A BIOECONOMETRIC ANALYSIS OF THE
GULF OF MEXICO COMMERCIAL REEF FISH FISHERY
BY
TIMOTHY GORDON TAYLOR
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR
THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1980
ACKNOWLEDGMENTS
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.
ii
TABLE OF CONTENTS
ACKNOWLEDGMENTS .. . . . . ...
LIST OF TABLES . . . . . .
LIST OF FIGURES .. .......... . .
ABSTRACT ...........
CHAPTERS
I INTRODUCTION .. . ...........
Objectives . . . . .
Scope . . . .
II THEORETICAL FOUNDATIONS ............
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 MultiSector Fishery ....
III EMPIRICAL MODEL .......
Introduction . . . . .
Catch Equation Specification and Estimation .
Specification of Fishing Effort .. ....
Within Region Specification Considerations
Stochastic Approximation of Resource Stock
Effects . . . . .
iii
Page
ii
vi
ix
xi
1
6
8
9
10
10
14
18
19
24
33
34
38
43
51
51
52
53
55
57
TABLE OF CONTENTS (Continued)
Page
CrossSectional 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
APPENDICES
A SPECIES COMPOSITION OF THE FISHERY AND THE
DISTRIBUTION OF FISHING ACTIVITY ........... 140
B DATA UTILIZED IN ANALYZING THE GULF OF MEXICO
COMMERCIAL REEF FISH FISHERY ......... . 144
iv
TABLE OF CONTENTS (Continued)
Page
C DERIVATION OF EFFORT LEVELS FOR MAXIMUM
SUSTAINABLE YIELD AND MAXIMUM ECONOMIC YIELD .... 154
D GRAPHICAL DERIVATION OF THE DOUBLE HUMPED
SUSTAINABLE REVENUE CURVE ...... ......... .157
E IDENTIFICATION OF THE ORDER OF THE AUTOREGRESSIVE
PROCESSES IN THE CATCH EQUATIONS ............ .159
Bartlett Test .. .. .......... 159
Max x2 Test .... . .. ..... 160
Akaike's Final Prediction Error (FPE) Test ..... 160
DurbinWatson Test ................. .161
F MARKETING AND PRICE INFORMATION CONCERNING GULF
OF MEXICO RED SNAPPER AND GROUPER ........... 163
G RESOURCE STOCK ADJUSTMENT FOR FIXED LEVELS OF
EFFORT . . . . . . 166
H THE GULF OF MEXICO OPTIMIZATION MODEL AND
RESULTS OF EXOGENOUS CHANGES IN FISHING POWER ..... 169
BIBLIOGRAPHY .... ... ................ ...... 172
BIOGRAPHICAL SKETCH .................. .. 176
LIST OF TABLES
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,
195775 ..... . ..... .. .. ... ... 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, 19571975 ...... 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
vi
LIST OF TABLES (Continued)
Table
13 Estimated species composition of MEY catch of reef
fish . . . . . .
14 Fishing power components for proportional increases
along rays defined by constant vessel sizecrew
size ratios . . . . .
A1 Species in the management unit ....
A2 Species included in the fishery but not in the
management unit . . . .
B1 Red Snapper catch by countries, 19701973 .. .
B2 Grouper catch by countries, 19701973 .. ...
B3 Estimated catch and effort in Gulf of Mexico
recreational reef fishery for selected years .
B4 Estimated number and weight of reef fish caught by
recreational fishermen in the Gulf of Mexico, 1975
B5 Estimated catch of reef fish per handline vessel
in the Gulf of Mexico, 19571975 .. ......
B6 Catch of reef fish by handline vessels in the Gulf
of Mexico Reef Fish Fishery, 19571975 .. ....
B7 Average crew size of reef fish vessels by state,
19571975 . . . . . .
B8 Average size of reef fish vessels by state, 1957
1975 . . . . . .
B9 Number of reef fish vessels by state, 19571975 .
F1 Domestic marketing of grouper and snapper by Gulf
of Mexico commerical fish dealers, 1977 .. ..
F2 Two stage Aitken's parameter estimates for Red
Snapper and Grouper price equations .. ......
H1 Maximum economic yield in the reef fishery given
a 10 percent increase in average fishing power
per vessel . . . . .
Page
. 115
S. 116
S. .. .140
. 141
S. 144
. ... 145
. 146
. .. 147
. 148
S. .. 149
S.... 150
S. ... 151
. 152
. 163
. 164
. 170
vii
LIST OF TABLES (Continued)
Table Page
H2 Maximum economic yield in the reef fishery given
a 15 percent increase in average fishing power
per vessel ................ .... .. 170
H3 Maximum economic yield in the reef fishery given
a 20 percent increase in average fishing power
per vessel ................... ..... 171
H4 Maximum economic yield in the reef fishery given
a 25 percent increase in average fishing power
per vessel .............. ...... .. 171
viii
* 0
Vill
LIST OF FIGURES
Figure
ix
Page
11
13
18
20
22
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 multisector 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 Isofishing 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
39
48
. 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
A BIOECONOMETRIC ANALYSIS OF THE
GULF OF MEXICO COMMERCIAL REEF FISH FISHERY
By
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 multisector 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 nonlinear optimization
model for the Gulf of Mexico Reef Fish Fishery was constructed through
xi
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.
xii
CHAPTER I
INTRODUCTION
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 19701975
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 statesFlorida,
Alabama and Mississippiexperienced 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
6
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
3
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.
Objectives
The passage of the Fishery Conservation and Management Act of 1976
(PL94265) 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.
Scope
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 multisector 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.
CHAPTER II
THEORETICAL FOUNDATIONS
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 multisector fishery with variable
product price and pecuniary externalities.
Industry here refers to the aggregate of all vessels operating in
a given fishery.
10
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 usedependent 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 nonexploited 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
11
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.
r(P)
C MP
I I
450 I I I
I I
p p* p p Population (t)
1 max N
Figure 1. Relationship between population size and mature
progeny
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
12
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
micax
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
below.
13
MSY
rJ
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),
I I
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 Ushape, 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
14
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.
15
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
3
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 nonequilibrium 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.
16
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
1
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
17
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 nonequilibrium catch function,
the function
C = qE (M E) (8)
K1
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.
18
50
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
19
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 usedependence 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
20
the marginal product of effort (VMPE) with its marginal (average) cost,
r. To demonstrate how the common property nature of an unregulated
fishery encourages nonoptimal 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
VMP VAP1 VMP2 j\VAP2
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
21
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 socalled 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 longrun
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
22
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
(10)
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
cost.
Effort
optm E
Figure 5. Cost and revenue in an unregulated fishery
with constant product price
23
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 firstorder 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
maximized.
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
24
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.
25
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
5
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.
26
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
d2f
interior maximum growth rate (MSY) exists, and finally 2 < 0 ruling
dx
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
as
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
27
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 longrun 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 ax > 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
zero.
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
28
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 firstorder
conditions:
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
29
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.
30
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
function.
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.
31
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)
(20)
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
xf
xp' (m) + 2 < c3
V_ 3 c
if < m = m
Vc2
f
S= P(m) x c = Vc f3x
V3
F(X, m, Vx) = 0
(20a)
(20b)
(20c)
(20d)
(20e)
32
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 openaccess 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 wellbehaved 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.
33
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
neoclassical economic tradition to derive the results of the undesira
bility of unregulated competition, the economic inefficiency of MSY
34
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 multisector fishery where each sector cor
responds to a subindustry 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 nonequilibrium 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
disciplines.
35
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.
36
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 nonequilibrium 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)
yield.
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
nonequilibrium yield functions. In this study, the term nonequilib
rium implies that no biological equilibrium condition of catch equal to
37
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 nonequilibrium 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 nonequilibrium 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. Nonequilibrium 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. Nonequilibrium
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, nonequilibrium 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 nonequilibrium in nature.
7The notion of derived equilibrium yield equations is developed in
the following chapters.
38
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.
C)
V
a)
w4
E E
m c
Figure 7.
Effort
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
39
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 "doubledhumped" (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.
,TC
C 2
I A TC1
1 TR
a,
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
8
A graphical derivation of the "doublehumped" sustainable revenue
function is presented in Appendix D for the case of a linear demand
function.
40
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 nonequilibrium 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
1
further assumed that f(X1, ..., X ) is such that
> 0 i = 1 ..., n (23a)
aXi
i
2
< 0 i = 1, ..., n (23b)
aX
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)
dP
where d < 0. Finally, the cost equation is defined by
dC
n
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
41
aK
Sr.
aX.
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
n
MAX 7 = P(C)C z r X.
i=1
(26)
s.t. C = f(X, ..., Xn)
Equation (26) can be seen to be a constrained maximization problem with
9
the constraint being the yield equation. Utilizing the method of
lagrange multipliers, equation (26) can be restated by
n
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
= + Cx = 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,
dP
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).
(25b)
(27a)
(27b)
(27c)
42
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
+th
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 wellknown
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 (27a27c). 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
43
P r/ i = .., n (30)
1 r X.
dP
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 MultiSector Fishery
The term fishery is somewhat synonymous with the traditional
definition of an industry. A sector is defined in this study to be a
subindustry 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 multisector fishery, assume there are
N sectors or regions, each facing a demand function defined by
P = P.(C1, ..., CN) i = 1, ... N (31)
th
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
44
aP.
< 0 all i, j(31a)
aC
th
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
u'
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
n
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
N N
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)
45
where the are the undetermined lagrange multipliers. Differentiation
1
of equation (35) with respect to C., X.. and x. yields the N(n + 2)
1 3frsto r c i
firstorder conditions
aCi
(35a)
N aP
k
P. + C = 0
k=1 aCi k 1
af.
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, .
aP.
+
3a. Ci
term on
revenue
output.
(MRi).
N aP
al k Ck'
k=l i
.., N. This expression for A. can be rewritten as X. = (P.
1 1 i
aP
) + z k C. In this form, it can be seen that the first
kfi aCi k.
the righthand side of the equality is precisely the change in
th th
in the i region with respect to variations in the i region's
th
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
region
aL
aX..
31
aL
ax.
(35b)
(35c)
(36)
46
aP af.
1R k#i C i k 1 aX Jji
af
This equation can be used to show that in equilibrium, r / = r. /
aff
Small i, j, and that these expressions are in turn equal to the
Xji
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
aP
on the lefthand 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 nonequality
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
47
aP
MR. MC. = Ck C(38)
S 1 kfi aC k
f.
where MCi has replaces the expression r / it can be seen that the
1
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 righthand 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
aP
is that 0, i k. In this case, independent maximization of
aC
regional profits is equivalent to maximizing profit in all regions
simultaneously.
.th
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
aP
(37). The distance OAOB 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 crossregional
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 multisector 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
48
Price in
Region i
P / MC.
P2i 
A I
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 multisector 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
1
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
49
Pi I *
correct price. Unless 0 i k, P. will not equal P., which will
k
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 crossregional 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
50
individual states or regions. Rather, management should take into
consideration all states simultaneously.
CHAPTER III
EMPIRICAL MODEL
Introduction
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 nonexistent. While some data of this type
could be collected for perhaps one or two years, this would not be suf
ficient given the longrun 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, 19571975). The data used thus cor
respond to aggregate crosssection 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
51
52
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
corsssectional 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 nonequilibrium 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
it
average vessel size (gross registered tonnage) and k is a constant
parameter.
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 CobbDouglas type function can capture nonlinear 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
by
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
nonlinear 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
form.
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 tl. 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,t2 + + Ppi UtP + 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 crosssectional specification of the catch
equations.
CrossSectional 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 nonequality 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 nonlinear 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
variance.
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, nonlinear 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 nonlinear 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
disturbances.
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 inputoutput 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
1
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. =
1
A. for all i, j.
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)
where
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
nonspherical. The precise form is conditioned by the exact form of the
autoregressive processes corresponding to each state's disturbance
vector.
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
autoregression.
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 = PiUitI + 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
N U N
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
it
period defined 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
*I
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 Tk 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
variancecovariance 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
8
the appropriate structure of the price equations. The choice of the
appropriate estimator and the estimation scheme utilized are then
discussed.
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, 195775). Over this period both have exhibited
8The term price in this section pertains to the nominal dockside or
exvessel 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
9
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
Ohio.
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 nonstorable
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, 195775). 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 exvessel 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 crosssectional 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 twostage 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
by
1 ^ ^
_i T E eit ejt. (73)
ij T jt
By replacing o.. in equation (71) with .ij, the twostage 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).
CHAPTER IV
EMPIRICAL RESULTS
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
1
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
GMRFF.
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 Uit1
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
1.4236
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
2.5
o 2.3
0
2.1
S1.9 
iu 1.7
1.5
4I)
S 1.3
1.1
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, 195775
Estimated fishing power index
Year
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, 19571975
Florida West Coast Alabama Mississippi Louisiana Texas
Year
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
aStandardized
(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 isofishing
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 i0.4792 (77)
li Po 2i
87
where Xli and X21 are, respectively, average crew size and average
th
vessel size in the ith state. Several isofishing 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
80
70 A'
6 60
N W
S. 50 
S. 40 A E = 3.0
"I ) p
os\
> 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. Isofishing 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
approximation.
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 ttest 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 nonequilibrium equations. Nonequilibrium functions
