responds to a sub-industry defined on regional or state basis.

Fishing Effort and Equilibrium Yield

Before pursuing these extensions, a brief digression on the concept

of fishing effort and equilibrium versus non-equilibrium production

functions will be useful. Prior to this discussion, a specific defini-

tion of fishing effort has been omitted, being defined only as some

measure of fishing activity. Traditionally, this measure has been

defined to be a composite of physical inputs in the fishery. Gulland

(1965) and Rothchild (1977) have noted that the notion of fishing effort

to the biologist and economist are different, especially in the long run.

This difference can be seen by comparing the effects of doubling effort

under the biological definition of effort as opposed to that of the

economist. Biological definitions of fishing effort are generally

couched in terms of catch. This results in the conclusion that a doubl-

ing of effort, other things being equal, must necessarily result in a

doubling of catch. In contrast, the economic definition of effort is

independent of catch. Under this definition a doubling of effort does

not necessitate a doubling of catch. Thus, it can be seen that the

concept of fishing effort can indeed be quite different to different



In spite of these apparent conceptual differences, effort is still

generally considered as a single composite input. The present analysis

diverges from this notion and considers that any measure of effort must

be composed of several components. Paralleling Gulland (1965), fishing

effort can be thought of as being composed of three basic components.

These are normal fishing effort, fishing power and fishing intensity.

Nominal fishing effort can be thought of as a unit of measure or perhaps

an industry size measure such as the number of vessels. Fishing power

is a measure of the input characteristics of firms (vessels) in the

fishery. Finally, fishing intensity can be thought of as some type of

time measure such as days fished. Fishing intensity is often implicit,

in the data, being defined by the observation interval.

To make this notion more explicit, assume that fishing intensity is

implicit in the interval of observation. Fishing power can then be

represented by

Ep = g(X1, ... Xn) (21)

when E denotes fishing power and the X i = 1, ..., n are input char-

acteristics of firms in the fishery. If g(X1, ..., Xn) is assumed to be

the same for all firms, total fishing effort is then given by

E = EN g(X1, ..., Xn) (22)

where EN denotes nominal effort. This notion of effort will be main-

tained throughout the remainder of this study. Further discussion will

follow in the empirical analysis to be presented.


Having briefly clarified the definition of fishing effort, it

remains to draw a distinction between the equilibrium yield functions

such as those developed by Pella and Thomlinson (1969) and Schaefer

(1957) and non-equilibrium yield functions. Equilibrium yield functions,

as shown above, are derived in such a manner as to produce a relation-

ship between fishing effort and sustainable yield. It is precisely

this relationship between catch and effort that has resulted in the

term equilibrium yield function. The catch resulting from any level of

effort along these functions corresponds to equilibrium (sustainable)


The significance of such functions in the analysis of fishery

production is twofold. First, the use of such functions implicitly

ensures that biological equilibrium is achieved. This means that catch

rates are always equated to sustainable yield. A note of caution should

go with such a strong statement, however. In the empirical estimation

of such functions, the degree to which such estimated equilibrium yields

and actual equilibrium yields coincide rests largely on the adherence of

certain underlying assumptions (Pella and Tomlinson, 1969). Thus,

empirically estimated equilibrium yield functions may not incorporate

the biological equilibrium condition of catch equals sustainable yield

to any reasonable degree. Secondly, equilibrium yield functions gen-

erally impose specific functional forms on the observed relationship

between catch and effort. This is significant in that the set of valid

equilibrium yield functions is fairly limited.

In contrast to the notion of equilibrium yield functions is that of

non-equilibrium yield functions. In this study, the term non-equilib-

rium implies that no biological equilibrium condition of catch equal to


sustainable yield is imposed on the relationship between catch and

effort. Very little attention has been given to equations of this type.

The most notable exception is a paper by Bell et al. (1973) wherein they

addressed the question of constant versus decreasing returns in an

essentially non-equilibrium framework. Within the framework of bio-

economics it may seem objectionable to consider analyzing fishery pro-

duction without implicitly incorporating biological concepts in the form

of population dynamics. A closer examination of non-equilibrium yield

functions may serve to lessen these objections.

One of the primary reasons for analyzing fishery production is to

develop analytic models which can be used in studying the effects of

various management alternatives. Non-equilibrium yield functions are

very amenable to such types of analysis for two reasons. First, this

class of functions provides a much wider range in the choice of func-

tional relationships between catch and effort. This is especially

significant in that equilibrium yield functions generally treat fishing

effort as a single variable or composite measure. Non-equilibrium

models, however, can be specified in several variables which serve to

decompose effort into components which greatly enhance the analytic

ability of the model with respect to management questions. Secondly,

with the appropriate stochastic incorporation of unobservable population

effects, non-equilibrium yield functions can be used to derive equilib-

rium yield relationships.7 Unless otherwise stated, all yield (catch)

equations in the ensuing analysis will be non-equilibrium in nature.

7The notion of derived equilibrium yield equations is developed in
the following chapters.


Variable Product Price

Since most fishery analyses are done at the aggregate or industry

level, the validity of a constant product price is questionable.

Relaxing the assumption of constant product prices complicates some of

the traditional theoretical results in the context of equilibrium yield

functions. Anderson (1976) has shown that both the derivation of maxi-

mum economic yield (MEY) and the results of unregulated competition are

obscured when product price is variable. Consider Figure 7 which illus-

trates the MEY and open access solutions under the assumption of a

constant product price, p.



m c

Figure 7.


Open access equilibrium and maximum economic yield
in a fishery with constant product price

The curve labeled TR is the "monetized" sustainable yield functions

and the line labeled TC corresponds to total cost. In that price is

constant, it is important to note that the sustainable yield function

retains its shape. There are unique effort levels corresponding to


MEY (E ) and the open access solution (Ec). Now consider Figure 8,

where price is variable. It can be seen that the total revenue function

no longer retains the shape of the sustainable yield function, but

rather has become "doubled-humped" (Anderson, 1973).8 If the relevant

total cost curve is TCI, there still is a unique open access solution.

However, there are now three effort levels (El, E2 and E3) wherein

marginal cost equals marginal revenue.

C 2


1 TR

E2 E3 E

Figure 8.

Open access equilibria and maximum economic yield
in a fishery with a variable product price

Thus, the task of finding the correct solution requires finding a

global optimum from several local solutions where marginal revenues and

costs are equated. If the relevant cost curve is TC2, in addition to

multiple MEY solutions, there now exists three points (A, B and C) where

A graphical derivation of the "double-humped" sustainable revenue
function is presented in Appendix D for the case of a linear demand


the open access result of total cost equals total revenue holds. Thus,

it can be seen that relaxation of the constant price assumption does

indeed confuse and complicate many of the theoretical results derived by

using equilibrium yield functions.

Utilizing non-equilibrium yield functions can avoid some of these

complications. Let the yield function for a fishery be defined by

C = f(X1, ..., Xn) (23)

where C denotes output and the X. corresponds to n inputs. It is
further assumed that f(X1, ..., X ) is such that

> 0 i = 1 ..., n (23a)

< 0 i = 1, ..., n (23b)

Equations (23a) and (23b) merely assert that the marginal product func-

tion is everywhere positive and declining. The price of output is now

defined to be a declining function of catch

P = P(C) (24)

where d < 0. Finally, the cost equation is defined by

K = r. X. (25a)
i=1 l 1

where the input prices, ri, are assumed constant. From equation (25a),

the marginal cost of Xi is then given by



i = 1, ..., n

Given these assumptions on the technical relationship between catch,

inputs and product and input prices, the profit maximization problem or

equivalently the MEY problem can be stated in the form

MAX 7 = P(C)C z r X.


s.t. C = f(X, ..., Xn)

Equation (26) can be seen to be a constrained maximization problem with
the constraint being the yield equation. Utilizing the method of

lagrange multipliers, equation (26) can be restated by

MAX L = P(C)C z r. x X[C f(X1, .., X )] (27)
i=l n

Differentiation of equation (27) yields the first order conditions

aL p + dP
= + C-x = o
aC dC

L = -r + X 0
ax. ax.
1 1

i = 1, ..., n

a = f(X .. X ) -C =0
ax 1 n

Examination of the first equation indicates that the lagrange multiplier,
X, is equal to P + -C C, which is precisely marginal revenue.

9 s section draws upon Intri gator (1971
This section draws upon Intrilligator (1971).






Substitution of P + P C into equation (27b) for X yields

(P+ CdP C- r. i 1 n (28)

The expression in equation (28) states that in equilibrium, the marginal
revenue product of the thinput must be equated to its price, or

equivalently, its marginal cost. Equation (28) can be rewritten in an

alternative and perhaps more illuminating fashion as

(P + C) r/i= ,..., n (29)
dC 1 X."

af / f
From equation (29) it is readily seen that r / = r. / for all
i e ji
i and j. Now, in equilibrium, ri / is precisely equal to the

marginal cost of output. Hence, equation (29) states the well-known

result that, in equilibrium, marginal cost equals marginal revenue.

Equation (29) provides a convenient way of examining some possible

consequences of various management goals. One can consider the implica-

tions with respect to input usage levels under various management goals.

In this case, the industry is treated as a single firm and the manage-

ment goal is defined to be profit maximization. The relevant equations

to be solved in this case are given by equations (27a-27c). Assume

that the input levels resulting from this solution are denoted by Xi,

i = 1, ..., n. Consider now the relationship between these input levels

and those that would result if the fishery was managed at price equals

marginal cost. Under this regime, the equilibrium equations analogous

to equation (29) would be


P r/ i = .., n (30)
1 r X.

From equation (24) it can be shown that (P + dC C) is always less than

P. This taken in conjunction with equations (23a) and (23b) can be

used to show that the input levels satisfying equation (30), say X ,

must be such that X' > X. for all i. Thus, the obvious result that

input levels under the management strategy of marginal cost pricing are

greater than those corresponding to profit maximization is obtained.

A Multi-Sector Fishery

The term fishery is somewhat synonymous with the traditional

definition of an industry. A sector is defined in this study to be a

sub-industry defined in terms of geographical location. Thus, if a

large fishery is composed of several states or distinct geographic

regions, under the above definition, each state or region can be con-

strued as a single sector.

To begin the analysis of a multi-sector fishery, assume there are

N sectors or regions, each facing a demand function defined by

P = P.(C1, ..., CN) i = 1, ... N (31)

where P. is the price received by producers in the i region and the C.

correspond to the outputs of the N regions. It should be emphasized

here that the C. are assumed to be the same product. The subscript

refers to regions rather than commodities. This form of the demand

equation will be discussed later. The demand equations given in equa-

tion (31) are assumed to be such that


< 0 all i, j(31a)

Turning to the yield equations, let the i region's catch function

be defined by

C = f (X, ..., Xni) (32)

where the X.. refer to the jth input used in the ith region. It is
further assumed that fi (Xi' ..., Xni) i = 1, ..., N satisfy the con-

ditions stated in equations (23a) and (23b). Finally, assume that the

cost equation for the ithregion is given by

Ki = z r.X.. i = 1, ..., N (33)
j=l j J1

where the r. are fixed input prices assumed the same for all regions.

The profit maximization problem for the entire fishery can be

stated as

MAX ~r = Pi(C1', ... CN) C. K. (34)
i=l i=l

s.t. Yi = f(Xli, ..., Xni) i = 1, ..., N

Once again, using the method of lagrange multipliers, equation (34) can
be restated in the form

MAX L = E Pi(C1, ..., CN) Yi K. + [f (X., .,.
i i i
Xi) Ci ] (35)


where the are the undetermined lagrange multipliers. Differentiation
of equation (35) with respect to C., X.. and x. yields the N(n + 2)
1 3frst-o r c i
first-order conditions



N aP
P. + C = 0
k=1 aCi k 1

-r. + *. 1 0
+j 1 aX i

fi (Xli' Xni) Ci

i 1, ..., N

i = 1, ..., N
j = 1, ..., n

= 0

i =1, ..., N

From equation (35a) it can be immediately seen that X. = P. +

i = 1, .
3a. Ci
term on




N aP
al k Ck'
k=l i

.., N. This expression for A. can be rewritten as X. = (P.
1 1 i
) + z k C. In this form, it can be seen that the first
kfi aCi k.

the right-hand side of the equality is precisely the change in
th th
in the i region with respect to variations in the i region's
Thus, this term is equal to the i region's marginal revenue

Now, substituting for Xi in (35b) yields

af. af.
(MR. + E 1 C) =- r.
1 kti Ci k aXji

j = l, ..., n
i = 1, ..., N

Rearranging terms in equation (36) results in the expression for the ith








aP af.
1R k#i C i k 1 aX Jji

This equation can be used to show that in equilibrium, r / = r. /

Small i, j, and that these expressions are in turn equal to the

marginal cost of output. From equation (37), it can then be seen that

the single result of within region marginal cost equals within region

marginal revenue does not necessarily hold when the industry is composed

of several regions of whose profit is jointly maximized.

The reason for this apparent divergence between each region's

marginal cost and revenue can be explained by examining the second term
on the left-hand side of equation (37), E k C This term is equal
kfi 3Ci k
to the sum of the change in total revenues in the N 1 regions induced

by a change in the output of the ith region. From this, the non-equality

of within region marginal costs and within region marginal revenues makes

more sense. The equating of within region marginal revenues and mar-

gional costs fails to account for interregional price effects. When

maximization deals with all sectors simultaneously, these price effects

are "internalized," resulting in the expression in equation (37).

Having seen that simultaneous maximization of profit in the above

situation does not result in the traditional result in the equality of

each region's marginal cost and marginal revenue, it is of interest to

determine the sign of difference between these two terms. Knowledge of

this sign will enable comparison of input levels obtained under the

above procedure and those obtained by maximization of each region's

profit independently of other regions. Rewriting equation (37) as


MR. MC. = Ck C(38)
S 1 kfi aC k

where MCi has replaces the expression r / it can be seen that the

sign of the difference between MR. and MC. is given by the sign of
P 9P k
-C Ck. From equation (31a), < 0 for all i, k and Y is
ki aCi k aC k

strictly positive. Therefore, the sign of the right-hand side of equa-

tion (38) must be greater than zero, implying that MRi is greater than

MC.. It should be noted here that a sufficient condition for MRi = MCi

is that 0, i k. In this case, independent maximization of

regional profits is equivalent to maximizing profit in all regions

The implications of these results on input levels for the i

region can be seen by examining Figure 9. The figure illustrates that

equating MR. and MC. results in an output of Cli, which is greater than

that produced, C2i, under the equilibrium conditions stated in equation
(37). The distance OA-OB is equal to kC- C Taken in conjunction
k=i i 1
with the nature of the yield function (equations (23a) and (23b)), it

can be concluded that inputs are at lower levels when cross-regional

price effects are taken into account.

As in the case of a single sector fishery, the question of produc-

ing at the point where price is equated to marginal cost must be con-

sidered in the multi-sector fishery. As with the single sector fishery,

assume that the desired yield levels have been determined for each

region. In that fixing yield levels results in fixing price, the

profit maximization problem reduces to a cost minimization problem with


Price in
Region i

P / MC.
P2i -

kY B I
-kti y K .
kfi i I MRi
I P. (Y ..., YN)

0 Y2iYi Catch in Region i
2 Yii Yl

Figure 9. Equilibrium in a multi-sector fishery with variable
price and pecuniary externalities

constant price and output. The main concern, in so far as the multi-

sector fishery is concerned, is the correct choice of price. Consider,

for example, that C. i = 1, ..., N are the desired yields for the N
regions. Now, if the relevant demand curve for region i is given by

P. = P (C., ., Cn) (39)

the resulting expected price will be given P. = E[ P. (Y' Y ) ].
1 1' n
This price reflects the interregional price effects and is used to

derive the appropriate levels of inputs. If, however, interregional

price effects are ignored and the demand function erroneously is assumed

to be of the form

Pi = Pi (Ci) (40)

the resulting expected price, P) ] will not be the


Pi I *
correct price. Unless 0 i k, P. will not equal P., which will

result in a different set of input levels being chosen. The reason this

is so is that P. fails to reflect between region price effects.

These results are not surprising. In a situation in which each

sector is being managed independently, only costs and revenues specific

to the region will be considered in the decision process. If, in fact,

there exist cross-regional price effects which are not considered, each

region in attempting to maximize its profits will tend to choose higher

levels of output and, hence, higher input levels than would be obtained

if the interregional effects were taken into account. If all regions

were under the control of one central management authority, these inter-

regional price effects would be "internalized" and the appropriate

levels of regional input levels and outputs would be obtained.

As a motivation behind how such a situation as described above

could arise, consider a fishery which is composed of several states

fishing over a fairly large geographic area. Furthermore, assume that

the fishery is such that each state's demand price is determined in

part by the within state supply of fish and in part by a national

market, supplied by shipments from all states in the fishery. Thus, the

price in each state is determined directly within state catch and in-

directly through a national market by the catch of all states in the

fishery. Now, if the management authority is extended over the fishery,

the appropriate method of incorporating management goals becomes a

relevant question. One possible goal of management could be to manage

the fishery in such a manner as to maximize the entire fishery's profit.

A key result of the above analysis is that under these circumstances,

management should not be undertaken on an independent basis by


individual states or regions. Rather, management should take into

consideration all states simultaneously.



This chapter presents the empirical model for the Gulf of Mexico

Reef Fish Fishery to be utilized in simulating the effects of various

management alternatives. The first section of this chapter deals with

the specification and estimation of the regional catch equations. The

second section contains the regional demand equations facing producers.

Before proceeding with the specification and estimation of the

various empirical relationships, a brief discussion concerning the

nature and type of data employed in this study is in order. As with

many fisheries, data on the GMRFF are extremely limited. Primary data

at the firm level are almost non-existent. While some data of this type

could be collected for perhaps one or two years, this would not be suf-

ficient given the long-run nature of this study. Furthermore, virtually

no consistent continuous set of biological data on resource stock sizes

suitable for econometric analysis exist. Such data could be collected,

but only at extremely high costs in terms of both time and dollars.

The major source of data used in this study is Fishery Statistics

of the United States (U.S. NMFS, 1957-1975). The data used thus cor-

respond to aggregate cross-section time series observations on states

participating in the GMRFF for the years 1957 to 1975 inclusive

(Appendix B). Because of the aggregate nature of these data, the



relationships discussed are necessarily aggregate in nature. Such

aggregation unfortunately limits the resulting empirical models in many

undesirable ways.

Catch Equation Specification and Estimation

In order to specify state catch equations, the form of the catch

equation for an arbitrary region is first developed in a deterministic

fashion. After developing the typical region's catch equation, the

corss-sectional specification is presented. Commensurate with this

discussion is the stochastic specification of the catch equation. The

presentation concludes with the choice of the appropriate estimator.

A general expression for a fishery catch equation is given by

C = f(E, S) (41)

where C refers to catch, E is effective fishing effort and S denotes the

resource stock size. Since stock size is seldom observable, the catch

equation stated in (41) is often modified for empirical analysis to

C = f*(E) (42)

In equation (42) aggregate catch is expressed as a function of only

effective fishing effort. The f (E) function is used to denote the fact

that the influence of the resource stock is not considered explicitly as

in equation (41) but rather indirectly. The equilibrium yield models

presented in the previous chapter are one such class of models.1

1Consider the Schaefer formula as an example. Equation (41) cor-
responds to C = KEP in the Schaefer model. In this context P is elimi-
nated through algebraic manipulation to derive the Schaefer type
equilibrium yield function C = AE BE2. Here, AE BE2 corresponds to
f*(E) in equation (42) above.

A second class of models which correspond to those defined by equation

(42) are non-equilibrium yield functions. In these types of models,

stock effects are incorporated through stochastic processes.

Specification of Fishing Effort

Central to.the development of an empirical representation of the

ith state's catch function is the notion of effective fishing effort.

Recall from equation (22) that effective fishing effort is primarily

composed of a nominal component and a fishing power component. Further,

fishing power was seen to be a function of input levels.

The aggregate nature of the data must be considered in specifying

the fishing power equation for the ith state in the GMRFF. As such, the

fishing power function relates to the average fishing power correspond-

ing to vessels operating out of ports located in each state. The fish-

ing power function for the ith state is thus given by

Sexp(k) Xl X i = 1, ..., 5 (43)
it lit 2it t = ..., 19

where EP denotes fishing power, Xlit is average crew size, X2it is
average vessel size (gross registered tonnage) and k is a constant


The choice of average crew size and average vessel size as the

relevant inputs for use in specifying the fishing power function was in

part determined by available data. The nature of the fishing activity

for vessels in the GMRFF suggest that these variables are appropriate

measures of labor and capital inputs which determine fishing power,

however. The general fishing process involves operating hand or power

driven reels which control the fishing line. Average crew size provides

a good aggregate measure of "gear contact" with the resource stock since

each crewman usually operates only one reel. The use of average vessel

size measured in gross registered tonnage is also a reasonable represen-

tation of the capital input in the fishing power equation. In the reef

fishery, factors such as sea conditions and weather can impair or pre-

vent altogether the fishing process. Vessel size is a factor which

affects the ability or certainty of the fishing process to be undertaken.

This measure also is related to the duration of fishing trips and to a

lesser degree, the distance a vessel can travel to fishing grounds.

Thus, the effect of vessel size on fishing power is seen to be related

to the ability of vessels to undertake and sustain the fishing process.

The choice of the functional form was in part due to the ease with

which the Cobb-Douglas type function can capture non-linear production

relationships without a large loss in degrees of freedom. In addition,

this functional form also facilitates testing the hypothesis that the

fishing power function exhibits constant returns to scale. Given that

fishing power is a theoretical construct, it may be that a doubling of

all inputs doubles fishing power. Equation (43) as specified permits a

direct test of this hypothesis. Finally, the ji, j = 1, 2, provide a

convenient means of judging the relative importance of each input with

respect to the "production" of fishing power. A priori, one would

expect that average crew size should have a larger influence on fishing

power than does vessel size, given its relationship to "gear contact"

with the resource stock.

Total effective fishing effort is defined to be the product of the

nominal measure of effort and fishing power.2 It should be noted that

fishing intensity is assumed to be implicit in the observation interval

of the time series data. The nominal effort measure used in this study

is defined to be the number of vessels. The expression for total effort

is then given by

Eit = exp(k) Vit X X (44)

where Eit is total effort and Vit refers to total vessels in state i and

time t. Examination of equation (44) reveals that total effort in

state i and time t is precisely the number of vessels operating in the

corresponding region and time period, "weighted" by the average fishing

power corresponding to those vessels.

Within Region Specification Considerations

The relationship in equation (44) serves to define total effort as

a function of the size of a fishery (number of vessels) and the cor-

responding average levels of capital and labor inputs (fishing power).

This equation, however, is definitional in nature and as such precludes

direct estimation of parameters independently of catch. Thus, it is

necessary to specify the ith region's catch equation.
The empirical form of the catch equation for the ith state is given


Ci exp[A(S)it] E = 1, ... 5 (45)
it .. ., 19

2Hereafter, the terms total effort and total effective fishing
effort are used interchangeably.

where Cit represents combined catch of red snapper and grouper, A(S)it

is a stochastic process generated by the resource stock, Eit is total

effort and .i is a constant. The aggregation of red snapper and

grouper into a single catch variable is an unfortunate result of limited

data. While separate catch series are available, input data are not

disaggregated to permit an analysis of the allocation of these inputs

between species.

The form of the catch equation, as with the fishing power equation,

was chosen in part due to the ability of such a function to capture a

non-linear relationship between catch and effort while retaining an

intrinsic linear form for estimation purposes. The form chosen does,

however, lend itself to an approximate test of an interesting hypothesis.

The notion of returns to scale is a somewhat misleading notion with

respect to fisheries production due to the dynamic nature of the

resource stock. It has been used, however, in fishery production

literature (Schaefer, 1957; Scott, 1955). More precisely, the non-

equilibrium catch equations utilized in stock production models gener-

ally assume constant returns to scale as exhibited by the ith state's

catch equation. Acceptance that the stochastic process, A(S)it, in

equation (45) adequately accounts for the effects of the resource stock

on catch allows the utilization of the pi parameter to conduct an ap-

proximate test of the "constant returns" assumption of the stock produc-

tion models of Schaefer (1957) and Pella and Tomlinson (1969).

3The specification of the stochastic nature of A(S)it will be
discussed more fully in the latter portion of this section.

The catch equation in (45) can be expressed in terms of nominal

effort and fishing power by substituting equation (44) into equation
(45) for Eit. The resulting reduced form catch equation is

I i Xli x 2i (46)
it = exp[A'(S)t] V Xlit (46)

where the term A'(S)it denotes the fact that the constant k in equation
(44) has been incorporated into the stochastic process A(S)it and the

reduced form parameters are ji = BiOi' j = 1, 2. To facilitate

further discussion, it is convenient to write equation (46) in an alter-
native form. By defining cit = In Cit, x = In Xi and so on, equation

(46) can be written in double log form as

cit = A'(S)it + i + i li Xlit + 2ix2it (47)

i = 1, ..., 5
t = ..., 19

The nature of the stochastic process A'(S)it can be analyzed in this

Stochastic Approximation of Resource Stock Effects

The expected presence of a stochastic process in the catch equation

derives from the nature of the omitted resource stock variable.4 From

the discussion contained in Chapter II, it is apparent that the change
in the resource stock over time is proportional to the difference
between catch and sustainable yield. An expression for the size of the

4The discussion that follows implicitly assumes that the resource
stock variable is uncorrelated with the included set of regressors.

the resource stock in any given time period can then be given by

St = S + X(C_1 Ct) (48)

where St is the stock size in time t, Ct is sustainable yield produced
by the resource stock in time t, Ct is the corresponding catch and X is
a constant of proportionality. Thus, equation (48) states that the

stock size in time t is equal to the stock size in the preceding time

period plus a proportion of the difference between sustainable yield and
catch in time t-l. While only Ct_1 is observable, equation (48) serves
to suggest that the resource stock variable is at least to a certain
degree, autocorrelated. Thus, the omission of the resource stock vari-
able is expected to generate some systematic variation in the distur-

bances of the catch equation which can be approximated by an
autoregressive process. In particular then, the stochastic process
corresponding to the ith region's catch equation is assumed to take the
general form

A'(S) = A' + U (49)
it 1 it

where A! is constant and the disturbance term, Uit, is postulated to be
characterized by a pth order autoregressive process. On substitution of
equation (49) into equation (47) the ith state's catch equation can be
expressed as

cit = A + Bivt + li xlit + 2ix2it + Uit (50)

where Uit= PliUt- + p2i Ui,t-2 + + Ppi Ut-P + eit and eit is

assumed to be white noise.5 Before proceeding to a more detailed

specification of the stochastic properties of equation (50), it is

necessary to consider the cross-sectional specification of the catch


Cross-Sectional Specification Considerations

The main consideration which must be addressed in the cross-

sectional specification of the catch equations involves determining what

restrictions exist on the catch equation parameters across states. Both

the nature of the fishing process and the nature of the data must be

considered in determining these restrictions.

Consider the equality or non-equality of the fishing power function

coefficients. In general, the fishing process and input characteristics

of vessels across states are very similar. Given this, it seems reason-

able to assume that the fishing power coefficients are constant across

states in the GMRFF. The fishing power function, incorporating this

restriction is given by

E = exp(k) X t = 1, ..., 19 (51)
it lit 2it
i = 1, ..., 5

The consequences of such a restriction are not without complications,

however. The assumption that aji = ajk for j = 1, 2 and all i, k

requires that non-linear restrictions be placed on the reduced form

parameters of equation (50). More explicitly, these restrictions take

5A white noise process is defined as a sequence of independent
identically distributed random variables with zero mean and constant

the form of Pk ji = Bj "jk for j = 1, 2 and all i, k. Thus, in the
absence of any a priori assumptions concerning the Bi, i = 1, ..., 5,

parameters across states, non-linear restrictions on the parameters must

be incorporated for correct estimation. If, however, Bi = Pj for all

i, j, this restriction is trivially satisfied and estimation difficul-
ties are greatly reduced.

The Bi parameters measure the marginal response of total state

catch to small changes in vessel numbers holding fishing power constant.

Given the homogeneity of the fishing process across states and the fact

that the fishing power function serves to "weight" vessels according to

the input characteristics of each state's vessels, the assumption that

i = Bj for all i, j may not be an unreasonable assumption to make. As
stated above, making the assumption Bi = .B for all i, j insures that

the non-linear restrictions on the reduced form parameters are met,

There is also an additional gain realized by assuming that the catch

equations for the GMRFF producing states to take the form

cit = Ai + Bvit + Xlit + 2 x2it + Uit (52)

Stated in the manner above, the data on vessels, crew sizes and vessel

sizes can be pooled across states. Not only does such pooling generate

considerably more variation in the vectors of regressors, which aids in
parameter estimation, it also creates a sizeable gain in degrees of

freedom. The final specification of the state catch equations given in

euqation (52) illustrates the cross equation restriction corresponding

6Recall that i.. = i a.. for all i, j. If .. ajk -for all i, k,
it follows that Bki ji = i "jk for all i, j, k.

to the equality of the reduced form (and structural) parameters across

states. It should be noted, however, that the mean of each region's

stochastic process, Ai, is unconstrained across equations. The specifi-

cation and intrepretation of the Ai parameters, in particular, and the

stochastic processes characterizing the regression disturbance, in

general, provide a convenient introduction to the discussion concerning

the choice of the appropriate estimator.

The geographic location of the primary reef fish stocks are

depicted in Figure 10. There is some indication, although there is not

an overwhelming amount of evidence, that the reef fish stocks do not

exhibit a great deal of migratory behavior (GMFMC, 1979). This fact,

taken in conjunction with the large geographic dispersion of fishing

grounds, suggests that the GMRFF is composed of several biologically

independent stocks of reef fish. Additional information on the general

fishing locations of vessels originating from various states' ports

(GMFMC, 1979), indicates that vessels originating from different states

fish on common grounds. This information suggests each state catch

function should have a stochastic process dominated by the stock most

frequently fished, and that these processes should be contemporaneously

correlated due to the intermixing of vessels from different states.

Thus, the overall structure of the system of catch equations is char-

acterized by a system of seemingly unrelated regression equations (SUR)

with cross equation parameter restrictions and autoregressive


The information above also serves to give an interesting interpre-

tation to the Ai parameters. These parameters serve to determine the

"location" of the catch equations for each state in input-output space.

Figure 10. Principal fishing grounds in the Gulf of Mexico
Reef Fish Fishery

Given that all other technical parameters are constrained to be constant

across states, the A. may serve to indicate the relative size or densi-
ties of the primary stocks fished by vessels for each state.

Furthermore, testing the difference between the A. constitutes an

approximate test for the degree to which various states' vessels fish

common grounds. The reasoning behind this is that if vessels from dif-

ferent states fish common grounds, the stock densities should eventually

become equal. This can be investigated by testing the hypothesis A. =
A. for all i, j.

Choice of Estimator for the Catch Equations-

As a prelude to the discussion regarding the estimation of the

catch equations, it is convenient to place the system of equations into

matrix form. This is accomplished by

C = .XB + U (53)


C = NT x 1 vector of logged catch variables;

g = (k + N) x 1 vector of parameters to be estimated; and

U = NT x 1 vector of disturbances with EU = 0, EUU = 1.

The NT x (k + N) matrix of regressors is of the form

X = [D X] (53a)

with the NT x N matrix, D, composed of appropriately defined state dummy

variables and X corresponding to the NT x K matrix of logged values of

regressors given in equation (52). Specification of the distrubance

term is given in general form to emphasize the covariance matrix is

non-spherical. The precise form is conditioned by the exact form of the

autoregressive processes corresponding to each state's disturbance


Estimation of the catch equation parameters must be done in two

basic steps. The first step involves the identification and estimation

of the autoregressive processes for each state generated by the unobserv-

able resource stock. Once this is accomplished, the appropriate form of

the covariance matrix of the disturbances can be ascertained and the

appropriate estimator for the reduced form parameters derived.

Due to the small sample size of each cross section (T = 19), many

of the standard time series identification techniques for determining

the autoregressive order parameter are unsatisfactory. This mainly

results from the fact that most statistical tests on the order parameter

are only asymptotically valid and utilize a variance measure that is

inversely proportional to the sample size. There are, however, several

techniques, such as Akaike's (1969) FPE criterion which do not suffer

from this limitation. Several alternative identification procedures

were used in the identification of the residual autoregressive process.

These procedures are outlined in Appendix E.

The estimated residuals used in the identification process were

generated by applying a two stage Aitken's estimation procedure to

equation (52). More precisely, the NT x 1 vector of residual estimates

is given by

0 = C (54)

where 3 = (X'(i1 aI)X)-" (X'(" a I)C) and E is the N x N matrix of

estimated contemporaneous covariances. The estimated residual vector

was then partioned into N, T x 1 vectors corresponding to the N cross-

sections. The results of the identification procedure indicated that

all sets of estimated residuals were characterized by first order


Having determined the order of the autoregressive processes for

each state catch equation, a complete specification of the distrubances

for the system of catch equations can now be made. Denote the stochas-

tic process of the ith region by

Uit = PiUit-I + e it (55)

The stochastic specification of U in equation (55)7 is then given by

E(Uit) = ii (55a)

E(UitUjt) = (55b)

4ij t = s
E(e ite) = { (55c)
o t t s

E(Uio Ujo) = ij /1 -Pi Pj (55d)

1ii) 1
for i, j = 1, ..., N and Uio N(O, 2) and eit N(O, i).
1 Pi
As a result of the residual identification process, the system of
catch equations can be characterized as a system of seemingly unrelated
regression equations with cross equation parameter restrictions.
Furthermore, the disturbances exhibit first order autoregression. A
great deal of literature pertaining to the estimation of this type of
equation system is available. Most notable is the work done by Parks
(1967), Kmenta (1971), Kmenta and Gilbert (1968) and Zellner (1962).
The most important of these insofar as this study is concerned is the
work done by Kmenta and Gilbert on the small sample properties of alter-
native estimators for systems of equations similar in nature to the
catch equations above. As mentioned previously, data on the catch equa-
tion variables is limited, resulting in rather small sample sizes.

The form of U is given by
U1 = [ Uil ..., UiT, ..., UN1, ..., UNT '

Given that all estimators for the above equation system possesses only

asymptotic properties, it is appropriate to use a relative efficiency
criterion in small samples as a basis for choosing the "best" estimator

for the system of catch equations given in equation (53).
Drawing on the results of Monte Carlo studies conducted by Kmenta

and Gilbert (1968), a four stage Aitken's estimator (FSAE) was chosen as

the appropriate estimator. The formation of this estimator proceeds in

two basic steps. Given that the disturbances in each equation in the
system are known to follow a first order autoregressive process, the

first step involves the application of the two stage Aitken's estimator

to equation (53) to generate a sequence of estimated residuals for each

state catch equation. These residuals correspond to those given in

equation (54). The use of this four stage estimator is the reason that
the residuals estimated using equation (54) were used in the order
parameter identification. The estimation of the autoregressive parame-
ters is accomplished by

p. it- / Z i = 1, N (56)
t=2 T 1 t=l T

where U is the estimated residual for the ith state and tth time
period defi-ned in equation (54).

The second step in deriving the FSAE involves a second application

of the Atiken's two stage estimator. Before this estimator is applied,
however, the data is transformed by

i = 1 p. C i = 1, ..., N (57a)
111 i1

c = Ct pCi i = 1, ..., N (57b)
it it 1 i ,t- t =2, ..., T

^ .
Xi = 1 p X all i, j (57c)

Xit = it pixjitl all j (57d)
jt =2, ..., T

where cit and xjit are defined as in equation (53a). In matrix form,

the transformed system can be written as

C = X + U (58)

where C is an NT x 1 vector of transformed logged catch values and X

is an NT x (N + K) matrix of transformed regressors in log form. The

effect of the transformation is to remove the autoregressive effects
from the NT x 1 disturbance vector, U Thus, EU U = I IT where

11 h12 "' 1N

S= 21 22 "' 2N (59)

N1l RN2 NN

corresponds to the contemporaneous covariance matrix of the transformed

disturbances. To estimate t, ordinary least squares was applied to

equation (58) to yield

^* *^ (60)
U = C X (60)

where B = (X* X*)- (X*' C*). Estimates of the ij were calculated by

1 T ^. ^.
ij T-k Uit U jt (61)

where Uj., t = 1, ..., T is the estimated residual sequence correspond-
ing to the ith state. The estimated covariance matrix, f, is formed by
replacing j.. with cij in equation (59). Finally, the FSAE for the
system of catch equation is given by

S= (X* (- a I)X)-1 X*( a I)C* (62)

where I is an identity matrix with rank T. Furthermore, the estimated
variance-covariance matrix for B is given by

*' 1 -1
COV e = (X (~ I)X ) (63)

The precise statistical properties of p are somewhat difficult to
ascertain. The primary reason for this relates to the stochastic

specification of the system of equations. Under the assumption that the
true stochastic specification is first order autoregression with con-
temporaneous correlation, the asymptotic covariance matrix for p is
consistent and asymptotically normal and efficient. Unfortunately, the
actual stochastic specification of the system constitutes a pretest.

This seriously clouds the precise statistical properties of the esti-
mated coefficients.

Price Equation Specification and Estimation

The latter part of Chapter II presented a scenario in which pro-
ducing states faced a variable product price. The price faced by
producers in any given state was dependent on the outputs of all

other states. The purpose of this section is to present the specifica-

tion and estimation of a system of interrelated price equations for the

GMRFF which are similar in nature to those discussed in the previous

chapter. The discussion that follows first addresses issues involving
the appropriate structure of the price equations. The choice of the

appropriate estimator and the estimation scheme utilized are then


Aggregation Across Species

As with the catch equations, the price equations must be specified

in aggregate terms. The basic reason for this relates to the data

limitations which required the catch equations to be specified in terms

of the aggregate catch of reef fish. Thus, to be compatible with the

catch equations, the price relationships must be specified in terms of

an aggregate "price" of reef fish. Within the model "price" of reef

fish serves as a measure of the average value per pound produced by reef

fish vessels.

The degree to which such a "price" can be used in deriving valid

price equation estimates depends on many factors. These factors include

the similarity of the prices and markets for red snapper and grouper,

the relative magnitudes of each species in total catch and the similarity

of the price responses to changes in catch for each species. Since

1957, the dockside price of red snapper has been about twice that of

grouper (U.S. NMFS, 1957-75). Over this period both have exhibited

8The term price in this section pertains to the nominal dockside or
ex-vessel price.

fairly consistent price increases. Both species have exhibited similari-

ties in product form when shipped from dockside although the type and

location of the markets to which they are sent differ. In 1977, 81.6

percent of grouper and 93.7 percent of all red snapper taken in the

GMRFF were shipped from dockside in fresh iced form (GMFMC, 1979).

Similarly, over half of each of these species was shipped to wholesalers.

In terms of market location, 58.4 percent of the red snapper caught was
shipped to Northeastern markets, and 24.1 percent was shipped to South-

eastern markets.0 In contrast, only 15 percent of the grouper catch

went to Northeastern markets while 77 percent was shipped to markets in

the Southeast (Appendix F). Thus, while the absolute prices of the two

species differ as do the location of terminal markets, the basic trends

in prices as well as the product form for grouper and red snapper

shipped from dockside are very similar.

In estimating price equations using price of reef fish as the

dependent variable, some bias in the parameter estimates will be

incurred. The degree of bias is related to both the relative magnitudes

of each species in total catch and the similarity of price responses to

changes in catch of each species. Florida is the dominant producer of

grouper in the GMRFF. The proportion of grouper in the total catch of

states other than Florida has been relatively insignificant. No state

other than Florida has accounted for more than 4 percent of the total

grouper catch since 1970 (GMFMC, 1979). Thus, only the price equation

for Florida appears susceptable to significant aggregation bias.

Includes New York, Illinois, Michigan, Maryland, Pennsylvania and

10ncludes South Carolina, Georgia and Florida.

The degree to which this bias is incurred rests on the similarity

or dissimilarity of the price responses to changes in catch for each

species. To see this, let

Pkt = aok + YkCkt k = S, G (65)

denote the price equations for red snapper, (S), and grouper, (G). Now,

if instead of equation (65) the price equation is written as

Pt = a + YCt (66)

where Pt = w1 Pst + W2 P w + 2 = 1 and C = C the estimated

parameters will correspond to a = W aoG + W2aoG and y = wls + W2Yg.

If, however, s = Yg = Yo, the price response parameter in equation (66)

will be y = (wl + w2)Yo = Yo since w1 + w2 = 1. To test the equality

of the price response parameters for grouper and snapper caught by

Florida vessels, separate price equations were estimated for each

species. The results of estimation indicated that the price response

parameters were of very similar magnitude (Appendix F).

The necessity to aggregate across species in specifying the reef

fishery price equations is unfortunate. However, the similarity between

grouper and red snapper as food fish at dockside, the relatively small

size of groupers in the reef fish catches of states other than Florida

and the similarity of the price response parameters of both species for

Florida makes the expected consequences of such aggregation small.

Thus, in the analysis that follows, the price equations are defined in

terms of the aggregate variables, price of reef fish and total catch of

reef fish.

The main product form of reef fish when shipped from dockside is

fresh iced. Thus, reef fish can be considered to be non-storable

products. Furthermore, the direction of causality is such that quantity

produced determines price at dockside. The implication of this is that

the price (demand) equations should be specified in price dependent

form. This specification is harmonious with fishery demand analyses

conducted by others (Cato, 1976; Doll, 1972). Within this context,

price can be considered as determined by factors such as quantity pro-

duced, quantity of substitute products, income and tastes and preferences

of consumers (Tomek and Robinson, 1972). Since the price equations

considered here relate to dockside prices, the empirical price equations

derived below abstract from many of these causal factors.

The price equations in the latter part of Chapter II were presented

in general form and in such manner that the price in each state was a

function of the quantities produced in all states. In spite of this

general form, there is considerable information on the structure of

prices in the GMRFF which suggests some parameter restrictions. Cato

and Prochaska (1976a) have shown that Florida is the dominant producer in

the GMRFF. Their findings suggest that Florida is the only state which

has a significant effect on prices in other participating states. This

result is not surprising in that Florida's catch since 1957 has

accounted annually for an average of 45 percent of all red snapper and

84 percent of all grouper caught in the Gulf of Mexico region.

Furthermore, it appears that all states are net exporters of reef fish.

Since 1975, the GMRFF has accounted annually for approximately 98

percent of all red snapper and 92 percent of all grouper produced in the

United States (U.S. NMFS, 1957-75). Data for 1977 also indicate that

93.3 percent and 89.0 percent of the red snapper and grouper catch,

respectively, was shipped to areas other than the GMRFF states (Appendix

F). The implication of this data is that there should be very little,

if any, interregional trade in reef fish among GMRFF states. This is

significant in that it implies the absence of any systematic price dif-

ferentials across states based on transportation costs, thus simplifying

the price model greatly.

The information above serves to provide the basis for specification

of the empirical price equations. The price equation for Florida is

assumed to have the form

Plt = ^01 + 11 C1t + 21t + elt (67)

where Plt is the ex-vessel Florida "price" of reef fish, Clt is the

corresponding catch and t is a time trend variable. Specification of

the disturbance component, e1t, is discussed below. For other states,

the general form of the price equation for the ith state is given by

it = YOi + li Cit + Y2i Cit + '3it + eit (68)

i = 2, ..., 5

where Pit and Cit are, respectively, the "price" and catch of reef fish

in state i. Once again, the specification of the disturbance term, eit,

is discussed below.

Incorporation of the trend variable in equations (67) and (68) was

done to account for demand shifts over time. This variable is a com-

posit proxy for effects such as population and income. The basic reason

these variables were not explicitly incorporated into the price equation

relates back to data limitations and the utilization of the "price" of

reef fish as the dependent variable. Although grouper and red snapper

are very similar products at dockside, they tend to move through dif-

ferent markets both with respect to type as well as location. Further,

as evidenced by Appendix F, there are also several marketing levels

through which these species pass in moving from dockside to retail.

Thus, any attempts to infer income flexibilities for reef fish would be

necessarily crude. In an effort to avoid such potentially misleading

inference and still capture the effects of such factors, the use of a

time trend variable was employed.

Choice of Estimator for the Price Equation

As with the catch equations, the choice of the appropriate esti-

mator for equations (67) and (68) is oriented toward obtaining desirable

statistical properties for the parameter estimates. Choosing the

appropriate estimator largely rests on the stochastic specifications of

the disturbance terms in equations (67) and (68). The delineation of

the price equations on a state basis is done on the basis of recorded

data rather than on the basis of some other economic factors which would

serve to delineate the appropriate cross-sectional units. In relation

to the economic structure of price determination in the GMRFF, such

division of units on the basis of geographic boundaries is admittedly

arbitrary with respect to the actual economic structure of the GMRFF.

Thus, a considerable degree of contemporaneous correlation in the dis-

turbances of the price equations is anticipated. The disturbance

specification for the price equations in equations (67) and (68) is

given by

E(eit) = 0 for all i, t (69a)

E.. t = s and all i, j
E(eit e) = { 'J (69b)
i s 0 t f s and all i, j

The stochastic specifications given in equations (69a) and (69b)

serve to characterize the price equations as a system of seemingly un-

related regression equations. Zellner (1962) has shown that the best

estimator for this type of equation system in terms of relative effi-

ciency is a two-stage Aitken's estimator. This estimator was utilized

in estimating the price equation parameters. Before proceeding with a

presentation of this estimator, it is convenient to place the price

equations in matrix form. Let P be an NT x 1 vector of prices, Zi be

T x ki matrix of exogenous variables corresponding to the independent

variables given in equations (67) and (68) for the ith region and e be

an NT x 1 vector of disturbances. The price equations in matrix form

are then given by

P = Zy + e (70)

where Z is a NT x (zKi) block diagonal matrix with Zi i = 1, ..., N

constituting the diagonal blocks. Further, E(e) = 0 and E(e e') = Qo IT

where IT is a T x T identity matrix and Q has the form

a11 021 ." alN

n = 21 22 ". 02N (71)

N1 aN2 aNN

The Aitken's estimator for y is then given by

y = (Z'(-I IT)Z) (Z'( l IT) P). (72)

Although this estimator is consistent, asymptotically normal and effi-
cient (Kmenta, 1971), it is not feasible in that n is unknown. The
covariance matrix can, however, be estimated as follows. Let eit be the

estimated residuals from the ordinary least squares regression of the
price equation for the it state alone. The aij can then be estimated


1 ^ ^
_i T E eit ejt. (73)
ij T jt

By replacing o.. in equation (71) with .ij, the two-stage Aitken's
estimator (TSAE) for y

S= (Z'(n1 IT)Z)- (C'("-1 IT)P) (74)

is obtained. Zeller (1962) has shown that y has the same asymptotic
properties as y given in equation (72).


Previous chapters in this study have developed a theoretical model

indicative of the GMRFF and presented the specification of the empirical

equations to be utilized in analyzing the fishery. The various esti-

mators for the empirical equation were also discussed and derived. This

chapter presents a discussion of the empirical results obtained.

The first section contains an analysis of the estimated state catch

equations. The following section presents a similar analysis for the

estimated state price equations. The third section develops the com-

plete reef fishery model used to determine maximum economic yield.

Further, the results of profit maximization in the fishery are presented

and discussed in detail. The final section of this chapter compares the

results of this analysis with those obtained by the Gulf of Mexico

Fishery Management Council.

Analysis of Production in the
Gulf of Mexico Reef Fish Fishery

The state catch equations were characterized in equation (52) as a

system of seemingly unrelated regression equations with autoregressive

disturbances and cross equation parameter restrictions. As such, a four

stage Aitken's estimator (FSAE) was utilized in estimating the catch

equation parameters. To a certain extent, the validity of the a priori

specification of the catch equations can be measured by the gains in

efficiency obtained by using the FSAE as opposed to the ordinary least

squares with dummy variables (OLSDV) estimator.

An examination of the parameter estimates obtained from the two

estimators indicates that both estimators yield parameter values of

similar magnitude with the exception of the parameter estimate for

vessel size (Tables 1 and 2). Furthermore, both estimators yield

parameter estimates of reasonable magnitude and the expected sign. The

gains from using the systems estimator become apparent when the standard

errors of the parameter estimates are examined. All standard errors

obtained utilizing the FSAE are substantially lower than the correspond-

ing estimated standard errors obtained from the OLSDV estimator with the
exception of the standard error of the vessel size parameter. Finally,

examination of the estimated autoregressive parameters illustrates that

the estimated equations for all states, except Louisiana, are charac-

terized by significant first order autoregression (Table 2).

This brief comparison of estimators for the reef fishery catch

equations gives considerable support to the a priori specifications of

the preceding chapter. Given the small sample size employed, the gains

in efficiency obtained by the FSAE coupled with the reasonable magnitude

of the estimated parameters and the strong presence of autoregression,

serve to give heuristic confirmation of the overall specification and

choice of estimator in deriving the empirical catch equations for the


It should be noted that the slight increase in the standard error
of the vessel size parameter must be considered in light of the fact
that the FSAE estimate is approximately three times larger than the
OLSDV estimate.

Table 1. Ordinary least squares with
Fishery catch equations

dummy variables parameter estimates for the Gulf of Mexico Reef Fish

Dependent variable Intercept an vessels in crew size An vessel sizeb

An Florida catch 3.72581 0.848001 0.756031 0.111163 R2 = 944
(.71085) (.08539) (.22132) (.16841)
An Alabama catch 3.0625 0.848001 0.756031 0.111163
(.86631) (.08539) (.22132) (.16841)

An Mississippi catch 3.3531 0.848001 0.756031 0.11163
(.8558) (.08539) (.22132) (.16841)
An Louisiana catch 1.1514 0.848001 0.75601 0.111163
(.74639) (.08539) (.22132) (.16841)

An Texas catch 2.007 0.848001 0.75601 0.11163
(.7782) (.08539) (.22132) (.16841)

aCatch is measured in thousands of pounds.

Vessel size is measured in gross registered tons.

Table 2. Four state Aitken's parameter estimates for the Gulf of Mexico Reef Fish Fishery catch equations

Dependent variable Intercept an vessels an crew size An vessel sizeb Uit-1

An Florida catch 3.15533 0.740230 0.713178 0.340649 0.44048
(.68466) (.067263) (.18169) (.17306) (.036391)
An Alabama catch 2.374897 0.740230 0.713178 0.340649 0.85468
(.80167) (.067263) (.18169) (.17306) (.022373)
An Mississippi catch 2.747624 0.740230 0.713178 0.340649 0.74216
(.76746) (.067263) (.18169) (.17306) (.028931)
An Louisiana catch 0.52701 0.740230 0.713178 0.340649 0.40764
(.73240) (.067263) (.18169) (.17306) (.40089)
An Texas catch 1.62417 0.740230 0.713178 0.340649 0.44820
(.74008) (.067263) (.18169) (.17306) (.037976)

aCatch is measured in thousands of pounds.
Vessel size is measured in gross registered tons.

Fishing Power

The estimated catch equations represent reduced form expressions.

It can be recalled from equation (44) that total effort was composed of

nominal effort and fishing power, and that the estimated fishing power

function can be derived from the estimated reduced form catch equations.

The estimated fishing power function corresponding to equation (43) for

an arbitrary state in the GMRFF is given by

.9635 .4601 (75)
Pi li 2i

where Xli and X21 are average crew size and average vessel size in the

ith state, respectively.2 Examination of equation (75) indicates that

average crew size has a much larger effect on fishing power than does

average vessel size. This was expected, however, as crew size is a

direct measure of "gear contact" with the resource stock. The estimated

fishing power elasticity corresponding to crew size is 0.9635, implying

a 10 percent increase in average crew size would increase fishing power

approximately 9.6 percent. The corresponding elasticity for average

vessel size is estimated to be 0.4601. Thus, a 10 percent increase in

average vessel size increases fishing power 4.6 percent. To the extent

that vessel size measures the ability of vessels to undertake and

sustain the fishing process, this elasticity may be interpreted as the

effect on fishing power of increased fishing time. This interpretation

It will be recalled that the fishing power function's constant
term was subsumed in the intercept of the catch function. In that the
ensuing discussion proceeds in relative terms, the constant in equation
(75) has been set equal to one with no loss of generality.

is reasonable since factors such as weather and sea conditions can

impair or prevent fishing.

The scale elasticity for the fishing power function in equation

(75) is estimated to be 1.4236. The implication of this elasticity is

that a proportionate increase of average vessel and crew size by some
factor, X, would increase fishing power by X .4 Thus, the fishing

power function exhibits increasing returns to scale. The effects on

fishing power of increasing average vessel size and crew size from 10 to

100 percent are shown in Figure 11. The result of increasing returns in

the fishing power function are somewhat surprising. A priori, returns

to scale in the neighborhood of unity were anticipated. This expecta-

tion rested mainly on the definitional nature of the fishing power

o 2.3

S1.9 -
iu 1.7


S 1.3

Proportionate Increase
7 lin Average Crew and
.1 .3 .5 .7 .9 1,1 Vessel Size

Figure 11. Estimated relative fishing power for proportionate
increases in average crew size and vessel size3

In Figure 11, average crew size and vessel size are assumed to
take an initial value of 1. While no actual vessels exhibit such input
proportions, choosing such levels alters only the scale of the figure.

expression. The interpretation of the output elasticities, however,

serve to make the appearance of increasing returns a reasonable result.

The primary role of the fishing power function in relation to the

empirical catch equations involves weighting the nominal effort compo-

nent (vessels). The basic notion here is that a standardized measure of

fishing effort can be derived by weighting vessels in the fishery by

certain input characteristics. Such a standardized measure of fishing

effort is extremely important, not only in analyses such as the current

study, but also in estimating Schaefer (1954) type sustainable yield

functions. By weighting vessels according to their relative fishing

power with respect to some base period, a standardized measure of fish-

ing effort such as standardized vessels can be obtained.

The estimated fishing power indices for each state participating in

the GMRFF for the years 1957 to 1975 are shown in Table 3. The fishing

power index is defined by

I Xlit .9635 it .4601(76)
I = ( X X ) (76)
Xlb X2b

where Xlb and X2b are average crew and vessel size in the base year and

state. In Table 3, Florida's 1960 input composition serves as the base.

Examination of the fishing power indices illustrates that Florida

vessels in 1975 are characterized by the lowest average fishing power

per vessel in the fishery. Perhaps more surprising is the fact that in

1975, Florida vessels possessed only about 75 percent of the fishing

power of vessels in the base period. Since 1967, Mississippi vessels,

on average, have had the greatest fishing power. Mississippi vessels in

1975 had slightly over four times the fishing power of Florida vessels.

Table 3. Estimated relative fishing power indices by state, 1957-75

Estimated fishing power index
Florida Alabama Mississippi Louisiana Texas
West Coast

1957 1.064 2.271 1.210 1.315 0.970
1958 1.176 2.361 1.297 1.281 1.154
1959 0.932 2.468 1.514 1.366 0.915
1960 1.000 2.471 1.971 1.423 1.165

1961 1.001 2.729 2.133 1.278 1.102
1962 0.984 2.785 2.487 1.604 1.153
1963 0.994 2.870 2.527 1.623 1.327
1964 0.959 2.755 2.571 1.594 1.585
1965 0.909 2.970 3.057 1.461 1.553

1966 0.937 3.251 3.148 1.554 1.836
1967 0.030 3.119 3.529 1.073 1.657
1968 0.911 2.973 3.328 1.098 1.538
1969 0.960 2.973 3.325 1.098 1.491
1970 0.875 2.404 3.268 1.268 1.377

1971 0.893 2.407 3.299 1.172 1.723
1972 0.783 2.458 3.295 0.963 1.628
1973 0.774 2.447 3.322 1.153 1.547
1974 0.860 2.612 3.309 1.250 1.504
1975 0.766 2.612 3.313 1.229 1.592

aThe fishing power index is calculated by Ip = X95 X45 / X196
Xb where Xl and Xi are, respectively, average crew size and
2b where lit 2it th th
average vessel size in the i state and the tth time period. The 1960
input composition for Florida constitutes the index base.

Utilizing the fishing power indices to create a standardized mea-

sure of fishing effort in the GMRFF has significant implications in

relation to both stock assessment and management questions. A compari-

son of fishing effort measured in nominal terms (vessels) and fishing

effort measured in standardized terms (standardized vessels) is pre-

sented in Table 4. The standardization of vessels results in substantial

Table 4. Estimated number of standardized reef fish vesselsa and actual number of reef fish vessels
in the Gulf of Mexico Reef Fish Fishery by state, 1957-1975

Florida West Coast Alabama Mississippi Louisiana Texas
Standardized vessels Standardized vessels Standardized vessels Standardized vessels Standardized
Vessels vessels vessels vessels veVesse vessels vessels

1957 108 115 11 25 5 6 2 3 129 125
1958 120 141 11 26 7 9 5 6 89 103
1959 300 280 12 30 8 12 12 16 158 145
1960 180 180 12 30 11 -22 13 19 118 138

1961 219 221 13 36 12 26 30 38 151 166
1962 232 228 15 42 12 30 36 58 152 175
1963 280 278 22 63 13 33 30 49 119 158
1964 334 317 22 61 14 36 23 37 93 147
1965 337 306 20 59 14 43 23 34 85 132

1966 274 257 22 72 17 54 13 20 64 113
1967 267 248 19 59 20 71 6 6 66 109
1968 256 233 12 36 21 70 5 6 50 77
1969 242 236 12 36 20 67 5 6 46 69
1970 257 225 11 26 19 62 6 8 23 32

1971 282 252 11 27 20 66 7 8 30 52
1972 306 240 12 30 21 69 11 11 45 73
1973 331 256 11 27 19 63 13 15 41 64
1974 353 304 11 29 18 60 13 16 40 60
1975 425 326 11 29 18 60 14 17 34 54

(Table 3).

reef fish vessels are calculated by multiplying the actual number of vessels by the corresponding fishing power index

increases in estimated fishing effort in all states except Florida.

Standardization of vessels in Florida leads to a downward adjustment in

measured fishing effort for all years since 1962. Thus, it can be seen

that the failure to adjust nominal effort (vessels) by the fishing power

indices can result in serious overestimations of stock assessment mea-

sures such as catch per unit of effort.

The similarity of the fishing power function to the traditional

economic production function facilitates an analysis of the substituta-

bility of average crew and vessel size in "producing" fishing power.

The importance of such substitution relates primarily to questions

involving appropriate management levels of effort. Consider, for

example, a situation where a specified number of vessels with a given

average fishing power per vessel are determined to constitute of appro-

priate amount of effort by management authorities. Any change in average

fishing power per vessel will create a change in effective fishing

effort, even though the actual number of vessels may remain constant.

The implication here is that changes in fishing power determinants

(average crew and vessel size) must be managed such that changes in

these factors do not change average fishing power, or the number of

vessels must be adjusted to reflect these changes.

Constant levels of fishing power may be analyzed with iso-fishing

power contours calculated from the fishing power function in equation

(75). The expression for these contours for a given level of fishing

power, Ep is given by

X E1.0417 i-0.4792 (77)
li Po 2i


where Xli and X21 are, respectively, average crew size and average
vessel size in the ith state. Several iso-fishing power contours are

shown in Figure 12. The rate of substitution necessary to maintain a

constant level of fishing power depends not only on the ratio of vessel

size to crew size on a given contour, but also on the location of the

contour (Figure 12). Consider point A on the Ep = 1.5 contour. An in-

crease in crew size of one man requires a decrease in vessel size of

approximately 10 gross registered tons to maintain fishing effort at a

constant level (point B). Next, consider point A' on the Ep = 2.0

contour. A one man increase in crew size now requires a decrease in

70 -A'

6 60
S. 50 -

S.- 40 A E = 3.0
"I ) p
> 30 = 2.5

0 Ep = 1.5
0 Ep = 1.5
SEEp = 1.0

0 2 3 4 5 6 7 8 9 10 11 Crew Size

Figure 12. Iso-fishing power contours for selected levels of
relative fishing power4

The contours are expressed in terms of the fishing power index
described in equation (76). This changes only the scale of measure.
The contours shown in the figure ignore any technological limitations on
substitution. Thus, the ranges of substitution shown in all probability
exceed the limits of the feasible range of substitution. For example, a
10 ton vessel with 11 crew is clearly an infeasible input composition
for the fishery.

in vessel size of 20 gross registered tons to maintain fishing power at

the same level (point B).

These aspects of the substitutability between crew size and vessel

size are significant in regards to managing the GMRFF. Management mea-

sures must focus on regulating nominal fishing effort (vessels), fishing

power or both. Figure 12 demonstrates that significant changes in the

average input composition in each state may be required to maintain

fishing power at constant levels. Furthermore, given the substantial

differences in the average fishing power of vessels across states in the

GMRFF, it can be seen that management measures aimed at maintaining

fishing power at constant levels must be formulated on an individual

state basis.

Catch Equations

The catch equations derived in equation (45) expressed catch as a

function of effective fishing effort, with effective fishing effort

defined to be the product of nominal effort (vessels) and fishing power.

This section centers on catch equations conditioned by fixed levels of

fishing power. Thus, each state's catch becomes a function of the

number of vessels fishing power being fixed.

The output elasticity of vessels with given fishing power is esti-

mated to be 0.74023 (Table 2). Recalling that this parameter is con-

strained to be constant across states, it may be interpreted to estimate

a 7.4 percent increase in catch in each state given a 10 percent increase

in vessels holding fishing power in each state at a fixed level. Given

the manner in which fishing power has been defined, this output elas-

ticity is synomous with returns to scale in the fishery. The notion of

returns to scale must be used with caution within the context of fishery

production, however.

Scale elasticities measure the percentage change in output given a

1 percent change in all inputs. Within the context of fishery produc-

tion, the resource stock constitutes an unobservable input. A simulta-

neous increase in all physical inputs which serves to increase catch

must necessarily alter the resource stock size. Thus, any true measure

of returns to scale in terms of only measured physical inputs is con-

founded by unobserved stock size changes. Given the incorporation of

the autoregressive process to account for such unobserved changes, the

estimated scale elasticity of 0.74023 can be considered as a reasonable


The catch equations underlying all stock production models have

assumed constant returns to fishing effort as pointed out in Chapter II.

If one is willing to accept that the autoregressive processes in the

estimated catch equations adequately account for changes in the resource

stock, the estimated scale elasticity for vessels may be used to conduct

an approximate test of the "constant returns" hypothesis. A t-test of

the null hypothesis of 0.74023 equal to one versus the alternative of

less than one can be rejected at the .05 level of significance. Given

the rejection of this hypothesis and the large absolute difference

between the estimated parameter and unity, it is apparent that the GMRFF

is characterized by diminishing returns to scale.

Derived Equilibrium Catch Equations

The estimated catch equations in the form presented in Table 2

correspond to non-equilibrium equations. Non-equilibrium functions

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