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The Dynamics of an Open Access Fishery

Trond Bjørndal; Jon M. Conrad

The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 20, No. 1. (Feb., 1987), pp. 74-85.

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The dynamics of an open access fishery TROND BJBRNDAL Norwegian School of Economics JON M. CONRAD Cornell University

Abstract. A discrete time non-linear deterministic model for an open access fishery is developed and the equilibrium is characterized. The open access exploitation of North Sea herring during the period 1963-77 is analysed. Alternative production functions are considered and estimated for the Nonvegian purse seine fishery. The bionomic equilibrium and approach dynamics are presented when prices and costs are changing. The results indicate that the resource stock was saved from possible extinction by the closure of the fishery at the end of the 1977 season.

Sur la dynumique d'une zone de pgches quund l'entrke est libre. Les auteurs developpent un modele deterministe non-lineaire en temps discret d'une zone de p&ches ou l'entree est libre et definissent les caracteristiques de l'equilibre. L'exploitation du hareng de la Mer du Nord qui s'est faite sans entraves B l'entrte pendant la periode 1963-1977 est analysee avec ce modele. Des fonctions de production de rechange sont examinees et calibrkes pour la p&che B l'essaugue par la flotte norvegienne. L'kquilibre bionomique et la dynarnique de l'approche A cet Cquilibre sont examines dans un univers ou les p i x et les cofits sont changeants. Les resultats de l'analyse montrent que le stock de ressource a echappe a la dispasition possible gr%ce a la fesmeture de la zone de phhes a la fin de la saison de 1977.

INTRODUCTION

Open access exploitation of common property fish resources frequently causes severe stock depletion. Indeed, the question whether open access may cause stock extinction has been analysed by several authors (Smith, 1968 and 1975; Berck, 1979; Hartwick, 1982). Moreover, as Smith (1968) has pointed out, although stock equilibrium under open access may be positive, the stock may be driven to extinction along the path of adjustment. Stock equilibrium may also be stable and positive with fixed prices and technology and still drift towards extinction over time, since these fixed variables drift in the long run.

Canadian Journal of Econormcs Relue canadienne d'Economique, xx. No. 1 February fkvrier 1987 Printed In Canada Imprimk au Canada

0008-4085 / 87 / 74-85 $1.50 @ Canadian Economics Association

Dynamics qf a11 open access J l h e r y 75

With the exception of Wilen (1 976) the work on the dynamics of open access or free entry fisheries is mainly theoretical. The purpose of this paper is to provide an empirical application, based on the North Sea herring fishery, with special reference to the question of stock extinction under open access. Herring is a schooling species. The schooling behaviour has permitted the development of very effective harvesting techniques. With modern fish-finding equipment, harvesting can remain profitable even at low stock levels. Open access exploitation of a number of schooling species has caused severe stock depletion (Murphy, 1977). The question of possible stock extinction thus takes on special importance for schooling species.

In the second section we shall develop a deterministic model for an open access fishery based on Smith (1968) and give a characterization of open access equilibrium. In the following section open access exploitation of North Sea herring during the period 1963-77 will be analysed. Alternative production functions are considered and estimated for the Norwegian purse seine fishery. The bionomic equilibrium and approach dynamics are presented when prices and costs are changing. Finally, the work is summarized and some policy implications are discussed.

THE OPEN ACCESS MODEL

In this section we construct a simple open access model to discuss steady state (equilibrium) conditions and system dynamics. The model will be specified in discrete time as a system of difference equations. Tinie is partitioned into annual increments, a procedure consistent with the data used to estimate production and growth functions and the equation for capital (vessel) dynamics. It is also consistent with the observation that vessel owners are reluctant to incur the cost of regearing once a decision has been made to enter the herring fishery which, in the North Sea, has a seasoil running from May until September. While the steady state equilibria for differential equation systems and their difference-equation analogues are usually equivalent, the stability and thus approach dynamics can be qualitatively different. The distinction becomes more than a mathematical curiosity in resource systems, where discrete-time and possibly lagged adjustment to biological and economic conditions can lead to overshoot and greater potential for overharvest and possibly species extinction.

The model presumes an industry production function

where Y, is yield (harvest) in year t , Kt are the number of vessels in the fishery during year t , and St is the fishable stock at the beginning of year t.

The number of vessels, Kt, may be a crude measure of actual fishing effort. In demersal fisheries the best measure might be the volume of water 'screened' by nets during the season (Clark, 1985). However, in a fishery on a schooling

76 Trond Bjorndal and Jon M. Conrad

species like herring, search for schools of herring is of predominant importance. Accordingly, in such fisheries the number of participating vessels may be an appropriate measure of effort.

For schooling stocks, like herring, there is some question as to 'elasticity' of yield with respect to stock size, St. If as a population declines it continues to concentrate in (fewer) schools of the same approximate size, and if these schools can be located with relative ease by electronic search, then yield may be essentially determined by effort, independent of stock, until the population declines to a small number of schools. If this were the case, the produc- tion function H ( . ) might depend strictly on Kt, and catch per unit effort, often used to estimate stock, would not predict the collapse of the fishery (Clark and Mangel, 1979; Ulltang, 1980).

Assuming that vessel numbers are an appropriate measure of effort and that yield is stock dependent, the standard open access model proceeds by defining industry profit (net revenue) in year t as

where p and c are the per unit price for yield and cost per vessel, respectively. Two additional assumptions are implicit in equation (2). First, the fishery must be one of several sources of the species in question; otherwise price would depend on yield, that is, p, = p ( y ) where p(.) is an inverse demand function. Cost per vessel is also assumed given. Second, the unit prices and costs are assumed constant through time. Neither assumption is likely to hold in 'real world' fisheries, but their maintenance permits the estimation of an open access or bionomic equilibrium, which may give an indication of the extent of overfishing.

Vessels are assumed to enter a profitable fishery and exit an unprofitable fishery according to

where n > 0 is an adjustment parameter (unit: vessels/$). With n positive, it will be the case that (a) K t + , > K, if m, > 0; (b) K t + , < K, if a, < 0; and (c) K t + , = K, if a = 0. It is possible that the rates of entry and exit may differ, in which case n' might apply if a, > 0 and n might apply if a, < 0 where n + , n- > 0, M + * n-.

Finally, the resource stock is assumed to adjust according to

where F(S,) is a net natural growth function. It is often assumed that there exist stock levels S and S where F(S) = F(S) = 0, F(S) < 0 for 0 < S < S, F ( S ) > 0 for S < S < 3, and F ( S ) < 0 for S > 3.

Taken together equations (3) and (4) constitute a dynamical system (or an iterative map). More specifically, with given values for Soand KO the system

Dynamics of an open access fishery 77

can be iterated forward in time. The trajectory (St, Kt) may be plotted in phase-space. A stationary point (S, K ) is one for which K t + , = Kt = K and S ~ + I= St = S for all future t . Such a point must satisfy K = pH(K, S ) / c and N ( K , S ) = F(S).

For the Gordon-Schaefer model (Clark, 1976) where F(S,) = r S t ( l - S t /L ) and N ( K , , S,) = qK,S,, the differential equation system takes the form

where r is the intrinsic growth rate, L is the environmental carrying capac- ity. and q is the catchability coefficient. The system has an equilibrium at S, = c / ( p q ) and K, = r ( l - S,/L)/q, whlch is the focus of a stable spiral (see figure la). The difference equation analogue might be written

and is capable of more complex behaviour, including limit cycles (see figure lb)!

T H E N O R T H SEA H E R R I N G F I S I I E R Y 1963-77

The North Sea herring fishery takes place in the central and northern North Sea, with the main season in the months May to September. In the present case study data for the Norwegian purse seine fleet will be used to estimate production functions and vessel dynamics. The fishery, utilizing this technolo- gy, started in 1963. In the middle of the 1970s, however, the stock was severely depleted under an open access regime and the fishery war; closed at the end of 1977. Severe regulations have been in effect ever since to allow the stock to recover.

Table 1 contains data on stock size, Norwegian purse seine harvest and the number of Norwegian purse seiners for the period 1963-77. Other countries (Denmark, the Netherlands, Germany, and the United Kingdom) were also harvesting the herring stock using a variety of gear, including single and pair

1 For system (6) with S, == c l ( p q ) > 0 and K , = r(1 - S,IL)Iq > O the open access equilibrium is stable (a node or spiral) and limit cycles are precluded by the Rendixon-du Lac test (see Clark. 1976, 2 0 3 4 ) . For the difference equations in system (7). simulation for p = 1.000, r == 3,000, q = 3.8 X l o r 5 , n = 0.0001, r = 0.5, and L = 250,000 from So = 250,000 and KO = 1,000, results in a convergent spiral. Changing n to 0.000175 leads to a limit cycle and with n = 0.000175 and r = 2.6. ceteris paribus, an invariant circle is obtained. The difference equation system, with its inherent lag, is capable of ~ n u c h more complex, possibly 'chaotic' behaviour.

78 Trond Bj~jrndal and Jon M. Conrad

F I G U R E l a : Phase plane analysis of system (6). The point (S,, KW) is the focus of a stablc spiral.

F I G U R E Ib: Phase plane analysis of system (7). The point (S,. ti,) is the focus of a lirnit cyclc.

Dj~nurnicsoj' at7 open uccess flfi'slzerv 79

North Sea hcrritig stock. Not~bcglan pursc seine harvest, and the t~unlber o f N o s w e g i a ~ ~pursc seiners

Stock aizc Norwegian lian~est Number of participating Year S,(tonnes) 1; (tonties) p u r c heinel.\ K ,

trawl and drift nets. After 1963, however, the purse seine technology became the dominant gear, and lacking data on the number and harvest of other gear types, we used the Norwegian purse seine data to estimate parameters for several alternative production forms. The stock estimates ( S , )were obtained by virtual population analysis (Ricker, 1975). Unrestricted or.s regressions were run, and table 2 shows four estimating equations (a)(iHd)(i) and four associated production functions (a)(iiHd)(ii). The exponents on K, in (a)(ii) and (b)(ii) would indicate a yield/vessel elasticity greater than one. This is presumably the result of economies of scale in searching for schools of herring, since inforlnation about locations of schools tends to be shared between boats in this fishery. The yield-stock elasticity in (b)(ii) and (d)(ii) are both significantly positive and less than one. Thus, as stock declines, catch per vessel will decline and there will be a stock-dependent incentive to exit from the industry, as indicated by the rather rapid departure of Norwegian purse seiners from the fishery after 1968. The relnaining vessels, however, seemed more than adequate to continue harvest in excess of natural growth and recruitment, and from inspection of table 1 it is still not clear whether exit would have been rapid enough for the stock to increase.

The expression for profits was specified as

where c, = e,?, + j,; e, is the average number of days spent fishing herring, ZI is the operating cost per day in year r , a n d j , are the fixed and opportunity costs incurred during the herring season.

80 Trend Bj@rndal and Jon M. Conrad

Estimates of production function parameters for the Norwegian purse seine fleet: all regressions oLs with r-statistics in parentheses"

( b ) ( i ) In = -2.7876 + 1.3556 In ti, -t 0.5621 In S , (adjusted R2 = 0.96) (2.11 ) (16.39) (5.84)

( i i ) I; = 0.O6157ti,' 3 5 h . ~ , " "' ( c ) ( i ) In ( S , - 1 ; ) = -0.5683" + 1.0398' In .TI - 0.001 1 ti, (adjusted R' = 0.99)

( 1.29) (30.81) (3.74)

( i i ) k; = <S,(I - cJ itcalin;1

( d ) ( i ) In ( );/ti,) = - 1.6718"+ 0.6086 In S, (adjusted R' = 0.54) (0.84) ( 4 16)

( i i ) 7; = .s,"""K,

" Autocorrelation Lvas indicated only in cquatiotlh ( a ) and ( d ) . First-order correction did not significantly alter the magnitude of the estimated coefficients. Two-stage least squares did not indicate the presence of simultaneous equations bias which can occur if estimates of S, are bahed o n current period harvest. This is leas of a problem when stock estimates are obtained by virtual population analysis.

" Not slgnificantl! diffcrcnt from zero: parameter assumed to be zero in the associated production function. Not significantly different from 1.00: parameter set equal to one in the associated production function.

Vessel dynamics were assumed to occur according to

Equation (9) assumes that entry or exit will depend on the sign of normalized profit per boat. This form was employed to take advantage of previous analysis by Bj~lrndal and Conrad (1985). Estimates of n ranged between 0.08 and 0.10.

A discrete-time analogue to the logistic growth function might be written as

where estimates of r and L were 0.8 and 3.2 X lo6 metric tonnes. Equation (10) is an approximation to a more complex delay-difference equation discussed in Bjorndal (1984).

Of the four production models the Cobb-Douglas form = a ~ , h ~ , " , resulted in the most plausible values for the bionomic equilibrium and open access dynamics. The open access system may be written as

Dynumics o f an open uccess jishery 8 1

TABLE 3

Costs ( p e r \eason per vessel) and herring price (per tonne): figures in Norwegian kroticru

Year ( ' I h

" Price figures have been adjusted by a factor of 0.6, which represents the boat owner's share of income. Costs cover only costs incurred by the boatowtier. S O U K C ~ S :p,: The Directorate of Fisheries, Norway

c,: The Budget Committee for the Fishing Industl-1. Norway

If c, = c and p, = p , then one obtains the following equations for the bionomic equilibrium

While it is not possible to solve for explicit expressions for S , and K,, it is possible to solve for S , and K, numerically. By making an initial guess for K,, the first equation in (12) provides a value for S,. Substituting t h s value into the second equation one obtains a value for K,, consistent with growth and yield. Calling the initial guess Z,, one can evaluate /Z, - K,l. If this is not within an arbitrary E , readjust the guess according to Z , = (Z, + K,)/2. This process will converge to the bionomic equilibrium from above or below K,.

During the period 1963-77 prices and costs were changing as indicated in table 3. If the 1975 values of c = 556,580 and p = 735 (both in Norwegian kroner) were somehow fixed into the future and all other parameters remained unchanged, then the bionomic equilibrium is calculated at S , = 430,191 (tonnes), K, = 393 (boats), and Y , = 297,887 (tonnes). When c, and p, are allowed to vary as per table 3, the time paths for S , and K, are given in table 4

82 Trond B j ~ r n d a l and Jon M. Conrad

TABLE 4

Rionomic equilibrium and open access dynamics

Oprl? cic.cess dytlcimic~ With cr and as given in table 3. S,, = 2.325.000. KO = 120. then

Year Sr k', Y

' After 1972 harvest exceed5 S r but 17or S r plus growth. This is possible, since growth to the resource occurs before harvesting (see equation for St.+,, above).

and plotted in phase-space in figure 2. The values for K, might be interpreted as an estimate of 'purse seine equivalents' fishing herring in the entire North Sea. Thus K, is larger than the number of Norwegian purse seiners that participated in the fishery during the period. The stock actually increases until 1965 and then decreases monotonically. The estimates of the herring stock in table 1 are a bit more ragged, lower than the simulated estimates until 1973 and higher thereafter. Of particular interest is the overshoot 'past' the 1975-based bionomic equilibrium and the continued decline in stock. In contrast to the results of Wilen, there is no increase in the stock and the 'first loop' of a convergent spiral has not been completed.

In 1977 Norway and the EEC agreed to close the fishery. There are no official

Dyrzamics of an open uccess f z s h e ~ 83

HERRING STOCK (million tonnes) F I G C J R E2. Sirnulatiotl of North Sea herring fisher:\

prices nor data to estimate costs after this year. One can only speculate what the future evolution of stock and vessel numbers would have been. It seems entirely plausible that with declining harvest, relative price increases would have exceeded relative cost increases with species extinction the result. If the price in 1978 were increased to 2,000 NoK/metric tonne and costs held steady, the species 'simulates' to extinction in 1983. IJnder the moratorium which lasted until 1981 the stock was allowed to recover, and fisheries scientists estimated the 1983 stock level at 600,000 metric tonnes.

CONCL,USIONS A N D POLICY IMPLICATIONS

In the empirical analysis of open access systems it is important to note that

84 Trond B j ~ r n d a l and Jon M. Conrad

non-linear difference equations, with or without longer lags, are capable of more complex dynamic behaviour than their continuous-time (differential equation) analogues. The lag in adjustment by both the exploited species and the harvesters themselves is often a more accurate depiction of dynamics, and the differential equation systems are best viewed as theoretical approxima- tions.

If adjustment in an open access system is discrete, there is a greater likelihood of overshoot, severe depletion, and possibly extinction. When discrete adjustment takes place in a system where the species exhibits schooling, declining stocks may fail to reduce profits rapidly enough to turn the critical 'first corner' in an approach to bionomic equilibrium. The fact that the economic and natural environments are subject to fluctuations places greater importance on modelling the dynamics of non-autonomous systems as opposed to the calculation of equilibria based on long-run or average values.

The analysis of the North Sea herring fishery would seem to support many of the above points. During the 1963-77 period the resource (1) was subject to open access exploitation by Norway and members of the EEC; (2) exhibited a weak yield-stock elasticity (because of schooling) which failed to encourage a rapid enough exit of vessels from the fishery; and (3) was saved from more severe overfishing and possibly extinction by the closure of the fishery at the end of the 1977 season.

Recent analysis by Bjorndal (1985) indicates that the optimal stock level is likely to be in the range 1.0-1.4 million tonnes, supporting a harvest of 550,00M00,000 tonnes. With the recovery of the resource, the stock might be initially managed through a system of internationally assigned but inhanation- ally transferable quotas. In the longer run a system allowing fisheries managers from one country to purchase or lease the quota rights of another should permit the total allowable catch (TAC) to be harvested at least cost. The theory and institutions for the management of transboundary resources are still at an early stage of development, but likely to be of critical importance if the value of fisheries resources are to be maximized among coastal countries.

Berck, P. (1979) 'Open access and extinction.' Econometrica 47, 877-82 Bjsrndal, T. (1984) 'The optimal management of an ocean fishery.' Unpublished PH D

dissertation, Department of Economics, University of British Columbia -(1985), 'The optimal management of North Sea herring.' Working Paper

No. 2/ 1985, Centre for Applied Research, Norwegian School of Economics and Business Administration, Bergen

Bjsrndal, T. and J.M. Conrad (1985) 'Capital dynamics in the North Sea herring fishery.' Working Paper No. 01/85, Institute of Fisheries Economics, Norwegian School of Economics and Business Administration, Bergen

Clark, C.W. (1976) Mathematical Bioeconomics: The Optimal Management of Renewable Resources (New York: Wiley)

-(1982) 'Concentration profiles and the production and management of marine fisheries.' I n W. Eichhorn, et al. eds, ficotlomic Theoty o f 'Natural Resources (Wiirzburg: Physica Verlag)

-(1985) Bioeconomic Modelling and Fisheries Management (New York: Wiley) Clark, C.W. and M. Mangel (1979) 'Aggregation and fishery dynamics: a theoretical

study of schooling and the purse seine tuna fisheries.' Fi.shery Bulletin 77, 317-37 Hartwick, J.M. (1982) 'Free access and the dynamics of the fishery.' I n L.J. Mirman

and D.F. Spulber, eds, E.rsays in the Economics of'Rer~ewable Resources (Amsterdam, New York, Oxford: North-Holland)

Murphy, G.I. (1977) 'Clupeids' I n J.A. Gulland, ed., Fish Populatron Dynanlrts (New York: Wiley)

h c k e r , W.E. (1975) Computation and Ir~terprerarion of Biological Statistrcs o f f i s h Populations (Ottawa: Environment Canada)

Smith, V.L. (1968) 'Economics of production from natural resources.' Anlerican Econonzic Review 58. 409---3 1

-(1975) 'The primitive hunter culture. Pleistocene extinction, and the rise of agricul- ture.' Journal of Polrtrcal Economy 83, 727--55

Ulltang, 0 . (1980) 'Factors affecting the reactions of pelagic: fish stocks to exploita- tion and requiring a new approach to assessment and management.' Kapp. P.-l/: Rbun. Cons. Int. Eixplor. Mer. 177, 489--504

Wilen, J.E. (1976) 'Common property resources and the dynamics of over-exploita- tion: the case of the North Pacific ful seal.' Paper No. 3 in the Programme in Resource Economics. Department of Economics, University of British Columbia

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The Dynamics of an Open Access Fishery Trond Bjørndal; Jon M. Conrad The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 20, No. 1. (Feb., 1987), pp. 74-85. Stable URL:

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References

Open Access and Extinction Peter Berck Econometrica, Vol. 47, No. 4. (Jul., 1979), pp. 877-882. Stable URL:

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Economics of Production from Natural Resources Vernon L. Smith The American Economic Review, Vol. 58, No. 3, Part 1. (Jun., 1968), pp. 409-431. Stable URL:

http://links.jstor.org/sici?sici=0002-8282%28196806%2958%3A3%3C409%3AEOPFNR%3E2.0.CO%3B2-3

The Primitive Hunter Culture, Pleistocene Extinction, and the Rise of Agriculture Vernon L. Smith The Journal of Political Economy, Vol. 83, No. 4. (Aug., 1975), pp. 727-756. Stable URL:

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