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Optimal fisheries investment Charles, Anthony 1982

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OPTIMAL FISHERIES INVESTMENT by ANTHONY TREVOR CHARLES B.Sc. Carleton University, 1978 THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of Mathematics We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1982 © Anthony Trevor Charles, 1982 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mathematics The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: October 5, 1982 i i Abstract This thesis explores problems of optimal investment a r i s i n g in f i s h e r i e s and other renewable resource industries. In such industries, two simultaneous investment problems must be addressed: investment in the resource stock (the biomass) and investment in the c a p i t a l stock (harvesting capacity). Each of these investment problems faces a key complication; investment in the resource is constrained by the natural population dynamics, while in many cases investment in the physical c a p i t a l stock suffers from i r r e v e r s i b i l i t y , since c a p i t a l used in natural resource industries is often non-malleable. In addition, a l l investment decisions must be made within an uncertain environment; f u l l information is never av a i l a b l e . Building upon the work of Clark, Clarke and Munro (E.conometrica, 1979), we develop a two-state two-control model which incorporates investment delays and stochastic resource fluctuations within a seasonal (discrete-time) framework. A dynamic programming approach is used to analyse the model h e u r i s t i c a l l y and to obtain numerical results, beginning with a study of the ideal deterministic case and proceeding to a f u l l stochastic analysis. The key assumption of i r r e v e r s i b l e investment i s maintained throughout the thesis. We have examined the q u a l i t a t i v e and quantitative effects on optimal investment strategies of several economic and ecological factors: (i) delays in investment, ( i i ) population dynamics parameters, ( i i i ) s e l l i n g price, (iv) c a p i t a l cost and operating cost, (v) depreciation rate, (vi) discount rate, and ( v i i ) the le v e l of uncertainty in the resource stock. We have found that the key cost parameter for the investment problem is the r a t i o of unit c a p i t a l costs to unit operating costs. Depreciation can play a rather counter-intuitive role; in some circumstances optimal investment lev e l s can increase with the depreciation rate, contrary to the usual treatment of depreciation as an additional cost of c a p i t a l . The introduction of uncertainty in the form of stochastic resource fluctuations can substantially change the optimal investment policy, but thi s tends to have l i t t l e e f f e c t on the value of the fishery. We analyse the factors which determine the role of randomness in optimal f i s h e r i e s investment, and discuss in some d e t a i l the implications for management. Solution of the stochastic optimization problem studied here requires the use of rather complicated numerical methods, which are described in d e t a i l in the thesis. These methods are quite general, and should prove useful in analysing other related stochastic models. T a b l e of C o n t e n t s A b s t r a c t i i Acknowledgement v I . I n t r o d u c t i o n 1 I I . O p t i m a l F i s h e r i e s Investment: The D e t e r m i n i s t i c Case ....7 A. The Model 8 B. A n a l y t i c R e s u l t s 17 C. H e u r i s t i c A n a l y s i s 25 D. N u m e r i c a l Method 33 E. N u m e r i c a l R e s u l t s 40 F. Summary and D i s c u s s i o n 57 I I I . O p t i m a l F i s h e r i e s Investment under U n c e r t a i n t y 61 A. The Model 64 B. A n a l y t i c R e s u l t s 69 C. H e u r i s t i c A n a l y s i s 77 D. N u m e r i c a l Method 81 E. N u m e r i c a l R e s u l t s 108 F. Summary and D i s c u s s i o n 128 IV. C o n c l u s i o n s 130 R e f e r e n c e s 144 Appendix A 150 T a b l e s 154 F i g u r e s 1 58 V Acknowledgement Throughout my y e a r s of grad u a t e s t u d i e s , I have b e n e f i t t e d enormously from the i d e a s , e n t h u s i a s m and o v e r a l l g u i d a n c e of my t h e s i s s u p e r v i s o r , C o l i n C l a r k . My s i n c e r e t h a n k s t o Gordon Munro f o r many s t i m u l a t i n g d i s c u s s i o n s and f o r c o n t i n u a l s u p p o r t . I am i n d e b t e d t o Donald Ludwig b o t h f o r e x p l a i n i n g the i n t r i c a c i e s of n u m e r i c a l a n a l y s i s , and f o r p o i n t i n g out some of the i n t e r e s t i n g problems t o examine. I am g r a t e f u l t o the o t h e r members of my t h e s i s committee, U l r i c h Haussmann, Ray H i l b o r n , C a r l W a l t e r s and F r e d e r i c k Wan, f o r s u s t a i n e d i n t e r e s t i n my work, and t o the Head of the U.B.C. Mathematics Department, Dr. B. M o y l s , f o r h e l p i n g t o make my l i f e a t U.B.C. e n j o y a b l e . A l l i n a l l i t has been a v e r y s t i m u l a t i n g r e s e a r c h environment. My a p p r e c i a t i o n a l s o t o Mary Margaret D a i s l e y f o r t w i c e t y p i n g Chapter I I of t h i s t h e s i s . I would l i k e t o acknowledge the f i n a n c i a l s u p p o r t of the N a t u r a l S c i e n c e s and E n g i n e e r i n g R esearch C o u n c i l of Canada t h r o u g h the 1967 S c i e n c e S c h o l a r s h i p . F i n a l l y i t i s a p l e a s u r e t o acknowledge the e f f o r t s of my w i f e , E l i z a b e t h A b b o t t , who has done a w o n d e r f u l j o b of p u t t i n g up w i t h me throughout the c o u r s e of my t h e s i s r e s e a r c h . 1 Chapter I. Introduction The problem of determining optimal investment levels and optimal c a p i t a l stocks is a pervasive one in the economics l i t e r a t u r e . Indeed investment decisions together with decisions regarding the optimal levels of production are key to the e f f i c i e n t operation of an economy. [See Gordon (1961) for a discussion on the importance of c a p i t a l investment in resource industries, and Freidenfelds (1981) for a comprehensive treatment of simple capacity expansion models.] In recent years, considerable research has been undertaken in expanding the t r a d i t i o n a l analysis of investment, both at the le v e l of the firm and of the society. C l a s s i c a l theory, including such standard topics as marginal e f f i c i e n c y of c a p i t a l , the investment m u l t i p l i e r and IS-LM analysis, suffers in p a r t i c u l a r from i t s s t a t i c deterministic nature. Led by K.J. Arrow and his co-workers (Arrow and Lind, 1970; Arrow and Kurz, 1970), modern Capital Theory has adopted optimal control theory as a tool to study, as Arrow puts i t , "investment as optimization over time". From the outset i t has been realized that perhaps the two major complications in determining optimal investment strategies are the lack of m a l l e a b i l i t y of c a p i t a l and the uncertain environment in which investment decisions must be made. On the one hand, investment decisions tend to be i r r e v e r s i b l e , while on the other hand such decisions can never be based on f u l l 2 information, since the appropriate l e v e l of investment for long-lived c a p i t a l depends on the state of the world several years from the present. Uncertainty in investment has been addressed by many researchers, including Arrow and Lind (1970), who discussed the general question of risk and risk-sharing in public investments, Baumol (1968), Brock and Mirman (1978) and H i r s h l e i f e r (1966). Recently, McDonald and Siegel (1982) have undertaken an interesting application of option theory to the problem of investment under uncertainty. Their model allows for the p o s s i b i l i t y that firms w i l l shut down i f production i s uneconomical; th i s approach seems p a r t i c u l a r l y useful in considering natural resource industries. The non-malleability of c a p i t a l , and the corresponding problem of i r r e v e r s i b l e investment, was discussed in general terms by Arrow (1968). Campbell (1980) and Lasserre (1982) examined i r r e v e r s i b l e investment models of exhaustible resource production, while Margolick, Charles and H e l l i w e l l (1981) studied the problem of optimal investment in hydro-electric capacity under demand uncertainty. Perhaps nowhere is the optimal i r r e v e r s i b l e investment problem more d i f f i c u l t to analyse than in the area of renewable resource management, and f i s h e r i e s management in p a r t i c u l a r , where one is faced with a strongly fluctuating resource, for which the underlying dynamics are known only approximately, together with v o l a t i l e prices and a complicated i n d u s t r i a l structure of fishermen and processors. This may explain in part 3 why most models of f i s h e r i e s management problems to date have concentrated on the dynamics of the natural c a p i t a l , the f i s h stock, rather than the manmade c a p i t a l , in the form of boats and gear. The prototypical control problem (see for example Clark and Munro, 1975) tends to involve a d i f f e r e n t i a l or difference equation governing the size of the f i s h stock, an objective function to be maximized, and a control (usually instantaneous harvesting e f f o r t or end-or-year escapement). It i s often assumed that the fi s h i n g e f f o r t , E ( t ) , i s subject to a constraint 0<E(t)<K at any time t, where K is a fixed constant representing a given c a p i t a l stock, or "capacity" of the f i shery. Certainly, many authors have noted that the c a p i t a l stock K is unlikely to remain constant over time. For example, Smith (1968) considered the case where c a p i t a l follows an uncontrolled d i f f e r e n t i a l equation, with the l e v e l of c a p i t a l responding to fishery rents. In a control-theoretic context, the common assumption that escapement can be set at any value less than or equal to the recruitment i m p l i c i t l y assumes that unlimited amounts of c a p i t a l are available, i f needed, to harvest the stock. In both these cases, c a p i t a l i s considered to be malleable, so that both investment and disinvestment are possible. While this i s not unreasonable i f the fishery can u t i l i z e the c a p i t a l stock of a larger neighbouring fishery (as is the case with the B r i t i s h Columbia herring fishery in rel a t i o n to the salmon fishery) i t is unlikely to hold in general. 4 Given that a vessel entering a particular fishery may have e s s e n t i a l l y no alternative uses, one must return to the question of optimal i r r e v e r s i b l e investment. For the p a r t i c u l a r case of the fishery, Clark and Kirkwood (1979), Dudley and Waugh (1980) and S i l v e r t (1977) have addressed the question of how many vessels ( or in general how much harvesting capacity) should be optimally allocated to a fishery by i t s sole owner or s o c i a l manager. In each of these papers, however, the models being considered allow the investment problem to be reduced to a once-and-for-all decision at the outset. A more general analysis has been undertaken by Clark, Clarke and Munro (1979), who obtained a f u l l analytic solution to the continuous-time deterministic investment problem with 2 state variables (biomass and f l e e t capacity) and 2 controls (fishing e f f o r t and investment). Recently McKelvey (1982) has studied a similar model, involving the optimal mix of " s p e c i a l i s t " and "generalist" vessels in a fishing f l e e t . This thesis attempts to extend the study of optimal c a p i t a l investment in renewable resource industries by expanding the work of Clark, Clarke and Munro (1979) in a number of di r e c t i o n s . In Chapter II :, we analyze a model similar to theirs but more r e a l i s t i c in a number of respects: (i) time is considered to be discrete between fishing seasons although continuous within each season, ( i i ) the decision variables are end-of-season escapement and yearly investment (as opposed to instantaneous fishing e f f o r t and investment i»n the CCM continuous time case), and ( i i i ) delays are allowed between the 5 time at which investment decisions are made and the time at which these investments come on-line. A dynamic programming approach i s u t i l i z e d , allowing us to study a r b i t r a r y stock-recruitment functions, including the Beverton-Holt and Ricker forms, and to obtain detailed comparative dynamics res u l t s . In p a r t i c u l a r we describe the effect on optimal investment/escapement p o l i c i e s of the following factors: (i) discrete-time vs. continuous-time analysis, ( i i ) investment delays, ( i i i ) fecundity and carrying capacity of the stock, (iv) f i s h price, (v) c a p i t a l cost, (vi) discount rate, and ( v i i ) depreciation rate. In Chapter III, this model i s modified to incorporate stochastic fluctuations in the resource stock, and a study is made of optimal investment under uncertainty. We examine the appearance of an optimally-managed stochastic fishery, the role of economic and ecological parameters in determining how uncertainty affects optimal investment and escapement p o l i c i e s , and the r e l a t i v e performance of deterministic vs. stochastic strategies. F i n a l l y , Chapter IV highlights the more important results, discusses the implementation of optimal investment programs, and offers some concluding remarks regarding the effects of economic and ecological parameters, and the role of uncertainty, in determining optimal investment strategies. Our method of analysis involves analytic, h e u r i s t i c and numerical approaches. Dynamic programming is the primary tool of 6 both Chapters II and I I I : useful insights can be obtained from an analytic treatment of a s i m p l i f i e d model and from h e u r i s t i c analysis of the general case, but i t i s the numerical method and results which occupy most of our attention. A word on notation: t h i s thesis was produced using the documentation program FMT on the University of B r i t i s h Columbia computer system, together with a preprocessor developed by Donald Ludwig of U.B.C. This program was capable of producing a l l symbols required for th i s thesis apart from three: the symbol for i n f i n i t y (denoted instead as " i n f " ) , the square root sign ("sqrt"), and the integral sign, which i s denoted "|" herein. 7 Chapter I I . Optimal Fisheries Investment: The Deterministic Case We begin our examination of optimal f i s h e r i e s investment by considering a deterministic model which, in i t s simplest form, represents the discrete-time analogue of the Clark, Clarke and Munro (1979) model (referred to here as "CCM"). Throughout the chapter, our model and results are compared to those of CCM; we s h a l l see that while the q u a l i t a t i v e behaviour is similar, some important differences do a r i s e . In addition, the dynamic programming approach used here allows several additional features to be considered. We w i l l be able to describe in d e t a i l the role of (i) discrete-time analysis, ( i i ) investment delays, ( i i i ) fecundity and carrying capacity of the stock, (iv) f i s h price, (v) c a p i t a l cost, (vi) discount rate, and ( v i i ) depreciation rate in determining optimal investment/escapement p o l i c i e s . The outline of the chapter is as follows: Section A describes the model, section B presents simple analytic results, while in section C a h e u r i s t i c analysis of the model is undertaken. Section D discusses the numerical method and in section E the results are presented. F i n a l l y , in section F, a summary of the results together with a discussion completes the chapter. (The reader wishing to arrive more quickly at the main results in section E w i l l find that sections B and D can be omitted without loss of continuity.) 8 A. The Model (i) The f i s h stock is assumed to be represented by a single quantity, the biomass. This common assumption ignores multi-species and multi-cohort problems, but s i m p l i f i e s the model in order to place primary emphasis on the role of investment and f l e e t capacity in our analysis. The biomass i s governed by a stock-recruitment relationship R = F(S ), which in our applications w i l l be either pure n+1 n compensatory (F'>0,F"<0) or overcompensatory, using the Beverton-Holt or Ricker forms respectively (Beverton & Holt, 1957; Ricker, 1954). Intraseasonal growth of individuals in the population is neglected, so that the natural population dynamics (reproduction and natural mortality) is assumed to occur during the off-season. However, fishing mortality occurs continuously during the season, so that during the course of a single season, the biomass follows the common d i f f e r e n t i a l equation dx/dt= -h(t)= -qE(t)x(t) where h(t) i s the harvest rate, E(t) i s the instantaneous aggregated fishing e f f o r t , q is a constant c a t c h a b i l i t y c o e f f i c i e n t and i n i t i a l l y x(0)= R . The escapement n T is then S =R exp{-q|E(t)dt} where T i s a fixed maximum season n n 0 length. ( i i ) The c a p i t a l stock, or fl e e t capacity, K, is represented by the maximum instantaneous fishing e f f o r t : at any point t in the season n, we must have 0<E(t)<K . Hence K depicts the catching n n 9 power of the f l e e t , a highly aggregated measure including such elements as actual vessels, nets, machinery, engines and t r a i n i n g of the fishermen. This ignores questions of gear type; the type of vessel, or mix of vessel types, chosen for a fishery can a f f e c t operating costs, the proportions of factor inputs (labour, f u e l , etc.) and even the health of the f i s h stock. (In the B r i t i s h Columbia fishery, g i l l n e t t e r s , seiners and t r o l l e r s each claim that the other two gear types are more damaging to the resource.) On the other hand, use of catching power as the measure of capacity is superior to measuring c a p i t a l simply by the f l e e t size; i t is often observed that control of the number of boats in a fishery (through licensing) tends to have l i t t l e e f f ect on the o v e r a l l harvesting p o t e n t i a l . R e s t r i c t i n g entry merely inspires existing fishermen to upgrade their own catching power. The cost per unit of new f l e e t capacity i s assumed to be a constant, 6 , irrespective of the current l e v e l of capacity. Furthermore i t is assumed that this cost must be paid in f u l l at the time the new capacity is ordered. Depreciation is assumed to occur at the end of each season, with a constant fraction r (the depreciation rate) of the current c a p i t a l stock wearing out or otherwise being removed from the fishery at that time. Perhaps the most important assumption in our model, as in the CCM model, i s the i r r e v e r s i b i l i t y of investment. We assume here that the f l e e t capacity cannot be decreased at w i l l , but only through the process of depreciation. Hence the dynamics of 10 the c a p i t a l stock, K, can be expressed as follows: K = 0-r)K +1 I >0 n+1 n n+1 n+1 where the investment I becomes available in year n+1. [This n+J key assumption could be relaxed somewhat i f we were to allow a positive scrap value for f i s h i n g c a p i t a l (see CCM, pp35-37, and the discussion in ( i i i ) below) or the p o s s i b i l i t y of bringing outside vessels, either domestic or foreign, into the fishery on a temporary basis (see, for example, McKelvey, 1981). In any case, investment is not e n t i r e l y i r r e v e r s i b l e in our model since c a p i t a l depreciates annually. Of course, this w i l l also be the situation in real-world f i s h e r i e s , so that the above expression is a reasonable r e f l e c t i o n of the true i r r e v e r s i b i l i t y problem.] A further consideration in dealing with investment p o l i c i e s is the p o s s i b i l i t y of a delay existing between the time an investment decision is made and the time the corresponding new capacity becomes available. Such delays may arise due to the time necessary to construct new vessels and/or transport them to the f i s h i n g grounds. In a stochastic fishery, the existence of delays w i l l increase the l e v e l of uncertainty in planning f i s h e r i e s investment. On the other hand, investment delays in a deterministic world are of minor importance to the analysis, increasing the e f f e c t i v e c a p i t a l cost but not a f f e c t i n g the substance of the management problem. We have chosen to undertake most of our analysis using the more r e a l i s t i c delayed investment assumption, both for the sake of realism and to compare our deterministic results with those obtained using the stochastic 11 model of Chapter 111. To simplify the structure of the model while incorporating a reasonable delay, we s h a l l assume that, in any given year, the decision regarding next season's optimal capacity must be made before the end of the current season and f u l l payment for any new investment must be made in the current season. ( i i i ) The fishery management problem involves yearly escapement and investment decisions. The timing of the two decisions depends on the assumption regarding delays in bringing investment on-line. The following applies to the delayed investment case. Given the recruitment R , the optimal n escapement S* is chosen, subject to the constraint n R exp{-qTK }<S*<R where the lower l i m i t i s reached by fi s h i n g n n n n at maximum e f f o r t throughout the season. Then the optimal investment for next year, I >0, i s chosen, based on the value n+1 of S* : payment is made in year n for this new capacity, n To t h i s point, we have not discussed the i n d u s t r i a l structure of the fishery. There are two primary components of a fis h i n g industry, the harvesting and the processing sectors; incorporation of the l a t t e r in a f i s h e r i e s model can be quite complicated (Clark and Munro, 1980), p a r t i c u l a r l y i f processors have considerable monopsonistic power. A f u l l analysis of the optimal capacity problem would have to include both f l e e t capacity and processing capacity as separate variables, with 1 2 their corresponding investment rates as controls. In addition, variable processing costs must be dependent on actual harvest rather than on e f f o r t per se. We avoid these complications here by omitting e x p l i c i t reference to the processing sector in the model. On the other hand, one may think of the model as applying to the entire fishing industry (for a p a r t i c u l a r species), with an appropriate fraction of the processors' c a p i t a l costs included in the unit capacity cost, 6. This may be reasonable, at least for a given f i s h stock, since presumably one unit of harvesting capacity w i l l require a certain fixed amount of processing capacity to handle i t s output. In any case, we s h a l l make the additional assumption that the fishery faces a given constant s e l l i n g price, p, and a known cos t - o f - e f f o r t function c(E), with convex (U-shaped) marginal and average cost curves. In p a r t i c u l a r , this implies perfect e l a s t i c i t y of demand for f i s h , an assumption which s i m p l i f i e s the economics in our model, but i s l i k e l y to be reasonable only i f the fishery does not enjoy too large a market share. The interesting and r e a l i s t i c problem of price uncertainty w i l l be considered in future work. Our model allows for the p o s s i b i l i t y of an "alternative fishery", which provides a backup for the p r i n c i p a l fishery. For s i m p l i c i t y we assume this alternative fishery i s capable of producing a constant net revenue p per unit e f f o r t per unit, time. In addition we assume that p< {1 - a(1-y)}6/aT, so the alternative fishery is not worth developing on i t s own (capital costs 6 cannot be covered by the t o t a l present value of rents, 13 namely <j/>T[ 1 + a ( 1 - y ) +a2 ( 1 - y ) 2 + . . . ] =c/»T/{ 1 -<j( 1 - y ) } ) . The o v e r a l l effect of the alternative fishery i s to provide a use for excess c a p i t a l (for which the c a p i t a l cost i s a by-gone), while also increasing intraseasonal opportunity costs. (The role of an alternative fishery i s somewhat related to the existence of a market for scrap, since both provide a use for excess capacity a r i s i n g in the p r i n c i p a l fishery. However, scrapping of capacity is in i t s e l f i r r e v e r s i b l e , whereas'an alternative fishery plays the part of a " s i d e l i n e " , to be used whenever desirable.) Left out from the model are such complications to the fishery management problem as: (i) international (transboundary) problems, ( i i ) the p o s s i b i l i t y of set-up costs and fixed (non-capital) costs , and ( i i i ) d i f f e r e n t i a t i o n between labour and c a p i t a l as inputs to the fishery, together with questions of multispecies e f f e c t s , gear type, processing, demand e l a s t i c i t y , nonlinear costs and mixed f l e e t s discussed above. Despite these drawbacks, the model appears to contain at least the essence of the optimal f i s h e r i e s investment problem. The yearly rents accruing to the f l e e t , as a function of recruitment, capacity, escapement and investment, are given by: T T rr(R,K,S,l) = | [ pqEx-c (E) ]dt +|/>(K-E)dt -61 0 0 T = p(R-S) -|c(E(t))dt -(/>/q)log(R/S) + />TK -61 0 using dx/dt= -qEx, x ( 0 ) = R, x(T)= S. [For given capacity K and T fishing e f f o r t E, the quantity |/>(K-E)dt represents the rent 0 14 obtained from using surplus capacity in the alternative fishery.] This rents function i s determined once we have found an expression for the cost-minimizing e f f o r t path and season length which reduce the biomass from R to S within the maximum season length T. Consider the problem: T T Min{ |c(E(t))dt } subject to 0<E(t)<K and |E(t)dt=q~.1 log(R/S). Note f i r s t that whatever the optimal season length, e f f o r t can be considered to be constant within the season: th i s follows from our assumption of a convex average cost curve. If e f f o r t varies during the season, costs could be reduced by using an averaged e f f o r t l e v e l . Hence the cost-minimization problem can be re-stated: Min{ T - C ( E ) } subject to 0<E<K and T=(qE)- 1 log(R/S) where 0<r<T. This in turn can be si m p l i f i e d to the problem of minimizing c(E)/E subject to (qT)- 1log(R/S)<E<K . Define E 0 to be the minimum average cost e f f o r t l e v e l , so that c ( E 0 ) / E 0 = c'(Eo). If E 0 l i e s in the interval [(qT)- 1 log(R/S),K] then the optimal e f f o r t i s E* = E 0 . If E0<(qT)""1 log(R/S) then E*=(qT)' 1log(R/S) and i f E0>K then E*=K. Using this result together with the d e f i n i t i o n of r above, we can now write the minimum yearly variable costs as a function of R, K, and- S : 0 0 -qTK -qTE 0 Tc([qT]- 1 log[R/S]) K>E Re <S<Re -qTE 0 r-c(E)= V log(R/S)-c(E 0)/qE 0 K>E o i Re <S<R (j.og(R/S) -c (K)/qK K<E0 1 5 The yearly rents function i s then given e x p l i c i t l y when this expression is inserted in the equation: ir(R,K,S,l)= p(R-S) -r-c(E) - ( p/q) log (R/S ) + ,oTK -61 The "objective" of the fishery manager in r e a l i t y is l i k e l y to be a complicated composite of such considerations as rent maximization, harvest maximization, harvest and employment s t a b i l i t y , conservation of the f i s h stock and d i s t r i b u t i o n of the rents (ACMRR Working Party on the S c i e n t i f i c Basis of Determining Management Measures, 1980). There i s , of course, no reason to exclude any of these objectives and indeed the appropriate balance between c o n f l i c t i n g objectives i s an e t h i c a l and p o l i t i c a l decision. To simplify the analysis here we s h a l l adopt the common economic objective of present value rent maximization. (In Chapter III, we b r i e f l y consider the case of risk aversion, replacing yearly rents, n, with U ( i r ) , where the u t i l i t y function U is increasing and concave [U'>0, U"<0].) Hence our problem can be stated as follows; n-1 Max [ Z a rr(R ,K ,S ,1 )] subject to: {S,;I 2;S 2;...} n>1 n n n n+1 R =F(S ), K = U-r)K +1 , R exp{-qTK }<S <R , I <0, n+1 n n+1 n n+1 n n n n n+1 where o is the annual discount factor. It is worthwhile noting at t h i s point that we have posed here an overa l l opimization problem, without addressing the d i f f i c u l t task of determining how to reach t h i s optimum. Regulatory measures to achieve optimal e f f o r t or optimal harvests within the fishing season have been the subject of much 16 discussion [see for example Pearse (1979), Scott and Neher (1981), and Clark (1980)]. The problem of achieving the optimal f l e e t capacity through regulation has not been as widely discussed, but i s perhaps not as d i f f i c u l t a problem conceptually; we s h a l l return to thi s question in Chapter IV. (iv) The dynamic programming equation for the value of the fishery in state (R ,K ) at the start of a season n i s given by: n n V(R ,K )= Max Max U ( R , K , S , I ) + cV(R ,K )} n n S I n n n n+1 n+1 n+1 n n where R = F(S ), K = (1~r)K +1 , the outer maximization i s n+1 n n+1 n n+1 subject to R exp{-qTK }< S <R and the inner maximization is n n n n over the range I >0. This i s simply a statement, using n+1 Bellman's (1957) p r i n c i p l e of optimality, that the value of the fishery is given by the maximum value of the sum of current rents plus the discounted future value of the fishery, where the escapement and investment levels are chosen from the set of a l l feasible values. Removing the subscripts on the variables, this can be rewritten: (1) V(R,K)= Max Max W(R,K,S,I) +oV(F(S),(1-y)K+I)} R-exp{-qTK}<S<R I>0 where ir(R,K,S,I)= p(R-S) -r-c(E) - ( P/q) log (R/S ) + pTK -61, with T'c(E) being determined as above. This equation (1) w i l l form the basis for most of the analysis and results presented in this 1 7 chapter. The following three sections study equation (1) from ana l y t i c , h e u r i s t i c and numerical perspectives, respectively. B. Analytic Results To obtain some analytic insights into the optimal investment problem, we f i r s t consider a s i m p l i f i c a t i o n of the model described above. The stock- recruitment function F(S) i s given by F(S)= aS/(l+aS/b) in the Beverton- Holt case. If the maximum i n t r i n s i c growth rate, a, i s very large, recruitment is p r a c t i c a l l y independent of the previous year's escapement. Indeed as a tends to i n f i n i t y , we obtain the approximation F(S)= b =constant. This rather t r i v i a l stock-recruitment function allows us to concentrate on the c a p i t a l investment problem, and in part i c u l a r to obtain an analytic upper bound on the true optimal f l e e t capacity (since the assumption F(S)=b is as optimistic as one can be regarding the productivity of the fi s h e r y ) . The assumption that recruitment i s independent of escapements may be considered a reasonable approximation in some circumstances; Australian prawn stocks appear to f i t this assumption, a factor which has enabled f a i r l y detailed models of these f i s h e r i e s to be analyzed (Clark & Kirkwood,1979; Dudley & Waugh,l980). On the other hand, Walters & Ludwig (1981) have shown that errors in measuring f i s h stocks can lead to the appearance of independence when in fact there may be considerable compensation in the stock-recruitment relationship. If bionomic equilibrium occurs at a s u f f i c i e n t l y high stock size one need not be overly concerned with over-harvesting and the 18 threat of extinction. However, to allow for the possible risk of overly low escapements we sha l l a r b i t r a r i l y include in the objective function a po s i t i v e , increasing function of escapement, b(S), representing a measure of the economic or so c i a l benefits of an escapement S. In a sense, this function corrects for the extreme assumption of constant recruitment. Thus the model we consider here i s id e n t i c a l to that of section A, except that we specify R =F(S )=b=constant (which n+1 n we s h a l l write simply as R) and add a benefit function b(S) to the rents ir (R, K, S , I )•. The l a t t e r function becomes: i r ( R , K , S , I ) = p(R-S) - T - C ( E ) - ( p/q) log (R/S ) +/>TK -61 +b(S) The dynamic programming equation (1) can now be written: (2) V(R,K)= Max Max [ IT (R,K,S , I ) +oV(R, ( 1 -y )K + I ) ] I>0 R-exp{-qTK}<S<R where R is constant. Letting v(K)=V(R,K) and d i f f e r e n t i a t i n g with respect to I, we see that the optimal capacity for next season is given by v ( K * ) = 6 / a where we have used the equation K TT (R, K, S , I ) =-6 . Assuming that v(') i s an increasing function of I capacity, K, i t is clear that there i s a unique optimal l e v e l of fle e t capacity, K*. Hence whatever the i n i t i a l values of R and K, the optimal policy w i l l be to invest up to the capacity level K*, which can be chosen as a one-time i r r e v e r s i b l e decision at the outset. (If i n i t i a l l y K>K*, one must allow depreciation to take place u n t i l K has dropped below K* and then invest up to K*). Whatever the lev e l of capacity in a given year, the optimal -qTK escapement S* is chosen subject to Re <S<R and based on cost 19 minimization in harvesting the stock from R to S. Since the le v e l of escapement has no ef f e c t on the fishery in future years, the optimal escapement can be found simply by maximizing the yearly rents. Each year's decision is i d e n t i c a l , so that the problem reduces to optimizing over one season, with c a p i t a l costs suitably amortized. Assuming risk n e u t r a l i t y , U ( i r ) = i r , the problem can be stated as follows: T ( 3 ) Max Max [p(R-S) -Min | c(E)dt +b(S) K>0 R-exp{-qTK}<S<R T;E 0 -(p/q)log(R/S) -U-/>T)K] where the t o t a l variable cost is to be minimized subject to T 0<r<T, 0<E(-)^K, and q|E(t)dt=log[R/S]. The constant K is found 0 by adding the annual "rental" cost of c a p i t a l [that i s , the n amortized c a p i t a l cost = 6/ Z a ] to the annual depreciation n > 1 costs, 7 6 . Hence K = [ ( 1 - O ) / O +y]6. The cost minimization, given R and S, was performed in section A, where we obtained the res u l t : -qTK -qTE 0 ( T c ( [ q T ] - 1 log[R/S]) ; K>E0,Re <S<Re T \ -qTE 0 Min |c(E)dt = r»c(E)= <{ log(R/S) .c(E 0)/qE 0 ; K>E0,Re <S<R ^log(R/S)-c (K)/qK ; K<E0 We wish now to determine the optimal S* for any combination of R and K. Let us assume i n i t i a l l y that K>E0. Then, defining -qTE 0 S'=Re , we have: -qTK Re <S<S'; ir=p(R-S) -Tc ( [ 1 /qT ] log [ R/S ]) - ( p/q) log [ R/S ] +b (S) S'<S<R ; rr= p(R-S) -q." 1 log [R/S] • [c (E 0 )/E 0 +/>] +b(S) 20 D i f f e r e n t i a t i n g each of these equations and setting d i r / d S = 0 we obtain: -qTK Re <S<S'; [ p-b' (S) ] q S = c ' ([ 1 /qT ] log [ R/S ] ) +/> S'<S<R ; [p-b'(S)]qS= c'(E0)+/» Define S=s(R) to be the solution of the f i r s t of these equations and S 0 to be the solution of the second, so that [p-b'(S 0) ]qS0= c'(E0)+/>. Def ine R +(K) as the solution of: -qTK -qTK ( 4 ) [p-b' (Re )]qRe = c'(K) + />. -qTK Note that i f R=R+(K), we have s(R +(K))=R +(K)e . Now S'<S<R qTE 0 i f f S 0^R^S 0e , so that the optimal escapement is S*=S0 in qTE 0 this case. Si m i l a r l y i f S 0e <R<R+(K) then S*=s(R). It is reasonable to assume, as we s h a l l , that the benefits of escapement increase with escapement, but at a decreasing rate, so that b ' ( S ) >0 but b"(S) <0. Hence b'(S) is decreasing, so that [p-b'(S)]qS is increasing in S. For S<R, we then have that Max{[p-b'(S)]qS} occurs at S=R. Furthermore for given R, c'([l/qT]log[R/S]) i s decreasing in S. Thus one of the equations dir/dS=0 must have a solution S*, unless either: (i) [p-b'(R)]qR < c'(E0)+p so that R<S0, in which case the optimal S*=R. -qTK -qTK ( i i ) [p-b' (Re ) ]qRe > c'(K)+p so that R>R+(K) and the -qTK optimal escapement is S*=Re For K<E0, yearly rents ir(R,K,S,l) are given by : ir= Max{p(R-S)-q" 1 log [ R/S ] • c (K) /K + b( S ) - ( p/q) log [ R/S ] - (tc-pT) K] 21 -qTK over S in the int e r v a l Re £S<R. Setting the derivative of this expression (with respect to S) equal to zero defines a desired escapement S=e(K) s a t i s f y i n g the equation [p-b'(S)]qS= c(K)/K +fi . Incorporating the constraints on S and defining R**(K)=e(K)exp{qTK], we conclude that i f K<E0 the optimal escapement i s : fR-exp{-qTK) ; R>R**(K) S*= < e(K) ; otherwise [R ; R<e(K) -qTE 0 Defining R*=S0e , we can now summarize the optimal escapement and the yearly rents function, for any values of R and K, in the following table: (a) K>E0 Recruitment s* Yearly Rents tr R<S0 R f 1 = b(R) S0<R<R* So "2 = p(R-S 0)-q- 1log(R/S 0)c(E 0)/E 0 -(/)/q)log(R/S 0)+b(S 0) R*<R<R+(K) S(R) "3 = p(R-s(R))-Tc([l/qT]log[R/s(R)] -(/>/q)log[R/s(R) ]+b(s(R) ) R>R+(K) -qTK Re tr„ -qTK -qTK = pR(l-e )-Tc (K)-/>TK+b(Re (b) K<E0 Rec rui tment s* Yearly Rents ir R<e(K) R "5 = b(R) e(K)<R<R**(K) e(K) "6 = p(R-e(K))-q- 1log[R/e(K)]c(K)/K -(/>/q)log[R/e(K) ]+b(e(K) ) R>R**(K) -qTK Re "7 -qTK -qTK = pR(l-e )-Tc (K)-/>TK+b(Re 22 Now our objective becomes: Max [n(R,K)-(K-pT)K] . In other K>0 words, we wish to maximize the yearly fishery rents net of c a p i t a l (depreciation and 'rental') costs. Consider f i r s t the range of values K>E0. Note that, since [p-b'(S)]qS is increasing in S and c'(K) is increasing in K, d i f f e r e n t i a t i n g equation ( 4 ) shows that (d/dK)R +(K)-qTR +>0, so that (d/dK)R +(K)>qTR +>0. Hence R+ i s monotone increasing and we can define an inverse function k(R) such that K=k(R) i f f -qTK R=R+(K), and R<R+(K) i f f K>k(R). Since s(R +(K))=R +(K)e , we -qTk(R) qTE 0 have that s(R)=Re for any R>S0e . Now the optimal escapement s(R) can be attained with K=k(R), so we conclude that qTE 0 i f R>S0e i t w i l l never be optimal to have K>k(R). Optimizing qTE 0 over the range K<k(R) [R£R +(K)], with R>S0e , we have: -qTK -qTK d»r/dK= [p-b'(Re ) ]qRe -T• c ' (K) -pT= K ~ P T Simplifying t h i s expression produces: -qTK -qTK ( 5 ) [p-b'(Re )]qRe = c'(K) + K/T Now we have assumed that K>pT, so that, for given K, the right hand side in this equation i s greater than that of equation ( 4 ) . Since K=k(R) solves ( 4 ) and the optimal capacity K*(R) solves ( 5 ) , i t follows that we must have the optimal capacity K*<k(R). Recall that we assumed K>E0. If in fact -qTE 0 -qTE 0 [p-b'(Re )]qRe < c'(E 0) +K/T then ( 5 ) cannot be solved for K*>E0 and so the optimal capacity must be K*<E0. 23 qTE 0 If S0<R<S0e , jr(R,K) i s independent of R, so that the optimal capacity amongst values K>E0 i s Min{K : K>E0}=E0. If R<S0, the same result holds; K*=E0. However in both cases this is merely the optimum for K>E0; a value K*<E0 w i l l be preferable i f drr/dK(R,E 0 )<K-/)T. We now consider the p o s s i b i l i t y K*<E0. Unfortunately analysis in thi s case is complicated by the ambiguity resulting from the lack of concavity in the rents function; for example qTK the sign of the derivative of e(K)e is ambiguous. Hence we sha l l content ourselves with p a r t i a l r esults. F i r s t , note that since c ( E 0 ) / E 0 = c ' ( E 0 ) , we must have e(E 0)=S 0, since both sides of t h i s expression s a t i s f y the same equation. Suppose that qTE 0 R>S0e and the solution K* of ( 5 ) is such that K*<E0 and qTK* R>e(K*)e =R**(K) (this w i l l c e r t a i n l y be the case i f K* is only s l i g h t l y less than E 0 ) . Then K* is the optimal capacity. qTK* If, however, R>e(K*)e , then K* l i e s outside the range for which equation ( 5 ) applies, and hence the optimum must instead l i e in the interval [ 0 , K ' ] , where K' s a t i s f i e s e(K')exp{qTK'}=R . In addition, note that for K<E0, c(K)/K decreases with K, so that e(K) must also decrease with K. Hence i f R<S0, R<e(K) for a l l K<E0, and so K*=0 in thi s case. To.summarize, the optimal capacity K* is given by: qTE 0 K*= solution of ( 5 ) ; i f either (i) K*>E0 and R>S0e qTK* or ( i i ) K*<E0 and R>e(K*)e 24 K*=0 ; i f R<S0 0<K*<Eo ; otherwise If we assume linear costs, c(E)=cE, then we can l e t E o=0 and the solution s i m p l i f i e s to the following: -qTK -qTK K*= solution of [p-b' (Re ) ]qRe = c +tc/T or K*= 0 i f [p-b'(R)]qR< c +K/T If in addition we ignore benefits of escapement,, so that b(S)=0 for a l l S, we have: K*= [qT]" 1 log[R/x$ ] where x$=(c +ic/T)/pq This simply states that s u f f i c i e n t capacity should be in place to harvest the biomass R down to bionomic equilibrium x$, where both f i s h i n g costs and time-scaled c a p i t a l costs are included. In cases where recruitment i s thought to be approximately constant from year to year, the results presented in this section can be used to provide an estimate of the optimal capacity l e v e l . Even for f i s h e r i e s with more complicated stock-recruitment relationships, these results provide a simple upper bound on the optimal f l e e t size (assuming the population dynamics parameters are known and stochastic v a r i a b i l i t y is s u f f i c i e n t l y low). In f i s h e r i e s with more extensive randomness, a stochastic analysis i s necessary - in Chapter III.B we consider such a stochastic analogue. 25 C. Heuristic Analysis We now return to the general model, and in p a r t i c u l a r the general stock-recruitment function F(S), introduced in Section A. Our objective in this section is to gain q u a l i t a t i v e information about the f u l l optimal investment and escapement problem from an h e u r i s t i c study of the dynamic programming equation (1). Let us assume for now that the f i s h stock displays pure compensatory population dynamics (as in the Beverton-Holt model) and that V >0, V >0, V >0, V <0 over a l l non-zero R K RK KK values of R and K for which V is twice d i f f e r e n t i a b l e . The l a t t e r assumption simply states that more f i s h and more c a p i t a l increase the value of the fishery, that more f i s h are more desirable the larger the c a p i t a l stock, and that the fishery has decreasing marginal returns to c a p i t a l . D i f f e r e n t i a b i l i t y of the value function w i l l not be addressed in any d e t a i l here. Instead, we s h a l l assume that V(R,K) is at least C 2 in both variables, except at points where the defining equations for the control variables change abruptly. With this assumption, we s h a l l see below that such non-differentiable points can be expected to l i e on 3 curves in the biomass/capacity plane, and therefore constitute a set of measure zero in R2. This "almost everywhere" d i f f e r e n t i a b i l i t y arises frequently in optimal control theory, and in fact was proved a n a l y t i c a l l y by Clark, Clarke and Munro (1979) for the continuous-time version of the i r r e v e r s i b l e investment model. In our case, the p a r t i a l derivatives w i l l be evaluated at points [F(S),h(S)] lying away from the 3 "switching curves", and hence 26 are assumed to e x i s t . (In p a r t i c u l a r , V and V are assumed to R K be piecewise d i f f e r e n t i a b l e . ) (a) Optimal investment Performing the inner maximization in ( 1 ) , for fixed S, we obtain the optimality equation for investment: (6) V (F(S),(1- r)K+I)= 6/a or 1=0 i f V (F(S),(1-y)K)<6/c K K This states that, unless the f l e e t is temporarily overcapitalized, next year's optimal capacity, (l-r)K +1, should be set such that the marginal benefit of an extra unit of c a p i t a l equals i t s marginal cost. Define K= h(S) to be the solution of the i m p l i c i t equation V (F(S),K)= 6/a , so that h(S) is next season's optimal K capacity, and V (F(S),h(S))= 6/a. Performing the t o t a l K derivative of t h i s l a t t e r equation produces: » F'(S)V (F(S),h(S)) + h'(S)V ( F ( S ) , h ( S ) ) = 0 RK RK Rearranging this expression, we find that: h'(S)=F'(S). [V (F(S),h(S))]/[-V (F(S),h(S))]>0 , RR KR by our above assumptions. Hence the optimal capacity target h(S) is an increasing function of escapement. While the concavity of h(S) cannot be deduced from the expression for h'(S), i t can be expected since compensatory recruitment produces decreasing marginal returns to escapement. Thus i f (l-r)R>h(S), the optimal investment is I*=0 (capital is already s u f f i c i e n t l y abundant), while otherwise I* 27 i s c h o s e n s o t h a t ( 1 - r ) K + i * = h ( S ) . T h i s c a n be w r i t t e n : (7) I * ( S , K ) = M a x { h ( S ) - ( 1 - r ) K , 0 ) . s o t h a t i n v e s t m e n t d e s i r e d f o r t h e n e x t s e a s o n i s a f u n c t i o n o f b o t h c a p a c i t y a n d e s c a p e m e n t i n t h e c u r r e n t s e a s o n . (b) O p t i m a l e s c a p e m e n t Now i n s e r t i n g I * ( S , K ) i n t o ( 1 ) , p e r f o r m i n g t h e o u t e r m a x i m i z a t i o n by t a k i n g a t o t a l d e r i v a t i v e w i t h r e s p e c t t o S , a n d n o t i n g t h a t f o r a n y S a n d K , e i t h e r I * ( S , K ) = 0 o r S oV ( F ( S ) , ( 1 - c ) K + I * ( S , K ) ) - 6 = 0 , we o b t a i n t h e o p t i m a l i t y K e x p r e s s i o n e q u a t i n g t h e m a r g i n a l b e n e f i t a n d m a r g i n a l c o s t o f a n i n c r e m e n t a l i n c r e a s e i n e s c a p e m e n t : ( 8 ) < * F ' ( S ) V ( F ( S ) , ( 1 - O ) K + I * ( S , K ) ) R fp O - x 0 / S ) i f K < E * - q T E * p ( l - x § / S ) i f K > E * ; Re <S<R - q T K - q T E * P ( 1 - x g ( R , S ) / S ) i f K > E * ; Re <S<Re w h e r e we d e f i n e : x 0 = [ c ( K ) / K + />]/pq, x g = [ c ( E * ) / E * +p]/pq, q T K x g ( R , S ) = [ c ' ( [ 1 / q T ] l o g ( R / S ) + / > ] / p q , a n d t h e c o n s t r a i n t Re <S<R h a s b e e n n e g l e c t e d - t e m p o r a r i l y . T h e r e i s no c o n c e p t u a l d i f f i c u l t y i n u s i n g t h i s o p t i m a l i t y e q u a t i o n t o d e r i v e an o p t i m a l e s c a p e m e n t p o l i c y , w h i c h i n g e n e r a l w i l l d e p e n d on b o t h c u r r e n t e s c a p e m e n t a n d c u r r e n t 28 capacity; S=s(R,K). However, note that i f costs were linear in e f f o r t , c(E)=c«E, then we would have x 0=x§=x^ and in pa r t i c u l a r X Q would be independent of R, since in the linear cost case c(E)/E=c'(E)=c for a l l values of E. This assumption would allow us to consider an escapement target curve S=s(K) rather than a surface s(R,K). Hence for descriptive s i m p l i c i t y and to compare our results to those of Clark, Clarke and Munro (where a similar assumption was used), we s h a l l set c(E)=cE, and denote by x 0=(c + /))/pq the common quantities x£=x§=x§, for the remainder of this work (apart from III.B). We s h a l l assume in addition that S=s(K) i s the unique solution of ( 8 ) , representing the target escapement for any given level of the c a p i t a l stock, R. Consider the equation: ( 9 ) a F ' ( S ) V ( F ( S ) , h ( S ) ) = p(1 -x 0/S) R If ( 9 ) has a solution (assumed unique), denote i t by S~ . Then s(K)= S" is the optimal escapement for a l l K < ( 1/[ 1 - y ] )h (S"" ) , since we have 0 - y ) K +I*(S,K)= h(S) for a l l S near S" , but amongst these values of S, only the value S" s a t i s f i e s the optimality equation ( 9 ) . Hence S~ is the "low-capacity l i m i t " for the optimal escapement s(K). Denote by S + the solution of: ( 1 0 ) o F ' ( S ) V (F(S),inf)= p(1 -x 0/S) . R This S + is simply the optimal equilibrium escapement for the discrete-time abundant-capital problem (c.f. 5T in CCM). In other words, S + is the "high-capacity l i m i t " of s(K). To see t h i s , define the function v(R)=V(R,inf) and note that for K very large 29 and R>s(inf)=S +, we have: v(R+dR)= ir(R+dR, inf ,S +,0) +av(F(S +)) = p(R-S +) -(c/q)log(R/S +) +pdR -(c/q)log((R+dR)/R) + O V ( F ( S + ) ) = v(R) +pdR -(c/q)(dR/R) . Therefore v (R)= p -c/qR= p ( l -x0/R) i f R>s(inf)=S +. R In fact, we wish to derive an expression for V (F(S),inf)=v (F(S)), for S near S +. Note f i r s t that i f S° i s R R the unexploited equilibrium biomass, given by F(S°)=S°, and assuming that S°>x 0, then an optimal value of S + must be such that S + < S ° . [Otherwise the harvestable, and p r o f i t a b l e , biomass between S + and S ° would be wasted.] Hence for any S near S +, we must have F(S)>S, so that indeed F ( S ) > S + in a neighbourhood of S +, and we can write: v (F(S))=p(1-x 0/F(S)). Then for such R values of S, equation ( 1 0 ) becomes: a F ' ( S ) ( 1 - x 0 / F ( S ) ) = p ( 1 - x 0 / S ) , and hence S + is given as the solution of the Modified Golden Rule equation: F ' ( S ) . [ ( 1 - X O / F ( S ) ) / ( 1 - X 0 / S ) ] = 1 / a . Note that our assumption V (',O>0 implies RK V (F ( S),inf)>V ( F ( S ) , h ( S ) ) for any S. Also, (p/aF'(S))(1-x 0/S) R R is an increasing function of S . Hence we can conclude that S"<S+; in other words, the optimal escapement target is higher at larger f l e e t capacities. We wish now to consider the behaviour of the target escapement curve s(K). For ( 1 - y ) K < h ( S " ) , we have s(K)= S" so 30 that ds/dK= 0 . For (1-y)K>h(S") , S= s(K) is defined by: (11) V (F(S) , ( l-y)K)= (p/a)F'(S)(1-Xo/S) . R D i f f e r e n t i a t i n g this equation with respect to K and rearranging terms we obtain: (12) S'(K)= ( l- r)F ' ( S)V (F ( S ) , ( l-y ) K ) RK .{(p/a)[x 0/S 2-(F"(S)/F' ( S ) )(1-Xo/ S ) ] -F'(S) 2V (F(S),(l-y)K)}" 1 RR With the assumptions to date, the sign of s'(K) i s ambiguous. However we w i l l be assured that s'(K)>0 i f V(R,K) i s either concave or not too convex in R , so that V (*,•) is not too RR large. This would seem to be a l i k e l y s i t u a t i o n , although i t proves to be faulty to assume that V <0 in a l l cases, as we RR s h a l l see in section E. The value of. s(K) is the target escapement given capacity -qTK K. Due to the constraint Re <S<R , this target may not be attainable. Hence the optimal escapement S* = S*(R,K) must be defined as follows: ("R-exp{-qTK} ; R>s (K)exp(qTK] (13) S*(R,K)= -j s(K) ; R intermediate R ; R<s(K) giving the desired (feasible) escapement for any pair (R,K). (c) Summary We have now h e u r i s t i c a l l y derived a synthesis of the optimal harvesting/investment policy, in the form of the two policy functions s(K) and h(S), giving the optimal action 31 ( S * , I * ) a s a f u n c t i o n o f t h e s t a t e ( R , K ) . T h e o p t i m a l c a p a c i t y m h ( S ) i s l i k e l y t o be z e r o b e l o w some m i n i m u m e s c a p e m e n t S >0 a n d m c o n c a v e i n c r e a s i n g f o r S>S . T h e o p t i m a l e s c a p e m e n t i s e x p e c t e d t o be n o n - d e c r e a s i n g a s a f u n c t i o n o f c a p a c i t y , a p p r o a c h i n g an a s y m p t o t e S + a s K t e n d s t o i n f i n i t y . A t t h e b e g i n n i n g o f a s e a s o n , g i v e n r e c r u i t m e n t R a n d c a p a c i t y K , t h e f i s h s t o c k i s f i r s t h a r v e s t e d down t o S * ( R , K ) . T h e n d e p r e c i a t i o n a n d i n v e s t m e n t o c c u r , s u c h t h a t i f t h e d e p r e c i a t e d c a p a c i t y i s ( 1 - r ) K < h ( S * ) , i n v e s t m e n t b r i n g s t h e c a p a c i t y t o h ( S * ) by t h e s t a r t o f t h e n e x t s e a s o n . T h e p r o c e s s i s t h e n r e p e a t e d f r o m t h e new b i o m a s s / c a p a c i t y p o i n t ( F ( S * ( R , K ) ) , M a x [ ( 1 - r ) K , h ( S * ( R , K ) ) ] ) . T h e r e s u l t i n g t r a j e c t o r y e v e n t u a l l y c o n v e r g e s on a l o n g r u n e q u i l i b r i u m p o i n t , d i s c u s s e d i n s e c t i o n E ( s e e f i g u r e s 1 a n d 2 ) . ( d ) T h e i n s t a n t a n e o u s i n v e s t m e n t c a s e T h e a b o v e h e u r i s t i c d i s c u s s i o n a p p l i e d t o t h e d e l a y e d i n v e s t m e n t c a s e . I f i n s t e a d i n v e s t m e n t i s a s s u m e d t o o c c u r i n s t a n t a n e o u s l y , a n d t h e a s s u m p t i o n o f l i n e a r c o s t s i s m a i n t a i n e d , e q u a t i o n (1) must be c h a n g e d t o b e c o m e : (14) V ( R , K ) = Max Max {p (R-S ) - [ (c + p ) / q ] l o g ( R / S ) +/>TK-6I I>0 {S} + a V ( F ( S ) , ( 1 - y ) ( K + I ) ) } w h e r e t h e i n n e r m a x i m i z a t i o n i s now o v e r t h e r a n g e - q T ( K + I ) Re <S<R. P e r f o r m i n g t h i s i n n e r m a x i m i z a t i o n we o b t a i n : (15) o F ' ( S ) V ( F ( S ) , ( 1 - y ) ( K + I ) ) = P ( 1 - ( x 0 ) / S ) R u n l e s s t h e c o n s t r a i n t on S i s b i n d i n g . 32 Equation (14) can be rewritten: (14') V(R,K)= Max Max Max { ir (R, S , J) + cV( F (S) , ( 1 -y ) (K+I+J) )-61} I>0 J>0 {S} = Max [ V ( R , K + I ) - 6 l ] I>0 where now the maximization with respect to S is r e s t r i c t e d to -qT(K+I+J) Re ^S<R. The optimal investment is then obtained by maximizing the right hand side to obtain: (16) V (R,K+I)= 6 or 1= 0 i f V (R,K)<6 . K K Note that (16) can be written V (F(S),K*)= 6 , giving this K year's optimal capacity K* i m p l i c i t y as a function of the previous season's escapement. Clearly (16) and (6) are equivalent, apart from the reduced unit c a p i t a l cost in the instantaneous case (where investments do not have to be paid for in advance). Likewise, (15) and (8) are equivalent, giving the optimal escapement as a function of this year's optimal capacity, K+I . We can conclude therefore that the delayed investment case with unit c a p i t a l cost 6 and the instantaneous investment case with unit c a p i t a l cost 06 should produce comparable res u l t s , as expected for a deterministic model. 33 D. Numerical Method While the h e u r i s t i c analysis of the previous section provided considerable information about optimal investment and escapement p o l i c i e s , numerical methods are required to produce quantitative r e s u l t s . We therefore turn now to a numerical solution of the dynamic programming equation (1). The two primary techniques for obtaining solutions in dynamic programming are policy i t e r a t i o n and value i t e r a t i o n , both of which are i t e r a t i v e procedures which, under appropriate conditions, converge to the optimal policy and value functions. While value i t e r a t i o n is useful for solving f i n i t e time horizon optimization problems, the policy i t e r a t i o n approach is more helpful for the i n f i n i t e time case considered here, and has the advantage of concentrating e x p l i c i t l y on the objects of interest, namely the management p o l i c i e s . Procedures have been developed for speeding the convergence of policy i t e r a t i o n (Puterman and Shin, 1978), but in our case, reasonably rapid convergence is obtained with simple policy i t e r a t i o n based on equations (6) and (8) in the previous section. The procedure used i s as follows: (a) Determining the value function F i r s t , a set of uniformly-spaced mesh points was chosen for each dimension of the positive quadrant of the R-K plane. .. The upper and lower l i m i t s for biomass and capacity were chosen a r b i t r a r i l y but so as to include the R-K region of interest: the lower capacity l i m i t was K,=0 in a l l cases. Variation of these l i m i t s did not substantially a f f e c t the optimal policy or value 34 functions. The effect of the number of mesh points used is discussed later in this section. An i n i t i a l guess i s made for the policy functions s(K) and h ( S ) and the value function corresponding to these p o l i c i e s i s determined for each mesh point by solving a set of simultaneous equations of the form: (17) V(R ,R )= r ( R ,K ) + c V ( F ( S * ( R ,K )),K'(R ,K )) i j i j i j i j - - T T ( R ,K ) + 0{6 1V ( R ,K ) +6 2V ( R ,K ) i j 1 m 1 m+1 +6 3V ( R ,K ) +6„V ( R ,K )} 1+1 m 1+1 m+1 where R = Max{R > F ( S * ( R ,K )) : k=1,--.,M}, K' ( R ,K ) is next 1 k i j i j year's capacity, K = Max {K <K' (•,•): k=1,--',N] and the 6 are m k i 4 determined by linear interpolation (6 ^ 0 and I 6 =1). The i i = 1 i system (17) contains MN equations in the MN unknowns V ( R ,K ) . i j In vector-matrix form, the system can be rewritten: (18) V= ir +cQV or (19) (l-aQ)V= ir , where I i s the identity matrix and Q is the appropriate t r a n s i t i o n matrix. For a<1 , the matrix A=I-aQ is diagonally dominant and hence (19) has a unique solution. The system (19) may be solved in a straightforward manner by Gaussian elimination, but for MN r e l a t i v e l y large, two alternative procedures seem to be superior. F i r s t l y , i f the depreciation rate r= 0 , the c a p i t a l stock cannot decrease from one year to the next, so K ' ( R ,K ) >K and i j j 35 hence m>j in (17). If we can assume that the maximum capacity value, K , i s s u f f i c i e n t l y large that no investment occurs N given t h i s capacity then the equations in (17) can be divided into subsystems (with K fixed within each subsystem) and solved j consecutively, beginning with the M equations having K = K , j N followed by the equations for which j=N-1 and so on. I f , for par t i c u l a r values of i and j , we have m>j the equation (17) can be solved d i r e c t l y for V(R ,K ) using previously determined i j values on the right hand side. Furthermore i f m= j the values of V(',K ) can be immediately inserted into (17), having been m+ 1 determined in the previous step. Thus the problem is reduced from solving one MN by MN system to solving N systems of maximum size M by M . A second approach is to notice that, for any values of i and j , at most 5 of the MN entries in row ( i j ) are non-zero. Hence i t e r a t i v e sparse matrix methods" are highly suitable for this problem. The diagonally dominant nature of the matrix A= I - a Q enabled the use.of a sparse symmetric over-relaxation (SSSOR) package written in /370 Assembler and available on the University of B r i t i s h Columbia computer system (Conrad & Wallach, 1977). The scheme can reduce the number of calculations per i t e r a t i o n below that required by standard SOR methods. As with other over-relaxation techniques, one must choose a relaxation parameter o (1.0<U<2.0) , with u = 1.0 corresponding to the basic Gauss-Seidel method. The rate of convergence to the 36 value function did not appear sensitive to the choice of o : any value 1.0£u^1.2 seemed suitable. In our numerical work, the SSSOR method proved quite e f f i c i e n t , being as inexpensive in the 7 = 0 case as the above step-by-step procedure. (b) Policy improvement Once the value function at the mesh points has been determined, a C 2-cubic spline approximation method is used to f i t a smooth V(R,K) surface and hence to obtain the f i r s t p a r t i a l derivatives V and V at the mesh points. Tension was R K not introduced into the spline approximation so that in theory inaccuracies in the solution for the V(R ,K ) value could lead i j to unwanted curvature in the surface V(R ,K ) . However, a i j f a i r l y fine mesh (M=30, N=30) was used and no unreasonable behaviour of the V and V values could be detected. R K The values of V and V are then used to solve equations R K (6) and (8), again resorting to linear interpolation where necessary. The solutions are two new (and improved) policy functions h(S) and s(K) respectively. These are then used to derive a new value function, and the process is repeated u n t i l the policy functions become stationary. S p e c i f i c a l l y , for a prescribed fracti o n f, 0<f<1 , and mesh sizes AR, AK, st a t i o n a r i t y of the p o l i c i e s is considered to have been achieved when: (20) Max |h(i) - h ( i ) | < f • AR and Max |s(j) - s ( j ) | < f-AK i new old j new old 37 In most cases, depending on the i n i t i a l guesses for the policy functions, no more than 5 i t e r a t i o n s were necessary to obtain convergence, indicating that the numerical scheme i s well-behaved. While an examination of the Modified Golden Rule equation for optimal escapement in the case of Beverton-Holt dynamics [see equation (22) in the following section] shows that the optimal escapement at high capacities, S +, i s unique, the he u r i s t i c analysis of section C does not rule out multiple solutions for the value of S"= s(0) . To check for such a p o s s i b i l i t y in our base case prawn fishery, the value of S" was varied, while keeping S constant at i t s optimal value, and s(K) was formed by an appropriate continuous connection between s(0)= S~ and s(inf)= S + . For each S", the value function was calculated (using the optimal capacity function). This value function reached i t s maximum when S" was set at i t s optimal value, S"*, although i t changed very l i t t l e with variations in S" near S"*. It therefore seems reasonable to assume that our numerical scheme is determining a unique optimal escapement curve s(K) . (c) The instantaneous investment case The above numerical method applies to the case of delayed investment, but the procedure with instantaneous investment is very sim i l a r . The only difference arises in solving for the new h(R) function. In the instantaneous case, for any K,<K2^h(R), we have V(R,K 2)= V(R,K,) +6-(K 2-K,). Hence (16) i s s a t i s f i e d for any K+I<h(R)= Max{K+I : V (R,K+I)= 6}. Determining h(R) requires K 38 finding the value of K at which V begins to decrease from the K l e v e l 6. However values of V are only obtained at mesh points, K and h(R) cannot be determined by interpolation in t h i s case, so an a r b i t r a r y linear extrapolation technique is used, based on the f i r s t two mesh points having V <6. S p e c i f i c a l l y , i f K V (R,K )= 6 and V (R,K )<6 , then K j K j + 1 h(R)= K - [ ( 6 - V )/(V -V )](K -K ), j+1 j+1 j+1 j+2 j+2 j+1 where V.=V(R,K.), unless this l i e s outside the i n t e r v a l [K ,K ] in which case h(R) is set equal to the appropriate j j + 1 endpoint. For the two f i s h e r i e s considered here, 'this method performed well in one (the whale fishery -see section E) but proved ill-behaved in the other (the base case prawn f i s h e r y ) . In the l a t t e r case, i t was not possible to achieve an accurate solution in a reasonable number of i t e r a t i o n s . It is not clear, however, whether the numerical scheme is non-convergent or merely subject to very slow convergence. As we are only interested in the instantaneous investment case as a bridge between the CCM results and our discrete-time delayed investment results, we have not investigated in depth the cause of these numerical d i f f i c u l t i e s . Instead we note that each policy i t e r a t i o n step produced h(R) and s(K) curves which agreed q u a l i t a t i v e l y with the h e u r i s t i c analysis of section C. Furthermore, the best results we were able to obtain for the 39 prawn fishery, when averaged over a number of policy i t e r a t i o n s , agree closely with the transformation of the corresponding delayed investment curves (see section C for a discussion of this transformation). Hence we have reported these averaged results as an indicator of the optimal instantaneous investment p o l i c i e s for the prawn fishery, and in p a r t i c u l a r , the true optimal capacity curve in figure 1 should be considered to l i e somewhere in the range h(R)±2. (d) Testing the numerical method In a l l the results obtained in thi s paper, the number of mesh points for biomass and capacity were set at M=30 and N=30 respectively. The linear interpolation method used in our numerical scheme can lead to inaccuracies which decrease as the mesh becomes f i n e r . To check the adequacy of the meshes used in our analysis, results for our base case fishery were also obtained with a finer mesh, M=N=60 , and a coarser one, M=N=10. With the finer mesh, no change in the value function could be seen, and the optimal capacity function changed by less than 1 percent. The maximum change in the escapement function s(K) was 4 percent but in most cases i t changed by 1-1.5 percent. The coarse mesh performed surprisingly well, with reductions in the value function of approximately 1 percent compared with our normal mesh. The function h(S) d i f f e r e d by up to 4 percent while changes of up to 13 percent occured in the values of s(K). While i t appears that a coarse mesh is useful to obtain good approximate res u l t s , the accuracy of our M=N=30 mesh makes i t of more use in studying policy implication and parameter 40 s e n s i t i v i t y . The M=N=60 case proved too expensive for the small increase in accuracy i t affords. E. Numerical Results. We are now in a position to provide a f u l l analysis of the deterministic investment and escapement model embodied in equation (1). Using the procedure of section D, numerical results have been obtained, based on two f i s h e r i e s ; (i) the Australian Gulf of Carpenteria banana prawn fishery (Clark & Kirkwood, 1979) and ( i i ) the aggregated pelagic whaling fishery (Clark & Lamberson, 1980). The purpose of our numerical study was not to derive detailed solutions for these f i s h e r i e s in p a r t i c u l a r but rather to gain an understanding of the optimal investment problem. To this end, the received data were substantially s i m p l i f i e d . In the case of the prawn fishery, a homogeneous f l e e t was assumed, no alternative fishery was allowed (/>= 0), an average prawn weight was used in l i e u of intraseasonal growth, and natural mortality was constrained to occur at the end of each season. The whaling data used by Clark & Lamberson was converted from continuous to discrete-time and, as in the continuous-time case, delays in recruitment were neglected (cf. Clark, 1976b). The stock-recruitment function F(S) was given by R= F(S)= aS/d+aS/b) or R= F(S)= aSexp{ - (a/eb) S} , for the Beverton-Holt and Ricker cases respectively, where S i s the escapement after fishing has taken place. The maximum possible recruitment, b, for the prawn fishery was set equal to the 41 sample mean of recruitment data from Kirkwood (1980). The rate of growth, or fecundity, of the prawn stock (a) was set a r b i t r a r i l y . The data used for each fishery are presented in Table ( I ) , where the values given for a and b correspond to the Beverton-Holt stock-recruitment function. If S is the escapement -mT after f i s h i n g , Se is taken to be the end-of-year escapement after both fi s h i n g and natural mortality. An examination of the -mT stock-recruitment functions indicates that the factor e can be d i r e c t l y incorporated by changing the value of a given in -mT Table (I) to ae ; t h i s i s done in our analysis. [Note that although our results are based on two f i s h e r i e s and two stock-recruitment functions, they are in fact quite general. With the choice of Beverton-Holt or Ricker functions, one can capture the q u a l i t a t i v e features of most f i s h e r i e s , and by varying the parameters in our two base case f i s h e r i e s , a r b i t r a r y economic and ecological conditions can be considered.] It w i l l be of interest to compare our optimal p o l i c i e s with the open-access scenario resulting from uncontrolled f i s h e r i e s development. If we assume that in the open-access case, investment continues u n t i l the average net revenue (per unit capacity) just covers the unit c a p i t a l cost, then in equilibrium we have: ( o / [ 1-o] ) -ir(R,K,S,I )/K= 6 where the l e f t hand side represents the t o t a l present value of discounted rents, per unit of c a p i t a l . Setting I=rK to hold the 42 c a p i t a l s t o c k c o n s t a n t i n e q u i l i b r i u m , a n d a s s u m i n g f u l l u t i l i z a t i o n o f t h e f l e e t , t h i s c a n be w r i t t e n : ( o / [ 1 - a ] ) • [ p ( R - S ) / K - c T - r 6 ] = 6 qTK w i t h R = F ( S ) = S e . T h i s s i m p l i f i e s t o : q T K (e - 1 ) S / K = [ ( [ l - a ] / a + r ) 6 + c T ] / p (=RHS) qTK S o l v i n g t h i s e q u a t i o n s i m u l t a n e o u s l y w i t h F ( S ) / S = e p r o d u c e s t h e o p e n - a c c e s s e q u i l i b r i u m c a p i t a l s t o c k a n d b i o m a s s . I f B e v e r t o n - H o l t s t o c k r e c r u i t m e n t i s a s s u m e d , s o t h a t F ( S ) = a S / t 1 + ( a S / b ) ] , t h i s s o l u t i o n c a n be s i m p l i f i e d . I n q T K e q u i l i b r i u m we h a v e F ( S ) / S = a / [ 1 + ( a S / b ) ] = e , s o t h a t - q T K S= ( b / a ) ( a e - 1 ) . H e n c e t h e o p e n - a c c e s s c a p a c i t y c a n be r e s t a t e d a s t h e s o l u t i o n o f t h e e q u a t i o n : - q T K qTK (21) ( b / a ) ( a e - l ) ( e - 1 ) / K = RHS w h e r e RHS i s a s a b o v e . T h i s e q u a t i o n c a n be s o l v e d i t e r a t i v e l y , a n d i s a p p l i e d i n p a r t (b) b e l o w . R e s u l t s i n t h e f o r m o f f e e d b a c k c o n t r o l d i a g r a m s a r e g i v e n f o r t h e a b o v e p r a w n f i s h e r y d a t a i n f i g u r e s 1 a n d 3 , c o r r e s p o n d i n g t o t h e c a s e s o f i n s t a n t a n e o u s a n d d e l a y e d i n v e s t m e n t r e s p e c t i v e l y . S i m i l a r r e s u l t s a r e shown i n f i g u r e s 2 a n d 4 f o r t h e w h a l e f i s h e r y . [ A s d i s c u s s e d i n s e c t i o n D , t h e i n s t a n t a n e o u s i n v e s t m e n t r e s u l t s f o r t h e p r a w n f i s h e r y a r e o n l y a p p r o x i m a t e , d u e t o n u m e r i c a l d i f f i c u l t i e s . H o w e v e r t h i s d o e s n o t a f f e c t t h e q u a l i t a t i v e d i s c u s s i o n b e l o w . ] 43 (a) The Instantaneous Investment Case Interpretation of these results is f a c i l i t a t e d by comparing them with those obtained by CCM, who assumed that investment occurs instantaneously. Let us f i r s t concentrate on our figures 1 and 2, which are precisely the discrete-time analogues of the CCM re s u l t s , in pa r t i c u l a r their figure 2. Our s(K) and h(S) curves correspond closely with their switching curves, <r, and e2 respectively. To the l e f t of the s(K) curve, no harvesting should take place, while above and to the l e f t of the h(R) curve, no investment should be undertaken. If K<h(R), immediate investment should occur u n t i l K= h(R). To the right of the s(K) curve, harvesting should take place u n t i l the stock i s reduced to s(K), or as close as possible to that escapement, given the le v e l of capacity available. A sampling of possible t r a j e c t o r i e s i s shown in each figure. Note that a l l t r a j e c t o r i e s eventually converge on a single long-run equilibrium point, (R,K), given in terms of recruitment and capacity (after depreciation and reinvestment). The equilibrium point corresponds to the point (x*,K*) in CCM and represents the optimal equilibrium in the case where c a p i t a l is t o t a l l y malleable but not "abundant", so that the rental cost of c a p i t a l must be included in variable costs. This is discussed in more d e t a i l in Appendix A. Unlike the CCM case, this equilibrium point is not apparent from the synthesis diagram. In the continuous-time case, when a trajectory crosses the lin e x= x* below (x*,K*), the optimal policy is an instantaneous investment to the capacity l e v e l K= K*, thereafter remaining at 44 (x*,K*). In discrete time, however, almost a l l t r a j e c t o r i e s approach and/or cross R= R , rather than touch that l i n e , so the use of a single f i n a l impulse control at R is not s u f f i c i e n t to reach the equilibrium point. Furthermore, whereas equilibrium i s reached in f i n i t e time with the continuous-time CCM model, in our discrete-time situation the approach to equilibrium i s asymptotic. This can be seen by setting r =R -R, k =K -K , l i n e a r i z i n g about the n n n n equilibrium and assuming that k << ( r / [ l - r ] ) K . We obtain the n system r =a,r - aib,r = ( a 1 - a 2 b 1 ) r where n+1 n n n a,= F'(R•exp{-qTK])•exp{-qTK], a 2 = qTR.exp{-qTK]•F'(R•exp{-qTK}) and b!= h'(R). Hence (r ,k ) tends asymptotically to ( 0 , 0 ) as n n long as a,-a 2b,= expf-qTK}F'(R•exp{-qTK}•[1-qTRh' (R) ] <0, that is h'(R) > 1/qTR. This asymptotic convergence can be seen in the sample t r a j e c t o r i e s : indeed, equilibrium points were v e r i f i e d by determining the common point of convergence of a number of sample t r a j e c t o r i e s . Apart from the differences mentioned above, the behaviour of our instantaneous investment model and that of CCM are quite similar, r e f l e c t i n g the pure compensatory nature of both the Beverton-Holt function and the continuous-time growth function used by CCM. 45 (b) Delays in Investment As discussed in section C, the introduction of a delay between the time an investment decision is made and the time the corresponding new capacity becomes available produces l i t t l e change in the desired escapement and capacity in any given year, except in as much as payment for the new capacity must be made e a r l i e r than would be the case for instantaneous investment, and hence e f f e c t i v e c a p i t a l costs are higher. However the appearance of the optimal p o l i c i e s can be substantially d i f f e r e n t , with optimal capacity given as a function of escapement rather than recruitment. Figures 3 and 4 depict the optimal p o l i c i e s for the prawn and whale f i s h e r i e s respectively, with delayed investment, but otherwise unchanged parameters. The policy curves for the prawn fishery were derived using both our usual mesh of M=N=30 and a finer mesh M=N=60. As discussed in section D, the differences are minimal: changes which occur when the finer mesh i s used are indicated in figure 3. The introduction of delays in investment does not change the s(K) curves, r e f l e c t i n g the common optimality equations for escapement, derived in section C. L i t t l e q u a l i t a t i v e change in the h(S) curve i s noticeable between the instantaneous and delayed investment cases for a fishery based on a slow-growing stock (whales). However, in the case of fast-growing prawns stocks, a low escapement this year can s t i l l produce a large recruitment next year. With delayed investment, i t may be optimal to plan and pay for investment this year, even though 46 s t o c k s s e e m l o w , s i n c e t h e new c a p a c i t y w i l l n o t b e a v a i l a b l e u n t i l n e x t y e a r , when a h i g h e r c a p a c i t y c a n b e u s e d t o h a r v e s t t h e m u c h l a r g e r s t o c k . T h i s c a n l e a d t o t h e s i t u a t i o n s h o w n i n f i g u r e 3, w h e r e p o s i t i v e i n v e s t m e n t c a n be o p t i m a l e v e n i n some s i t u a t i o n s w h e r e n o h a r v e s t i n g t a k e s p l a c e . I n s i t u a t i o n s w h e r e t h e o p t i m a l e s c a p e m e n t a n d o p t i m a l c a p a c i t y p o l i c y f u n c t i o n s i n t e r s e c t , t h e o p t i m a l e s c a p e m e n t s ( K ) m u s t be " c o n s t a n t , i . e . a v e r t i c a l l i n e , b e l o w t h e i n t e r s e c t i o n p o i n t , a s i n d i c a t e d i n f i g u r e 3. M a t h e m a t i c a l l y t h i s c a n b e s e e n f r o m t h e d e r i v a t i o n o f t h e v a l u e S" i n s e c t i o n C. I n t u i t i v e l y , t h e r a t i o n a l e f o r t h i s i s a s f o l l o w s . I f , f o r a g i v e n e s c a p e m e n t S, t h e c u r r e n t c a p a c i t y i s r e l a t i v e l y l o w , i n v e s t m e n t w i l l t a k e p l a c e u p t o t h e c a p a c i t y l e v e l h ( S ) , a l e v e l d e p e n d e n t s o l e l y on t h e e s c a p e m e n t . N e x t y e a r ' s r e c r u i t m e n t , F ( S ) , i s a l s o d e p e n d e n t o n t h e e s c a p e m e n t . H e n c e c u r r e n t c a p a c i t y i s i r r e l e v a n t t o t h e d e t e r m i n i a t i o n o f o p t i m a l e s c a p e m e n t , w h i c h i s t h e r e f o r e i n d e p e n d e n t o f K; t h a t i s , s ( K ) = c o n s t a n t . A p a r t f r o m t h e d i f f e r e n c e s i n a p p e a r a n c e a n d i n t e r p r e t a t i o n o f t h e o p t i m a l p o l i c i e s d e s c r i b e d a b o v e , t h e c a s e s o f i n s t a n t a n e o u s a n d d e l a y e d i n v e s t m e n t a r e c o m p a r a b l e . I n p a r t i c u l a r , t r a j e c t o r i e s a p p r o a c h a l o n g r u n e q u i l i b r i u m , d e r i v e d i n a m a n n e r s i m i l a r t o t h a t o f t h e i n s t a n t a n e o u s i n v e s t m e n t c a s e - s e e A p p e n d i x A. T h e s e l o n g r u n e q u i l i b r i u m p o i n t s a r e s h o w n i n f i g u r e s 3 a n d 4; c o m p a r i n g w i t h t h e c o r r e s p o n d i n g o p e n - a c c e s s r e s u l t s o b t a i n e d u s i n g e q u a t i o n ( 2 1 ) , we f i n d t h a t f o r t h e p r a w n f i s h e r y , t h e o p e n - a c c e s s c a p a c i t y , 16.8 s t a n d a r d i z e d v e s s e l s , i s r o u g h l y d o u b l e t h e o p t i m a l l e v e l . 47 In the whale fishery, however, equilibrium biomass i s very sensitive to the c a p i t a l stock. Hence the open-access and optimal capacities cannot d i f f e r by much in t h i s case; in fact they are almost i d e n t i c a l , at 2505 and 2250 catcher days /year respectively. The indication of these results i s that the extent by which open-access conditions leads to over-investment can vary considerably. Of course, the actual open-access investment behaviour may be quite complicated, so that our model only approximates the true s i t u a t i o n . For the remainder of this paper we s h a l l concentrate on the delayed investment model, turning now to results r e l a t i n g to the comparative dynamics for several economic and ecological parameters. (c) Fecundity and Carrying Capacity of the Stock Using the Beverton-Holt stock-recruitment function -mT -mT R = F(S) =ae S/(1+ae S/b) a measure of the maximum fecundity -mT ( i n t r i n s i c growth rate) is F'(0)= ae . As a increases, holding the maximum recruitment b constant, recruitment becomes less and less dependent on escapement. One would expect that the higher is the fecundity, the better off is the fishery and hence the higher i s the optimal capacity. This i s confirmed for the prawn fishery in figure 5, where optimal policy functions are shown for each of a= 3.5, 14, 42 and 560, with b= 7.0xl0 6 fixed. With -mT a= 3.5, we have ae = 0.95<1, so the stock size w i l l decline towards zero even without f i s h i n g . Not surprisingly, a zero investment l e v e l is optimal in this case and harvesting should 48 take place down to s(K)= x 0 i f possible. The optimal policy functions for the case a= 14 resemble those of the r e l a t i v e l y low-fecundity whale fishery, while a= 42 corresponds to our base case for the prawn fishery. As fecundity increases, the upward-shifting of the h(S) optimal capacity curve continues. The l i m i t i n g case where recruitment is independent of escapement is approximated here by setting a= 560; the optimal capacity curve i s f a i r l y f l a t , with h(S)-12, for a l l but the lowest escapements. Note that as fecundity decreases, the optimal escapement at low f l e e t capacity approaches x 0, i . e . s(0) tends to x 0. This confirms the reasoning that with a slow-growing stock and a low le v e l of capacity one has l i t t l e incentive to conserve the current stock: the future is not rosy in such a s i t u a t i o n . On the other hand, at high capacity levels the optimal escapement is determined from the Modified Golden Rule equation (see section C), which for the Beverton-Holt F(S) function described above can be written: -mT -mT (22) (b +ae S) 2(S-x 0) -oae b(b~x 0)S +ax 0b 2 = 0 where b= 7.0X10 6, x 0= 0.993X10 6 and c,M,T are as before. For each value of a , there is a unique positive root of (22). This optimal escapement s(K), for large K, depends on a in a rather complicated way, as indicated in figure 5. As in the low capacity case, s(K)- x 0 i f a is very small. As fecundity increases, the reproductive potential of the stock is improved, and higher escapements s(K) are desirable. When a becomes very large, recruitment becomes less dependent on escapement, so that 49 s ( K ) c a n b e l o w e r e d w i t h l i t t l e e f f e c t o n f u t u r e s t o c k s . T h e m a x i m u m r e c r u i t m e n t l e v e l , b , s e r v e s a s a s u i t a b l e i n d i c a t o r o f t h e c a r r y i n g c a p a c i t y o f a f i s h s t o c k w i t h B e v e r t o n - H o l t d y n a m i c s . A s s t a t e d e a r l i e r , o u r b a s e c a s e u s e d t h e v a l u e b= 7 . 0 x 1 U 6 , d e r i v e d f r o m t h e s a m p l e mean o f K i r k w o o d ' s ( 1 9 8 0 ) r e c r u i t m e n t d a t a . T h i s d a t a h a d b e e n s u b s t a n t i a l l y r e v i s e d a n d e x t e n d e d f r o m t h a t u s e d i n t h e a n a l y s i s o f C l a r k a n d K i r k w o o d : t h e i r o l d e r d a t a p r o d u c e s t h e v a l u e b= 1 1 . 3 x 1 0 6 . F i g u r e 6 i n d i c a t e s t h e e f f e c t o f t h e r e v i s i o n i n t h e v a l u e o f t h e c a r r y i n g c a p a c i t y . T h e f e c u n d i t y i s a = 4 2 . 0 f o r b o t h c a s e s . C l e a r l y t h e c h a n g e p r o d u c e s s u b s t a n t i a l m o v e m e n t i n b o t h o p t i m a l p o l i c y f u n c t i o n s . T h e o p t i m a l e q u i l i b r i u m c a p a c i t y d e c l i n e s f r o m 14. 5 t o 8.2 i f t h e new d a t a i s u s e d i n p l a c e o f t h e o l d . F u r t h e r r e s u l t s ( n o t s h o w n h e r e ) i n d i c a t e t h a t t h i s r e l a t i v e d e c r e a s e i n o p t i m a l c a p a c i t y h o l d s a l s o f o r h i g h e r v a l u e s o f f e c u n d i t y , e . g . a = 5 6 0 . I n s u m m a r y , t h e f e c u n d i t y a n d c a r r y i n g c a p a c i t y o f t h e f i s h s t o c k c a n h a v e s u b s t a n t i a l e f f e c t s o n t h e o p t i m a l p o l i c i e s , i n p a r t i c u l a r t h e o p t i m a l c a p a c i t y f u n c t i o n . T h i s i s e s p e c i a l l y o f i n t e r e s t i n s u c h c a s e s a s t h e b a n a n a p r a w n f i s h e r y , w h e r e l i t t l e i s k n o w n a b o u t t h e s t o c k - r e c r u i t m e n t r e l a t i o n s h i p . D e a l i n g w i t h p a r a m e t e r u n c e r t a i n t y i n t h e s e f i s h e r i e s b e c o m e s a n i m p o r t a n t p r o b l e m f o r f u r t h e r r e s e a r c h , ( d ) T h e D e p r e c i a t i o n R a t e T h e v a l u e o f t h e d e p r e c i a t i o n p a r a m e t e r r u s e d b y C l a r k a n d L a m b e r s o n ( 1 9 8 2 ) , n a m e l y r= 0 . 1 5 , was u t i l i z e d i n o u r b a s e c a s e r u n s f o r t h e w h a l i n g f i s h e r y . T h e same v a l u e w a s c h o s e n a s 50 reasonable for the prawn fishery as well. In a c a p i t a l investment model, i t i s c e r t a i n l y of interest to examine the effect on optimal p o l i c i e s of variations in the depreciation rate.- Figure 7 indicates the optimal s(K) and h(S) curves for the (a=42) prawn fishery with y= 0, 0.05, 0.15 and 0.20. The results for the fishery are i n t u i t i v e l y appealing. A decrease in the depreciation rate leads to an upward s h i f t in the investment curve h(S), r e f l e c t i n g the increased l i f e and hence the increased value of a new unit of capacity. On the other hand, an increase in r increases the desire to use capacity before i t depreciates, leading to a s h i f t in the s(K) curve to lower escapements. This l a t t e r s h i f t is less pronounced at high capacity values, where c a p i t a l is r e l a t i v e l y abundant in the "n-ear future" even for y = 0.20. With no depreciation (r=0), the capacity K can never decrease. Hence any point (S,K) which s a t i s f i e s -qTK S= Max{s(K),F(S)e } (see Appendix A) and which l i e s above the curve K= h(S) is an equilibrium point. These points form an equilibrium curve (shown in figure 7) upon which a l l t r a j e c t o r i e s w i l l converge. For the p a r t i c u l a r case shown in figure 7, the optimal capacity curve h ( S ) i s very f l a t for S>2.2X10 6, with h(S)-15.2. If i n i t i a l l y K<15.2, the prawn stock w i l l eventually reach a point ( S , K ) , with S>2.2X10 6, and investment w i l l occur u n t i l K-15.2. Thereafter an equilbrium point, close to that shown for K= 15.2, w i l l be approached. The corresponding results for the whale fishery are shown in figure 8, where the optimal policy functions for y- 0 and 51 r= 0.15 are given. The variation of the s(K) curves with y i s q u a l i t a t i v e l y similar to that of the prawn fishery. In the y- 0 case, the long run equilibrium w i l l again l i e on that part of -qTK the curve defined by S= Max {s(K), solution of S= F(S)e } which l i e s above the curve K= h(S). However the h(S) curve i s now s u f f i c i e n t l y steep that i f capacity is i n i t i a l l y low, a wide range of equilibrium points may be reached, depending on the i n i t i a l recruitment value. The equilibrium curve for y= 0 and equilibrium point for 7 = 0.15 are indicated in figure 8. The unusual aspect of the whaling fishery results is the intersection of the two h(S) investment curves and in particular the fact that for S> 12.2x10", the investment curve for 7 = 0.15 l i e s above the 7 = 0 curve. As described above, one would expect that i f a unit of investment is pr o f i t a b l e at any fixed point in time, given a r e l a t i v e l y high depreciation rate., then that same unit of capacity is even more desirable i f there is no depreciation. Indeed, this i s the case with the prawn fishery results above. To examine this phenomenon further, consider figure 9, depicting the effect of depreciation on a lower-fecundity (a=14) prawn fishery (with otherwise unchanged parameters). The change in fecundity has not altered the relationship between the 7 = 0 and 7 = 0.15 investment curves determined in figures 7. However, note that to this point, no change has been made in the other key investment parameter, unit c a p i t a l cost. Consider the re l a t i v e magnitudes of unit c a p i t a l costs, 6, and maximum yearly unit variable costs, c«T. For the prawn fishery the ra t i o 6/cT 52 =11.3 while 6/cT= 2.0 for the whale fishery. If we reduce the c a p i t a l cost in the prawn fishery so that 6/cT= 2.0, implying 6= $0.0832xl0 6, we obtain the optimal p o l i c i e s shown in figure 10, which are q u a l i t a t i v e l y similar to those of the whale fishery. An analysis of t r a j e c t o r i e s for the policy functions of figure 10 indicates that the r e l a t i v e heights of the r= 0 and r= 0.15 optimal capacity curves are determined not simply by the r a t i o 6/c'T but rather by a more complicated comparison of the present values of investment costs vs. rents for various investment strategies. The no-depreciation h(S) curve represents a balance between investment costs and the natural preference for a larger c a p i t a l stock to enable more rapid accumulation of rents as the stock i s harvested down to equilibrium. Depreciation introduces two new factors: (i) the need for future investment to overcome depreciation and ( i i ) the desire to "beat" depreciation by harvesting the stock before the f l e e t "wears out". It is this l a t t e r effect which appears responsible for • the r=0.15 h(S) curve lying above the corresponding zero depreciation curve at high escapement l e v e l s . However as depreciation increases beyond 15 percent, the optimal capacity curve drops as the yearly costs of overcoming depreciation predominate. When y=1.00, so that vessels l a s t for only one season, the h(S) curve l i e s completely below i t s no-depreciation counterpart. This rather complicated response to the depreciation rate seems to depend c r i t i c a l l y on actual parameter values, necessitating careful treatment of the data in s p e c i f i c applications. Nevertheless, consideration of the magnitudes of 5 3 the i n t r i n s i c growth rate, a, and the r a t i o of c a p i t a l to operating costs, 6/cT, provides an indication of the role that depreciation might play in a p a r t i c u l a r fishery, (e) Capital Cost, Fish Price and Discount Rate One would expect i n t u i t i v e l y that the attractiveness of developing a fishery i s enhanced by low unit c a p i t a l costs ( r e l a t i v e to operating costs), a high s e l l i n g price, and a low discount rate. These expectations are confirmed here, and we deduce the s e n s i t i v i t y of optimal investment and escapement levels to these parameters. The result of a decrease in unit c a p i t a l cost (with unit variable cost fixed) is shown in figure 11 for our base case prawn fishery, and by comparing figures 9 and 10 for an a= 14 (low fecundity) prawn fishery. In the l a t t e r case, a reduction to almost one-sixth the usual c a p i t a l cost, from $0.47X106 to $0.0832X106, produces a 3.5-fold increase in the equilibrium capacity (with y=0.15). In the former case, a halving of the c a p i t a l cost resulted in a 1.7-fold increase in equilibrium capacity. In both cases, optimal escapement at low capacities increased as c a p i t a l cost decreased, r e f l e c t i n g the increased benefit in saving more of the f i s h stock for the future, at which time capacity w i l l be higher. The variation of. the optimal policy functions with f i s h price is shown in figure 12 for our usual (a=42) prawn fishery. A doubling of the price, from i t s actual l e v e l of $0.9/kg to $1.8/kg, produced more than a doubling in equilibrium capacity, while a halving of the price made investment e n t i r e l y 54 uneconomic, so that depreciation w i l l slowly reduce the f l e e t size to zero. However, harvesting s t i l l takes place in this low-price case, whenever R>s(K), although the s(K) curve i t s e l f has shifted to the right r e l a t i v e to the base case. A harvesting domain w i l l exist as long as the price is high enough that x 0= c/(pq) <b, the maximum possible recruitment with our Beverton-Holt model. The optimal policy functions obtained for the (a=42) prawn fishery with discount factors (and corresponding discount rates) of a= 0.99 (1 percent), 0.90 (11 percent) and 0.8 (25 percent) are shown in figure 13. Naturally, the lower the rate of discounting, the higher is the benefit from investing in capacity for the future (to become available next year) and the higher is the desired escapement, s(K), to be l e f t at the end of the current season. It is interesting to note that while optimal escapements (for fixed K) increase with o , the equilibrium escapement decreases with a, r e f l e c t i n g the optimality of using the increased capacity which becomes available with low di scount i ng. (f) Ricker Stock-Recruitment While the concave, monotone properties of the Beverton-Holt stock-recruitment function make i t pleasant to study, i t is but one of several commonly-used reproduction functions. The Ricker form, R= F(S)= aS•exp{-(a/eb)S}, is used extensively in studying salmon stocks (Ricker, 1975), provides reasonable f i t s to data for gadoid stocks, including haddock and Arcto-Norwegian cod (Cushing and Harris, 1973) and has recently been applied to 5 5 northeast P a c i f i c herring populations (Walters, 1981). It has the property that recruitment attains a maximum value of R=b at S=eb/a, and thereafter declines roughly exponentially. Since K=h(S) is determined from the equation V (F(S),K)= 6 / c i t would K be expected that the optimal capacity w i l l follow the behaviour of F(S), i n i t i a l l y increasing to a maximum and thereafter decreasing. Numerical results, shown in figure 14, confirm this expectation. The parameters of the Ricker function used in thi s example, namely a= 11.639 and b= 7.0x106, were chosen so that the maximum recruitment is id e n t i c a l to that of the Beverton-Holt form for our base case prawn fishery, and occurs at S= 6.0x106 (to produce a reasonable agreement with the Beverton-Holt curve for low and medium escapements). Other model parameters are unchanged from those given in Table (I) for the prawn fishery, and delayed investment is assumed. Note that there is a threshold l e v e l of escapement, and hence of F(S), below which no investment occurs. At high escapements, F(S) decreases and eventually drops below i t s threshold value: thereafter we have h(S)=0. (The horizontal scale in figure 14 has been changed from that used in previous results in order to include this upper cutoff.) Comparing figure 14 with figure 3, i t can be seen that optimal capacity levels are approximately equal in the neighbourhood of S= 6.0X106, where the Ricker curve peaks. In figure 14, the h(S) curve increases rapidly for S>6.0xl0 6 and declines rather slowly to the right of the maximum value, 56 mimicking the behaviour of the Ricker curve i t s e l f . The optimal escapement curve s(K) behaves s i m i l a r l y to those of Beverton-Holt cases, except the optimal high-capacity escapement has substantially increased, r e f l e c t i n g reduced fecundity at low escapements for our pa r t i c u l a r Ricker curve. Further results (not shown here) indicate that as the Ricker curve s h i f t s to lower escapements (with i t s maximum value unchanged), the optimal policy functions s h i f t s i m i l a r l y . It becomes desirable to leave escapements which w i l l produce high recruitments next season, and to invest only i f the current escapement w i l l result in such high recruitments. In addition, optimal capacity levels increase, r e f l e c t i n g the importance of harvesting s u f f i c i e n t l y to reduce high recruitments down to r e l a t i v e l y low escapements, (g) The Value Function To this point we have dwelt on deriving and examining the optimal policy functions, s(K) and h(S), under various assumptions and parameter combinations. However the dynamic programming approach produces not only the optimal p o l i c i e s but also the optimal value function. Indeed for any p o l i c i e s s(K) and h(S), the corresponding value function is the solution.of the equation: V(R,K)= ir(R,K,S*(R,K) , Max {h (S* (R, K) ) -(1-y)K,0} +aV(F(S*(R,K)), Max{h(S*(R,K)), (l-y)K}) where S*(R,K) depends on s(K) through equation (13). A sample value function, corresponding to the optimal policy functions for our base case prawn fishery, is represented 57 i n T a b l e ( I I ) . A t l o w l e v e l s o f t h e c a p i t a l s t o c k , t h e v a l u e f u n c t i o n i s q u i t e i n s e n s i t i v e t o t h e l e v e l o f r e c r u i t m e n t , R. T h i s r e f l e c t s t h e f a c t t h a t , w i t h l o w f l e e t c p a c i t y , i n c r e a s e d r e c r u i t m e n t h a s l i t t l e e f f e c t o n r e n t s f o r t h e c u r r e n t s e a s o n , a n d s i n c e t h e p r a w n s t o c k i s f a s t - g r o w i n g , d i f f e r e n c e s i n t h i s y e a r ' s s t o c k s i z e t e n d t o s u b s t a n t i a l l y d i s a p p e a r b y n e x t y e a r . An e x a m i n a t i o n o f T a b l e ( I I ) s h o w s t h a t f o r B e v e r t o n - H o l t p o p u l a t i o n d y n a m i c s , we h a v e V >0, V >0 , V >0 a n d V <0, a s R K RK KK e x p e c t e d , t h r o u g h o u t t h e R-K r a n g e c o n s i d e r e d . T h e c u r v a t u r e o f t h e v a l u e f u n c t i o n i n t h e R - d i r e c t i o n , m e a s u r e d b y V , v a r i e d RR c o n s i d e r a b l y , f r o m n e g a t i v e t o p o s i t i v e v a l u e s . H o w e v e r V was n o t " t o o c o n v e x " i n R s i n c e t h e f u n c t i o n s ( K ) w a s i n e v e r y c a s e n o n - d e c r e a s i n g ( t o w i t h i n t h e a c c u r a c y o f t h e n u m e r i c a l s c h e m e ) . A s e x p e c t e d , t h e v a l u e f u n c t i o n b e h a v e d a p p r o x i m a t e l y l i n e a r l y i n R w h e n b o t h R a n d K w e r e l a r g e . O f p a r t i c u l a r i n t e r e s t i s t h e s e n s i t i v i t y o f t h e v a l u e f u n c t i o n t o t h e o p t i m a l p o l i c y f u n c t i o n s : t o w h a t e x t e n t c a n t h e p o l i c y c u r v e s b e a l t e r e d ( p e r h a p s t o t a k e i n t o a c c o u n t o t h e r o b j e c t i v e s ) w i t h o u t s e r i o u s l y r e d u c i n g r e n t s f r o m t h e f i s h e r y ? T h i s i m p o r t a n t q u e s t i o n w i l l b e c o n s i d e r e d i n some d e t a i l i n C h a p t e r I I I . F. S ummary a n d D i s c u s s i o n U s i n g a d y n a m i c p r o g r a m m i n g a p p r o a c h , h e u r i s t i c a n a l y s i s a n d n u m e r i c a l m e t h o d s , we h a v e s t u d i e d a d e t e r m i n i s t i c m o d e l o f o p t i m a l f i s h e r i e s i n v e s t m e n t . T h e m o d e l , w h i l e s i m i l a r t o t h a t 58 of Clark, Clarke and Munro, d i f f e r s in a number of respects: in p a r t i c u l a r , our model is based on a seasonal, discrete-time fishery and allows for delays in bringing new investment into service. In addition, the f l e x i b i l i t y of the dynamic programming method permitted consideration of the important Ricker stock-recruitment function as well as the Beverton-Holt form. However the key assumption of CCM regarding the i r r e v e r s i b i l i t y of capital' investment has been retained. Results in the form of optimal investment and escapement policy functions have been obtained numerically using policy i t e r a t i o n . These results were compared to those of Clark, Clarke and Munro, and the effects of a discrete-time analysis, delayed investment and variations in a number of model parameters were examined, using 1 data from the Gulf of Carpenteria prawn fishery and the aggregated whaling fishery. In the instantaneous investment case, our results were found to correspond closely to those of CCM, apart from natural differences between discrete- and continuous-time analysis. In pa r t i c u l a r their important conclusion regarding optimality of "a complex pattern of expansion, overcapacity, and gradual contraction via depreciation" towards an "optimal sustained y i e l d " equilibrium holds for our discrete-time model as well. The introduction of delays in investment added further realism to our model and changed the appearance of the optimal capacity function. However for the deterministic model of t h i s chapter, i t was shown that a simple transformation exists between the optimal policy functions in the cases of delayed and 59 instantaneous investment. Delays in bringing new investment on-line necessitate advance planning, so that i t can be optimal to invest even in years of low stock size, knowing that stocks w i l l have recovered when the new c a p i t a l becomes available. The effects of changes in the values of fecundity, maximum recruitment, f i s h price, c a p i t a l cost, discount rate and depreciation rate were investigated. In most cases the qu a l i t a t i v e effec'ts of these parameters were as expected. The optimal capacity function proved to be p a r t i c u l a r l y sensitive to the f i s h price and stock-recruitment parameters, indicating the potential importance of including parameter uncertainty in the analysis. The effect of variations in the depreciation rate was found to be rather complicated, depending both on the magnitude of unit c a p i t a l costs ( r e l a t i v e to operating costs and other economic factors) and on the actual values of the depreciation rate being considered. It was suggested that i f c a p i t a l i s r e l a t i v e l y inexpensive and i f the depreciation rate i s positive but not too large, the optimal capacity at high stock sizes can be greater than in the zero depreciation case. An explanation was put forward for this e f f e c t , which runs counter to the usual idea that depreciation, as a type of fis h i n g cost, should lead to lower investment. The use of a Ricker stock-recruitment form produced l i t t l e change in the structure of the optimal escapement curve. However, the optimal capacity function adopted an appearance mimicking that of the stock-recruitment curve i t s e l f : increasing 60 rapidly at low escapements and declining r e l a t i v e l y slowly at higher escapements. Clearly t h i s effect i s important for investment strategies in those f i s h e r i e s where over compensation ex i s t s . 61 Chapter I I I .  Optimal Fisheries Investment Under Uncertainty It has long been observed that f i s h e r i e s , perhaps as much as any natural resource, exhibit remarkably high levels of uncertainty, a r i s i n g from economic and b i o l o g i c a l factors, and a f f e c t i n g not only the participants in the fishery but the resource managers as well. Hence i t has seemed obvious that attempts should be made to incorporate uncertainty into fishery management models. Walters & Hilborn (1978) distinguish three general classes of uncertainty in f i s h e r i e s management: 1. Random effects whose future frequency of occurrence can be determined from past experience; 2. Parameter uncertainty that can be reduced by research and acqu i s i t i o n of information through future experience; 3. Ignorance about the appropriate variables to consider and the appropriate form of the model. Most studies of the optimal management of f i s h e r i e s under uncertainty have concentrated on the f i r s t type of uncertainty. Typically the corresponding deterministic dynamics of the fishery are transformed to a stochastic analogue, and the resulting stochastic optimization problem i s analyzed using dynamic programming. In a r e a l i s t i c multi-species, multi-cohort multi-parameter fishery, stochastic effects can enter in a number of ways. Typ i c a l l y , the source of uncertainty is taken to be 62 environmental fluctuations a f f e c t i n g the population dynamics of the f i s h stocks, although Lewis (1975) and Andersen (1982) allow for price v a r i a b i l i t y as well. Dudley & Waugh (1980) study a simulation-optimization model in which yearly recruitment, mortality rate and c a t c h a b i l i t y fluctuate simultaneously. While fluctuations are usually taken to be independent from one point in time to another, they may in fact form a Markov process (Spulber,1981). Each species or each age-cohort may respond to randomness d i f f e r e n t l y (Mendelssohn,1978,1980a; Spulber, 1978). Nonconvexities and risk aversion can affect substantially the role of uncertainty (Lewis,1981). Andersen (1981) reviews research on the behaviour of competitive firms under uncertainty, and applies these results to optimal management of fi s h i n g firms faced with stock and price uncertainties. Pindyck (1982) examines the interaction of ecological uncertainty, demand e l a s t i c i t y and the biomass growth function in the context of renewable resource markets, extending the analysis of the exhaustible resource case in Pindyck (1980). Methods of solving such stochastic optimization problems have also varied considerably. Jaquette (1972,1974) and Reed (1974,1978,1979) used analytic approaches to study the effects of uncertainty on discrete-time dynamic programming models of f i s h e r i e s . Walters (1975) and Walters and Hilborn (1976), using a dynamic programming approach with Kalman f i l t e r techniques, examined both stochastic effects and problems of parameter uncertainty. Beddington and May (1977) studied the effect of uncertainty on Maximum Sustainable Yi e l d p o l i c i e s using 63 c h a r a c t e r i s t i c return times and the c o e f f i c i e n t of variation in f i s h e r i e s y i e l d s . Ludwig (1979a) formulated a continuous-time stochastic control model to which he applied perturbation techniques, while Ludwig & Varah (1979) used numerical methods to study the same model. Smith (1978) looked at continuous time models where the optimal policy was not a bang-bang control. Mendelssohn & Sobel (1980) placed the f i s h e r i e s problem in the context of c a p i t a l accumulation and used discrete-time dynamic programming techniques to obtain theoretical results for a f a i r l y general f i s h e r i e s model. Both May e t . a l . (1978) and Spulber (1978) have emphasized the role of steady-state p r o b a b i l i t y d i s t r i b u t i o n s for optimally harvested f i s h stocks. Aron (1979) formulated and analysed a compromise harvesting policy which performed well according to a number of indicators and which in addition was robust to lack of knowledge of yearly biomass l e v e l s . Two useful surveys of the l i t e r a t u r e on f i s h e r i e s under uncertainty are Andersen & Sutinen (1981) and Spulber (1982). There is also a growing body of work in the areas of behavioural modelling with uncertainty (see, for example, Bochstael & Opaluch, 1981) ; and predictive modelling (Eswaran and Wilen, 1977). In t h i s chapter we examine the role of stochastic biomass fluctuations in determining optimal f i s h e r i e s investment strategies. Section A describes the stochastic model, which extends the deterministic model of Chapter II by allowing the resource stock to fluctuate randomly from year to year. As in Chapter II, sections B,C,D and E contain, respectively; analytic 64 results, h e u r i s t i c analysis, numerical method and numerical r e s u l t s . The chapter ends with a summary and discussion in section F. Again, for the reader wishing to move quickly to the main results of the chapter, sections B and D can be bypassed without loss of continuity. A. The Model The stochastic model presented here is similar to that described in the previous chapter, with the exception that the biomass is assumed to follow a stochastic stock-recruitment relationship. S p e c i f i c a l l y the recruitment in year n, R is n governed by a lognormal probability d i s t r i b u t i o n with mean F(S ), .where S i s the previous year's escapement and F(-) n - 1 n - 1 is the corresponding deterministic stock-recruitment function. Use of the lognormal d i s t r i b u t i o n is motivated by two factors. F i r s t , i t i s the natural d i s t r i b u t i o n to r e f l e c t the large number of independent m u l t i p l i c a t i v e effects facing the growth of f i s h from the egg to the adult stage. Second, the lognormal d i s t r i b u t i o n reproduces q u a l i t a t i v e features of f i s h e r i e s data, where one sees a large number of low-to-medium recruitments and occasional very large recruitments. If the biomass has been aggregated over a number of f i s h stocks, one is faced with the p o s s i b i l i t y that environmental fluctuations w i l l affect the various stocks to d i f f e r e n t extents ( p a r t i c u l a r l y i f stocks are geographically separated). We s h a l l assume here that the randomness is perfectly correlated across 65 stocks, or a l t e r n a t i v e l y that some average noise l e v e l has been determined. If in fact the stocks behaved independently, we would expect that the ove r a l l effect of environmental fluctuations on the aggregated biomass would tend to be r e l a t i v e l y small. The intraseasonal dynamics of the f i s h stock and the yearly dynamics of the c a p i t a l stock are as in the deterministic case. In p a r t i c u l a r , the biomass x in year n i s governed by the d i f f e r e n t i a l equation dx/dt= -h(t)=-qE(t)x(t) with x(0)=R and n the instantaneous fi s h i n g e f f o r t E(t) subject to 0<E(t)<K . [For n additional realism, the c a t c h a b i l i t y c o e f f i c i e n t q might be treated as stochastic (Dudley & Waugh, 1980), r e f l e c t i n g random search for f i s h during the season. However, this additional complication i s unlikely to be as important to investment planning as year-to-year fluctuations in the resource, and w i l l not be pursued further here.] Hence the optimal escapement S* must be chosen subject. to n R exp{-qTK }<S*^R , where T is the maximum season length. Given n n n n S*, the optimal addition to fl e e t capacity K desired to become n available at the beginning of the next season (namely I ) is n+1 determined before the end of the current season. Again the resale value of the vessels is assumed to be zero, so the ca p i t a l stock can be reduced only through depreciation, which occurs at the end of each season. Note that the inherent delay 6 6 i n b r i n g i n g new i n v e s t m e n t o n - l i n e , w h i l e o f l i t t l e c o n s e q u e n c e i n t h e d e t e r m i n i s t i c c a s e , c o n t r i b u t e s t o t h e f i s h e r y m a n a g e r ' s u n c e r t a i n t y i n a s t o c h a s t i c e n v i r o n m e n t . A m a j o r a s s u m p t i o n i n t h i s m o d e l , a s i n p r a c t i c a l l y a l l o t h e r f i s h e r y o p t i m i z a t i o n m o d e l s , i s t h e o b s e r v a b i l i t y o f t h e y e a r l y r e c r u i t m e n t . I n t h e r e a l w o r l d , t h e r e i s o f c o u r s e n o way o f k n o w i n g how many f i s h a r e a v a i l a b l e a t t h e b e g i n n i n g o f a f i s h i n g s e a s o n , u n l e s s some t e s t f i s h i n g h a s t a k e n p l a c e . T y p i c a l l y a r o u g h g u e s s i s made o f t h e a v a i l a b l e s t o c k , o p e n i n g s a r e made a c c o r d i n g l y , a n d t h e s t o c k i s r e - e s t i m a t e d p e r i o d i c a l l y o n t h e b a s i s o f new i n f o r m a t i o n . S t u d i e s h a v e b e e n made c o n c e r n i n g t h e r o l e o f u n c e r t a i n t y i n m e a s u r e m e n t s o f r e c r u i t m e n t a n d e s c a p e m e n t [ L u d w i g a n d W a l t e r s ( 1 9 8 1 ) , W a l t e r s a n d L u d w i g ( 1 9 8 1 ) ] a n d c o n c e r n i n g t h e e f f e c t s o f a d a p t i v e i n - s e a s o n m a n a g e m e n t [ M a n g e l & C l a r k ( 1 9 8 2 ) ] . W h i l e s u c h c o n s i d e r a t i o n s a r e c e r t a i n l y o f i m p o r t a n c e , i t i s d i f f i c u l t t o i n c l u d e t h e m i n a n i n t e r s e a s o n a l m o d e l o f o p t i m a l i n v e s t m e n t a n d e s c a p e m e n t d e c i s i o n - m a k i n g . We s h a l l h e n c e f o r t h make t h e s i m p l i f y i n g a s s u m p t i o n t h a t , b y some m e a n s s u c h a s a t e s t f i s h e r y , e a c h s e a s o n ' s r e c r u i t m e n t i s k n o w n p r e c i s e l y . I n C h a p t e r I I , we a d o p t e d a s t h e s o c i a l o b j e c t i v e m a x i m i z a t i o n o f t h e p r e s e n t v a l u e o f t h e f i s h e r y . H o w e v e r , i n f o r m u l a t i n g t h e o b j e c t i v e o f t h e f i s h e r y m a n a g e r o n e s h o u l d c o n s i d e r t h e a t t i t u d e o f s o c i e t y t o w a r d s r i s k . M e n d e l s s o h n ( 1 9 8 0 b ) h a s s t u d i e d t h e P a r e t o o p t i m a l t r a d e o f f b e t w e e n e c o n o m i c r e t u r n s a n d v a r i a n c e i n t h e s e r e t u r n s , w h i l e L e w i s ( 1 9 7 7 ) h a s p o i n t e d o u t t h e m e r i t s o f i n c o r p o r a t i n g r i s k p r e f e r e n c e o r r i s k 67 aversion in a u t i l i t y function of yearly rents accruing to the fishery, risk aversion being r e f l e c t e d in a concave u t i l i t y function. [In fact such a u t i l i t y function might be expected to depend also on past rents (Ryder & Heal, 1973), as well as past and present catches.] We s h a l l adopt the u t i l i t y function approach here, with U=U(ir), but assume that society is risk neutral with respect to investment costs. Hence the u t i l i t y function applies to yearly rents net of investment expenditures, a quantity which serves as a proxy for the current health of the fishery. This r e a l i s t i c a l l y r e f l e c t s society's aversion to low stock levels and to low incomes for fishermen, rather than to low yearly rents per se. Naturally, u t i l i t y must be measured in the same units as investment costs, since yearly benefits are now given by: B(R,K,S,I )= U( ir(R,S) ) -61 using the same notation as in Chapter II, except with ir(R,S)= p(R-S)-(c/q)log(R/S) , where we have assumed linear costs, c(E)=cE, and no alternative fishery, in order to make the description more transparent. Note that the intraseasonal fishery is assumed to be deterministic, and hence is unaffected by year-to-year biomass fluctuations (given the i n i t i a l recruitment). The stochastic control problem can now be stated as follows: n-1 Max [ l a E { B ( R , K , S , I ) } ] {S,;I 2;S 2;...} n>1 n n n n+1 subject to dynamics R ~<t> (•) and K =(1-y)K +1 , n+1 F ( S ( n ) ) ; t f n+1 n n+1 together with control constraints R •exp{-qTK}<S<R and I ^0. n n n+1 6 8 H e r e <t>_ ( R ) = ( i / * R ) - 1 e x p { - ( l o g R - l o g R +a2/2)2/2e2} i s a R;<y l o g n o r m a l d e n s i t y w i t h mean R a n d u n c e r t a i n t y p a r a m e t e r <s [ s o t h a t t h e v a r i a n c e i s R 2 ( e - 1 ) ] , a i s t h e a n n u a l d i s c o u n t f a c t o r , a n d i t i s u n d e r s t o o d t h a t S a n d I a r e t o b e c h o s e n n n+1 i n y e a r n , g i v e n t h e s t a t e ( R , K ) i n t h a t y e a r . ( T h r o u g h o u t t h e t h e s i s , we d e f i n e v t o be t h e s q u a r e r o o t o f 2 i r . ) T h e d y n a m i c p r o g r a m m i n g e q u a t i o n c o r r e s p o n d i n g t o t h i s p r o b l e m i s t h e f o l l o w i n g : ( 1 ) V ( R , K ) = Max Max [ B ( R , K , S , I ) + a E { V ( R ' , ( 1 - y ) K + I ) } ] R - e x p { - q T K } < S < R I > 0 w h e r e ( R , K ) i s t h e ' s t a t e ' t h i s y e a r , ( S , l ) a r e t h e c o n t r o l s ( d e c i s i o n v a r i a b l e s ) , a n d R' i s n e x t y e a r ' s r e c r u i t m e n t ( l o g n o r m a l l y d i s t r i b u t e d a s a b o v e ) . C l e a r l y t h i s e q u a t i o n h a s a f o r m v e r y s i m i l a r t o e q u a t i o n 11 ( 1 ) . A s i n C h a p t e r I I , we now p r o c e e d t o c o n s i d e r a n a l y t i c , h e u r i s t i c a n d n u m e r i c a l t r e a t m e n t s o f e q u a t i o n ( 1 ) , i n t h e f o l l o w i n g t h r e e s e c t i o n s . 69 B. A n a l y t i c R e s u l t s . B e f o r e e x a m i n i n g t h e f u l l m o d e l we s h a l l f o l l o w t h e p r o c e d u r e o f C h a p t e r I I a n d f i r s t l o o k a t some s i m p l i f i c a t i o n s o f t h e m o d e l w h i c h p e r m i t a m o r e a n a l y t i c t r e a t m e n t . I n p a r t i c u l a r , r e c r u i t m e n t w i l l b e c o n s i d e r e d h e r e t o f o l l o w a s t a t i o n a r y t i m e - i n d e p e n d e n t p r o b a b i l i t y d i s t r i b u t i o n ; t h i s i s e q u i v a l e n t t o a s s e r t i n g t h a t r e c r u i t m e n t i s i n d e p e n d e n t o f p r e v i o u s e s c a p e m e n t l e v e l s , a n a s s u m p t i o n w h i c h was u s e d f o r t h e d e t e r m i n i s t i c c a s e i n I I . B . A s b e f o r e , we s h a l l c o m p e n s a t e f o r t h i s e x t r e m e a s s u m p t i o n b y a d d i n g a " b e n e f i t s o f e s c a p e m e n t " f u n c t i o n b ( S ) t o t h e y e a r l y r e n t s ; t h i s p r o v i d e s a n i n c e n t i v e t o a v o i d o v e r l y l o w b i o m a s s l e v e l s . A s a f u r t h e r s i m p l i f i c a t i o n , we a s s u m e r i s k n e u t r a l i t y , U ( j r ) = i r , b u t i n t h i s s e c t i o n we i n c l u d e n o n l i n e a r c o s t s a n d a n a l t e r n a t i v e f i s h e r y . H e n c e t h e t w o - s t a g e o p t i m i z a t i o n p r o b l e m c a n be s t a t e d a s f o l l o w s [ e q u a t i o n ( 2 ) ] : Max E{ Max [ p ( R - S ) - T C ( E ) + b ( S ) - ( p / q ) l o g ( R / S ) - ( K - p T ) K ] } K>0 R R - e x p { - q T K } < S < R w h e r e t h e e x p e c t a t i o n i s w i t h r e s p e c t t o a n a r b i t r a r y d e n s i t y f u n c t i o n #(R) ( n o t n e c e s s a r i l y l o g n o r m a l ) . F o r g i v e n R we h a v e , a s b e f o r e , x ( 0 ) = R a n d h = - q E x , w i t h o t h e r n o t a t i o n a s i n I I . B ( i n p a r t i c u l a r , K = [ ( 1 - O ) / C + r ] 6 ) . T h e i n n e r o p t i m i z a t i o n ( w h i c h i s s t r i c t l y d e t e r m i n i s t i c d u e t o o u r a s s u m p t i o n o f p e r f e c t k n o w l e d g e o f t h e s t o c k s i z e a t t h e b e g i n n i n g o f t h e s e a s o n ) was s o l v e d i n C h a p t e r I I , w h e r e we o b t a i n e d t h e o p t i m a l e s c a p e m e n t a n d t h e y e a r l y r e n t s f u n c t i o n i n t e r m s o f - R a n d K: (a) K>E0 Recrui tment . S* Yearly Rents n R<S0 R f 1 = b(R) S0<R<R* So ir 2 = p(R-S 0)-q- 1log(R/S 0)c(E 0)/E 0 -(/>/q)log(R/S 0)+b(S 0) R*<R<R+(K) S(R) * 3 = p(R-s(R))-Tc([l/qT]log[R/s(R)] -( P/q)log[R/s(R)]+b(s(R)) R>R+(K) -qTK Re f « -qTK -qTK = pR(l-e )-Tc (K)-/>TK+b(Re (b) K<E0 Rec rui tment s* Yearly Rents ir R<e(K) R = b(R) e(K)<R<R**(K) e(K) f 6 = p(R-e(K))-q- 1log[R/e(K)]c(K)/K -( P/q)log[R/e(K)]+b(e(K)) R>R**(K) -qTK Re TT7 -qTK -qTK = pR(l-e )-Tc (K)-/)TK+b(Re where as before; -qTK -qTK (i) R+ is the solution of [p-b'(R +e )]qR +e = c'(K)+ ( i i ) S 0 is the solution of [ p-b' (S 0 ) ]qS 0 = c'(E0)+/> qTE 0 ( i i i ) R*= S 0e (iv) s(R) i s the solution of [p-b'(S)]qS=c'([qT]" 1log(R/S)) (v) e(K) solves [p-b'(S)]qS= c(K)/K +p for K<E0, (vi) R**(K)=e(K)exp{qTK} and ( v i i ) E 0 is the solution of c'(E 0)=c(E 0)/E 0. In addition, note that S 0 and R* are independent of K. Then the optimization problem reduces to: 71 (3) Max [ E{ ir ( R , K) } - ( K - / » T ) K ] K>0 w h e r e E { i r ( R , K ) } i s g i v e n i n t h e c a s e K > E 0 b y : (4) K>E S 0 R* E { i r ( R , K ) } = | T r , * ( R ) d R + | i r 2 * ( R ) d R 0 S 0 R + + | 7 r 3 # ( R ) d R +| r r « 0 ( R ) d R R* R>R + a n d i n t h e c a s e K < E 0 b y t h e f o l l o w i n g : e ( K ) R** (5 ) K < E 0 ; E { i r ( R , K ) } = | i r _ s * ( R ) d R + | i r 6 * ( R ) d R + | rr 7 # ( R ) dR 0 e ( K ) R>R** Now i f K > E 0 we h a v e : B u t b y d e f i n i t i o n o f R+, n3=n^ a t R=R+,E=R. F u r t h e r m o r e , b y i n s p e c t i o n , ir 3 i s i n d e p e n d e n t o f K, s o t h a t d j r 3 / d K = 0 . H e n c e t h e o r i g i n a l p r o b l e m i s s o l v e d e i t h e r b y a n i n t e r i o r m a x i m u m i n t h e i n t e r v a l [ E 0 , i n f i n i t y ) o r b y some v a l u e 0 < K * < E o ; i n o t h e r w o r d s , K* i s e i t h e r < E 0 o r i s g i v e n b y t h e s o l u t i o n o f : ( 6 ) | (diu/dK) *(R)dR=ic-/>T T h e q u a n t i t y d e f i n e d b y dP={ dir„ i f R>R + , 0 i f R<R + .} r e p r e s e n t s t h e i n f i n i t e s i m a l g r o s s y e a r l y b e n e f i t s t o a n i n c r e a s e i n f l e e t c a p a c i t y o f d K . T h e q u a n t i t y icdK i s t h e e f f e c t i v e y e a r l y p a y m e n t f o r t h e a d d i t i o n a l c a p i t a l d K , w h i l e />TdK i s t h a t p a r t o f t h e c a p i t a l c o s t icdK w h i c h c o u l d b e o f f s e t b y r e n t s f r o m t h e a l t e r n a t i v e f i s h e r y . T h u s e q u a t i o n ( 1 ) c a n be d E { n } / d K = ir 3 ( R + , K ) * ( R + ) ( d R + / d K ) + | (dtr 3 / d K ) <t> ( R ) dR So - T T 4 ( R + , K ) 0 ( R + ) ( d R + / d K ) + ( d i r n / d K ) * ( R ) d R R>R + R>R + 72 written: E{ marginal benefit } = marginal (capital) cost This result i s as expected, being the usual economic optimality condition. Equation (6) can be expressed as follows: -qTK -qTK (7) | {[p-b' (Re )]qRe -c' (K)-/>} • *(R)dR= K/T -p R>R + -qTK -qTK Note that dir„/dK= {[p-b'(Re ) ]qRe -c'(K)-/>}-T >0 for R>R+. Furthermore dR+/dK >0 and for fixed R, d(dn«/dK)/dK <0. Hence the l e f t hand side of (7) i s a decreasing function of K, so that (7) has at most one solution K*. If we set K=E0 in the l e f t hand side of equation (7) and find that: -qTE 0 -qTE 0 | {[p-b'(Re )]qRe -c'(E 0)}•*(R)dR< K / T -p R>R + then the optimal capacity K* must be less than E 0 . In that case we wish to solve dE {tr} / dK = K - pT for K<E0 . Note f i r s t that t r 5 = i r 6 at R=e (K) , i r 6 = i r 7 at R=R**(K) and e(K) was constructed such that the p a r t i a l derivative of ir 6 with respect to e(K) i s 0. Hence we find that for K<E0: R** (8) ( l/T)dE{ir}/dK= ( 1 / qTK) [ c (K) /K-c ' (K) •] | log [ R/e (K) ] * (R) dR e(K) -qTK -qTK + | {[p-b'(Re )]qRe -c '(K )-/>}•*( R) dR = tc/T -p R>R** If we know that K*<E0, but (8) has no non-negative solution, then we conclude that the optimal capacity is K*=0; the fishery is not worth developing. A simpler form of the solution can be obtained by 73 neglecting benefits of escapement, so b( •)=(), and assuming linear variable costs, c(E)=cE. In thi s case Eo=0 so that we do not have to be concerned about the regions K<E0 and qTE 0 S 0^R^S 0e , and the solution becomes: -qTK (9) | [pqRe -c]*(R)dR= K / T R>R + The term R+ i s as before but is now given by the simpler form qTK R +=x 0e , where x 0=c/(pq). Equation (9) can be rewritten: (10) ( R j " 1 ! (R-R +)*(R)dR = K / C T = e R>R + or equivalently: (11) ( R + ) - 1 | R#(R)dR +*(R+)= 1+K/CT = 1+0 R>R + Here *(•) is the cumulative d i s t r i b u t i o n function for *(R). Note that 0= K / C T = [(1-C)/C +r](6/cT)= [r+y](6/cT) where r i s the discount rate. Hence 9 i s the r a t i o of yearly fixed costs to maximum yearly operating costs. In this simple case we can note that: (i) The depreciation rate and the discount rate play i d e n t i c a l roles in determining the optimal capacity, ( i i ) The r a t i o of unit c a p i t a l cost to maximum yearly variable costs (cT) is the c r i t i c a l cost parameter, and ( i i i ) D i f f e r e n t i a t i n g (11) with respect to 9=[y+r](6/cT), using qTK* R +=x 0E , and simplifying, we obtain: dK*/de=-l/[qT(e+1-*(R +))] <0 so that, as expected, the optimal capacity decreases with the 74 depreciation rate, the discount rate and the r e l a t i v e cost of c a p i t a l . The more interesting question i s the effect of uncertainty on the optimal capacity. D i f f e r e n t i a t i n g (10) with respect to <s, we obtain: - ( R + ) - 2 | [R-R ++R +]*(R)dR(dR +/dtf) +(R +)" 1 | (R-R+)# (R)dR = 0 R>R+ R>R+ <J where <t> is the p a r t i a l derivative of <t> with respect to e. e Simplifying and using (10) as well as the relationship dR+/dK*=qTR+, we have: dK*/dtf= {qTR +[1+G-*(R +)]" 1} | ( R - R j * (R)dR R>R+ a Since 1+0 £1, we can conclude that the optimal capacity increases with uncertainty whenever | (R-R+)# (R)dR >0. R>R+ a Examples (1) As a f i r s t example of the use of equation (11), consider a f i s h stock following a uniform probability d i s t r i b u t i o n on [0,R], so that i t s mean value is R/2 and i t s variance i s R2/12. (This i s not claimed to be a r e a l i s t i c s i tuation but w i l l allow us to obtain an algebraic solution.) Then we have the following: *(R+)={ R+/R i f R+<R ; 1 i f R+>R } and (R + ) " 1 | R#(R)dR= { (R 2-R + 2)/(2R +R) i f R+<R ; 0 i f R+>R } R>R + If 9>0, substitution of these quantities into equation (11) shows that we must have R+^R, and R+ must s a t i s f y : (R 2-R + 2)/(2RR +) + R + /R = 1+G This i s a quadratic equation for R+ which can be easily solved 75 t o p r o d u c e : R + = [ ( 1 + 0 ) - s q r t { ( 1 + 9 ) 2 - 1 } ]R T h e n , f r o m t h e d e f i n i t i o n o f R + , t h e o p t i m a l c a p a c i t y i s g i v e n b y K * = ( q T ) " 1 l o g ( R + / x 0 ) . I n C h a p t e r I I we s h o w e d t h a t t h e o p t i m a l c a p a c i t y i n a d e t e r m i n i s t i c e n v i r o n m e n t w i t h c o n s t a n t y e a r l y r e c r u i t m e n t R/2 i s g i v e n b y K * ( d e t ) = ( q T ) - 1 l o g [ ( R / 2 ) / { x 0 ( 1 + e ) } ] . C o m p a r i s o n o f t h e e x p r e s s i o n s f o r K* a n d K * ( d e t ) s h o w s t h a t : K * = K * ( d e t ) + ( q T ) - 2 l o g ( X ) w h e r e X = 2 [ 1 + G ] / ( [ 1 + 9 ] + s q r t ( [ 1 + e ] 2 - 1 ) ) >1 s i n c e [ 1 + 6 ] 2 - 1 < [ 1 + 0 ] 2 . T h e r e f o r e i n t h i s e x a m p l e we h a v e K * > K * ( d e t ) i n a l l c a s e s ( a s s u m i n g © > 0 ) . I n p a r t i c u l a r , i t c a n b e s h o w n t h a t f o r e v e r y s m a l l ( r e l a t i v e l y c h e a p c a p i t a l o r l o w d i s c o u n t a n d d e p r e c i a t i o n r a t e s ) , we h a v e t h e r e s u l t : ( K * - K * ( d e t ) ) / K * ( d e t ) ~ l o g 2 / l o g { R / 2 x 0 } s o t h a t t h e r e l a t i v e i n c r e a s e i n t h e o p t i m a l c a p a c i t y f r o m t h e d e t e r m i n i s t i c t o t h e s t o c h a s t i c c a s e c a n be s u b s t a n t i a l i f t h e mean r e c r u i t m e n t i s n o t t o o h i g h r e l a t i v e t o x 0 . ( 2 ) N e x t , l e t u s c o n s i d e r t h e m o r e r e a l i s t i c c a s e o f a l o g n o r m a l p r o b a b i l i t y d e n s i t y w i t h mean E { R } = R a n d s t a n d a r d u n c e r t a i n t y p a r a m e t e r a. I n t h i s c a s e we h a v e : - ( x - b ) 2 / 2 t f 2 x | R * ( R ) d R = | (ve)" 1 e e d x R>R + x > l o g ( R + ) - z 2 / 2 =R | i / " 1 e d z w h e r e z + = l o g ( R + - b - t f 2 )/e z > z + = R [ 1 - * ( l o g ( R + /R) A - < r / 2 ) ] N = R [ 1 - * ( - x / c r - f f / 2 ) ] N 76 -<r2/2 where b=log(Re ), x= log(R/R +)= log(R/x 0)-qTK*, * (z)=Pr{N(0,1)<z} ( i . e . the cumulative d i s t r i b u t i o n function N for a standard normal random variable) and G i s as before. x S i m i l a r l y , *(R +)= * (-x/tf+tf/2) and (R +)' 1=e /R. Hence (11) N s i m p l i f i e s to: x (12) e [1-* (-x/<y-ff/2) ]+* (-x/«y + tf/2) = 1+e N N Now | (R-R+)*(R)dR= | R*(R)dR - R+ | *(R)dR R>R+ R>R+ R>R+ =R[ 1 (-x/<r-tf/2)] - R +[1~* ( - x A + tf/2)] N N The p a r t i a l derivative of this quantity with respect to e is then as follows: (13) - R ( X / * 2 - 1 / 2 ) * ( - x A-tf/2) + R + ( x / t f 2 + l / 2 ) « (-x/«+#/2) N N x - ( x 2 A 2 + x + tf2/4)/2 = (R + /i/){ -e (x/ t f 2-1/2)e -(x 2/ t f 2-x+ t f 2/4)/2 + (x/tr 2"+l/2)e } - ( x 2 A 2 - x + * 2/4)/2 = (R + /i/)e [x/ t f 2-x/ t f 2 + (l/2) + d/2)] = R+* (-x/a+a/2) >D N Hence we can conclude that in the sim p l i f i e d model of this section, the optimal capacity must increase with uncertainty i f recruitment follows a stationary lognormal d i s t r i b u t i o n . As we sh a l l see, no such simple statement can be made in the general stock-recruitment s i t u a t i o n . The following section returns to this general model, deducing q u a l i t a t i v e behaviour of optimal 77 investment p o l i c i e s under uncertainty. C. Heuristic Analysis Our analysis in th i s section proceeds in a manner analogous to that of section II.C . We begin by performing the inner maximization in equation (1), for fixed S , to obtain the optimality equation for investment: (14) E{ V (R,(l-y)K+l) }= 6/a or 1=0 i f E{V (R,(1 -7)K)}<6/O K K where the expectation over R i s with respect to the lognormal density <t> (•). This states that, unless the f l e e t is F(S),c temporarily overcapitalized, next year's optimal capacity, (l-y)K+I*, should be set such that the expected marginal benefit of an extra unit of c a p i t a l equals i t s marginal cost. Now define K = h ( S ) to be the solution of the equation E{V (R',K)}=(1/a)6, so that h ( S ) represents next season's K optimal capacity. We observe that t h i s equation can be written Uz-<r 2/2) E{V ( F ( S ) e ,K)}= 6/a and note the s i m i l a r i t y to equation z K (6) in Chapter I I . D i f f e r e n t i a t i n g with respect to S, we obtain: (15) h'(S)= [F'(S)/F(S)]-EV (R',h(S))/-EV (R',h(S)) >0 RK KK given the reasonable assumptions (for the Beverton-Holt model) that: (i) there are decreasing marginal benefits to increasing the c a p i t a l stock [V <0] KK 78 ( i i ) the marginal benefit of increasing the c a p i t a l stock increases with the current resource stock size [ V >0] , and RK ( i i i ) F ( 0 is a positive increasing function. Hence the optimal capacity for next season i s an increasing function of the current escapement, as in the deterministic case. If c a p i t a l remaining from the current season exceeds the desired capacity for the following year, that i s (l-r)K>h(S), then the optimal investment i s I*=0 (capit a l is already s u f f i c i e n t l y abundant), while otherwise I* is chosen so that (1-r)K+I*=h(S). This i s precisely equation (7) of Chapter II.C, which we repeat here: (16) I*(S,K)=max{h(S)-(1-r)K,0} Now performing the outer maximization we obtain the optimality expression equating the expected marginal benefit with the marginal cost of an incremental increase in escapement: (17) ... o [ F ' (S)/F(S) ] E { R ' -V (R' , 0-y)K+I*(S,K) )}= p ( 1 ~x 0/S ) U' (ir (R, S ) ) R where x 0=c/(pq) represents bionomic equilibrium and the (-qTK) constraint Re <S<R has been neglected temporarily. Note that equation (17) involves R,S and K (unless U' is i d e n t i c a l l y a constant). Hence the introduction of a n o n t r i v i a l u t i l i t y function implies that i t is no longer possible to deduce a target escapement curve S=s(K). Instead we w i l l have S=s(R,K). While this more complicated case could s t i l l be analyzed numerically, we s h a l l opt in this study for the descriptive s i m p l i c i t y a r i s i n g from the existence of the target curve s(K). 79 T h e r e f o r e , f o r t h e r e m a i n d e r o f t h i s c h a p t e r , l e t U ( i r ) = r r f o r a l l j r . T h e n (17) t a k e s a f o r m s i m i l a r t o t h a t o f e q u a t i o n 1 1 ( 8 ) ; we s h a l l a s s u m e a s b e f o r e t h a t (17) h a s a u n i q u e s o l u t i o n S = s ( K ) . I n C h a p t e r I I , t h e f o l l o w i n g r e s u l t s w e r e o b t a i n e d f o r t h e d e t e r m i n i s t i c m o d e l : ( i ) I f K = h ( S ) a n d S = s ( K ) i n t e r s e c t a t S = S " , t h e n t h e o p t i m a l e s c a p e m e n t i s i n d e p e n d e n t o f f l e e t c a p a c i t y i f K i s r e l a t i v e l y s m a l l , t h a t i s ; s ( K ) = S _ f o r a l l K < ( 1 / [ 1 - r ] ) • h ( S - ) . T h u s S" r e p r e s e n t s t h e o p t i m a l e s c a p e m e n t a t l o w f l e e t c a p a c i t i e s . ( i i ) A s f l e e t c a p a c i t y K b e c o m e s v e r y l a r g e , t h e o p t i m a l e s c a p e m e n t s ( K ) a p p r o a c h e s o r r e a c h e s t h e l e v e l S + r e p r e s e n t i n g t h e a b u n d a n t - c a p i t a l e q u i l i b r i u m a n d c o r r e s p o n d i n g t o X i n C l a r k , C l a r k e a n d M u n r o ( 1 9 7 9 ) . ( i i i ) S " < S + ( i v ) s ( K ) i s " l i k e l y " t o be an i n c r e a s i n g f u n c t i o n o f K t h r o u g h o u t . R e s u l t s ( i ) , ( i i i ) a n d ( i v ) c a r r y o v e r t o t h e s t o c h a s t i c c a s e by e n t i r e l y a n a l o g o u s r e a s o n i n g , r e p l a c i n g V , V , V b y R K E { V } , E { V } , E { V } r e s p e c t i v e l y , w h e r e v e r t h e y a r i s e . To d e r i v e R K t h e s t o c h a s t i c e q u i v a l e n t o f r e s u l t ( i i ) , we p r o c e e d i n a m a n n e r a n a l o g o u s t o t h a t o f I I . C , l e t t i n g v ( R ) = V ( R , i n f ) a s b e f o r e a n d n o t i n g t h a t i f K = i n f i n i t y t h e n S = s ( i n f ) f o r a n y R - v a l u e , s o we h a v e : v ( R + d R ) = r r ( R + d R , i n f , s ( i n f ) , 0 ) + a v ( F [ s ( i n f ) ] ) a n d v ( R ) = j r ( R , i n f , s ( i n f ) , 0 ) + c v ( F [ s ( i n f ) ]) H e n c e v ( R + d R ) - v ( R ) = p d R - ( c / q ) l o g ( [ R + d R ] / R ) = p ( ! - x 0 / R ) d R 80 so that v (R)= p(l-x 0/R) it R>S +=s(inf) R Setting R=F(S) and substituting into (17) t h i s expression for v produces: R a[F'(S)/F(S)]E{R'.p(1-x 0/R')} =p(1-x0/S) Now E{R'}=F(S), so this equation reduces to: pa[F*(S)/F(S)](F(S ) - X o ) =p(l-x 0/S) or: F ' ( S ) • [ 1 - x 0 / F ( S ) ] / [ 1 - X o / S ] =1/C which i s again the Modified Golden Rule equation. However, this is incorrect since the expression for v derived above applies R only for R>S+. While this inequality was v a l i d for a deterministic fishery, i t cannot be assumed in the stochastic case. Instead, we can write E[R'•v (R')l as follows: R E{R'-v (R')}= E{R' •p(1-x 0/R' )} R S + +| [R'-v (R')-p(R'-x 0)]*(R')dR' 0 R Now substituting into (17) and rearranging, we have the optimality expression: F ' ( S ) [ 1 - x 0 / F ( S ) ] / [ 1 - x 0 / S ] = S (l/o)-[F'(S)/pF(S)(1 - X o/S)]|[R'-V (R',inf)-(R'-x 0)]*(R')dR' 0 R For the Beverton-Holt F(S) function, the l e f t hand side of this equation is a decreasing function of S, so we can conclude that the abundant-capital optimal escapement in the stochastic case, S +, is greater (less) than i t s deterministic counterpart 81 according as the integral on the right hand side of th i s equation, evaluated at S=S+, is p o s i t i v e (negative). -qTK Now incorporating the constraint Re <S£R, we obtain equation (13) of Chapter II, namely: R«exp{-qTK} (18) S*(R,K)= <\ s(K) R>s(K)exp{qTK} R intermediate R<s(K) R Thus the two policy functions h(S) and s(K) are s u f f i c i e n t to determine the synthesis of our stochastic control problem. Since the he u r i s t i c analysis has followed so closely the deterministic case, resulting in similar policy functions h(S) and s(K), we expect the numerical results to mimic the deterministic case, at least q u a l i t a t i v e l y . The. important question then, is the extent to which randomness affects the quantitative aspects of the optimal p o l i c i e s ; t h i s w i l l be addressed in section E. F i r s t , we describe the numerical scheme used to produce our main re s u l t s . D. Numerical Method The dynamic programming equation (1) cannot be solved a n a l y t i c a l l y , so one must resort to numerical methods. As in the previous chapter, policy i t e r a t i o n is used to derive the optimal functions h(-) and s(«). However in t h i s chapter we develop a more accurate numerical method, to obtain a consistent solution using more of the information contained in the dynamic programming equation. In Chapter II, a 2-dimensional mesh was set up in the positive quadrant of the R-K plane, and the f i r s t objective was 82 to determine the matrix of values V , corresponding to any i j given pair of policy functions. A smooth surface was then formulated to f i t these points, and the matrices of p a r t i a l derivatives, V and V , were calculated. These were used (R)ij (K)ij in the policy improvement stage to determine the optimal p o l i c i e s for the current value function. The new (improved) p o l i c i e s were used in turn to calculate a new value function, and the process continued u n t i l convergence was achieved. This method, while accurate for a s u f f i c i e n t l y dense mesh, suffers from the disadvantage that not a l l available information i s being u t i l i z e d . In t h i s chapter, we obtain equations involving the p a r t i a l derivatives of the value function by d i f f e r e n t i a t i n g the dynamic programming equation (1) with respect to R and K (in turn). Three equations arise at each mesh point; the quantities V , V and V can then be i j ( R)ij ( K ) i j determined simultaneously by solving a suitable set of linear equations. This is a consistent method for c a l c u l a t i n g the p a r t i a l derivatives d i r e c t l y , rather than deducing them i n d i r e c t l y as in Chapter I I . A second change in the numerical method is made by dealing with the natural logarithm of the biomass as a state variable in place of the biomass per se. As pointed out by Ludwig (1979a), this change of variables is convenient when dealing with lognormal stochastic effects and when faced with integration over an i n f i n i t e i n t e r v a l . Naturally such a change does not aff e c t the results - indeed the results presented in the 83 following section have been tranformed back to the o r i g i n a l variables for ease of understanding. A f i n a l improvement can be obtained by writing V(•,•) as a product of cubics in x=log(R) and in K, with c o e f f i c i e n t s depending on the mesh point being considered. This ensures that a C 1 surface w i l l be generated in R-K space, (a) Change of variables We begin by examining h e u r i s t i c a l l y the expected optimal p o l i c i e s obtained using x= log(R) in place of R, log(S) in place of S, and defining the transformed log(stock)-log(recruitment) function f ( t ) = log(F(e ) ) . The value function can then be written V(x,K) and is given by the equation: (19) V(x,K)= Max Max [ rr (x ,K, £ , I ) +oE{V(x ' , ( 1 -y )K+I ) } ] x-qTK<£<x I>0 X £ where rr(x,K,£,I) = p(e -e ) - (c/q)(x-^) - 61, and the expectation is with respect to x'~N(f(£)-a 2/2,e 2). Performing the maximizations on the right hand side, we obtain the optimality relationships: (20) E{V ( fU)-« 2/2+z,(l- r)K+I)}= 6/a or 1=0 i f E{V }< 6/a ; z K z K E{V ( fU)-* 2/2+z,(1-y)K+I*)}= p(e - x 0 ) / f ' U ) Z X By the reasoning of Section C, we deduce the existence of optimal policy functions u(t) and w(K) such that ~x-qTK ; x>w(K)+qTK ^*(x,K)= 4 w(K) ; otherwise and K' =1 * (t\, K) =Max {u (t\) - ( 1 -y )K, 0 } x ; x<w(K) 84 represent the optimal log(escapement) as a function of the state (x,K) and the optimal investment as a function of log(escapement) and current capacity, respectively. The behaviour of the policy function u ( - ) can be described in part by setting (1-y)K+I=u(t) in (20) and d i f f e r e n t i a t i n g with respect to to produce: (21) u* U ) = f ' U )-E{V (x' ,u($) ) }/-E{V ( x ' , u U ) ) } > 0 xK KK where again we have assumed V >0 and V <0. The concavity of xK KK u ( » ) cannot be deduced d i r e c t l y from (21), but would seem to be l i k e l y i f f"<0 (c.f. Chapter I I ) . However, since f(^)= log(F(e )), i t can be shown that: f " U ) = (S 2/F(S)){F'(S)[(F(S)-S)/SF(S)] + F"(S)} where S=e . This indicates that the concavity of f i s ambiguous, since the f i r s t term in the parentheses i s positive while the second is negative, unless S is so large that F(S)<S, in which case we c l e a r l y have f " ( t ) < n . For the Beverton-Holt stock-recruitment function, t h i s expression for f" (t) can be considerably s i m p l i f i e d . Using F(S)= aS/d+dS), where d=a/b, i t i s e a s i l y shown that: f"(t)= -{d 2S 2+(2+a)dS-(a-1)}/[a(1+dS) 2] Since a>1, we can conclude that f w i l l be concave whenever the quadratic within the parentheses is p o s i t i v e . This w i l l be the case for a l l S>S., the positive solution of d 2S 2+(2+a)dS-(a-1)=0. For example, in the base case prawn fishery, f w i l l be concave for a l l S>0.877xl0 6, and with a=14 85 th i s result holds for a l l S>0.48 ( i . e . e s s e n t i a l l y a l l S-values). Hence in general we can expect to find the optimal capacity function increasing throughout, but concavity may be r e s t r i c t e d to some range S>S., which may in fact include the relevant range of escapements. We return now to a detailed discussion of the numerical method used to obtain the value function corresponding to a given pair of policy functions, and to determine the optimal p o l i c i e s for a given value function. These two steps performed consecutively w i l l converge to the optimal p o l i c i e s and value function. (b) Manipulation of V and i t s p a r t i a l derivatives We s h a l l f i r s t describe the procedure to solve for the value function given any pair of policy functions. The i n i t i a l step is to form a mesh of M values in the x-direction and a mesh of N values in the K-direction; these mesh points can be chosen more or less arb i t r a r i l y . , but numerical d i f f i c u l t i e s might be expected i f the meshes are very non-uniform. As in Chapter II, the lower l i m i t of the K-values was set at K.=0 in every case. The upper K-value was set at a convenient value above the optimal capacity level determined in Section B, since t h i s represents the maximum possible l e v e l to which a manager would invest, occuring only in the extreme case of stock/recruitment independence. The upper and lower l i m i t s of the x-mesh must be determined by examining the maximum and minimum "reasonable" recruitments which can arise given stock-recruitment parameters 86 and the maximum l e v e l of noise to be considered. For example, in the case of the stochastic (<y=0.58) prawn fishery discussed in section E , suitable l i m i t s for the x-mesh were x.=12.206 and x + = 16.811 (corresponding to R.=0.2X10 6 and R + = 20.0x1 0 6 ). While one wishes to include "most" of the x-line within the mesh, in theory a r b i t r a r i l y large and/or small recruitments are possible (with lognormal noise), so i t becomes necessary to approximate the value function for x-values above x + and below x.. These extrapolation schemes are described below. Varying the x-mesh l i m i t s x. and x + naturally has some effect on the precise location of the optimal p o l i c i e s h(S) and s(K). However, experiments with several x-meshes indicated that quantitative changes in the results were minor, and the q u a l i t a t i v e conclusions were never affected. Given any policy functions w(K) and u(£), the value function s a t i s f i e s the following integral equation: (22) V(x,K)= ir(x,K,€(x,K),Max{u(€(x,K))-(i-y)K,0}) + cE{V(f [ * (x,K) ]-<r2/2 + z,Max{uU (x,K) ) , ( 1-y)K} ) } z "x-qTK ; x>w(K)+qTK where z~N(0,<y2) and t(x,K)= - w(K) ; otherwise x ; x<w(K) This equation must hold for a l l values of x and K, but may not be everywhere d i f f e r e n t i a b l e . However, carrying over our smoothness assumption of section C, we suppose that V i s at least C 2 almost everywhere, for suitably smooth u and w functions. Hence both sides of (22) may be d i f f e r e n t i a t e d with respect to x and with respect to K, to obtain two new equations (where, for s i m p l i c i t y , we define K'=Max{u(t(x,K)),(1-r)K} and 87 I U , K ) = M a x { u U ) - ( 1 - r ) K , 0 } ) : ( 2 3 ) V ( x , K ) = (ir +ir £ + rr I £ ) x x £ x I t x + a[f ' U U , K ) ) < ; ] - E { V ( f ( 0 - ( t f 2 / 2 ) + z , K ' ) } X Z X + c [ I * ] - E { V ( f U ) " U 2 / 2 ) + z , K ' ) } £ x z K ( 2 4 ) V ( x , K ) = (it i. +TT I £ +ir I ) K £ K I £ K I K + o [ f ' ( t ( x , K ) ) * ] - E { V ( f ( 0 - ( t f 2 / 2 ) + z , K ' )} K z x + c [ ( l - r ) + I t +1 ] - E { V ( f U ) - U 2 / 2 ) + z , K ' ) } t K K z K F r o m t h e d e f i n i t i o n o f t\, we o b s e r v e t h a t : f l C~qT f > w ( K ) + q T K £ ( x , K ) = - 0 a n d e, ( x , K ) = - d s / d K i f x i s i o t h e r w i s e x 1 K 0 <w(K) f 0 CO r > u ( t ) / d - r ) I U , K ) = d h / d t a n d I U , K ) = ^ - ( 1 - r ) i f K i s | < u ( « ) / ( l - r ) O u r s m o o t h n e s s a s s u m p t i o n s i n d i c a t e t h a t ( 2 3 ) a n d ( 2 4 ) a r e v a l i d , e v e r y w h e r e e x c e p t p o s s i b l y a l o n g t h e c u r v e s x=w(K)., x = w ( K ) + q T K a n d K=u ( e. ( x , K) ) / ( 1 -y ) , w h e r e t h e p o l i c y f u n c t i o n s c h a n g e t h e i r f o r m ( c . f . CCM, p . 4 5 ) . S i n c e e q u a t i o n s ( 2 2 ) - ( 2 4 ) a r e a s s u m e d v a l i d f o r a l m o s t a l l p o i n t s i n t h e p o s i t i v e x-K q u a d r a n t , o n e w o u l d h o p e t h a t t h e y w o u l d h o l d i n p a r t i c u l a r a t e v e r y m e s h p o i n t ( x ,K ) . I f we a r e s o u n l u c k y a s t o h a v e o n e o f i j t h e 3 " s w i t c h i n g c u r v e s " p a s s i n g t h r o u g h a m e s h p o i n t , we h a v e a d o p t e d t h e r a t h e r a r b i t r a r y c o n v e n t i o n o f u s i n g ( 2 2 ) - ( 2 4 ) w i t h t h e a p p r o p r i a t e l e f t - h a n d o r r i g h t - h a n d p a r t i a l d e r i v a t i v e s i n 88 place of d , t\ , I and I . (A p a r t i a l inspection of the x K £ K numerical results did not uncover occasions where this measure had to be taken.) We thus have 3MN equations, from which we desire to obtain the 3MN values representing V(x ,K ), V (x ,K ) and V (x ,K ) at i j x i j K i j each mesh point. This apparently simple task is complicated by the existence of a further set of MN unknowns, to be discussed below. Note at t h i s point that these equations can be e a s i l y modified to the deterministic case by setting e=0, z=0 and removing the expectation operator. In order to compare results for the stochastic and deterministic cases, we s h a l l develop the numerical method for the two p o s s i b i l i t i e s concurrently; few changes w i l l be required. Consider the rectangle in x-K space defined by the 4 corners (x ,K ) , (x ,K ) , (x ,K ) and (x ,K ). Our i j i+1 j i j+1 i+1 j+1 f i r s t objective is to define the value function on this rectangle in terms of data at the corners and such that V is continuous and has continuous f i r s t p a r t i a l derivatives across each edge; this is desired so that the policy improvement process makes sense. Following Ludwig (1979b), we define functions g (x) as follows: n i 89 g ( x ) = ( 3 / A 2 ) ( x - x ) 2 + ( 2 / A 3 ) ( x - x ) 3 1i i+1 i+1 g ( X ) = ( 1 / A 2 ) ( X - X ) 3 + ( 1 / A ) ( X - X ) 2 2 i i+1 i+1 g ( x ) = ( 3 / A 2 ) ( x - x ) 2 - ( 2 / A 3 ) ( x - x ) 3 3 i i i g (X) = ( 1 / A 2 ) ( X - X ) 3 - d / A ) ( x - x ) 2 4 i i i where A= x -x . Then g (x )=1, g' (x )=0, g (x )=0 and i + l i 1 i i 1 i i 1i i +1 g' (x )=0, and s i m i l a r l y f o r the o t h e r f u n c t i o n s . These 1 i i + 1 g (x) f u n c t i o n s can be r e w r i t t e n as f o l l o w s : n i 4 4-r g (x)= I g ( i , n , y ) x n i 7=1 x f o r i=1,...,M and n=1,2,3,4. A s t r a i g h t f o r w a r d e x e r c i s e i n a l g e b r a produces the v a l u e s of the g (•,•,•) terms, d i s p l a y e d i n x Tab l e ( I I I ) . Now we d e f i n e f u n c t i o n s h (K) i n an e x a c t l y n j analogous way, s u b s t i t u t i n g K f o r x , j f o r i , and l e t t i n g A = K ( j + 1 ) - K ( j ) . F i n a l l y f o r n o t a t i o n a l c o n v e n i e n c e we adopt a c o n v e n t i o n f o r d e p i c t i n g the unknowns V, V , V and V a t the x K xK mesh p o i n t s on the r e c t a n g l e : v 1 1 = V ( i j ) , v12=V ( i j ) , v 1 3 = V ( i , j + 1 ) , v14=V ( i , j + l ) K K v21=V ( i j ) , v22=V ( i j ) , v23=V ( i , j + l ) , v24=V ( i , j + l ) x xK x xK and the e q u i v a l e n t d e f i n i t i o n s f o r v3> and v4«, w i t h i r e p l a c e d by i + 1 . Let us note a t t h i s stage t h a t i f our s u r f a c e i s t o be 90 r e p r e s e n t e d b y a p r o d u c t o f c u b i c s i n x a n d i n K, e a c h c u b i c c o n t a i n i n g 4 u n k n o w n c o n s t a n t s , we m u s t o f n e c e s s i t y d e a l w i t h 4 x 4 = 1 6 u n k n o w n s o n t h e r e c t a n g l e , r e p r e s e n t i n g 4 u n k n o w n s p e r c o r n e r . T h i s c o r r e s p o n d s t o 4MN u n k n o w n s i n a l l , a n d y e t we h a v e o n l y 3MN e q u a t i o n s . I n t h e o r y t h i s p r o b l e m c a n b e o v e r c o m e i n a v e r y s t r a i g h t f o r w a r d m a n n e r b y d i f f e r e n t i a t i n g e q u a t i o n ( 2 4 ) w i t h r e s p e c t t o x t o o b t a i n a f o u r t h s e t o f e q u a t i o n s f o r t h e c r o s s p a r t i a l d e r i v a t i v e . U n f o r t u n a t e l y t h i s p r o d u c e s a v e r y c o m p l i c a t e d e q u a t i o n i n v o l v i n g t h e s e c o n d p a r t i a l d e r i v a t i v e w i t h r e s p e c t t o x , s o t h a t i n e f f e c t o n e m u s t o b t a i n e q u a t i o n s f o r a l l t h e s e c o n d p a r t i a l d e r i v a t i v e s . T o a v o i d t h i s p r o b l e m , we s h a l l k e e p t h e n u m b e r o f e q u a t i o n s a t 3MN a n d i n s t e a d r e d u c e t h e n u m b e r o f u n k n o w n s b y a p p r o x i m a t i n g V t e r m s b y t h e s l o p e x K o f c u b i c s i n t h e x - d i r e c t i o n f i t t e d t o v a l u e s o f V . T h i s i s K d e s c r i b e d f u r t h e r b e l o w . W i t h t h e a b o v e d e f i n i t i o n s we a r e now i n a p o s i t i o n t o d e f i n e t h e v a l u e f u n c t i o n o n t h e ( i , j ) t h r e c t a n g l e : 4 4 4 4 4 - r 4-6 (op) ( 2 5 ) V ( x , K ) = I I I I g ( i , c , r ) g ( j , e , 6 ) x K v i j o=1 0=1 y=1 6=1 x K By s i m p l y d i f f e r e n t i a t i n g t h i s v a l u e f u n c t i o n w i t h r e s p e c t t o e i t h e r x o r K, we c a n d e d u c e t h e e x p r e s s i o n s f o r V a n d V on x K t h e r e c t a n g l e : 91 4 4 3 4 3 - r 4-6 U p ) V ( x , K ) = I E I I g ( i , c , r ) g ( j,»,6)(4~r)x K v ( x ) i j a=1 p=1 7=1 6=1 x K 4 4 4 3 4 - 7 3-6 (afi) V ( x , K ) = E E I E g ( i , a , 7 ) g ( j , ft, 6) ( 4 - 6 ) x K v ( K ) i j c=1 0=1 7=1 6=1 x K T h e d e r i v a t i o n o f t h e a b o v e 3 e q u a t i o n s was i n d e p e n d e n t o f t h e s t o c h a s t i c n a t u r e o f t h e p r o b l e m a n d h e n c e t h e e q u a t i o n s a p p l y i n t h e d e t e r m i n i s t i c c a s e a s w e l l . I n f a c t , i f i a n d j a r e c h o s e n s u c h t h a t x < f U ) < x a n d K <K' = ( 1 -y ) K + I ( £ ,K)<K , t h e n i i+1 j j+1 t h e f u t u r e v a l u e o f t h e f i s h e r y , g i v e n c u r r e n t e s c a p e m e n t t\ a n d c u r r e n t c a p a c i t y K, i s g i v e n b y V ( f ( ^ ) , K ' ) i n t h e a=0 c a s e . i j ( c ) E x t r a p o l a t i o n s B e f o r e p r o c e e d i n g f u r t h e r we t a k e u p t h e q u e s t i o n o f e x t r a p o l a t i o n s t o h i g h ( x > x + ) a n d l o w ( x < x . ) r e c r u i t m e n t v a l u e s i n t h e s t o c h a s t i c c a s e . F i r s t c o n s i d e r t h e r a n g e x > x + . S o l e l y f o r t h e p u r p o s e s o f t h i s e x t r a p o l a t i o n , we make 3 a s s u m p t i o n s t o s i m p l i f y t h e d y n a m i c s . T h e f o l l o w i n g a p p l y f o r x > x + a n d K l y i n g b e t w e e n K. a n d K + ; ( i ) £ ( x , K ) = x - q T K ( s o t h a t h a r v e s t i n g i s a t f u l l c a p a c i t y i f t h e r e c r u i t m e n t i s s u f f i c i e n t l y h i g h ) . ( i i ) u ( x - q T K ) - u ( x + - q T K ) ( i i i ) f ( x - q T K ) - f ( x + - q T K ) T o g e t h e r t h e s e a s s u m p t i o n s i m p l y t h a t t h e a v e r a g e r e c r u i t m e n t , t h e o p t i m a l i n v e s t m e n t a n d t h e f l e e t c a p a c i t y f o r n e x t s e a s o n a r e g i v e n b y t h e c o r r e s p o n d i n g v a l u e s f o r x = x + . T h e l a t t e r t w o a p p r o x i m a t i o n s a r e r e a s o n a b l e f o r a B e v e r t o n - H o l t m o d e l , s i n c e t h e f u n c t i o n f a p p r o a c h e s a n a s y m p t o t e a t h i g h x - v a l u e s . N o t e , 92 however, that the assumption that f and u are f l a t for x-qTK>x+-qTK, while l i k e l y to be accurate for low K-values, w i l l be less accurate as K i s increased. Hence the values x +,K + must be chosen so that x +-qTK + i s s u f f i c i e n t l y large. (In fact, the above assumptions are stronger than necessary, but serve our purpose of obtaining a rational yet simple extrapolation.) Equations for V(x,K) and V(x +,K) can now be written: V(x,K) = ir(x,K,I ) + a E U • V ( • , K ' ( x + , K) ) } (x +K) *(x +,K) V(x +,K)= ir(x +,K,I ) + o E { * • V ( • , K ' (x + , K) ) } (x +K) t(x +,K) x x-qTK where rr i s given by ir(x,K,l)= p(e -e )-cTK~6I, and I =Max{u(x+-qTK)-(1-r)K,0}. Hence V(x,K) can be approximated (x + K) by: -qTK x x + (26) V(x,K) - V(x +,K)+p(1-e ) ( e - e ) -qTK x Clearly i t also follows that V (x,K)~ p(1-e )e and x -qTK x x + V (x,K)~ V (x +,K) +pqTe (e -e ). The integral from x + to K K i n f i n i t y i m p l i c i t in equation (22) now becomes: (27) | *(x')[V(x +,K*)+p(1-exp{-qTK'})(exp{x'}-exp{x +})]dx* x>x + =[!-•((x +-x)/ t f)]V(x +,K') +p( 1 -exp{-qTK' } ) {exp{ x + *2/2 } [ 1 - *( (x + -x-« 2)/« ) ] -e [ l - * ( (x + - x ) A ) ]} where x= f U ) - c r 2 / 2 . 93 S i m i l a r l y , t h e i n t e g r a l s i n v o l v i n g V a n d V b e c o m e : x K ( 2 8 ) # ( x ' ) V ( x ' , K ' ) d x ' X>X + X =p( 1 - e x p { - q T K ' } ) e x p { x + <r 2/2} [ 1 - * ( ( x + - x - t f 2 ) / c ) 3 ( 2 9 ) <t> ( x ' ) V ( x ' , K ' ) d x ' x > x + k\ K = [ 1 ~ # ( ( x + - x ) / t f ) ] V ( x + , K ' ) + p q T e x p { - q T K ' } • K { e x p { I + t f 2 / 2 } [ 1-4>((x + - x - t f 2 ) / < r ) ] - e [ 1 - * ( ( x + - x ) / * ) ] } Now c o n s i d e r t h e a p p r o x i m a t i o n o f t h e v a l u e f u n c t i o n f o r v a l u e s o f x < x . . A s s u m e t h a t x . i s c h o s e n s u f f i c i e n t l y l o w s o t h a t f o r a n y x < x . , t h e e s c a p e m e n t f u n c t i o n s a t i s f i e s t ( x , K ) = x a n d t h e r e q u i r e d i n v e s t m e n t I ( ^ , K ) = 0 f o r a l l v a l u e s o f K. [ T h e l a t t e r c o n d i t i o n i m p l i c i t l y a s s u m e s t h a t , f o r g i v e n x , , t h e p a r a m e t e r a i n t h e s t o c k - r e c r u i t m e n t f u n c t i o n i s n o t t o o l a r g e , s o t h a t u ( ^ ) = 0 f o r f ; < x 1 . T o d e a l w i t h l a r g e a - v a l u e s , t h e v a l u e o f x , c a n be d e c r e a s e d a c c o r d i n g l y . F o r t h e e x t r e m e c a s e o f a = i n f i n i t y , t h e a p p r o x i m a t i o n w i l l b e s o m e w h a t i n a c c u r a t e , b u t t h i s w i l l h a v e n e g l i g i b l e e f f e c t i f x , i s s u f f i c i e n t l y s m a l l . ] T h e n t h e d y n a m i c p r o g r a m m i n g e q u a t i o n f o r V i n t h i s c a s e i s : F o r x s m a l l e n o u g h , x' w i l l a l s o b e s m a l l f o r ' m o s t ' z - v a l u e s . T h u s , f o l l o w i n g L u d w i g a n d W a l t e r s ( 1 9 8 2 ) we a s s u m e l i n e a r p o p u l a t i o n d y n a m i c s : F ( S ) = a S / d + a S / b ) "=" a S . T h i s p r o d u c e s t h e a p p r o x i m a t i o n f ( x ) = l o g ( a ) + x , o r f ( x ) = x + ( f ( x . ) - x . ) s i n c e f ( x . ) = ( 3 0 ) V ( x , K ) = o | { * ( x ' ) - V ( x ' , ( 1 - r ) K ) } d x ' f U ) = f ( x ) 94 X.x l o g ( a ) + x . . Assume t h a t V c a n be r e p r e s e n t e d by V ( x , K ) = v ( K ) e f o r s m a l l x. Then we have t h e f o l l o w i n g ( s e e L u d w i g a n d W a l t e r s , 1982) : Xx (31) v ( K ) e = a | * ( x ' ) v ( [ 1 - r ] K ) e x p { X x 1 } d x 1 f ( x ) = av( [ 1-r ]K) | <* e x p { x [ z + x-x.+f ( x . ) - < r 2 / 2 ] } d z N(0,tf 2) where t h e i n t e g r a l s a r e o v e r t h e r e a l l i n e . S i m p l i f y i n g , we o b t a i n : exp{ X 2 t f 2 / 2 - X t f 2 / 2 + X [ f ( x . )-x. ] } = v ( K ) / a v ( [ 1 - r ] K ) U n l i k e t h e s i t u a t i o n i n L u d w i g and W a l t e r s , o u r m o d e l i n v o l v e s c a p i t a l d y n a m i c s , w h i c h l e a d s t o t h e p r o b l e m of d e a l i n g w i t h t h e t e r m v ( K ) / v ( [ 1 - y ] K ) . F o r t h e p u r p o s e s of t h i s e x t r a p o l a t i o n we use t h e s i m p l i f i c a t i o n ( o n l y r e q u i r e d a t low x - v a l u e s ) t h a t v ( K ) i s p r o p o r t i o n a l t o K; h e n c e we o b t a i n v ( K ) / v ( [ 1 - y ]K)= 1 / ( 1 — r ) . Thus t h e c o e f f i c i e n t X. s a t i s f i e s : ( e2/2 ) X 2 + [ f ( x . )-x . - a 2 / 2 ] X + l o g [ c( 1 -y ) ] = 0 w h i c h c a n be s e e n t o h a v e e x a c t l y one p o s i t i v e r o o t X* a s d e s i r e d . The i n t e g r a l b e t w e e n - i n f i n i t y a nd x. c a n now be a p p r o x i m a t e d a s f o l l o w s : (32) | * ( x ' ) V ( x ' , K ' ) d x ' x<x . =v(K') | 0_ ( x 1 ) e x p { X * x ' } d x ' x<x . x ; cr = v ( K ' ) e x p { ( < r 2 / 2 ) X * 2 + x X * } * ( ( x . 2 ) /a ) =V(x. ,K' ) e x p { ( t f 2 / 2 ) X * 2 + ( x - x . )X*}4>( (x .-x-X* t f 2 )/«) where we have u s e d t h e p r o p e r t y : 95 - X x . v ( K ) = e V ( x . , K ) T h e c o r r e s p o n d i n g i n t e g r a l s i n v o l v i n g V a n d V a r e a l s o g i v e n x R b y ( 3 2 ) , b u t w i t h V r e p l a c e d b y V , a n d ' V r e s p e c t i v e l y . T h i s c a n x R b e s e e n b y u s e o f t h e f o l l o w i n g c o n s e q u e n c e s o f o u r a p p r o x i m a t i o n m e t h o d : Xx ( i ) V ( x , K ) = [ X v ( R ) ] e x Xx ( i i ) V ( x , K ) = v ' ( K ) e R - X x . ( i i i ) [ X v ( R ' ) ] = e V ( x . , R ) x - X x . ( i v ) v ' ( R ) = e V ( x . , K ) K T h i s c o m p l e t e s t h e e x t r a p o l a t i o n s t o h i g h a n d l o w r e c r u i t m e n t v a l u e s . We r e m a r k a g a i n t h a t t h e u s e o f a l o g a r i t h m i c v a r i a b l e x = l o g ( R ) h a s t h e a d v a n t a g e t h a t v e r y l o w a n d v e r y h i g h r e c r u i t m e n t s c a n be i n c l u d e d i n a f a i r l y u n i f o r m m e s h b y j u d i c i o u s c h o i c e o f x. a n d x t , a n d h e n c e t h e x - r e g i o n s w h e r e t h e e x t r a p o l a t i o n s a p p l y c a n be o n e s o f v e r y l o w p r o b a b i l i t y . N o t e t h a t t h e e x t r a p o l a t i o n s f o r t h e i n t e g r a l s f r o m - i n f i n i t y t o x . a n d f r o m x + t o i n f i n i t y i n c l u d e o n l y t h e u n k n o w n v a l u e s V ( x . , - ) , V ( x . , - ) , V ( x _ , - ) , V ( x + , - ) , V ( x + , - ) a n d x K x V ( x + , 0 , t o g e t h e r w i t h some kn o w n c o n s t a n t s . H e n c e t h e s e R e x t r a p o l a t i o n s c a n be i n c l u d e d w i t h i n t h e s y s t e m o f e q u a t i o n s 96 f o r V, V a n d V a t t h e m e s h p o i n t s b y s u i t a b l y a l t e r i n g t h e x K c o e f f i c i e n t s f o u n d a b o v e . T h e n u m e r i c a l c o n s t a n t s c a n be i n c l u d e d i n t h e e x p r e s s i o n s f o r i r , rr a n d tr . T h u s t h e s e r e g i o n s x K o f h i g h a n d l o w x n e e d n o t c o n c e r n u s f u r t h e r . ( O f c o u r s e , i n t h e «r=0 c a s e , t h e s e e x t r a p o l a t i o n q u a n t i t i e s a r e s e t e q u a l t o z e r o a s d e s i r e d . ) ( d ) F o r m u l a t i o n o f a s y s t e m o f l i n e a r e q u a t i o n s T h e e q u a t i o n s f o r V, V a n d V c a n now b e s u m m a r i z e d a s : x K M-1 x ( k + 1 ) ( 3 3 ) o " 1 V ( x , K ) = a - 1 r r + E | <f> ( x ' ) V ( x ' , K' ) d x ' k=1 x ( k ) M-1 x ( k + 1 ) ( 3 4 ) o " 1 V ( x , K ) = a - 1 r r + E [ T, | * ( x ' ) V ( x ' , K ' ) d x ' x x k=1 x ( k ) x x(k+1 ) + T 2 | * ( x * ) V ( x ' , K ' ) d x ' ] x ( k ) K M-1 x ( k + 1 ) ( 3 5 ) o ' 1 V ( x , K ) = a - ' i r + E [ T 3 | <*(x')V ( x ' , K ' ) d x ' K K k=1 x ( k ) x x ( k + 1 ) + T a | * ( x ' ) V ( x ' , K ' ) d x ' ] x ( k ) K w h e r e T 1 , T 2 , T 3 , T 1 | a r e g i v e n b y : T,= f ' ( t ( x , K ) ) t x T 2 = I a e. x T 3 = f ' ( « ( x , K ) ) ^ K T„= { ( 1 - y ) + I a +1 } C K K [ I n t h e d e t e r m i n i s t i c c a s e <t> i s g i v e n b y t h e d e l t a f u n c t i o n 97 c e n t r e d a t x = f U ( x , K ) ) . ] D e f i n i n g 1 b y K <K'<K a n d r e p l a c i n g 1 1 + 1 j w i t h 1 i n e q u a t i o n ( 2 5 ) a n d t h e c o r r e s p o n d i n g e x p r e s s i o n s f o r V a n d V , t h e i n t e g r a l s o f </>V, #V a n d $V i n t h e ( x ) i j ( K ) i j x K n i n t e r v a l ( x ,x ) c a n be w r i t t e n i n t e r m s o f i n t e g r a l s o f #x k k+1 f o r n = 1 , 2 , 3 , 4 . F o r e x a m p l e : 4 4 4 4 ( 3 6 ) | * ( x ' ) V ( x ' , K ) d x ' = E I E E g ( i , o , r ) g ( j , * , 6 ) o = 1 0 = 1 r = i 6=1 x K . 4-6 (ap) 4 - r • ( K ' ) v | * ( x ' ) ( x ' ) d x ' i D e f i n e X I N T ( k , i ) = | * ( x ' ) ( x ' ) d x ' w h e r e t h e l o w e r a n d u p p e r l i m i t s o f i n t e g r a t i o n a r e x a n d x r e s p e c t i v e l y . T h e s e k k+1 q u a n t i t i e s c a n be e v a l u a t e d e x p l i c i t l y , i n t e r m s o f c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n v a l u e s . T o a c c o m p l i s h t h i s , m ake t h e f o l l o w i n g d e f i n i t i o n s : y ( k ) = ( x ( k ) - x ) / t f f o r e a c h k. A ( k ) = # ( y ( k + 1 ) ) - * ( y ( k ) ) B ( k ) = e x p { - y ( k ) 2 / 2 } - e x p { - y ( k + 1 ) 2 / 2 ] C ( k ) = y ( k ) . e x p { - y ( k ) 2 / 2 } - y ( k + 1 ) . e x p { - y ( k + 1 ) 2 / 2 } D ( k ) = y ( k ) 2 . e x p { - y ( k ) 2 / 2 } - y ( k + 1 ) 2 - e x p { - y ( k + 1 ) 2 / 2 } T h e n we h a v e : 98 XINT(k,0)= A(k) XINT(k,D= ( t f/i/)B(k) +xA(k) XINT(k,2)= U 2 / i / ) C ( k ) + ( 2 <r x / v ) B ( k ) + ( c 2 + I 2 ) A( k ) XINT(k,3)= U 3/i/)D(k) + (3<r2x/V)C(k) + ( ( 2* 3 + 3<rx 2 ) /v/) B ( k ) +(3ff 2x+x 3)A(k) i (Note that in the deterministic case we have XINT(k,i)=x i f x <x<x and =0 otherwise. From this point on, the k k+1 deterministic and stochastic numerical methods for evaluating the value function are identic a l . ) Now define G, GPR, H, HPR, GH, GPRH and GHPR as follows: 4 G(k,o) = E g (k,o , y ) X I N T(k,4-r) 7 = 1 X 3 GPR(k,o) = I g ( k , c , r ) X I N T ( k , 3 - r ) « ( 4 - r ) 7=1 X 4 4-6 H(l,p) = I g (1,0,6) (K' ) 6=1 K 3 3-6 HPR(1,0) = E g U , 0 , 6 ) ( K ' ) -(4-6) 6=1 K GH(k,l,a , 0 ) . = G(k,a)H(l,*) GPRH(k,l,a,0)= GPR(k,a)H(l ,0) GHPR(k,l,a,p)= G(k,o)HPR(l ,0) Then for any values of x and K we have the equations: 99 M-1 4 4 ( 0 0 ) (37) o- 1V(x,K)= a " 1 * + E E E GH(k,a,0)v k=1 o=1 0=1 k l M-1 4 4 iafi) (38) o" 1V (x,K)= a-'n + E [ T, E E GPRH(k,o,0)v x x k=1 c=1 0=1 k l 4 4 ( 0 0 ) +T 2 I E GHPR(k,o,0)v ] o=1 0=1 k l M-1 4 4 ( 0 0 ) (39) o" 1V (x,K)= a " 1 * + E [ T 3 E E GPRH(k,o,0)v K K k=1 o=1 0=1 k l 4 4 ( 0 0 ) +Ta E E GHPR(k,0,0)v ] o=1 0=1 k l where the T v a l u e s a re as above. For co m p l e t e n e s s , we must s e t i GH(M,o,0)=O, GHPR(M,o,0)=O and GPRH(M,a, 0 ) = 0 f o r a l l o and 0. The next s t e p i s t o combine the GH, GPRH and GHPR terms s u i t a b l y ( 0 0 ) so t h a t each v term o c c u r s o n l y once on the r i g h t hand s i d e s k l of each of e q u a t i o n s ( 3 7 ) - ( 3 9 ) . T h i s i s the purpose of the f o l l o w i n g , where 0 = 1 , 2 , 3 , 4 and a=1,2 : c o d , k, 0 , 0 ) = GH(k,c,0) + GH(k-1 ,c+2,0) c o ( 2 , k , a , * ) = GPRH(k,0,0) +GPRH(k-1,0+2,0) c o ( 3 , k , a , 0 ) = GHPR(k,0,0) +GHPR(k-1,a+2,0) f o r k=2,...,M w h i l e f o r k=1 we have: c o ( 1 , 1 , a , 0 ) = GH(1,0,0) c o ( 2 , 1 , c , 0 ) = GPRH(1,a,0) c o ( 3 , 1 , 0 , 0 ) = GHPR(1,0,0) To t h i s p o i n t we have m a i n t a i n e d 4mn unknowns, i n c l u d i n g the c r o s s p a r t i a l d e r i v a t i v e s of V a t the mesh p o i n t s . We now 100 a p p r o x i m a t e V a t ( k , l ) b y t h e s l o p e o f t h e c u b i c a t ( k , l ) ( x K ) f i t t e d t h r o u g h t h e p o i n t s ( x ,V ) , ( x ,V ) a n d k-1 ( K ) k - 1 , l k ( K ) k , l ( x ,V ) . T h e b a s i c e q u a t i o n u s e d f o r k=2,...,M-1 i s t h e k+1 ( K ) k + 1 , l f o l l o w i n g : V - c l ( k ) V + c 2 ( k ) V + c 3 ( k ) V ( x K ) k l ( K ) k - 1 , l ( K ) k , l ( K ) k + 1 , l w h e r e , l e t t i n g A ( i ) = x ( i + 1 ) - x ( i ) , t h e c i ( k ) v a l u e s a r e g i v e n b y : d ( k ) = - A ( k ) / ( A ( k - 1 ) [ A ( k - 1 ) + A ( k ) ] ) c 2 ( k ) = ( 1 / A ( k - 1 ) ) - ( 1 / A ( k ) ) c 3 ( k ) = A ( k - D / ( A ( k ) [ A ( k - l ) + A ( k ) ] ) A t t h e e n d p o i n t s x . ( k = l ) a n d x + ( k = M ) , t h e c u b i c o b t a i n e d u s i n g t h e e n d p o i n t a n d i t s t w o n e a r e s t n e i g h b o u r s i s u s e d . I n e a c h c a s e t h e c o e f f i c i e n t s c o ( • , • , • , • ) m u s t b e m o d i f i e d a c c o r d i n g l y . Now a s i m p l e r e o r d e r i n g o f t h e c o e f f i c i e n t s i s a p p l i e d f o r n o t a t i o n a l c o n v e n i e n c e : c o ' ( j , k , 1 , 1 ) = c o ( j , k , 1 , 1 ) c o ' ( j , k , 1 , 2 ) = c o ( j , k , 2 , 1 ) c o ' ( j , k , 1 , 3 ) = c o ( j , k , 1 , 2 ) c o ' ( j , k , 2 , l ) = c o ( j , k , 1 , 3 ) c o ' ( j , k , 2 , 2 ) = c o ( j , k , 2 , 3 ) c o ' ( j , k , 2 , 3 ) = c o ( j , k , 1 , 4 ) w h e r e j=1,2,3 a n d k=1,...,M. F i n a l l y we d e f i n e t h e c o e f f i c i e n t s c o e f ( • , • , • , • ) a s f o l l o w s : 101 c o e f ( 1 , k , m , n ) = c o ' ( 1 , k , m , n ) c o e f ( 2 , k , m , n ) = T , c o ' ( 2 , k , m , n ) + T 2 c o ' ( 3 , k , m , n ) c o e f ( 3 , k , m , n ) = T 3 c o ' ( 2 , k , m , n ) +T„co'(3,k,m,n) f o r k=1,...,M a n d m=1,2 a n d n = 1 , 2 , 3 . T h i s p r o d u c e s t h e f i n a l f o r m o f t h e e q u a t i o n r e l a t i n g V a t a n y p o i n t ( x , K ) t o t h e 6M u n k n o w n s v , v , v , v , v a n d v k l ( x ) k l ( K ) k l k , l + 1 ( x ) k , l + 1 ( K ) k , l + 1 ( k = 1 , . . . , M ) , w h e r e 1 i s s u c h t h a t K < K ' ( x , K ) < K : 1 1+1 M ( 4 0 ) c - 1 V ( x , K ) = c - 1 r r + I { c o e f ( 1 ,k , 1 , 1 ) v + c o e f ( 1 , k , 1 , 2 ) v k = l k l ( x ) k l + c o e f ( 1 , k , 1 , 3 ) v + c o e f ( 1 , k , 2 , 1 ) v ( K ) k l k,1+1 + c o e f ( 1 , k , 2 , 2 ) v + c o e f ( 1 , k , 2 , 3 ) v } ( x ) k , l + 1 ( K ) k , l + 1 P r e c i s e l y a n a l o g o u s e q u a t i o n s f o r o"'V ( x , K ) a n d a " 1 V ( x , K ) a r e x K o b t a i n e d s i m p l y b y r e p l a c i n g rr b y it o r rr r e s p e c t i v e l y , a n d x K r e p l a c i n g c o e f ( 1 ) w i t h c o e f ( 2 ) o r c o e f ( 3 ) r e s p e c t i v e l y t h r o u g h o u t e q u a t i o n ( 4 0 ) . C h o o s i n g x=x a n d K=K f o r i = 1 , . . . , M a n d j = 1 , . . . , N now g i v e s u s i j 3MN e q u a t i o n s i n t h e 3MN u n k n o w n s V , V a n d V ; t h e i j ( x ) i j ( K ) i j s o l u t i o n o f p a r t ( i ) o f o u r n u m e r i c a l p r o b l e m i s now d r a w i n g n e a r . ( e ) S o l v i n g t h e s y s t e m o f e q u a t i o n s f o r V C o n s i d e r now t h e c h o i c e o f M a n d N, t h e n u m b e r o f m e s h p o i n t s i n t h e x - a n d K- d i r e c t i o n s r e s p e c t i v e l y . I n c h a p t e r I I , we u s e d M=N=30, w h i c h p r o v i d e d a s u i t a b l e b a l a n c e b e t w e e n c o s t 1 02 a n d a c c u r a c y w i t h t h e n u m e r i c a l m e t h o d u s e d t h e r e . On t h e o t h e r h a n d , L u d w i g ( 1 9 7 9 b ) f o u n d t h a t M=5 was s u f f i c i e n t f o r h i s o n e - d i m e n s i o n a l c u b i c s p l i n e m e t h o d . I n t h i s c h a p t e r we a d o p t t h e v a l u e s M=N=8, w h i c h r e s u l t s i n c o m p u t i n g c o s t s a n d a c c u r a c y c o m p a r a b l e ( o r s u p e r i o r ) t o t h e c a s e M=N=30 o f c h a p t e r I I . T h e s e n s i t i v i t y o f t h e s o l u t i o n t o t h e v a l u e s o f M a n d N w a s t e s t e d b y p r o d u c i n g r e s u l t s w i t h M=N=12. T h e s e r e s u l t s w e r e v e r y e x p e n s i v e t o o b t a i n , r e q u i r i n g u p t o 80 s e c o n d s o f C P U t i m e a n d 1000 p a g e - m i n u t e s o f CPU s t o r a g e VMI , t o o b t a i n o n e p a i r o f o p t i m a l p o l i c y f u n c t i o n s ( a s o p p o s e d t o 17 s e c o n d s C PU t i m e a n d 50 p a g e - m i n u t e s o f C P U s t o r a g e VMI i n t h e M=N=8 c a s e ) . F o r t h e b a s e c a s e p r a w n f i s h e r y , t h e maximum d i f f e r e n c e b e t w e e n t h e M=N=8 a n d M=N=12 r e s u l t s f o r t h e u ( ^ ) c u r v e s w as a p p r o x i m a t e l y 1.4 p e r c e n t f o r b o t h er=0 a n d ^ = 0 . 5 8 , w h i l e t h e w ( K ) c u r v e s d i f f e r e d n e g l i g i b l y . F o r t h e w h a l e f i s h e r y , d i f f e r e n c e s w e r e m o r e s u b s t a n t i a l : u p t o 6 p e r c e n t f o r t h e u ( t ) c u r v e a n d 11 p e r c e n t f o r t h e w ( K ) f u n c t i o n . I t a p p e a r e d t h a t a n M=N=8 m e s h was s u f f i c i e n t f o r a l l b u t t h e w h a l e f i s h e r y r e s u l t s , f o r w h i c h M=N=12 w a s u s e d . T h e 3MN b y 3MN s y s t e m o f l i n e a r e q u a t i o n s w h i c h m u s t now be s o l v e d h a s t h e f e a t u r e t h a t i n a n y r o w a t m o s t 2-3-M+1 o f t h e e n t r i e s c a n be n o n z e r o , s i n c e f o r g i v e n ( i , j ) o n l y e n t r i e s w i t h K=K o r K=K a r e o f i n t e r e s t , c o r r e s p o n d i n g t o t h e V, V a n d 1 1 + 1 x V t e r m s a t n e x t s e a s o n ' s l e v e l o f c a p i t a l s t o c k . I f 3MN i s K l a r g e c o m p a r e d t o 6M+1, s p a r s e m a t r i x m e t h o d s c a n be u s e d t o a d v a n t a g e t o s o l v e t h e s y s t e m . H o w e v e r 6 M / ( 3 M N ) = ( 2 / N ) s o t o 1 03 h a v e a s f e w a s 10 p e r c e n t o f t h e e n t r i e s n o n z e r o w o u l d r e q u i r e N >20. S i n c e t h e x - m e s h i s a t l e a s t a s c r i t i c a l a s t h e K - m e s h ( s i n c e we a r e a p p r o x i m a t i n g a d o u b l y i n f i n i t e x - i n t e r v a l ) , we w o u l d w a n t M>N>20. T h e r e s u l t i n g s y s t e m o f e q u a t i o n s w o u l d b e a t l e a s t 1 2 0 0 x 1 2 0 0 , w h i c h w h i l e m a n a g e a b l e i s r a t h e r c u m b e r s o m e . I n a n y c a s e , a s d i s c u s s e d a b o v e , t h e e x p e c t e d a c c u r a c y o f o u r c u b i c s p l i n e m e t h o d ( a s s u g g e s t e d b y t h e r e s u l t s o f L u d w i g a n d W a l t e r s , 1 9 8 2 ) i n d i c a t e s t h a t a m u c h c o a r s e r m e s h p r o v i d e s s u f f i c i e n t a c c u r a c y , s o t h a t s p a r s e m e t h o d s a r e n o t p a r t i c u l a r l y s u i t a b l e . N o t e i n a d d i t i o n t h a t w h i l e t h e s i m p l e r s y s t e m d e s c r i b e d i n C h a p t e r I I , i n v o l v i n g o n l y t h e v a l u e f u n c t i o n a t e a c h m e s h p o i n t , V ( i , j ) , i s d i a g o n a l l y d o m i n a n t , t h i s i s n o t t r u e i n t h e p r e s e n t c a s e s i n c e t h e s i g n o f t h e t e r m s i n v o l v i n g V a n d V n e e d n o t be p o s i t i v e a n d c l e a r l y t h e sum o f t h e t e r m s x K o n t h e r i g h t h a n d s i d e n e e d n o t e q u a l o n e . A s w e l l , t h e s y s t e m i s n o t a n d c a n n o t be made b l o c k d i a g o n a l s i n c e b o t h x a n d K c h a n g e f r o m y e a r t o y e a r a n d t h e e x t e n t o f t h a t c h a n g e d e p e n d s o n t h e c u r r e n t v a l u e o f ( x , K ) i n a n o n - t r i v i a l m a n n e r . I n d e e d a p a r t f r o m l i m i t e d s p a r s e n e s s t h e r e a p p e a r t o be no a s p e c t s o f t h e s t r u c t u r e o f t h e s y s t e m t h a t c a n be u t i l i z e d t o a d v a n t a g e i n t h e s o l u t i o n ; we t h e r e f o r e r e s o r t t o a n e f f i c i e n t p a c k a g e r o u t i n e f o r s o l v i n g g e n e r a l s y s t e m s A x = b . T h e p a c k a g e u s e d , S L I M P / D S L I M P , i s a v a i l a b l e o n t h e U.B.C c o m p u t e r s y s t e m , a n d i s w r i t t e n i n b o t h A s s e m b l e r a n d F o r t r a n . I t u s e s G a u s s i a n e l i m i n a t i o n ( L U d e c o m p o s i t i o n ) w i t h p a r t i a l p i v o t i n g , t o g e t h e r w i t h f o r w a r d a n d b a c k w a r d s u b s t i t u t i o n . T h e r e i s a b u i l t - i n i t e r a t i v e p r o c e d u r e w h i c h c a l c u l a t e s t h e r e s i d u a l v e c t o r r = b - A x i 1 04 a t e a c h s t e p i , s o l v e s A e = r f o r t h e e r r o r v e c t o r e a n d o b t a i n s a n i m p r o v e d s o l u t i o n x =x +e. I t e r a t i o n s c o n t i n u e u n t i l t h e i + 1 i e r r o r v e c t o r s a t i s f i e s a u s e r - s p e c i f i e d c o n v e r g e n c e c r i t e r i o n . I t w a s f o u n d i n m o s t c a s e s t h a t f e w e r t h a n 5 i t e r a t i o n s w e r e r e q u i r e d t o o b t a i n a v e r y a c c u r a t e s o l u t i o n . A n i m p o r t a n t r e m a r k a t t h i s p o i n t i s t h a t o u r p r o g r a m f u n c t i o n s e n t i r e l y i n d o u b l e p r e c i s i o n ; i n i t i a l a t t e m p t s t o u s e s i n g l e p r e c i s i o n p r o d u c e d m i s l e a d i n g r e s u l t s w h i c h w e r e c o r r e c t i n some i n s t a n c e s b u t c o m p l e t e l y w r o n g i n o t h e r c a s e s . T h e p r o b l e m a p p e a r e d t o a r i s e i n t h e u s e o f S L I M P , t h e s i n g l e p r e c i s i o n v e r s i o n o f t h e r o u t i n e , w h i c h s e e m s t o p r o d u c e c o r r e c t s i n g l e p r e c i s i o n r e s u l t s o n l y f o r r e l a t i v e l y s m a l l s y s t e m s . ( f ) P o l i c y i m p r o v e m e n t O n c e t h e v a l u e s o f V, V a n d V h a v e b e e n o b t a i n e d a t e a c h x K m e s h p o i n t , t h e p o l i c y i m p r o v e m e n t s t a g e c a n b e g i n . I m p r o v e m e n t o f t h e u ( ^ ) c u r v e u t i l i z e s e q u a t i o n ( 2 0 ) ; f o r e a c h v a l u e £=x , t h e o b j e c t i v e i s t o d e t e r m i n e t h e K' w h i c h i s o l v e s t h e o p t i m a l i t y e q u a t i o n : E{ V ( x , K ' ) } = 6 / a , w h e r e x h a s x K mean v a l u e f ( £ ) . We t h e n s e t u ( t ) = K ' . T h e p r o c e d u r e u s e d w a s b a s e d o n t h e e x p e c t a t i o n t h a t E { V } d e c r e a s e s w i t h K; m e s h v a l u e s K K i , K 2 , K 3 , . . . w e r e c h e c k e d c o n s e c u t i v e l y u n t i l E { V } < 6 / a . I f t h i s K o c c u r s a t K,, we s e t K'=0. I f i t o c c u r s a t K , t h e n we know t h a t j K <R'<K . j - 1 j 1 05 T h e s o l u t i o n i s a p p r o a c h e d b y a l t e r n a t i n g l i n e a r i n t e r p o l a t i o n a n d i n t e r v a l - h a l v i n g m e t h o d s ; t h i s p r o c e d u r e w a s u s e d t o s p e e d u p t h e r a t h e r s l o w c o n v e r g e n c e o f t h e l i n e a r i n t e r p o l a t i o n m e t h o d a l o n e . A t e a c h s t e p , we h a v e a n u p p e r a n d a l o w e r b o u n d o n K', s a y K ( _ ) a n d K ( + > . A t e s t v a l u e o f K, K ( ? ) , i s o b t a i n e d e i t h e r b y t a k i n g K ( ? ) = [ K ( _ ) + K ( + ) ] / 2 o r b y f i n d i n g t h e p o i n t w h e r e t h e l i n e a r i n t e r p o l a n t b e t w e e n E { V ( - j K ' " ' ) } a n d K E { V ( ' , K ( + > ) } i s e q u a l t o 6/a. T h e n E { V } i s d e t e r m i n e d a t t h i s K K t e s t v a l u e a n d c o m p a r e d t o 6/a. D e p e n d i n g on t h e r e s u l t , K ( ? ) t h e n b e c o m e s t h e new u p p e r o r l o w e r b o u n d o n K'. A t t h e n e x t s t e p , t h e m e t h o d t o o b t a i n K ( ? ) i s a l t e r n a t e d ; t h i s p r o c e s s i s c o n t i n u e d u n t i l a n a c c u r a c y c r i t e r i o n |K' -K' |<e i s r e a c h e d . i + 1 i A t e a c h s t a g e , t h e q u a n t i t y E { V } i s f o u n d a s f o l l o w s : K M ( 4 1 ) E { V }= I { c o ' ( 3 , k , 1 , 1 ) v + c o ' ( 3 , k , 1 , 2 ) v K k=1 k l ( x ) k l +co' ( 3 , k , 1 , 3 ) v +co' ( 3 , k , 2 , 1 ) v ( K ) k l k , l + 1 + c o ' ( 3 , k , 2 , 2 ) v + c o ' ( 3 , k , 2 , 3 ) v } ( x ) k , l + 1 ( K ) k , l + 1 D e t e r m i n a t i o n o f a n i m p r o v e d w ( K ) c u r v e i s a c c o m p l i s h e d i n a s i m i l a r m a n n e r . F o r e a c h m e s h v a l u e K , £ i s i n c r e a s e d t h r o u g h j t h e v a l u e s x 1 , x 2 , x 3 , . . . u n t i l we h a v e : E{ V ( x ' , K ' ) } < ( p / o ) ( e - x 0 ) / f ' ( O x w h e r e K'= M a x { u ( t ) , ( l - r ) K }, u s i n g t h e new u ( ^ ) c u r v e . To 106 n a r r o w i n o n t h e s o l u t i o n , a l t e r n a t i o n o f l i n e a r i n t e r p o l a t i o n a n d i n t e r v a l - h a l v i n g i s a g a i n u s e d , u n t i l a s i m i l a r a c c u r a c y c r i t e r i o n i s a t t a i n e d . T h e q u a n t i t y E { V } i s c a l c u l a t e d u s i n g x e q u a t i o n ( 4 1 ) , w i t h c o ' ( 3 , • , • , • ) r e p l a c e d b y c o ' ( 2 , • , • , • ) t h r o u g h o u t . G i v e n t h e new u(«) a n d w ( - ) c u r v e s , we d e t e r m i n e t h e c o r r e s p o n d i n g V ( ' , 0 f u n c t i o n , a n d t h e n c h e c k t h e c o n v e r g e n c e c r i t e r i o n : ( 4 2 ) Max | u ( i ) - u ( i ) | <e, a n d Max | w ( j ) - w ( j ) | < c 2 i new o l d j new o l d w h e r e i = 1 , . . . , M a n d j = 1 , . . . , N ; "new" a n d " o l d " r e f e r t o t h e c u r r e n t a n d p r e v i o u s v a l u e s r e s p e c t i v e l y . I f t h i s c r i t e r i o n i s m e t , s o t h a t l i t t l e c h a n g e o c c u r s i n t h e p o l i c y f u n c t i o n s b e t w e e n i t e r a t i o n s , we h a v e f o u n d t h e o p t i m a l p o l i c i e s . I f n o t , a n o t h e r p o l i c y i m p r o v e m e n t i t e r a t i o n i s p e r f o r m e d , u n t i l c o n v e r g e n c e i s o b t a i n e d . ( g ) T e s t i n g t h e n u m e r i c a l m e t h o d T h i s n u m e r i c a l s c h e m e was t e s t e d u s i n g t h e s i m p l i f i e d s t o c k - r e c r u i t m e n t f u n c t i o n F ( S ) = b = c o n s t a n t , a n d c o m p a r i n g r e s u l t s w i t h t h o s e o b t a i n e d u s i n g t h e a n a l y t i c m e t h o d s o f I I . B a n d I I I . B . We p e r f o r m e d 6 t e s t r u n s , b a s e d e a c h t i m e o n t h e p a r a m e t e r s f o r t h e p r a w n f i s h e r y i n T a b l e ( I ) , b u t v a r y i n g t h e u n i t c a p i t a l c o s t , t h e d e p r e c i a t i o n r a t e a n d t h e n o i s e l e v e l . W i t h a=0, i t was f o u n d i n I I . B t h a t t h e o p t i m a l c a p a c i t y K* i s g i v e n a n a l y t i c a l l y b y K*= [ q T ] " 1 l o g [ b/x*, ] w h e r e x § = ( 1 + e ) x 0 , w i t h 9 = [ ( l - o ) / a + r ] ( 6 / c T ) . F o r t h e a>0 c a s e , t h e a n a l y t i c o p t i m a l c a p a c i t y K* was f o u n d i t e r a t i v e l y , u s i n g e q u a t i o n ( 1 2 ) 107 a n d a h a n d c a l c u l a t o r . T h e a n a l y t i c a n d n u m e r i c a l r e s u l t s , t o g e t h e r w i t h t h e p e r c e n t a g e d i f f e r e n c e b e t w e e n t h e t w o , a r e f o l l o w s : P a r a m e t e r V a l u e s A n a l y t i c K* N u m e r i c a l K* A P e r c e n t 6=0 . 4 7 0 x 1 0 6 , tf = 0 1 2 . 4 8 1 2.44 0.3 6 = 0. 4 7 0 X 1 0 6 , tf=0 .58 1 2 . 5 3 1 2 . 4 8 0.4 6 = 0 . 0 8 3 2 x 1 0 s , tf = 0 3 2 . 9 2 3 2 . 9 3 0.04 6 = 0 . 0 8 3 2 x l 0 6 , tf=0 .58 3 4 . 5 3 3 4 . 6 3 0.3 6 = 0 . 0 8 3 2 x 1 0 6 , tf = 0 r = 0 3 6 . 2 7 3 7 . 6 5 3.8 6 = 0 . 0 8 3 2 x 1 0 6 , <r = 0 .58 r = 0 4 0 . 2 4 4 2 . 1 7 4.8 C l e a r l y t h e n u m e r i c a l m e t h o d p e r f o r m e d v e r y w e l l f o r t h e f i r s t 4 r u n s . N o t e t h a t f o r t h e f i r s t p a i r o f r u n s , a l t h o u g h t h e n u m e r i c a l i n a c c u r a c y i s o f t h e same m a g n i t u d e a s t h e d i f f e r e n c e b e t w e e n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c r e s u l t s , n e v e r t h e l e s s t h e e f f e c t i s t o s h i f t b o t h o f t h e l a t t e r r e s u l t s b y r o u g h l y t h e same a m o u n t . H e n c e t h e d e s i r e d c o n c l u s i o n t h a t K * ( s t o c h a s t i c ) > K * ( d e t e r m i n i s t i c ) c a n s t i l l b e made. O b v i o u s l y , h o w e v e r , o n e c a n f e e l m o r e c o n f i d e n t w i t h t h e n u m e r i c a l r e s u l t s when t h e d i f f e r e n c e i s l a r g e r b e t w e e n t h e K* v a l u e s i n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s , a s i n r u n s 3 a n d 4. When d e p r e c i a t i o n i s o m i t t e d , s o t h a t 7 = 0 , we f i n d t h a t t h e r e s u l t s a r e s t i l l s a t i s f a c t o r y , b u t s u b s t a n t i a l l y l e s s a c c u r a t e . T h e r e s e e m s t o be a f u n d a m e n t a l p r o b l e m w i t h t h e n u m e r i c a l s c h e m e when r = 0 ; i n m o s t c a s e s we h a v e c o n s i d e r e d , c o n v e r g e n c e o f t h e p o l i c y i t e r a t i o n m e t h o d f o r 7=0 r u n s r e q u i r e d 108 c o n s i d e r a b l y m o r e i t e r a t i o n s . I n a d d i t i o n , a n o n - c o n v e r g e n t c y c l i n g p h e n o m e n o n was f o u n d i n some c i r c u m s t a n c e s , w i t h t h e p o l i c y i m p r o v e m e n t s c h e m e j u m p i n g b e t w e e n 2 p a i r s o f c o n t r o l f u n c t i o n s a t c o n s e c u t i v e i t e r a t i o n s . T h e l a t t e r e f f e c t o c c u r r e d f o r t h e o n e r = 0 c a s e we h a v e r e p o r t e d h e r e ( i n s e c t i o n E ( c ) ) , b u t t h e j u m p i n g o c c u r r e d o n l y f o r o n e o f t h e 16 p o l i c y v a l u e s b e i n g c a l c u l a t e d , a n d t h e d i f f e r e n c e b e t w e e n t h e t w o v a l u e s a t t h i s p o i n t w a s s u f f i c i e n t l y s m a l l t h a t i t d i d n o t c a u s e a n y s u b s t a n t i a l a m b i g u i t y i n t h e r e s u l t s , w h i c h a g r e e d w i t h t h o s e o b t a i n e d i n C h a p t e r I I . A l t h o u g h d i s c o n c e r t i n g f r o m a n u m e r i c a l p o i n t o f v i e w , i t i s a t l e a s t c o m f o r t i n g t o r e m i n d o n e s e l f t h a t t h e r=0 c a s e i s a n e x t r e m e , e c o n o m i c a l l y u n l i k e l y s i t u a t i o n . E. N u m e r i c a l R e s u l t s . T h e n u m e r i c a l m e t h o d s d e s c r i b e d i n t h e p r e v i o u s s e c t i o n e n a b l e u s t o o b t a i n a f u l l s o l u t i o n t o t h e j o i n t i n v e s t m e n t / e s c a p e m e n t m a n a g e m e n t p r o b l e m u n d e r u n c e r t a i n t y . F o l l o w i n g t h e a p p r o a c h o f C h a p t e r I I , we h a v e u s e d t h e g e n e r a l f o r m o f t h e p r a w n m o d e l a s t h e p r i m a r y s o u r c e o f d a t a b u t h a v e v a r i e d t h e p a r a m e t e r s t o s t u d y c o m p a r a t i v e d y n a m i c s . T h i s i s d o n e f o r c o n v e n i e n c e b u t t h e r e i s n o t h i n g i n t h e s t r u c t u r e o f t h e p r a w n m o d e l t o d e t r a c t f r o m t h e g e n e r a l a p p l i c a b i l i t y o f o u r r e s u l t s . T h e b a s e p a r a m e t e r s f o r b o t h t h e p r a w n a n d w h a l e f i s h e r i e s a r e a s i n T a b l e ( I ) , w i t h t h e v a l u e <r = 0.58 u s e d f o r t h e u n c e r t a i n t y p a r a m e t e r i n t h e p r a w n f i s h e r y ( r e p r e s e n t i n g t h e s t a n d a r d d e v i a t i o n o f t h e l o g a r i t h m o f r e c r u i t m e n t ) . T h i s v a l u e o f e was t h e m aximum l i k e l i h o o d e s t i m a t e o b t a i n e d b y f i t t i n g a l o g n o r m a l 1 09 d i s t r i b u t i o n t o r e c r u i t m e n t d a t a o f K i r k w o o d ( 1 9 8 0 ) , w i t h t h e mean v a l u e o f t h e d i s t r i b u t i o n s e t e q u a l t o t h e s a m p l e mean o f t h e d a t a ( i m p l i c i t l y a s s u m i n g n o d e p e n d e n c e o f r e c r u i t m e n t o n e s c a p e m e n t ) . O u r a p p r o a c h i n m o s t c a s e s was t o c o m p a r e t h e o p t i m a l p o l i c y f u n c t i o n s d e r i v e d f o r a f i s h e r y s u b j e c t t o t h i s f a i r l y h i g h d e g r e e o f u n c e r t a i n t y w i t h t h e c o r r e s p o n d i n g o p t i m a l p o l i c i e s i n t h e e x t r e m e c a s e o f no r a n d o m n e s s (<r=0). T h e d e t e r m i n i s t i c r e s u l t s c o r r e s p o n d t o t h o s e f o u n d i n C h a p t e r I I , b u t a r e o b t a i n e d u s i n g t h e m o r e a c c u r a t e n u m e r i c a l m e t h o d d e s c r i b e d i n I I I . D . T h e r e a r e n o q u a l i t a t i v e d i f f e r e n c e s , a n d o n l y m i n o r q u a n t i t a t i v e d i f f e r e n c e s , b e t w e e n t h e s e d e t e r m i n i s t i c r e s u l t s a n d t h o s e o f C h a p t e r I I . I n some c a s e s we h a v e a l s o c o n s i d e r e d o t h e r l e v e l s o f u n c e r t a i n t y ( b e s i d e s <y = 0 . 5 8 ) , a n d h a v e t h u s b e e n a b l e t o g a i n i n f o r m a t i o n r e g a r d i n g t h e r a t e a t w h i c h t h e o p t i m a l p o l i c i e s c h a n g e a s t h e d e g r e e o f u n c e r t a i n t y v a r i e s . T h e d e t e r m i n i s t i c a n d s t o c h a s t i c p o l i c i e s h a v e b e e n c o m p a r e d u n d e r a n u m b e r o f p a r a m e t e r c o m b i n a t i o n s . I n p a r t i c u l a r we h a v e c o n s i d e r e d t h e r o l e o f : ( i ) t h e i n t r i n s i c b i o m a s s g r o w t h r a t e , ( i i ) t h e c a p i t a l c o s t ( r e l a t i v e t o v a r i a b l e c o s t s ) , ( i i i ) t h e d i s c o u n t r a t e , a n d ( i v ) t h e d e p r e c i a t i o n r a t e . We b e g i n b y d e s c r i b i n g t h e b e h a v i o u r o f a n o p t i m a l f i s h e r y o p e r a t i n g i n a s t o c h a s t i c e n v i r o n m e n t . F i g u r e 15 d e p i c t s a c c u r a t e p i e c e w i s e l i n e a r a p p r o x i m a t i o n s t o t h e o p t i m a l p o l i c y f u n c t i o n s h ( S ) a n d s ( K ) f o r t h e b a s e - c a s e p r a w n f i s h e r y w i t h <r = 0 . 5 8 . A s i n t h e d e t e r m i n i s t i c c a s e , a n d a s d e d u c e d i n I I I . C , t h e c u r v e h ( S ) r e p r e s e n t s t h e o p t i m a l c a p a c i t y f o r n e x t s e a s o n 1 10 g i v e n e s c a p e m e n t S t h i s y e a r . I n o t h e r w o r d s , s i n c e i t i s n e c e s s a r y t o o r d e r new c a p a c i t y o n e s e a s o n i n a d v a n c e o f d e l i v e r y , o n e s h o u l d p u r c h a s e M a x { h ( S ) - ( 1 - r ) K , 0 } i n new f l e e t c a p a c i t y , e v e n t h o u g h i n t h e s t o c h a s t i c c a s e r e c r u i t m e n t n e x t s e a s o n c a n o n l y b e p r e d i c t e d r o u g h l y ( i . e . i n t h e m e a n ) a t t h e t i m e o f o r d e r i n g t h e i n v e s t m e n t . T h e s ( K ) o p t i m a l e s c a p e m e n t c u r v e i s e n t i r e l y a n a l o g o u s t o i t s d e t e r m i n i s t i c c o u n t e r p a r t ; g i v e n c a p i t a l s t o c k K, t h e o b j e c t i v e i s t o h a r v e s t down t o t h e e s c a p e m e n t s ( K ) , o r a s c l o s e t o t h a t t a r g e t a s p o s s i b l e . I f R< s ( K ) , n o h a r v e s t i n g t a k e s p l a c e . T h e p o i n t ( 4 . 3 x 1 0 s , 7 . 7 5 ) m a r k e d i n F i g u r e 15 r e p r e s e n t s t h e e q u i l i b r i u m p o i n t f o r t h e f i s h e r y i_f t h e r e w e r e no r a n d o m f l u c t u a t i o n s . A s p o i n t e d o u t b y May e t a l ( 1 9 7 8 ) a n d S p u l b e r ( 1 9 7 8 ) , t h e d e t e r m i n i s t i c e q u i l i b r i u m p o i n t t r a n s l a t e s i n t o t h e s t e a d y s t a t e p r o b a b i l i t y d i s t r i b u t i o n o v e r s t o c k l e v e l s i n t h e s t o c h a s t i c c a s e . I n o u r 2 - d i m e n s i o n a l m o d e l , a n y s t e a d y - s t a t e w o u l d a l s o be 2 - d i m e n s i o n a l . T h e e x i s t e n c e o f s u c h a n e q u i l i b r i u m d i s t r i b u t i o n f o r t h e o p t i m a l l y - m a n a g e d f i s h e r y h a s n o t b e e n e x a m i n e d f o r o u r m o d e l . I n s t e a d we h a v e s i m u l a t e d a s t e a d y - s t a t e d i s t r i b u t i o n b y p l o t t i n g t h e e n d p o i n t s o f a l a r g e n u m b e r ( 1 6 0 ) o f 4 0 - y e a r s a m p l e p a t h s e m a n a t i n g f r o m t h e q u a s i - e q u i l i b r i u m p o i n t ; t h e s e e n d p o i n t s a r e d e p i c t e d i n f i g u r e 1 5 . ( N o t e t h a t t h e b i o m a s s a n d c a p a c i t y s c a l e s h a v e b e e n c h a n g e d s u b s t a n t i a l l y f r o m p r e v i o u s f i g u r e s i n o r d e r t o i n c l u d e a l a r g e r r a n g e o f b i o m a s s v a l u e s a n d t o show i n m o r e d e t a i l t h e f l u c t u a t i o n s i n c a p a c i t y . ) T h e c l o u d o f p o i n t s i n f i g u r e 15 c a n be i n t e r p r e t e d a s 111 f o l l o w s : t h e d e n s e r t h e p o i n t s i n a g i v e n r e g i o n o f t h e S-K p l a n e , t h e m o r e l i k e l y i s t h e f i s h e r y t o l i e i n t h a t r e g i o n ( i . e . t o h a v e t h a t e s c a p e m e n t a n d t h a t c a p a c i t y ) o v e r t h e l o n g t e r m . One c a n o b s e r v e a c o n s i d e r a b l e s p r e a d b o t h i n b i o m a s s a n d c a p a c i t y v a l u e s a b o u t t h e q u a s i - e q u i l i b r i u m p o i n t . T h e s p r e a d i n b i o m a s s v a l u e s i s d u e s i m p l y t o t h e s t o c h a s t i c n a t u r e o f t h e r e s o u r c e . V a r i a t i o n i n t h e c a p i t a l s t o c k , o n t h e o t h e r h a n d , i s a n i n d u c e d p h e n o m e n o n ; f l u c t u a t i o n s i n r e c r u i t m e n t l e a d d i r e c t l y t o v a r i a t i o n s i n e s c a p e m e n t , w h i c h i n t u r n c a u s e d i s p e r s i o n i n f l e e t c a p a c i t y , t h r o u g h t h e i n v e s t m e n t f u n c t i o n K * = h ( S ) . T h i s e f f e c t w i l l b e e v e n m o r e p r o n o u n c e d w i t h s l o w e r - g r o w i n g s t o c k s , w h e r e p a r t i c u l a r l y g o o d o r b a d e s c a p e m e n t l e v e l s w i l l t e n d t o i n f l u e n c e t h e f i s h e r y f o r l o n g e r p e r i o d s o f t i m e , a n d w i l l t h e r e f o r e h a v e a g r e a t e r e f f e c t o n d e s i r e d f l e e t c a p a c i t y . S i n c e t h e r e s o u r c e i s f a i r l y f a s t - g r o w i n g ( a = 4 2 ) , f e w p o i n t s a r e f o u n d a t l o w [ S < s ( K ) ] e s c a p e m e n t l e v e l s . I n f a c t t h e d i s t r i b u t i o n o f p o i n t s r e s e m b l e s a l o g n o r m a l d i s t r i b u t i o n i n t h e S - d i r e c t i o n , t r u n c a t e d b e l o w a t S = s ( K ) . T h i s i s u n l i k e l y t o be t h e c a s e p r e c i s e l y , h o w e v e r , s i n c e ( S , K ) r a t h e r t h a n ( R , K ) p o i n t s h a v e b e e n p l o t t e d , r e s u l t i n g i n a t i g h t e r d i s t r i b u t i o n ( l o g n o r m a l l y d i s t r i b u t e d R - v a l u e s w h i c h a r e s u f f i c i e n t l y h i g h a r e r e d u c e d b y f i s h i n g p r e s s u r e t o r e l a t i v e l y l o w e r S - v a l u e s ) . I n a d d i t i o n , t h e s p r e a d o f p o i n t s i n t h e S - d i r e c t i o n c a n be s e e n t o b e s m a l l e r a t h i g h c a p a c i t i e s , s i n c e i n t h i s c a s e f i s h i n g e f f o r t i s s u f f i c i e n t t o r e d u c e e v e n h i g h r e c r u i t m e n t s down t o e s c a p e m e n t s f a i r l y n e a r t h e s ( K ) c u r v e . S i n c e t h e e n d - p o i n t s o f 4 0 - y e a r s a m p l e p a t h s h a v e b e e n 1 12 p l o t t e d , o n e w o u l d e x p e c t t h a t t h e s i m u l a t e d d i s t r i b u t i o n s o p r o d u c e d s h o u l d be i n d e p e n d e n t o f t h e s t a r t i n g p o i n t . T h i s i s v e r i f i e d i n f i g u r e s 16 a n d 1 7 , w h e r e s i m i l a r s i m u l a t e d d i s t r i b u t i o n s h a v e b e e n p l o t t e d , b u t o r i g i n a t i n g a t ( 2 . 0 x 1 0 6 , 6 . 0 ) a n d ( 6 . O x 1 0 6 , 9 . 0 ) r e s p e c t i v e l y . No s i g n i f i c a n t d i f f e r e n c e s c a n b e s e e n ; h e n c e o n e c a n c o n c l u d e t h a t f i g u r e 15 p r o v i d e s a r e a s o n a b l e a p p r o x i m a t i o n t o t h e s t e a d y - s t a t e d i s t r i b u t i o n . F i g u r e 18 s h o w s t h e e f f e c t o f r e d u c i n g t h e n o i s e l e v e l f r o m <y=0.58 t o <r=0.2 i n t h e p r a w n f i s h e r y . A s e x p e c t e d , t h e s t e a d y s t a t e d i s t r i b u t i o n c o l l a p s e s t o w i t h i n a m u ch s m a l l e r n e i g h b o u r h o o d o f t h e q u a s i - e q u i l i b r i u m p o i n t . One w o u l d e x p e c t s t o c h a s t i c e f f e c t s t o be r e l a t i v e l y u n i m p o r t a n t a t s u c h l o w < r - v a l u e s ; h o w e v e r , a s s h a l l be s e e n , t h e v a l u e s o f c w h i c h c a n b e c o n s i d e r e d " l o w " d e p e n d s o n t h e o t h e r f i s h e r y p a r a m e t e r s . I n t h e w h a l e f i s h e r y , <y = 0.2 c a n be a s u b s t a n t i a l l e v e l o f n o i s e . T o s e e m o r e v i v i d l y t h e a c t u a l p r o c e s s o f m a n a g i n g a f i s h e r y i n a s t o c h a s t i c e n v i r o n m e n t , we h a v e p l o t t e d i n f i g u r e 1 9 ( a ) a s e t o f 8 2 0 - y e a r s a m p l e p a t h s f o r a n o p t i m a l l y  m a n a g e d b a s e c a s e p r a w n f i s h e r y , w i t h t h e r e c r u i t m e n t c h o s e n e a c h y e a r f r o m a l o g n o r m a l ( c = 0 . 5 8 ) d e n s i t y c e n t r e d on F ( S ) , w h e r e S i s t h e p r e v i o u s y e a r ' s e s c a p e m e n t . L i n e s a r e d r a w n j o i n i n g s u c c e s s i v e ( S , K ) p o i n t s , b e g i n n i n g a t t h e q u a s i - e q u i l i b r i u m p o i n t . A s a b o v e , i t c a n be s e e n t h a t o p t i m a l r i s k - n e u t r a l m a n a g e m e n t r e s u l t s i n c o n s i d e r a b l e v a r i a t i o n i n t h e f l e e t c a p a c i t y , a s w e l l a s t h e b i o m a s s , o v e r t i m e . F i g u r e 1 9 ( b ) s h o w s t h i s e f f e c t i n m o r e d e t a i l , f o r a s i n g l e 1 13 1 5 - y e a r r e a l i z a t i o n o f t h e f i s h e r y ' s d e v e l o p m e n t . T h e 4 p r o c e s s e s o f ( s t o c h a s t i c ) r e c r u i t m e n t , h a r v e s t i n g , i n v e s t m e n t a n d d e p r e c i a t i o n c o m b i n e t o d e t e r m i n e t h e m o v e m e n t f r o m o n e ( S , K ) p o i n t t o t h e n e x t , g o v e r n e d b y t h e p o l i c y c u r v e s . S i n c e t h e r e s o u r c e i s f a s t - g r o w i n g a n d h i g h l y v a r i a b l e i n t h i s e x a m p l e , t h e r e i s n o a p p a r e n t t r e n d t o r e t u r n t o t h e q u a s i - e q u i 1 i b r i u m p o i n t . We now e x a m i n e t h e e x t e n t t o w h i c h e a c h o f t h e f i s h e r y p a r a m e t e r s o u t l i n e d a b o v e s e r v e s t o d e t e r m i n e how u n c e r t a i n t y a f f e c t s t h e f i s h e r y ' s o p t i m a l m a n a g e m e n t . I n p a r t i c u l a r we w i s h t o s t u d y w h e t h e r i n v e s t m e n t i n c r e a s e s o r d e c r e a s e s w i t h i n c r e a s i n g u n c e r t a i n t y , a n d how t h e p a r a m e t e r s o f t h e f i s h e r y a f f e c t t h i s b e h a v i o u r . ( a ) T h e c o s t o f c a p i t a l . C o n s i d e r F i g u r e 20 w h e r e t h e o p t i m a l c a p a c i t y c u r v e s h ( S ) a r e s h o w n f o r 10 d i f f e r e n t c a s e s , w i t h a=42 t h r o u g h o u t b u t w i t h v a r y i n g l e v e l s o f u n c e r t a i n t y a n d u n i t c a p i t a l c o s t : 6 = $0 . 1 1 7 5 x 1 0 6 ; <r = 0 , 0 . 58 6 = $ 0 . 2 3 5 0 x 1 0 6 ; a=0,0.29,0.58 6 = $0 . 4 7 0 0 x 1 0 6 ; <r = 0 , 0 . 58 , 0 . 80 6 = $0 . 7 0 5 0 x 1 0 6 ; <y = 0 , 0 . 5 8 To c o n c e n t r a t e o n t h e i n v e s t m e n t p r o b l e m , we h a v e o m i t t e d t h e o p t i m a l e s c a p e m e n t c u r v e s s ( K ) f r o m f i g u r e 2 0 . I n f a c t , t h e d i f f e r e n c e b e t w e e n o p t i m a l e s c a p e m e n t s i n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s was n e g l i g i b l e . T h i s i s i n d i c a t e d i n t h e f o l l o w i n g n u m e r i c a l r e s u l t s , w h e r e t h e o p t i m a l v a l u e s o f s ( K ) 1 14 a n d h ( S ) a r e g i v e n [ a t t h e m e s h p o i n t s ] f o r t h e b a s e c a s e p r a w n f i s h e r y w i t h e=0 a n d <r=0.58 ( n o t e t h a t b i o m a s s i s m e a s u r e d i n m i l l i o n s o f k i l o g r a m s a n d c a p a c i t y i n " s t a n d a r d i z e d v e s s e l s " ) . s h ( S ) u = o ) h ( S ) u = o . 5 8 ) 0.2 0 .0 0 .0 0.5 0 .0 0 .0 1 .0 0 .861 0 .928 2.0 5 .375 5 .310 3.0 7 .215 6 .850 4.5 8 .363 7 .809 7.0 9 . 1 50 8 .670 2 0 . 0 10 .076 9 .450 K s ( K ) U = 0) s ( K ) ( t f = 0 .58) 0.0 1 .21 1 .21 3.0 1 .23 1 .22 6.0 1 .35 1 .35 9.0 1 .46 1 .46 12.0 1 .56 1 .56 15.0 1 .65 1 .64 18.0 1 .73 1 .71 21 .0 1 .81 1 .77 T h i s i n s e n s i t i v i t y o f t h e o p t i m a l e s c a p e m e n t l e v e l t o t h e d e g r e e o f u n c e r t a i n t y was f o u n d f o r m o s t o f o u r p a r a m e t e r c o m b i n a t i o n s a n d i s i n a c c o r d a n c e w i t h t h e r e s u l t s o f o t h e r r e s e a r c h e r s . T h e r e a r e , h o w e v e r c a s e s w h e r e u n c e r t a i n t y d o e s a f f e c t t h e s ( K ) c u r v e s : t h e s e a r e d i s c u s s e d b e l o w . F o r t h e r e m a i n d e r o f t h i s c h a p t e r , s ( K ) c u r v e s a r e s h o w n o n l y i n s u c h c a s e s . I t was s e e n i n C h a p t e r I I t h a t t h e i m p o r t a n t c o s t p a r a m e t e r i n t h e i n v e s t m e n t p r o b l e m i s n e i t h e r c a p i t a l c o s t n o r o p e r a t i n g c o s t a l o n e , b u t r a t h e r t h e r a t i o o f t h e t w o . S p e c i f i c a l l y , a u s e f u l q u a n t i t y t o s t u d y a p p e a r s t o be 6 / c T , t h e r a t i o o f u n i t c a p i t a l c o s t t o m a x i m u m y e a r l y o p e r a t i n g c o s t ( p e r u n i t o f c a p i t a l ) . I n a s e n s e , t h i s m e a s u r e s t h e c a p i t a l i n t e n s i t y o f t h e 1 1 5 f l e e t , s i n c e t h e p r e s e n t v a l u e o f t o t a l f l e e t c o s t s p e r u n i t o f c a p a c i t y ( a s s u m i n g f u l l u t i l i z a t i o n o f c a p i t a l ) i s 6 +?'CT= c T [ ( 6 / c T ) w h e r e p= <*/[ 1-<x( 1 - 7 ) ] . T h e i m p o r t a n t a s p e c t o f F i g u r e 20 i s t h e r e l a t i v e p o s i t i o n o f t h e h ( S ) c u r v e b e t w e e n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s , a s t h i s r a t i o o f c o s t s , 6 / c T , v a r i e s . H o w e v e r , s i n c e c T i s f i x e d h e r e , i t s u f f i c e s t o s p e a k i n t e r m s o f v a r y i n g t h e c a p i t a l c o s t . I t c a n b e s e e n t h a t a t l o w c a p i t a l c o s t s , t h e o p t i m a l c a p a c i t y i s s u b s t a n t i a l l y h i g h e r i n a f l u c t u a t i n g e n v i r o n m e n t . A s t h e u n i t c a p i t a l c o s t i n c r e a s e s , t h e d i f f e r e n c e b e t w e e n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s d e c r e a s e s , s o t h a t a t 6 = $ 0 . 2 3 5 m i l l i o n ( 6 / c T = 5 . 6 5 ) , t h e o p t i m a l c a p a c i t y w i t h <r=0.58 s t i l l l i e s a b o v e i t s d e t e r m i n i s t i c c o u n t e r p a r t , b u t t h i s r e l a t i o n s h i p i s r e v e r s e d a t l o w e r l e v e l s o f u n c e r t a i n t y (<r = 0 . 2 9 ) . A s t h e u n i t c a p i t a l c o s t i s i n c r e a s e d f u r t h e r , t o 6 = $ 0 . 4 7 m i l l i o n ( t h e b a s e c a s e , 6 / c T = 1 1 . 3 ) o r 6 = $ 0 . 7 0 5 m i l l i o n ( 6 / c T = 1 6 . 9 5 ) , t h e o p t i m a l c a p a c i t y d e c r e a s e s w i t h i n c r e a s i n g u n c e r t a i n t y . T h e s m a l l c r o s s o v e r i n t h e ( 6 = $ 0 . 7 0 5 m i l l i o n ) h ( S ) c u r v e s , a s y m p t o m o f t h e c o m p l e x i n t e r a c t i o n b e t w e e n t h e e s c a p e m e n t l e v e l , t h e d e g r e e o f u n c e r t a i n t y a n d o t h e r m o d e l p a r a m e t e r s , i s d i s c u s s e d b e l o w . F i g u r e ( 2 0 ) a l s o i n d i c a t e s t h a t , u n d e r u n c e r t a i n t y , t h e o p t i m a l e s c a p e m e n t c u r v e s s ( K ) move t o l o w e r e s c a p e m e n t s a s 6 i n c r e a s e s ; t h i s r e s u l t c o r r e s p o n d s t o t h a t o f C h a p t e r I I . H o w e v e r , t h e d i f f e r e n c e b e t w e e n o p t i m a l e s c a p e m e n t s i n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c r e s u l t s was n e g l i g i b l e i n t h i s c a s e . F i g u r e 21 s h o w s s i m i l a r r e s u l t s , a g a i n v a r y i n g t h e l e v e l o f u n c e r t a i n t y a n d t h e u n i t c a p i t a l c o s t , b u t t h i s t i m e f o r a 1 16 s l o w e r g r o w i n g ( a = 1 4 ) f i s h e r y . T h e k e y r e s u l t s a r e u n c h a n g e d i n t h i s c a s e ; a t l o w r e l a t i v e c a p i t a l c o s t s ( 6 / c T = 2 . 0 ) , i n v e s t m e n t s h o u l d b e h i g h e r u n d e r u n c e r t a i n t y w h e r e a s t h i s i s r e v e r s e d i f t h e f i s h e r y i s s u b j e c t t o h i g h u n i t c a p i t a l c o s t s . A t t h e i n t e r m e d i a t e l e v e l 6 = $ 0 . 2 3 5 m i l l i o n , t h e o p t i m a l h ( S ) c u r v e s i n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s e x h i b i t a c r o s s o v e r , a s i n t h e 6 = $ 0 . 7 0 5 m i l l i o n c a s e o f f i g u r e 2 0 , s o t h a t t h e i n t r o d u c t i o n o f r a n d o m n e s s i n c r e a s e s o p t i m a l i n v e s t m e n t a t l o w b i o m a s s l e v e l s w h i l e d e c r e a s i n g i n v e s t m e n t a t h i g h e r s t o c k s i z e s . T h i s r e s u l t c a n be e x p l a i n e d b y c o n s i d e r i n g t w o o p p o s i n g e f f e c t s : t h e " d o w n s i d e " p r o b l e m o f s u f f e r i n g i d l e e x c e s s c a p a c i t y i n b a d y e a r s , a n d t h e " u p s i d e " p r o b l e m o f l a c k i n g s u f f i c i e n t c a p a c i t y t o t a k e f u l l a d v a n t a g e o f g o o d y e a r s . W i t h t h e s e p a r t i c u l a r p a r a m e t e r c o m b i n a t i o n s , i t a p p e a r s t h a t t h e r o l e o f u n c e r t a i n t y d e p e n d s o n t h e e s c a p e m e n t l e v e l ; a t h i g h b i o m a s s l e v e l s , t h e v a r i a n c e i n r e c r u i t m e n t i s a l s o h i g h , s o t h a t t h e d o w n s i d e p r o b l e m p r e d o m i n a t e s . H e n c e i n v e s t m e n t i s l o w e r u n d e r u n c e r t a i n t y ; t h e b a l a n c i n g a c t t i l t s t o w a r d s c a u t i o n i n i n v e s t m e n t . On t h e o t h e r h a n d , i f e s c a p e m e n t i s a l r e a d y r e l a t i v e l y l o w , a n d t h e s t o c k t e n d s ( i n t h e m e a n ) t o g r o w r e a s o n a b l y r a p i d l y , t h e n t h e p o t e n t i a l b e n e f i t s t o e x t r a ( " u p s i d e " ) i n v e s t m e n t o u t w e i g h t h e d o w n s i d e r i s k . I n c r e a s e d i n v e s t m e n t u n d e r u n c e r t a i n t y b e c o m e s o p t i m a l . W h e r e t h e c r o s s o v e r w i l l o c c u r , i f i t d o e s a t a l l , s e e m s r a t h e r d i f f i c u l t t o p r e d i c t . I n d e e d i n m o s t c a s e s w h e r e a c r o s s o v e r a p p e a r s , t h e d i f f e r e n c e b e t w e e n t h e h ( S ) c u r v e s i n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s t e n d s t o be s m a l l . 1 1 7 W i t h r e g a r d t o t h e s ( K ) p o l i c y f u n c t i o n s , t h e s l o w e r - g r o w i n g a=14 f i s h e r y s h o w s a s l i g h t l y g r e a t e r e f f e c t o f r a n d o m n e s s o n t h e o p t i m a l e s c a p e m e n t l e v e l s t h a n i n t h e a =42 c a s e : t h i s i s i n d i c a t e d f o r t h e a = 1 4 , 6 = $ 0 . 0 8 3 2 x 1 0 6 f i s h e r y i n f i g u r e 23 a n d i s d i s c u s s e d f u r t h e r b e l o w . T o s u m m a r i z e t h e i n t e r p l a y b e t w e e n t h e l e v e l o f u n c e r t a i n t y a n d t h e u n i t c a p i t a l c o s t i t c a n b e s a i d t h a t , c e t e r i s p a r i b u s , t h e o p t i m a l f l e e t c a p a c i t y i n c r e a s e s w i t h u n c e r t a i n t y g i v e n a r e l a t i v e l y l o w r a t i o o f c a p i t a l t o o p e r a t i n g c o s t s . T h e b a l a n c e i n t h i s c a s e i s t i l t e d t o w a r d s o v e r c o m i n g t h e u p s i d e p r o b l e m b y i n v e s t i n g i n e x t r a c a p a c i t y a t r e l a t i v e l y l o w c o s t i n o r d e r t o b e n e f i t f r o m e x c e p t i o n a l l y h i g h r e c r u i t m e n t s . H o w e v e r , i f u n i t c a p i t a l c o s t s a r e s u f f i c i e n t l y h i g h r e l a t i v e t o o p e r a t i n g c o s t s , i n v e s t m e n t w i l l d e c r e a s e w i t h t h e l e v e l o f u n c e r t a i n t y . I n t h i s c a s e t h e d o w n s i d e r i s k o f m o r e f r e q u e n t b a d y e a r s ( w h e n t h e r e i s l i t t l e o r n o r e t u r n o n t h e e x p e n s i v e i n v e s t m e n t ) now o u t w e i g h s t h e a d v a n t a g e s o f h a v i n g e x t r a c a p i t a l a v a i l a b l e t o p r o f i t f r o m g o o d y e a r s . ( b ) T h e i n t r i n s i c b i o m a s s g r o w t h r a t e . F i g u r e 22 s h o w s t h e o p t i m a l c a p a c i t y f u n c t i o n s f o r e a c h o f c = 0 a n d <J-=0.58 a n d f o r e a c h o f t h e g r o w t h r a t e p a r a m e t e r s a= 14, 4 2 , 1 4 0 , 5 6 0 , w i t h o t h e r p a r a m e t e r s a s i n t h e b a s e c a s e f i s h e r y . ( T h e v a l u e a = 5 6 0 , c o r r e s p o n d i n g t o a v e r y h i g h g r o w t h r a t e , was c h o s e n t o a p p r o x i m a t e a s i t u a t i o n o f i n d e p e n d e n c e b e t w e e n r e c r u i t m e n t a n d e s c a p e m e n t . ) E x a m i n i n g F i g u r e 2 2 , o n e c a n o b s e r v e a u n i f o r m p r o g r e s s i o n 1 18 f r o m h i g h t o l o w g r o w t h r a t e s . When a = i n f i n i t y , we h a v e s h o w n i n I I I . B t h a t t h e o p t i m a l c a p a c i t y m u s t i n c r e a s e w i t h t h e l e v e l o f u n c e r t a i n t y . T h e c a s e a = 5 6 0 f o l l o w s t h i s r e s u l t , a t l e a s t f o r s t o c k l e v e l s S > 1 . 3 X 1 0 6 , w h e r e t h e o p t i m a l c a p a c i t y a t tf=0.58 i s s l i g h t l y h i g h e r t h a n i n t h e d e t e r m i n i s t i c c a s e . H o w e v e r w i t h a = 1 4 0 , t h e o p t i m a l c a p a c i t y i s l o w e r u n d e r u n c e r t a i n t y , a n d t h i s e f f e c t i n c r e a s e s a s t h e g r o w t h r a t e i s d e c r e a s e d t o a=42 a n d t h e n t o ' a = 1 4 . ( A s b e f o r e , i n e a c h c a s e , t h e r e i s n e g l i g i b l e d i f f e r e n c e b e t w e e n t h e s t o c h a s t i c a n d d e t e r m i n i s t i c o p t i m a l e s c a p e m e n t s . ) I f t h e u n i t c a p i t a l c o s t i s r e d u c e d t o $ 0 . 0 8 3 2 X 1 0 6 , r e s u l t s r e m a i n q u a l i t a t i v e l y u n c h a n g e d . F i g u r e 23 s h o w s t h e d e t e r m i n i s t i c (tf = 0) a n d s t o c h a s t i c (<y = 0.58) o p t i m a l p o l i c y f u n c t i o n s f o r t h e t w o c a s e s a=14 a n d a = 3 . 8 2 , w i t h t h i s l o w e r c a p i t a l c o s t . When n a t u r a l m o r t a l i t y i s t a k e n i n t o a c c o u n t t h e l a t t e r a - v a l u e c o r r e s p o n d s t o a m aximum n e t g r o w t h r a t e o f 4 p e r c e n t a n n u a l l y . T h i s v a l u e was c h o s e n t o e q u a l t h a t o f t h e w h a l e f i s h e r y , f o r c o m p a r a t i v e p u r p o s e s d i s c u s s e d b e l o w . I n t h i s c a s e , e v e n a t s u c h a l o w a - v a l u e , t h e u n i t c a p i t a l c o s t i s s u f f i c i e n t l y l o w t h a t t h e o p t i m a l c a p a c i t y i s p o s i t i v e f o r b o t h t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s , a n d i s h i g h e r u n d e r u n c e r t a i n t y , a p a r t f r o m a v e r y s m a l l c r o s s o v e r r e g i o n . A s t h e b i o m a s s g r o w t h r a t e i s i n c r e a s e d t o a = 1 4 , t h e o p t i m a l c a p a c i t y u n d e r u n c e r t a i n t y r i s e s e v e n f u r t h e r a b o v e i t s d e t e r m i n i s t i c c o u n t e r p a r t . W h i l e o n e c a n d e t e r m i n e o p t i m a l p o l i c i e s f o r t h e a = 3 . 8 2 f i s h e r y , i n f a c t t h i s f i s h e r y w o u l d n o t b e e c o n o m i c a l l y 119 s u s t a i n a b l e i n t h e d e t e r m i n i s t i c c a s e ; t h e e q u i l i b r i u m o c c u r s a t R = S = 0 . 2 8 x 1 0 6 , w h i c h i s w e l l b e l o w b i o n o m i c e q u i l i b r i u m . H e n c e n o f i s h e r y w i l l e x i s t u n l e s s f o r some r e a s o n t h e b i o m a s s i s i n c r e a s e d t o a b o v e 3.15 m i l l i o n . I n t h e s t o c h a s t i c c a s e t h e r e i s a l w a y s a p o s s i b i l i t y o f t h e b i o m a s s r e a c h i n g a s u f f i c i e n t l y h i g h l e v e l t o w a r r a n t i n v e s t m e n t i n t h e f i s h e r y . I n t h e e x a m p l e s h o w n h e r e , i f t h e s t o c k i s a t i t s d e t e r m i n i s t i c e q u i l i b r i u m ( w h i c h w o u l d o n l y o c c u r i n t h e m e a n ) , s t o c k s w o u l d o n l y i n c r e a s e t o l e v e l s w a r r a n t i n g i n v e s t m e n t a p p r o x i m a t e l y o n c e i n e v e r y 1 / P r { R > 3.2X10 6 } = 1 / P r { z > 4 . 4 9 } =270000 y e a r s - n o t a p a r t i c u l a r l y p r o m i s i n g f i s h e r y . H o w e v e r , t h i s o b v i o u s l y r e p r e s e n t s a n e x t r e m e c a s e o f t h e g e n e r a l p r o p o s i t i o n t h a t f i s h e r i e s c a n e x i s t w h i c h s h o u l d o n l y b e d e v e l o p e d p e r i o d i c a l l y , w hen e x c e p t i o n a l l y h i g h r e c r u i t m e n t s a p p e a r . W i t h r e g a r d t o o p t i m a l e s c a p e m e n t l e v e l s , w hen a = 3 . 8 2 we h a v e s ( K ) = x 0 ( f o r a l l K) i n t h e d e t e r m i n i s t i c c a s e , a n d s ( K ) > x 0 f o r <r = 0.58 . T h e a=14 o p t i m a l e s c a p e m e n t c u r v e s e x h i b i t r a t h e r c o m p l i c a t e d b e h a v i o u r , w i t h t h e s t o c h a s t i c s ( K ) c u r v e l y i n g a b o v e t h e e=0 c u r v e , e x c e p t i n a n i n t e r m e d i a t e r a n g e b e t w e e n K=10 a n d K = 2 4 . T h i s i n t e r m e d i a t e p h a s e a p p e a r s t o be d u e t o t h e c o n s i d e r a b l e v a r i a t i o n i n t h e f l e e t i n v e s t m e n t p o l i c i e s w i t h u n c e r t a i n t y , s i n c e I * ( S , K ) e n t e r s i n t o t h e o p t i m a l i t y e q u a t i o n w h i c h d e t e r m i n e s s ( K ) . H o w e v e r i n a n y c a s e , t h e m a x i m u m d i f f e r e n c e b e t w e e n t h e s ( K ) c u r v e s w i t h a=14 i s o n l y A s = 0 . 1 5 x 1 0 6 . T h e i n t e r p l a y b e t w e e n t h e d e g r e e o f r a n d o m n e s s a n d t h e i n t r i n s i c g r o w t h r a t e c a n be e x p l a i n e d b y a p p e a l i n g t o t h e 120 " d o w n s i d e v s . u p s i d e " a r g u m e n t d i s c u s s e d a b o v e . T h e s l o w e r - g r o w i n g i s t h e r e s o u r c e s t o c k , t h e g r e a t e r i s t h e c o n n e c t i o n b e t w e e n r e c r u i t m e n t a n d t h e p r e v i o u s s e a s o n ' s e s c a p e m e n t . I n e f f e c t , t h e m emory o f t h e s y s t e m i s l o n g e r . H e n c e i f a l o w r e c r u i t m e n t o c c u r s i n a n y g i v e n y e a r , t h i s t e n d s t o p e r s i s t f o r a l o n g e r p e r i o d o f t i m e ( a l t h o u g h o f c o u r s e s t o c h a s t i c f l u c t u a t i o n s w i l l c a u s e d e v i a t i o n s f r o m t h i s t r e n d ) . T h u s t h e d o w n s i d e r i s k i s g r e a t e r t h e h i g h e r t h e l e v e l o f u n c e r t a i n t y , s i n c e t h i s r a i s e s t h e p r o b a b i l i t y o f s u f f e r i n g b a d y e a r s . On t h e o t h e r h a n d , a s l o w e r - g r o w i n g s t o c k d o e s n o t n e e d t o be h a r v e s t e d q u i c k l y i n g o o d y e a r s s i n c e a h i g h r e c r u i t m e n t w i l l t e n d t o - p e r s i s t o v e r s e v e r a l s e a s o n s , w h e r e a s i n t h e f a s t - g r o w i n g c a s e r e c r u i t m e n t i s m o r e i n d e p e n d e n t o f e s c a p e m e n t , s o t h a t a g o o d y e a r m u s t be u t i l i z e d i m m e d i a t e l y o r f o r e v e r l o s t . ( N o t e t h a t t h e k e y p a r a m e t e r t h a t d e c i d e s t h e e f f e c t i v e i n t r i n s i c g r o w t h r a t e i s a « e x p { - m T } ( s e e C h a p t e r I I ) , s o t h a t t h e n a t u r a l m o r t a l i t y r a t e i s a l s o o f i m p o r t a n c e . H o w e v e r m i s c o n s i d e r e d f i x e d h e r e . ) H e n c e t h e u p s i d e b e n e f i t s a r e l o w e r i n t h e s l o w - g r o w i n g c a s e . On b a l a n c e t h e r e f o r e , t h e s t o c h a s t i c o p t i m a l c a p a c i t y w i l l e x c e e d i t s d e t e r m i n i s t i c c o u n t e r p a r t i f a = i n f i n i t y , b u t t h i s d i f f e r e n c e w i l l d e c r e a s e a n d l i k e l y b e c o m e n e g a t i v e a s t h e i n t r i n s i c g r o w t h r a t e d e c r e a s e s . ( c ) T h e d e p r e c i a t i o n r a t e . F i g u r e 24 s h o w s t h a t i n t h e a = 4 2 c a s e , t h e d e p r e c i a t i o n r a t e p l a y s a r o l e s i m i l a r t o t h a t o f t h e u n i t c a p i t a l c o s t d e s c r i b e d a b o v e . I n t h e a b s e n c e o f d e p r e c i a t i o n ( r = 0 ) , t h e 121 o p t i m a l c a p a c i t y u n d e r u n c e r t a i n t y e x c e e d s t h a t o f t h e d e t e r m i n i s t i c c a s e . S i n c e c a p i t a l i s i n f i n i t e l y l o n g - l i v e d , t h e e f f e c t i v e y e a r l y r e n t a l c o s t o f c a p i t a l i s r e l a t i v e l y l o w . H e n c e t h e d o w n s i d e r i s k o f a n i n c r e a s e d c a p i t a l s t o c k ( a b o v e t h a t o f t h e d e t e r m i n i s t i c c a s e ) i s r e l a t i v e l y s m a l l . A s t h e r a t e o f d e p r e c i a t i o n i n c r e a s e s , t h e e f f e c t i v e t i m e h o r i z o n f o r a g i v e n u n i t o f c a p a c i t y i s s h o r t e n e d , s o t h a t t h e u p s i d e b e n e f i t s o f a n e x t r a u n i t o f c a p a c i t y a r e r e d u c e d , s i n c e t h e r e a r e l i k e l y t o be f e w e r y e a r s i n w h i c h t h e f i s h e r y c o u l d t a k e a d v a n t a g e o f a h i g h e r l e v e l o f c a p a c i t y . T h u s i t i s n o t s u r p r i s i n g t o s e e t h a t t h e o p t i m a l c a p a c i t y d e c r e a s e s w i t h u n c e r t a i n t y when r = 0 . 1 5 o r 7 = 0 . 2 0 . I n e a c h o f t h e t h r e e c a s e s , t h e o p t i m a l e s c a p e m e n t was i n s e n s i t i v e t o t h e l e v e l o f u n c e r t a i n t y . W i t h a s l o w e r g r o w i n g ( a = 1 4 ) , l o w e r c a p i t a l c o s t ( 6 = $ 0 . 0 8 3 2 X 1 0 6 , 6 / c T = 2 . 0 ) f i s h e r y , we f o u n d i n I I . E ( d ) t h a t f o r t h e d e t e r m i n i s t i c c a s e , o p t i m a l i n v e s t m e n t l e v e l s c a n a c t u a l l y i n c r e a s e w i t h t h e d e p r e c i a t i o n r a t e i f e s c a p e m e n t i s s u f f i c i e n t l y h i g h . We a r g u e d t h a t t h i s r e s u l t w a s d u e t o a n i n c e n t i v e t o h a r v e s t t h e r e s o u r c e r e l a t i v e l y q u i c k l y , b e f o r e t h e f l e e t d e p r e c i a t e s . T h i s i n c e n t i v e o u t w e i g h s t h e e f f e c t i v e i n c r e a s e i n u n i t c a p i t a l c o s t d u e t o d e p r e c i a t i o n , i f c a p i t a l i s c h e a p i n i t i a l l y . F i g u r e 25 s h o w s t h i s r e s u l t a g a i n , t h i s t i m e f o r 7 = 0 . 0 5 a n d 7 = 0 . 1 5 c a s e s . H o w e v e r , t h e same b e h a v i o u r d o e s n o t c a r r y o v e r t o t h e s t o c h a s t i c f i s h e r y ; f r o m f i g u r e 2 5 , o n e c a n s e e t h a t f o r <r = 0 . 5 8 , o p t i m a l c a p a c i t y d e c r e a s e s a s t h e d e p r e c i a t i o n r a t e i s i n c r e a s e d , f o r a l l e s c a p e m e n t v a l u e s . T h i s may be d u e t o o n e o f t w o r e a s o n s : e i t h e r ( i ) t h e s t o c h a s t i c 1 22 i n v e s t m e n t l e v e l s a r e a l r e a d y s u f f i c i e n t l y h i g h t h a t t h e r e s o u r c e c a n b e h a r v e s t e d a s r a p i d l y a s n e c e s s a r y , o r ( i i ) s t o c h a s t i c f l u c t u a t i o n s a d d s u f f i c i e n t u n p r e d i c t a b i l i t y t o t h e f i s h e r y t h a t e x t r a i n v e s t m e n t i n o r d e r t o h a r v e s t q u i c k l y i n t h e f a c e o f d e p r e c i a t i o n i s n o t w a r r a n t e d . R e s u l t s f o r t h i s f i s h e r y a l s o s u p p o r t t h e d i s c u s s i o n c o n c e r n i n g t h e b a s e c a s e f i s h e r y ; t h e l o w e r t h e d e p r e c i a t i o n r a t e , t h e m o r e l i k e l y i s i n v e s t m e n t t o b e h i g h e r u n d e r u n c e r t a i n t y . I n d e e d , we h a v e t h e r e s u l t t h a t : ( s t o c h a s t i c o p t i m a l c a p a c i t y ) - ( d e t e r m i n i s t i c o p t i m a l c a p a c i t y ) i n c r e a s e s a s t h e d e p r e c i a t i o n r a t e d e c r e a s e s . ( d ) T h e d i s c o u n t r a t e . F i g u r e 26 d e p i c t s t h e o p t i m a l p o l i c y c u r v e s w i t h a n d w i t h o u t r a n d o m n e s s f o r 4 v a l u e s o f t h e d i s c o u n t f a c t o r ( a ) a n d c o r r e s p o n d i n g d i s c o u n t r a t e ( r = [ l - c ] / o ) : o=0.80 ( r = 0 . 2 5 ) , ' o=0.90 ( r = 0 . 1 l ) , c = 0 . 9 7 ( r = 0 . 0 3 ) a n d a = 0 . 9 9 ( r = 0 . 0 l ) . T h e r e s u l t s a r e p e r h a p s m o s t i n t e r e s t i n g f o r t h e o b s e r v a t i o n t h a t t h e q u a l i t a t i v e e f f e c t s o f v a r y i n g t h e l e v e l o f u n c e r t a i n t y s e e m q u i t e i n d e p e n d e n t o f t h e r a t e o f d i s c o u n t i n g . W h i l e i t i s t r u e t h a t : ( i ) t h e r e l a t i v e d i f f e r e n c e b e t w e e n o p t i m a l d e t e r m i n i s t i c a n d s t o c h a s t i c c a p a c i t i e s d i m i n i s h e s a s t h e d i s c o u n t r a t e d e c r e a s e s t o w a r d s z e r o , a n d ( i i ) a s o i n c r e a s e s , t h e r e g i o n o f e s c a p e m e n t s f o r w h i c h i n v e s t m e n t s h o u l d b e g r e a t e r u n d e r u n c e r t a i n t y b r o a d e n s , n e v e r t h e l e s s t h e o p t i m a l s t o c h a s t i c c a p a c i t y i s n e v e r s u b s t a n t i a l l y a b o v e i t s d e t e r m i n i s t i c c o u n t e r p a r t . I t a p p e a r s t h a t , a t l e a s t f o r t h e s e p a r a m e t e r 1 23 c o m b i n a t i o n s , t h e r e i s l i t t l e i n t e r p l a y b e t w e e n t h e d i s c o u n t r a t e a n d t h e l e v e l o f u n c e r t a i n t y . H o w e v e r , t h e r o l e o f t h e d i s c o u n t r a t e i n r e s o u r c e m a n a g e m e n t i s a c o m p l i c a t e d o n e ; i f i t a f f e c t s o t h e r e c o n o m i c p a r a m e t e r s ( s u c h a s t h e c o s t o f c a p i t a l ) , i t s i n t e r a c t i o n w i t h t h e l e v e l o f u n c e r t a i n t y may b e c o n s i d e r a b l y c h a n g e d . H e n c e o u r r e s u l t s h e r e s h o u l d n o t b e e x t r a p o l a t e d t o o f a r . ( e ) R i c k e r S t o c k - r e c r u i t m e n t I f i n s t e a d o f u s i n g B e v e r t o n - H o l t p o p u l a t i o n d y n a m i c s , we a d o p t a R i c k e r r e p r o d u c t i o n f u n c t i o n , we saw i n I I . E ( f ) t h a t t h e o p t i m a l i n v e s t m e n t c u r v e h ( S ) m i m i c s t h e R i c k e r f o r m , r i s i n g t o a p e a k a n d t h e n d e c l i n i n g t o z e r o . A s i n d i c a t e d i n F i g u r e 2 7 , t h e i n t r o d u c t i o n o f u n c e r t a i n t y ( t f = 0 . 5 8 ) d o e s n o t c h a n g e ' t h i s q u a l i t a t i v e b e h a v i o u r . I n t e r m s o f t h e f u t u r e o f t h e f i s h s t o c k , v e r y l a r g e e s c a p e m e n t s a r e a s b a d a s v e r y s m a l l o n e s : i f K=0 b u t e s c a p e m e n t S > 1 4 . 0 X 1 0 6 , t h e e x p e c t e d v a l u e o f a u n i t o f i n v e s t m e n t i s l e s s t h a n i t s c o s t , a n d h e n c e I * = 0 . C o m p a r i n g t h e d e t e r m i n i s t i c a n d s t o c h a s t i c o p t i m a l c a p a c i t y f u n c t i o n s , we s e e t h a t i f t h e f u t u r e o f t h e f i s h s t o c k i s r e l a t i v e l y b r i g h t ( 3 . 2 5 x l 0 6 < S < 9 . 7 5 X 1 0 6 ) t h e n i n v e s t m e n t i s l o w e r u n d e r u n c e r t a i n t y , w h i l e t h e r e v e r s e i s t r u e i f e x p e c t e d f u t u r e s t o c k s i z e s a r e r e l a t i v e l y s m a l l . T h i s i s e q u i v a l e n t t o a s i n g l e c r o s s o v e r i n t h e h ( S ) c u r v e s f o r t h e B e v e r t o n - H o l t c a s e ( f o r e x a m p l e , c o m p a r e t h e a = 1 4 , 6=0 . 2 3 5 x 1 ( 3 6 r e s u l t s s h o w n i n F i g u r e 2 1 ) , a n d h e n c e c a n be e x p l a i n e d b y t h e same r e a s o n i n g a s a b o v e . 1 24 ( f ) T h e w h a l e f i s h e r y . T o t h i s p o i n t i n o u r s t o c h a s t i c r e s u l t s we h a v e c o n c e n t r a t e d o n m o d i f i c a t i o n s o f t h e p r a w n f i s h e r y d a t a . F o r c o m p l e t e n e s s we now l o o k a t t h e e f f e c t o f u n c e r t a i n t y on t h e b a s e - c a s e w h a l i n g f i s h e r y . F i g u r e 28 s h o w s t h e o p t i m a l c a p a c i t y a n d e s c a p e m e n t c u r v e s ( d e r i v e d u s i n g a n M = N = 1 2 m e s h ) f o r t h e c a s e s o f tf=0, c=0.1 a n d a=0.2, w h i c h c o v e r s a r e a s o n a b l e r a n g e o f u n c e r t a i n t y f o r t h e a g g r e g a t e d w h a l e f i s h e r y . I t c a n be s e e n t h a t i n c r e a s i n g t h e l e v e l o f u n c e r t a i n t y d e c r e a s e s t h e o p t i m a l c a p a c i t y f o r l o w s t o c k s i z e s [ S < 3 6 X 1 0 4 ] b u t i n c r e a s e s o p t i m a l c a p a c i t y f o r l a r g e r b i o m a s s l e v e l s . I n o t h e r w o r d s i n v e s t m e n t u n d e r u n c e r t a i n t y s h o u l d r e s p o n d t o t h e s t a t e o f t h e f i s h e r y : i f t h e s t o c k i s r e l a t i v e l y h e a l t h y i t i s w o r t h w h i l e t a k i n g a d v a n t a g e o f g o o d y e a r s b y i n v e s t i n g i n c a p a c i t y a b o v e t h e d e t e r m i n i s t i c l e v e l , w h i l e f o r l o w e r s t o c k s i z e s a m o r e c o n s e r v a t i v e i n v e s t m e n t p o l i c y i s t o b e p r e f e r r e d . N o t e , h o w e v e r , t h a t t h e d e t e r m i n i s t i c e q u i l i b r i u m p o i n t f o r t h e w h a l e f i s h e r y l i e s a t a l o w b i o m a s s l e v e l ( S = 1 1 . 4'x 1 0 * , K = 2 2 5 0 ) , s o t h a t e v e n w i t h <r = 0.2 o n e w i l l r a r e l y s e e e s c a p e m e n t s S > 3 6 X 1 0 " i n t h e l o n g t e r m . On t h e o t h e r h a n d , t h e u n e x p i o i t e d d e t e r m i n i s t i c e q u i l i b r i u m l i e s a t S = 4 6 . 2 X 1 0 " , s o t h a t f o r t h i s e x a m p l e t h e o p t i m a l s t o c h a s t i c p o l i c y f a c e d w i t h a v i r g i n s t o c k c a l l s f o r a n i n i t i a l i n v e s t m e n t a b o v e t h e d e t e r m i n i s t i c o p t i m u m , b u t a m o r e c o n s e r v a t i v e i n v e s t m e n t s t r a t e g y i n t h e l o n g t e r m . T h i s r e s u l t i s c o n s i s t e n t w i t h t h e d i s c u s s i o n p r e s e n t e d a b o v e , w h e r e i t was s u g g e s t e d t h a t f i s h e r i e s w i t h s l o w - g r o w i n g s t o c k s w i l l t e n d t o h a v e o p t i m a l c a p a c i t i e s w h i c h d e c r e a s e w i t h 125 u n c e r t a i n t y . N o t e t h a t b o t h t h e m a x i m u m b i o m a s s g r o w t h r a t e a n d t h e r a t i o o f u n i t c a p i t a l c o s t t o m aximum y e a r l y v a r i a b l e c o s t ( 6 / c T = 2 . 0 ) f o r t h e b a s e - c a s e w h a l e f i s h e r y i s t h e same a s t h a t o f a " p r a w n f i s h e r y " w i t h a = 3 . 8 2 a n d 6 = $ 8 3 2 0 0 ( b y c o n s t r u c t i o n -s e e C h a p t e r I I . E ) . C o m p a r i n g f i g u r e 28 w i t h r e s u l t s f o r t h e a = 3 . 8 2 , 6 = $ 0 . 0 8 3 2 x 1 0 6 p r a w n f i s h e r y s h o w n i n f i g u r e 2 3 , o n e c a n s e e t h a t t h e o p t i m a l p o l i c y c u r v e s a r e s i m i l a r l y b e h a v e d i n m o s t r e s p e c t s , a l t h o u g h t h e l o c a t i o n o f t h e c r o s s o v e r b e t w e e n t h e d e t e r m i n i s t i c a n d s t o c h a s t i c h ( S ) c u r v e s i s s o m e w h a t d i f f e r e n t i n t h e t w o f i s h e r i e s . T h i s may b e d u e t o t h e d i f f e r e n c e i n n o i s e l e v e l s b e i n g c o n s i d e r e d o r may c o n c e r n t h e c a r r y i n g c a p a c i t y o f t h e f i s h e r y [ i . e . t h e s o l u t i o n o f F ( S ) = S ] , s i n c e t h e w h a l e f i s h e r y h a s a c a r r y i n g c a p a c i t y e q u a l t o 4 6 . 2 x 1 0 " > > x o = 5 . 5 x 1 0 " w h i l e t h e " p r a w n f i s h e r y " d e s c r i b e d a b o v e h a s c a r r y i n g c a p a c i t y 0 . 2 8 x 1 0 6 < x o = 0 . 9 9 x 1 0 6 . I n a n y c a s e t h e g e n e r a l r e s u l t a p p e a r s t o b e t h a t i n v e s t m e n t i s e i t h e r z e r o o r d e c r e a s e s w i t h u n c e r t a i n t y f o r m o s t r e a s o n a b l e e s c a p e m e n t v a l u e s i n a s l o w - g r o w i n g f i s h e r y ; o n l y i n t h e c a s e o f e x c e p t i o n a l l y h i g h e s c a p e m e n t s i s i t o p t i m a l f o r t h e s o c i a l m a n a g e r t o i n v e s t i n c a p a c i t y a b o v e t h e d e t e r m i n i s t i c o p t i m u m ( a s s u m i n g a l o w u n i t c a p i t a l c o s t ) . T h e o p t i m a l e s c a p e m e n t f o r t h e w h a l e f i s h e r y i n c r e a s e s w i t h i n c r e a s i n g u n c e r t a i n t y , a s i n t h e a = 3 . 8 2 , 6 = $ 0 . 0 8 3 2 x 1 0 6 p r a w n f i s h e r y d i s c u s s e d a b o v e . I t a p p e a r s t h a t m a n a g e m e n t o f a v e r y s l o w l y g r o w i n g f i s h e r y s h o u l d be m o r e c o n s e r v a t i o n i s t t h e h i g h e r t h e l e v e l o f u n c e r t a i n t y ; t h i s i s i n a c c o r d a n c e w i t h p r e v i o u s r e s e a r c h r e s u l t s . H o w e v e r , t h e d i f f e r e n c e b e t w e e n t h e <J-=0 a n d cr = 0.2 s ( K ) c u r v e s i s n e v e r v e r y g r e a t , p a r t i c u l a r l y f o r K - v a l u e s 126 n e a r t h e ( q u a s i - ) e q u i l i b r i u m p o i n t . P e r h a p s t h e m o s t i n t e r e s t i n g r e s u l t i n c o m p a r i n g t h e d e t e r m i n i s t i c a n d s t o c h a s t i c w h a l e f i s h e r i e s i s t h e l o c a t i o n o f t h e s e ( q u a s i - ) e q u i l i b r i u m p o i n t s : w hen <r=0.2, t h e w h a l e s t o c k s t e a d y - s t a t e i s c e n t r e d on S = 1 4 . 8 X 1 0 " , 30 p e r c e n t h i g h e r t h a n t h e d e t e r m i n i s t i c e q u i l i b r i u m . H e n c e , a l t h o u g h t h e r e i s l i t t l e c h a n g e i n t h e s ( K ) c u r v e s , u s e o f t h e o p t i m a l s t o c h a s t i c p o l i c y c a n e f f e c t i v e l y l e a d t o a s u b s t a n t i a l l y l a r g e r s t o c k o f w h a l e s ( o n a v e r a g e ) , w h i l e d e c r e a s i n g t h e ( m e a n ) o p t i m a l c a p a c i t y b y o n l y 11 p e r c e n t , f r o m K = 2 2 5 0 t o K = 2 0 0 0 . ( g ) P e r f o r m a n c e o f d e t e r m i n i s t i c v s s t o c h a s t i c s t a t e g i e s . A t t h i s p o i n t we a d d r e s s t w o f u n d a m e n t a l q u e s t i o n s : How s e n s i t i v e i s t h e v a l u e o f t h e f i s h e r y t o c h a n g e s i n t h e p o l i c y c u r v e s s ( K ) a n d h ( S ) a w a y f r o m t h e i r o p t i m u m p o s i t i o n s ? How w e l l d o t h e p o l i c y f u n c t i o n s o b t a i n e d a s o p t i m a l f o r a d e t e r m i n i s t i c e n v i r o n m e n t c o m p a r e t o " t r u e " o p t i m a l p o l i c i e s f o r t h e s t o c h a s t i c f i s h e r y ? T h e v e r y n a t u r e o f o p t i m a l c o n t r o l s s u g g e s t s t h a t s m a l l v a r i a t i o n s i n t h e c o n t r o l s s h o u l d h a v e e v e n s m a l l e r e f f e c t s o n t h e v a l u e f u n c t i o n ( L u d w i g , 1 9 8 0 ) . T h i s i n d e e d a p p e a r s t o be t h e c a s e i n o u r m o d e l . U s i n g t h e d e t e r m i n i s t i c v e r s i o n o f t h e m o d e l ( I I . D ) , t h e o p t i m a l p o l i c y f u n c t i o n h ( S ) f o r t h e b a s e c a s e p r a w n f i s h e r y was p e r t u r b e d f i r s t u p w a r d s a n d t h e n d o w n w a r d s b y 10 p e r c e n t . T h e r e d u c t i o n i n t h e v a l u e f u n c t i o n w a s l e s s t h a n 1.0 p e r c e n t i n b o t h c a s e s , a r e s u l t i n a g r e e m e n t w i t h L u d w i g ' s p o i n t t h a t t h e v a r i a t i o n i n t h e v a l u e f u n c t i o n s h o u l d b e p r o p o r t i o n a l 127 t o t h e s q u a r e o f t h e d e v i a t i o n i n t h e p o l i c i e s . We h a v e s e e n t h a t p o l i c i e s w h i c h t a k e i n t o a c c o u n t f l u c t u a t i o n s i n t h e f i s h e r y ' s e n v i r o n m e n t c a n d i f f e r f r o m t h e i r d e t e r m i n i s t i c c o u n t e r p a r t s b y a s m u c h a s 3 0 - 4 0 p e r c e n t , f o r r e a s o n a b l e p a r a m e t e r c o m b i n a t i o n s . O p t i m a l f i s h e r i e s i n v e s t m e n t , t h e n , c a n b e s i g n i f i c a n t l y h i g h e r , o r s i g n i f i c a n t l y l o w e r , u n d e r u n c e r t a i n t y . H o w e v e r , a s a b o v e , t h e o p t i m a l v a l u e f u n c t i o n i s r a t h e r i n s e n s i t i v e t o c h a n g e s i n t h e p o l i c y f u n c t i o n s a w a y f r o m t h e i r o p t i m a l p o s i t i o n s . T a b l e ( I V ) g i v e s t h e v a l u e f u n c t i o n ( a t m e s h p o i n t s i n t h e x-K p l a n e ) f o r a n a = 1 4 , 6 = $ 0 . 0 8 3 2 x 1 0 6 f i s h e r y , a n d f o r t w o s e t s o f p o l i c y f u n c t i o n s : ( i ) t h e o p t i m a l p o l i c i e s f o r t h e s t o c h a s t i c (<r=0.58) f i s h e r y , a n d ( i i ) t h e o p t i m a l p o l i c i e s f o r t h e c o r r e s p o n d i n g d e t e r m i n i s t i c f i s h e r y , b u t e v a l u a t e d i n a s t o c h a s t i c (<r = 0 . 5 8 ) e n v i r o n m e n t . By c o m p a r i n g t h e s e v a l u e f u n c t i o n s p o i n t b y p o i n t , o n e c a n s e e t h a t t h e l o s s f r o m u s i n g t h e d e t e r m i n i s t i c p o l i c y i s n e v e r m o r e t h a n $ 0 . 2 0 m i l l i o n . F o r e x a m p l e , w i t h S = 2 . 1 X 1 0 6 ( t h e q u a s i - e q u i 1 i b r i u m e s c a p e m e n t ) a n d K=0, t h e o p t i m a l i n v e s t m e n t ' f o r t h e s t o c h a s t i c f i s h e r y i s I * = 1 2 . 5 , w h i l e u s i n g t h e d e t e r m i n i s t i c p o l i c y we w i l l h a v e 1 = 8 . 8 , a 30 p e r c e n t u n d e r - i n v e s t m e n t . H o w e v e r t h e r e d u c t i o n i n v a l u e o f t h e f i s h e r y c a u s e d b y u s i n g t h e d e t e r m i n i s t i c p o l i c y i s r o u g h l y 3 p e r c e n t , o r o n l y $ 0 . 1 6 m i l l i o n , a r a t h e r n e g l i g i b l e a m o u n t when o n e c o n s i d e r s t h e o v e r a l l l a c k o f p r e c i s i o n a t t a i n a b l e i n r e a l - w o r l d f i s h e r i e s . 128 F. S ummary a n d D i s c u s s i o n A s t o c h a s t i c v e r s i o n o f t h e o p t i m a l f i s h e r i e s i n v e s t m e n t m o d e l h a s b e e n c o n s t r u c t e d b y a l l o w i n g y e a r l y r e c r u i t m e n t t o f o l l o w , a l o g n o r m a l p r o b a b i l i t y d i s t r i b u t i o n , w i t h mean - v a l u e g i v e n b y t h e d e t e r m i n i s t i c s t o c k - r e c r u i t m e n t f u n c t i o n . H e u r i s t i c a n a l y s i s o f t h e s t o c h a s t i c m o d e l i n d i c a t e d t h a t o p t i m a l p o l i c i e s u n d e r u n c e r t a i n t y s h o u l d n o t d i f f e r q u a l i t a t i v e l y f r o m t h e i r d e t e r m i n i s t i c c o u n t e r p a r t s ; n u m e r i c a l r e s u l t s c o n f i r m e d t h i s e x p e c t a t i o n . H o w e v e r , b y s i m u l a t i n g s t o c h a s t i c s a m p l e p a t h s a n d s t e a d y s t a t e d i s t r i b u t i o n s , i t w a s s h o w n t h a t i n p r a c t i c e a n o p t i m a l l y m a n a g e d s t o c h a s t i c f i s h e r y a p p e a r s q u i t e d i f f e r e n t f r o m a d e t e r m i n i s t i c o n e , w i t h c o n s i d e r a b l e f l u c t u a t i o n s i n f l e e t c a p a c i t y b e i n g o p t i m a l i n t h e s t o c h a s t i c c a s e . I n a d d i t i o n , t h i s c h a p t e r h a s e x a m i n e d t h e r o l e o f t h e u n i t c o s t o f c a p i t a l , t h e i n t r i n s i c b i o m a s s g r o w t h r a t e , t h e d e p r e c i a t i o n r a t e a n d t h e d i s c o u n t r a t e i n d e t e r m i n i n g t h e e f f e c t o f u n c e r t a i n t y o n o p t i m a l i n v e s t m e n t a n d e s c a p e m e n t s t r a t e g i e s . I t a p p e a r s t h a t o f t h e s e , t h e k e y p a r a m e t e r s a r e 6, t h e u n i t c o s t o f c a p i t a l ( r e l a t i v e t o o p e r a t i n g c o s t s , c T ) a n d t h e i n t r i n s i c b i o m a s s g r o w t h r a t e ( a ) . F i g u r e 29 i s a s c h e m a t i c s h o w i n g ( a , 6 ) c o m b i n a t i o n s ( w i t h c T f i x e d ) f o r w h i c h o p t i m a l c a p a c i t y i s g e n e r a l l y h i g h e r (+) o r l o w e r (-) u n d e r u n c e r t a i n t y ( < f = 0 . 5 8 ) , t o g e t h e r w i t h a r o u g h c u r v e d i v i d i n g t h e t w o r e g i o n s . I n g e n e r a l i n v e s t m e n t w i l l b e h i g h e r u n d e r u n c e r t a i n t y i f t h e r e s o u r c e i s f a s t - g r o w i n g a n d c a p i t a l i s r e l a t i v e l y c h e a p . T h e r e v e r s e w i l l b e t r u e f o r a s l o w - g r o w i n g s t o c k w i t h e x p e n s i v e c a p i t a l . W i t h r e g a r d t o o t h e r m o d e l p a r a m e t e r s , o u r r e s u l t s 1 29 i n d i c a t e i n p a r t i c u l a r t h a t t h e l o w e r t h e d e p r e c i a t i o n r a t e , t h e m o r e t h e t e n d e n c y f o r o p t i m a l i n v e s t m e n t t o b e h i g h e r u n d e r u n c e r t a i n t y . F i n a l l y i n t h i s c h a p t e r we h a v e d i s c u s s e d t h e o p t i m a l i n v e s t m e n t a n d e s c a p e m e n t p o l i c i e s f o r t h e c a s e o f R i c k e r s t o c k - r e c r u i t m e n t a n d f o r t h e a g g r e g a t e d w h a l i n g f i s h e r y . I t w a s f o u n d t h a t f o r m o d e r a t e l y h i g h l e v e l s o f v a r i a b i l i t y , t h e r e l a t i v e d i f f e r e n c e b e w e e n s t o c h a s t i c a n d d e t e r m i n i s t i c o p t i m a l c a p a c i t i e s c o u l d r e a c h 3 0 - 4 0 p e r c e n t . T h i s r e s u l t s i n s u b s t a n t i a l o v e r - o r u n d e r - i n v e s t m e n t i n f l e e t c a p a c i t y w hen t h e d e t e r m i n i s t i c m o d e l i s u s e d i n p l a c e o f a f u l l s t o c h a s t i c m o d e l . T a r g e t e s c a p e m e n t s , o n t h e o t h e r h a n d , t e n d e d t o b e r e m a r k a b l y i n s e n s i t i v e t o t h e l e v e l o f u n c e r t a i n t y i n t h e f i s h e r y ; t h i s i s i n a g r e e m e n t w i t h p r e v i o u s r e s u l t s , a l t h o u g h t h e p o s s i b i l i t y t h a t o p t i m a l e s c a p e m e n t s c a n b e l o w e r u n d e r u n c e r t a i n t y i s c o n t r a r y t o m o s t p r e v i o u s r e s u l t s ( w h e r e i n v e s t m e n t was n o t c o n s i d e r e d ) . T h e r e l a t i v e p e r f o r m a n c e o f d e t e r m i n i s t i c v s . s t o c h a s t i c p o l i c i e s was a l s o d i s c u s s e d , a n d t h e d i f f e r e n c e f o u n d t o be v e r y s m a l l i n c o m p a r i s o n t o d i f f e r e n c e s i n t h e p o l i c i e s t h e m s e l v e s ; t h e i m p l i c a t i o n s o f t h i s r e s u l t a r e d i s c u s s e d i n C h a p t e r I V . 130 C h a p t e r I V , C o n c l u s i o n s ( a ) R e v i e w I n r e n e w a b l e r e s o u r c e i n d u s t r i e s , t w o s i m u l t a n e o u s i n v e s t m e n t p r o b l e m s m u s t be a d d r e s s e d : i n v e s t m e n t i n t h e r e s o u r c e s t o c k ( t h e b i o m a s s ) a n d i n v e s t m e n t i n t h e c a p i t a l s t o c k ( h a r v e s t i n g c a p a c i t y ) . T h e l a t t e r i n v e s t m e n t p r o b l e m h a s b e e n s u p p r e s s e d i n m o s t a n a l y s e s t o d a t e , b o t h d e t e r m i n i s t i c a n d s t o c h a s t i c , b y a s s u m i n g e i t h e r a f i x e d c a p i t a l s t o c k o r t h e p e r f e c t m a l l e a b i l i t y o f c a p i t a l . H o w e v e r , i n many f i s h e r i e s , a n d o t h e r n a t u r a l r e s o u r c e i n d u s t r i e s , c a p i t a l i s i n f a c t n o n m a l l e a b l e , w i t h f e w i f a n y a l t e r n a t i v e u s e s . I n t h i s c o n t e x t , i n v e s t m e n t i s ' i r r e v e r s i b l e , a n d o p t i m a l m a n a g e m e n t o f t h e i n d u s t r y m u s t i n v o l v e c o n t r o l o f b o t h s t a t e v a r i a b l e s , t h e r e s o u r c e s t o c k a n d t h e c a p i t a l s t o c k , t h r o u g h a p p r o p r i a t e i n v e s t m e n t s t r a t e g i e s . C l a r k , C l a r k e a n d M u n r o ( 1 9 7 9 ) h a v e s t u d i e d t h i s i r r e v e r s i b l e i n v e s t m e n t p r o b l e m , o b t a i n i n g a f u l l a n a l y t i c s o l u t i o n i n t h e c o n t i n u o u s - t i m e d e t e r m i n i s t i c c a s e . T h i s t h e s i s l i a s e x p l o r e d v a r i o u s a s p e c t s o f t h e j o i n t i n v e s t m e n t p r o b l e m , b u i l d i n g u p o n t h e w o r k o f C l a r k , C l a r k e a n d M u n r o , w h i l e m a i n t a i n i n g t h e k e y a s s u m p t i o n o f i r r e v e r s i b l e i n v e s t m e n t . We h a v e u s e d a s e a s o n a l ( d i s c r e t e - t i m e ) s t o c h a s t i c f i s h e r i e s m o d e l i n w h i c h t h e r e s o u r c e s t o c k a n d t h e c a p i t a l s t o c k v a r y o v e r t i m e , c o n t r o l l e d b y t w o d e c i s i o n v a r i a b l e s , e n d - o f - s e a s o n e s c a p e m e n t a n d y e a r l y i n v e s t m e n t . T h e m o d e l i n c l u d e s d e l a y s i n i n v e s t m e n t , r e f l e c t i n g t h e r e a l i t y t h a t 131 i n v e s t m e n t d e c i s i o n s m u s t u s u a l l y b e made w e l l i n a d v a n c e o f t h e t i m e a t w h i c h t h e c o r r e s p o n d i n g new c a p i t a l i s r e q u i r e d . I n C h a p t e r I I , t h e d e t e r m i n i s t i c v e r s i o n o f t h e m o d e l w a s s t u d i e d , u s i n g a n a l y t i c , h e u r i s t i c a n d n u m e r i c a l m e t h o d s t o p r o d u c e c o m p a r a t i v e d y n a m i c s r e s u l t s . I n C h a p t e r I I I , we r e c o g n i z e d t h a t t h e r e s o u r c e s t o c k f l u c t u a t e s r a n d o m l y f r o m y e a r t o y e a r . T h e r e s u l t i n g s t o c h a s t i c m o d e l r e q u i r e d a m o r e r e f i n e d n u m e r i c a l m e t h o d , b u t a n a l y t i c a n d h e u r i s t i c t e c h n i q u e s c a r r i e d o v e r f r o m t h e d e t e r m i n i s t i c c a s e i n a f a i r l y s t r a i g h t f o r w a r d m a n n e r . W h i l e n u m e r i c a l r e s u l t s h a v e b e e n b a s e d o n t w o s p e c i f i c f i s h e r i e s , t h e m e t h o d o l o g y a n d t h e q u a l i t a t i v e r e s u l t s c a n b e e x p e c t e d t o a p p l y i n many f i s h e r i e s , a s w e l l a s i n f o r e s t r y a n d a g r i c u l t u r a l i n v e s t m e n t p r o b l e m s . I t i s c l e a r f r o m b o t h q u a l i t a t i v e a n d q u a n t i t a t i v e r e s u l t s p r e s e n t e d h e r e t h a t a f u l l a n a l y s i s o f r e n e w a b l e r e s o u r c e m a n a g e m e n t m u s t i n c l u d e q u e s t i o n s o f o p t i m a l i n v e s t m e n t s t r a t e g i e s . I n d e e d f o r many o f t h e c a s e s s t u d i e d , t h e i n v e s t m e n t a s p e c t i s s u b s t a n t i a l l y m o r e c o m p l e x t h a n t h e m o r e w i d e l y s t u d i e d o p t i m a l h a r v e s t i n g p r o b l e m . A s i n t h e C l a r k , C l a r k e a n d M u n r o w o r k , we h a v e f o u n d t h a t f i s h e r i e s w i l l t e n d t o move b e t w e e n t h r e e p r i m a r y r e g i m e s : ( i ) a h i g h - b i o m a s s , l o w - c a p a c i t y r e g i m e , w i t h b o t h h a r v e s t i n g a n d i n v e s t m e n t b e i n g d e s i r a b l e , ( i i ) a h i g h - b i o m a s s , h i g h - c a p a c i t y s i t u a t i o n , w h e r e i n v e s t m e n t i s u n w a r r a n t e d b u t h a r v e s t i n g t a k e s p l a c e , a n d ( i i i ) a l o w - b i o m a s s c a s e i n w h i c h t h e f i s h e r y i s e s s e n t i a l l y s h u t d o w n , w i t h n e i t h e r h a r v e s t i n g n o r i n v e s t m e n t b e i n g d e s i r e d . 132 I n a d d i t i o n t h e e x i s t e n c e o f d e l a y s i n b r i n g i n g i n v e s t m e n t o n - l i n e l e a d s t o t h e p o s s i b i l i t y o f a f o u r t h r e g i m e , i n w h i c h t h e r e s o u r c e s t o c k i s t o o l o w t o p e r m i t h a r v e s t i n g , b u t i s e x p e c t e d t o r e c o v e r d u r i n g t h e " i n v e s t m e n t d e l a y " p e r i o d . H e n c e p l a n n i n g a n d p a y m e n t f o r i n v e s t m e n t b e c o m e s d e s i r a b l e i f c u r r e n t c a p a c i t y i s s u f f i c i e n t l y l o w . T h e r e a d e r i s r e f e r r e d t o t h e f i n a l s e c t i o n o f e a c h c h a p t e r f o r a d e t a i l e d s u m m a r y o f t h e r e s u l t s . H e r e we s h a l l s i m p l y l i s t t h e m o r e i m p o r t a n t o f t h e s e r e s u l t s a n d t h e n p r o c e e d t o a d i s c u s s i o n o f o u r m e t h o d s o f a n a l y s i s , i m p l e m e n t a t i o n o f o p t i m a l i n v e s t m e n t p r o g r a m s , a n d t h e r o l e o f u n c e r t a i n t y i n o p t i m a l f i s h e r i e s i n v e s t m e n t . ( 1 ) W i t h l i n e a r v a r i a b l e c o s t s a n d r i s k n e u t r a l i t y , o p t i m a l m a n a g e m e n t i s c h a r a c t e r i z e d b y c a p a c i t y a n d e s c a p e m e n t t a r g e t c u r v e s , h ( S ) a n d s ( K ) , r e p r e s e n t i n g t h e o p t i m a l c a p a c i t y f o r g i v e n e s c a p e m e n t S, a n d t h e o p t i m a l e s c a p e m e n t f o r a g i v e n f l e e t c a p a c i t y K, r e s p e c t i v e l y . T h e s e c u r v e s h a v e t h e same q u a l i t a t i v e f o r m i n b o t h t h e d e t e r m i n i s t i c a n d s t o c h a s t i c c a s e s . ( 2 ) T h e p r i m a r y d i f f e r e n c e b e t w e e n t h e u s e o f s e a s o n a l a n d c o n t i n u o u s - t i m e m o d e l s i n s t u d y i n g t h e f i s h e r i e s i n v e s t m e n t p r o b l e m i s t h e a s y m p t o t i c a p p r o a c h t o e q u i l i b r i u m i n t h e f o r m e r c a s e , a s o p p o s e d t o t h e f i n a l " i m p u l s e " i n v e s t m e n t w h i c h a r i s e s i n t h e c o n t i n u o u s - t i m e s o l u t i o n . H o w e v e r , a s t o c h a s t i c v e r s i o n o f t h e c o n t i n u o u s - t i m e m o d e l w o u l d be u n l i k e l y t o m a i n t a i n t h i s i m p u l s e i n v e s t m e n t , s i n c e a s p e c i f i e d " e q u i l i b r i u m " b i o m a s s l e v e l c a n n o t be r e a d i l y a t t a i n e d , a s we saw w i t h o u r d i s c r e t e - t i m e m o d e l . 133 ( 3 ) T h e o p t i m a l c a p a c i t y f u n c t i o n m i m i c s t h e q u a l i t a t i v e b e h a v i o u r o f t h e u n d e r l y i n g s t o c k - r e c r u i t m e n t f u n c t i o n . T h i s e f f e c t i s d u e t o t h e d e l a y i n b r i n g i n g new i n v e s t m e n t o n - l i n e ; d e s i r e d c a p a c i t y f o r n e x t s e a s o n d e p e n d s o n t h e c u r r e n t e s c a p e m e n t , a c t i n g t h r o u g h t h e r e p r o d u c t i o n f u n c t i o n F ( S ) . ( 4 ) C o m p a r a t i v e d y n a m i c s r e s u l t s o b t a i n e d i n C h a p t e r s I I a n d I I I p r o v i d e i n s i g h t s i n t o b o t h q u a l i t a t i v e a n d q u a n t i t a t i v e b e h a v i o u r o f t h e e c o n o m i c a n d e c o l o g i c a l p a r a m e t e r s a r i s i n g i n f i s h e r i e s i n v e s t m e n t p r o b l e m s . O p t i m a l i n v e s t m e n t a p p e a r s t o be p a r t i c u l a r l y s e n s i t i v e t o f i s h p r i c e a n d s t o c k - r e c r u i t m e n t p a r a m e t e r s . I n a d d i t i o n we h a v e s h o w n t h a t d e p r e c i a t i o n m u s t be t r e a t e d a s m o r e t h a n s i m p l y a n e x t r a c o s t o f c a p i t a l , a n d m u s t be a n a l y s e d c a r e f u l l y i n d e t e r m i n i n g o p t i m a l i n v e s t m e n t p o l i c i e s . ( 5 ) I n a n o p t i m a l l y - m a n a g e d f i s h e r y w h e r e t h e b i o m a s s f l u c t u a t e s r a n d o m l y f r o m y e a r t o y e a r , c o n s i d e r a b l e v a r i a t i o n c a n a l s o be e x p e c t e d i n t h e o p t i m a l c a p i t a l s t o c k , w h i c h r e s p o n d s t o e n d - o f - s e a s o n b i o m a s s t h r o u g h t h e f u n c t i o n h ( S ) . ( 6 ) T h e i n t r i n s i c b i o m a s s g r o w t h r a t e a n d t h e c o s t o f c a p i t a l ( r e l a t i v e t o v a r i a b l e c o s t s ) s e e m t o be t h e k e y f i s h e r y p a r a m e t e r s i n d e t e r m i n i n g w h e t h e r i n v e s t m e n t i n c r e a s e s o r d e c r e a s e s w i t h t h e l e v e l o f u n c e r t a i n t y . I n g e n e r a l , we f o u n d t h a t i n v e s t m e n t w i l l b e h i g h e r u n d e r u n c e r t a i n t y i f t h e r e s o u r c e i s f a s t - g r o w i n g a n d c a p i t a l i s r e l a t i v e l y c h e a p . T h e r e v e r s e w i l l b e t r u e f o r a s l o w - g r o w i n g s t o c k w i t h e x p e n s i v e c a p i t a l . (7) W h i l e t a r g e t e s c a p e m e n t s t e n d e d t o be i n s e n s i t i v e t o t h e l e v e l o f u n c e r t a i n t y , t h e r e l a t i v e d i f f e r e n c e b e t w e e n s t o c h a s t i c 134 (<y=0.58) a n d d e t e r m i n i s t i c (tf=0) o p t i m a l c a p a c i t i e s c o u l d r e a c h a s m u c h a s 3 0 - 4 0 p e r c e n t . On t h e o t h e r h a n d , t h e " o p t i m a l " p o l i c i e s o b t a i n e d u s i n g t h e d e t e r m i n i s t i c a s s u m p t i o n p e r f o r m e d n e a r l y a s w e l l ( i n t e r m s o f t h e e x p e c t e d v a l u e o f t h e f i s h e r y ) a s t h e t r u e o p t i m a l ( s t o c h a s t i c ) p o l i c i e s . ( b ) M e t h o d s o f A n a l y s i s T h e g e n e r a l a p p r o a c h t o s o l v i n g o u r o p t i m a l i n v e s t m e n t a n d e s c a p e m e n t p r o b l e m h a s b e e n t o a d o p t a d y n a m i c p r o g r a m m i n g f o r m u l a t i o n , o b t a i n a n a l y t i c r e s u l t s f o r t h e s i m p l e c a s e o f d e n s i t y i n d e p e n d e n t s t o c k - r e c r u i t m e n t , a n a l y s e t h e m o d e l h e u r i s t i c a l l y , a n d t h e n p r o c e e d t o d e v e l o p a n u m e r i c a l s c h e m e t o s o l v e t h e p r o b l e m c o m p l e t e l y . I t was f o u n d t h a t t h e h e u r i s t i c a n a l y s i s , a l t h o u g h r e l a t i v e l y s i m p l e , was p a r t i c u l a r l y u s e f u l i n s u g g e s t i n g t h e q u a l i t a t i v e b e h a v i o u r o f t h e o p t i m a l p o l i c i e s . T h i s p r o v e d h e l p f u l on i t s own a n d a s a g u i d e l i n e f o r t h e n u m e r i c a l s c h e m e . Two a p p r o a c h e s t o t h e n u m e r i c a l s o l u t i o n o f o u r o p t i m i z a t i o n p r o b l e m h a v e b e e n d e v e l o p e d h e r e . T h e f i r s t , u s e d f o r t h e d e t e r m i n i s t i c c a s e i n C h a p t e r I I , was r e l a t i v e l y s t r a i g h t f o r w a r d , i n v o l v e d s i m p l e l i n e a r o r c u b i c i n t e r p o l a t i o n , b u t r e q u i r e d a 3 0 x 3 0 R-K m e s h f o r s u i t a b l e a c c u r a c y , a n d d i d n o t u s e t h e f u l l i n f o r m a t i o n c o n t a i n e d i n t h e d y n a m i c p r o g r a m m i n g e q u a t i o n . T h e s e c o n d a p p r o a c h , , d i s c u s s e d i n C h a p t e r I I I , i s a r a t h e r m o r e c o m p l e x a n d c o n s i s t e n t p r o c e d u r e b a s e d on a 2 - d i m e n s i o n a l C 1 c u b i c s p l i n e s u r f a c e f o r t h e v a l u e f u n c t i o n , a n d i n v o l v i n g a s i m u l t a n e o u s s o l u t i o n f o r t h e v a l u e f u n c t i o n a n d 135 i t s f i r s t p a r t i a l d e r i v a t i v e s . T h e l a t t e r m e t h o d r e q u i r e d o n l y a n 8 x 8 m e s h , b u t s i n c e 3 v a l u e s a r e d e t e r m i n e d a t e a c h m e s h p o i n t a n d s p a r s e m a t r i x m e t h o d s a r e n o t u s e f u l i n t h i s c a s e , t h e r e s u l t i n g c o s - t p e r r u n o f t h e d e t e r m i n i s t i c m o d e l w a s f o u n d t o b e r o u g h l y e q u i v a l e n t f o r b o t h a p p r o a c h e s ( a n d t h e c o r r e s p o n d i n g r e s u l t s w e r e i n v e r y c l o s e a g r e e m e n t ) . I n t h e s t o c h a s t i c c a s e , h o w e v e r , t h e s i m p l e r s c h e m e o f C h a p t e r I I f a i l e d t o p r o d u c e a c c u r a t e r e s u l t s ; h e n c e we c o n c l u d e t h a t t h e m o r e r e f i n e d p r o c e d u r e i s t o be p r e f e r r e d i n g e n e r a l . F i n a l l y , we n o t e t h a t a n i m p o r t a n t a d v a n t a g e o f t h e d y n a m i c p r o g r a m m i n g a p p r o a c h i s i t s v e r s a t i l i t y ; t h e n u m e r i c a l m e t h o d o l o g y d e v e l o p e d h e r e i n t o s o l v e o u r s p e c i f i c d y n a m i c p r o g r a m m i n g p r o b l e m c a n b e q u i t e e a s i l y a d a p t e d t o h a n d l e o t h e r r e l a t e d m o d e l s ( f o r e x a m p l e , t h e p o s s i b i l i t y o f a l t e r n a t i v e i n v e s t m e n t a s s u m p t i o n s ) . ( c ) I m p l e m e n t a t i o n A l t h o u g h t h e m o d e l d i s c u s s e d i n t h i s t h e s i s r e p r e s e n t s a n a b s t r a c t i o n o f r e a l - w o r l d f i s h e r i e s , i t i s s u f f i c i e n t l y g e n e r a l t o h a v e some a p p l i c a b i l i t y i n a v a r i e t y o f c i r c u m s t a n c e s . T h e p o s s i b i l i t y o f i m p l e m e n t i n g t h e o p t i m a l p r o g r a m s u g g e s t e d b o t h b y o u r r e s u l t s a n d t h e c o n t i n u o u s - t i m e r e s u l t s o f C l a r k , C l a r k e a n d M u n r o r a i s e s s e v e r a l q u e s t i o n s f o r f i s h e r i e s m a n a g e m e n t i n s u c h c a s e s . When t h e c a p i t a l s t o c k i s c o n s i d e r e d e x p l i c i t l y , i t b e c o m e s a p p a r e n t t h a t t h e t a r g e t e s c a p e m e n t s h o u l d d e p e n d o n t h e c u r r e n t l e v e l o f c a p i t a l i z a t i o n i n t h e f i s h e r y [ i . e . S * = s ( K ) ] , a n d t h a t , 136 p e r h a p s c o n t r a r y t o i n t u i t i o n , m a n a g e m e n t s h o u l d be s o m e w h a t m o r e c o n s e r v a t i o n i s t when t h e i n d u s t r y h a s a h i g h l e v e l o f c a p a c i t y . H o w e v e r , i n t h e c a s e s s t u d i e d h e r e , t h i s e f f e c t t e n d e d t o b e r a t h e r m i n o r , s o t h a t a f i x e d e s c a p e m e n t t a r g e t , e q u a l t o t h e a b u n d a n t - c a p i t a l M o d i f i e d G o l d e n R u l e e s c a p e m e n t , w i l l b e c l o s e t o o p t i m a l . On t h e o t h e r h a n d , f o r a l l b u t t h e m o s t r a p i d l y - g r o w i n g r e s o u r c e s t o c k s , t h e o p t i m a l c a p a c i t y o f t h e f i s h e r y d e p e n d s s t r o n g l y o n t h e c u r r e n t s t a t u s o f t h e f i s h s t o c k . I n o u r m o d e l t h i s i s m e a s u r e d b y t h e e s c a p e m e n t ; i n a n y c a s e w h a t i s r e q u i r e d i s s u f f i c i e n t i n f o r m a t i o n t o p r e d i c t t h e a v e r a g e f u t u r e s t o c k s i z e , f o r t h i s i s t h e q u a n t i t y w h i c h d e t e r m i n e s t h e d e s i r a b i l i t y o f i n v e s t i n g now. H e n c e t h e a c c u r a t e c o l l e c t i o n a n d a n a l y s i s o f e n d - o f - s e a s o n d a t a o n t h e s t a t e o f t h e f i s h r e s o u r c e b e c o m e s i m p o r t a n t f o r i n v e s t m e n t p l a n n i n g a s w e l l a s f o r t h e m o r e t r a d i t i o n a l c o n s e r v a t i o n p u r p o s e s . T h e i m p l e m e n t a t i o n o f o p t i m a l m a n a g e m e n t p r o g r a m s c l e a r l y r e q u i r e s a p p r o p r i a t e r e g u l a t o r y i n s t r u m e n t s . I f l e f t u n c o n t r o l l e d , o r e v e n p a r t i a l l y r e g u l a t e d , e x p e r i e n c e i n d i c a t e s t h a t f i s h e r i e s f r e q u e n t l y b e c o m e s u b s t a n t i a l l y o v e r - c a p i t a l i z e d ( i n t h e s e n s e t h a t i n v e s t m e n t o c c u r s a b o v e t h e o p t i m a l l e v e l s g i v e n b y t h e h ( S ) c u r v e ) . A s d e s c r i b e d a b o v e , t h e e s c a p e m e n t p r o b l e m c a n o f t e n be r e d u c e d t o d e t e r m i n a t i o n o f a s i n g l e b i o m a s s t a r g e t . S e v e r a l m e c h a n i s m s h a v e b e e n p r o p o s e d t o a c h i e v e s u c h a t a r g e t ; a l l o c a t e d c a t c h q u o t a s , t a x e s o n c a t c h , a n d l i m i t e d e n t r y a r e t h e m o s t f r e q u e n t l y d i s c u s s e d . W h i l e t h e u s e o f q u o t a s a n d t a x e s 1 37 i n a s t o c h a s t i c f i s h e r y h a s n o t y e t b e e n s u f f i c i e n t l y r e s e a r c h e d f r o m a t h e o r e t i c a l s t a n d p o i n t , i t c a n b e h o p e d n e v e r t h e l e s s t h a t a n a p p r o p r i a t e c o m b i n a t i o n o f t h e s e p o l i c y i n s t r u m e n t s w i l l b e a b l e t o a c h i e v e o p t i m a l e s c a p e m e n t t a r g e t s . C o n t r o l l i n g t h e c a t c h i n g p o w e r o f f i s h i n g f l e e t s h a s b e e n t h e s u b j e c t o f m u c h d e b a t e i n r e c e n t y e a r s . T h e n e e d f o r s u c h c o n t r o l i s w e l l - a c c e p t e d , b u t t h e m e a n s t o a c c o m p l i s h t h i s a r e n o t c l e a r . R e c e n t w o r k b y C l a r k ( 1 9 8 0 ) h a s s e t o u t a m e t h o d o l o g y f o r a c h i e v i n g t h e c o r r e c t l e v e l o f e f f o r t t h r o u g h t a x e s o r q u o t a s . C l a r k d e v e l o p e d a m o d e l o f e n t r y t o a n d e x i t f r o m t h e f i s h e r y b y i n d i v i d u a l f i s h e r m e n . H o w e v e r , t h i s m o d e l i s b a s e d o n t h e e x i s t e n c e o f a s u f f i c i e n t l y l a r g e " b a c k g r o u n d " c a p i t a l s t o c k f r o m w h i c h v e s s e l s c a n be d r a w n when r e q u i r e d . R e g u l a t o r y m e t h o d s o r o t h e r m e a n s a r e n e e d e d t o c o n t r o l t h e s i z e o f t h i s " c a p i t a l p o o l " i t s e l f . I n v e s t m e n t i n c e n t i v e s , t h r o u g h t a x m e a s u r e s , g r a n t s o r " f i s h e r i e s d e v e l o p m e n t l o a n s " , c a n be u s e d t o i n c r e a s e t h e c a p i t a l s t o c k . When t h e p r o c e s s i n g s e c t o r c o n t r o l s a c o n s i d e r a b l e s h a r e o f t h e h a r v e s t i n g c a p i t a l , t h r o u g h r e n t i n g o r l e a s i n g o f t h e i r v e s s e l s t o i n d i v i d u a l f i s h e r m e n , c o n t r o l o f f l e e t i n v e s t m e n t b e h a v i o u r c a n a l s o b e a c c o m p l i s h e d i n p a r t t h r o u g h d i r e c t r e g u l a t i o n s a p p l i e d t o t h e c o r p o r a t i o n s i n v o l v e d . I n c a s e s w h e r e p r i v a t e i n t e r e s t s h a v e n o t met t h e s o c i a l d e s i r e t o d e v e l o p a p a r t i c u l a r f i s h r e s o u r c e , d u e p e r h a p s t o d i f f e r e n c e s b e t w e e n p r i v a t e a n d s o c i a l c o s t s ( o r d i s c o u n t r a t e s ) , b o t h i n v e s t m e n t i n c e n t i v e s a n d d i r e c t g o v e r n m e n t a c q u i s i t i o n o f f l e e t c a p i t a l may be i m p o r t a n t . 138 T h e p r o b l e m o f f i s h e r i e s d e v e l o p m e n t , p a r t i c u l a r l y i n d e v e l o p i n g c o u n t r i e s , i n v o l v e s n o t o n l y t h e a c q u i s i t i o n o f s u f f i c i e n t p h y s i c a l c a p i t a l ( w h i c h i n i t s e l f may r e q u i r e f o r e i g n a i d ) b u t a l s o t h e p r o v i s i o n o f human c a p i t a l , t h r o u g h s u i t a b l e t r a i n i n g o f t h e f i s h e r m e n . T h i s r a t h e r o b v i o u s p o i n t h a s s i g n i f i c a n t i m p l i c a t i o n s f o r t h e i r r e v e r s i b l e i n v e s t m e n t p r o b l e m ; d e l a y s i n b r i n g i n g new i n v e s t m e n t o n - l i n e may be s u b s t a n t i a l l y i n c r e a s e d b y t h e n e e d f o r l e n g t h y t r a i n i n g p r o g r a m s . A t t h e v e r y l e a s t , o n e w o u l d e x p e c t t h e " c a t c h a b i l i t y " p e r u n i t o f f i s h i n g e f f o r t t o be r a t h e r l o w i n i t i a l l y , b u t t o i n c r e a s e o v e r t i m e a s s k i l l s o f t h e f i s h e r m e n a r e i m p r o v e d . W h i l e o u r r e s u l t s s u g g e s t t h a t t h i s e f f e c t w o u l d l e a d , c e t e r i s p a r i b u s , t o a l o w i n i t i a l c a p i t a l s t o c k , t h e p r o b l e m f a c e d b y s o c i e t y i s o n e o f b a l a n c i n g l o w i n i t i a l r e t u r n s p e r u n i t e f f o r t w i t h t h e n e e d f o r " l e a r n i n g b y d o i n g " on t h e p a r t o f t h e f i s h e r m e n . I f t h e f i s h e r y u n d e r c o n s i d e r a t i o n i s f a c e d w i t h a c a p i t a l s t o c k a b o v e t h e o p t i m a l l e v e l g i v e n b y t h e h ( S ) c u r v e , o u r m o d e l p r e s c r i b e s a m o r a t o r i u m on i n v e s t m e n t u n t i l t h e f l e e t h a s d e p r e c i a t e d s u f f i c i e n t l y . I n d e e d , a l m o s t b y d e f i n i t i o n , t h i s i s t h e o n l y p o s s i b l e s o l u t i o n g i v e n n o n m a l l e a b l e c a p i t a l ; a s p o i n t e d o u t b y C l a r k , C l a r k e a n d M u n r o , t h e r e i s no n e e d f o r m o r e d r a s t i c a c t i o n . S c r a p p i n g e x c e s s c a p a c i t y s e r v e s no p u r p o s e ( w i t h z e r o s c r a p v a l u e ) , a n d b u y - b a c k p r o g r a m s m e r e l y t r a n s f e r o w n e r s h i p o f t h e c a p i t a l f r o m p r i v a t e t o g o v e r n m e n t h a n d s . I n p r a c t i c e , h o w e v e r , b u y - b a c k p r o g r a m s may s e r v e a u s e f u l p u r p o s e ; c a t c h i n g p o w e r i s p e r m a n e n t l y r e m o v e d f r o m t h e f i s h e r y , m a k i n g 1 39 m a n a g e m e n t e a s i e r ( a s s u m i n g n o c o r r e s p o n d i n g a d d i t i o n s t o f l e e t c a p a c i t y o c c u r ! ) . F u r t h e r m o r e , t h e r e m o v e d v e s s e l s c a n o f t e n b e p r o f i t a b l y u t i l i z e d i n o t h e r f i s h e r i e s , p a r t i c u l a r l y t h o s e o f d e v e l o p i n g c o u n t r i e s . ( d ) T h e r o l e o f u n c e r t a i n t y T u r n i n g now t o e f f e c t s o f u n c e r t a i n t y i n f i s h e r i e s m a n a g e m e n t , we c o n s i d e r s e p a r a t e l y t h e t w o t y p e s o f u n c e r t a i n t y s t u d i e d i n t h i s t h e s i s , n a m e l y s t o c h a s t i c f l u c t u a t i o n s a n d p a r a m e t e r u n c e r t a i n t y . We h a v e s e e n t h a t e v e n i n a l o n g r u n s t e a d y s t a t e f i s h e r y , f l e e t c a p a c i t y i n a s t o c h a s t i c e n v i r o n m e n t s h o u l d b e e x p e c t e d t o f l u c t u a t e o v e r a f a i r l y w i d e r a n g e . T h i s r a n g e w i l l b e g r e a t e r t h e s l o w e r - g r o w i n g a n d t h e m o r e v a r i a b l e i s t h e r e s o u r c e s t o c k . I n p a r t i c u l a r , a n o p t i m a l i n v e s t m e n t p r o g r a m s h o u l d a l l o w t h e c a p i t a l s t o c k t o r e s p o n d p o s i t i v e l y t o u n u s u a l l y " g o o d " y e a r s , e i t h e r b y p e r m i t t i n g i n c r e a s e d e n t r y o r b y d i r e c t a c q u i s i t i o n o f e x t r a c a p i t a l . T h i s i s d o n e i n f u l l k n o w l e d g e t h a t i d l e c a p a c i t y w i l l t h e n b e g r e a t e r i n t h e " b a d " y e a r s . [ T h e p o s s i b i l i t y t h a t p r e s s u r e f r o m u s e r g r o u p s may l e a d t o t h e o v e r - u t i l i z a t i o n o f t h i s new c a p a c i t y i s a r e a l d a n g e r , b u t h a s n o t b e e n i n c l u d e d i n t h e m o d e l d i s c u s s e d h e r e . ] G i v e n a d e t e r m i n i s t i c i n v e s t m e n t m o d e l o f a r e a l - w o r l d f i s h e r y , o n e may w i s h t o know t h e q u a l i t a t i v e e f f e c t o f r a n d o m n e s s w i t h o u t u n d e r t a k i n g a f u l l s t o c h a s t i c a n a l y s i s . O u r r e s u l t s s u g g e s t t h e f o l l o w i n g g u i d i n g p r i n c i p l e : i f t h e r a t i o o f u n i t c a p i t a l c o s t t o y e a r l y o p e r a t i n g c o s t s s e e m s f a i r l y l o w , 1 40 a n d i f t h e r e s o u r c e i s r e a s o n a b l y f a s t - g r o w i n g ( a s w i t h p r a w n s t o c k s ) , t h e n i n v e s t m e n t i s l i k e l y t o b e h i g h e r u n d e r u n c e r t a i n t y . T h i s q u a l i t a t i v e i n f o r m a t i o n may be u s e f u l i n d e t e r m i n i n g w h e t h e r a f i s h e r y h a s i n d e e d e x p e r i e n c e d o v e r - i n v e s t m e n t , o r w h e t h e r a p p a r e n t e x c e s s c a p a c i t y i s i n f a c t o p t i m a l g i v e n t h e h i s t o r y o f t h e f i s h e r y ' s d e v e l o p m e n t i n t h e f a c e o f u n c e r t a i n f u t u r e s t o c k s i z e s . T h e i r r e v e r s i b i l i t y o f i n v e s t m e n t i n c r e a s e s t h e i m p o r t a n c e o f i n h e r e n t u n c e r t a i n t y i n t h e f i s h e r y . T h i s i s p a r t i c u l a r l y t h e c a s e f o r f i s h e r i e s w i t h s l o w - g r o w i n g r e s o u r c e s t o c k s , w h e r e t h e o c c u r r e n c e o f a n u n u s u a l l y " b a d " y e a r may l e a d t o c a p i t a l l y i n g i d l e f o r a s u b s t a n t i a l p a r t o f i t s e c o n o m i c l i f e . H o w e v e r , i n a c c o r d a n c e w i t h t h e w o r k o f o t h e r r e s e a r c h e r s , we h a v e f o u n d t h a t i n many l i n e a r - c o s t r i s k n e u t r a l f i s h e r i e s , o p t i m a l p o l i c i e s r e c o g n i z i n g t h e s t o c h a s t i c n a t u r e o f t h e f i s h e r y p e r f o r m o n l y s l i g h t l y b e t t e r t h a n p o l i c i e s b a s e d o n t h e c o r r e s p o n d i n g d e t e r m i n i s t i c m o d e l . I n o t h e r w o r d s , i n t h i s c a s e u s e o f d e t e r m i n i s t i c m o d e l s i s s u f f i c i e n t t o p r o d u c e p o l i c i e s w i t h n e a r - o p t i m a l p e r f o r m a n c e ( o n a v e r a g e ) . T h e l a t t e r r e s u l t c e r t a i n l y d o e s n o t i m p l y t h a t t h e e x i s t e n c e o f u n c e r t a i n t y i n t h e f o r m o f y e a r - t o - y e a r f l u c t u a t i o n s i s i r r e l e v a n t t o m a n a g e m e n t . R a t h e r , i t s i m p l y t e l l s u s t h a t a n y i n v e s t m e n t s t r a t e g y " n e a r " t h e o p t i m a l w i l l p e r f o r m a l m o s t o p t i m a l l y . A s p o i n t e d o u t b y L u d w i g ( 1 9 8 0 ) , t h e f r a c t i o n a l l o s s i n c u r r e d b y u s i n g a n o n - o p t i m a l p o l i c y i s g i v e n a p p r o x i m a t e l y b y t h e s q u a r e o f t h e f r a c t i o n a l d e v i a t i o n o f t h e p o l i c y a w a y f r o m o p t i m a l . I n o t h e r w o r d s , w i t h l i n e a r c o s t s a n d 141 r i s k n e u t r a l i t y , e c o n o m i c o p t i m i z a t i o n i s " f o r g i v i n g " ; o t h e r o b j e c t i v e s ( c o n s e r v a t i o n , j o b c r e a t i o n , e t c . ) c a n b e p u r s u e d w i t h l i t t l e l o s s i n t h e f i s h e r y ' s e c o n o m i c v a l u e . [ O f c o u r s e , t h e u n d e r l y i n g r e q u i r e m e n t i s t h a t t h e m o d i f i e d p o l i c y r e m a i n n e a r t h e o p t i m a l s t r a t e g y , w i t h a d e v i a t i o n o f r o u g h l y ±20 p e r c e n t b e i n g r e a s o n a b l e . ] L e w i s ( 1 9 8 1 ) h a s s h o w n t h a t t h i s " f o r g i v i n g " n a t u r e n e e d n o t a p p l y w h e n n o n l i n e a r i t i e s i n c o s t s o r u t i l i t y a r e i n c l u d e d . S i n c e o u r r e s u l t s s h ow t h a t , e v e n w i t h l i n e a r c o s t s a n d r i s k n e u t r a l i t y , i n v e s t m e n t p o l i c i e s a r e s t r o n g l y a f f e c t e d b y u n c e r t a i n t y , t h e i n c o r p o r a t i o n o f a d d i t i o n a l n o n l i n e a r i t i e s may make t h e u s e o f s t o c h a s t i c r a t h e r t h a n d e t e r m i n i s t i c p o l i c i e s p a r t i c u l a r l y i m p o r t a n t t o t h e f i s h e r y ' s p e r f o r m a n c e . T h i s w i l l be t h e t o p i c o f f u t u r e w o r k . W h i l e i t a p p e a r s r e a s o n a b l e t o p r o p o s e t h a t s t o c h a s t i c m o d e l s a r e u n n e c e s s a r y i n s t u d y i n g l i n e a r - c o s t r i s k n e u t r a l f i s h e r i e s , e v e n i n t h e s e c a s e s s u c h a s i m p l i f i c a t i o n may l e a d t o f a u l t y r e s u l t s w h e n e v e r a n a d d i t i o n a l c o m p o n e n t i s a d d e d t o t h e p r o b l e m , w h e t h e r e c o n o m i c ( m i x e d f l e e t s , p r o c e s s i n g s e c t o r ) o r e c o l o g i c a l ( a g e - s t r u c t u r e d f i s h s t o c k s , m u l t i - s p e c i e s e f f e c t s ) . I t s e e m s l i k e l y t h a t t h e c u r r e n t p a t t e r n a p p a r e n t i n f i s h e r i e s r e s e a r c h , o f f i r s t o b t a i n i n g i n i t i a l d e t e r m i n i s t i c r e s u l t s a n a l y t i c a l l y a n d t h e n u s i n g n u m e r i c a l m e t h o d s t o e x p a n d t h e s e r e s u l t s t o t h e s t o c h a s t i c c a s e , i n some s e n s e r e p r e s e n t s o p t i m a l b e h a v i o u r . A s m o r e a n d m o r e r e a l i s t i c m o d e l s a r e c o n s i d e r e d , a n a l y t i c r e s u l t s b e c o m e i m p o s s i b l e e v e n i n t h e d e t e r m i n i s t i c c a s e ; when n u m e r i c a l m e t h o d s a r e c a l l e d f o r , we h a v e f o u n d t h a t 142 o b t a i n i n g f u l l s t o c h a s t i c r e s u l t s c a n be a s s t r a i g h t f o r w a r d a s f i n d i n g t h e d e t e r m i n i s t i c s o l u t i o n . T h e t h r e e t y p e s o f u n c e r t a i n t y i m p o r t a n t i n f i s h e r i e s m a n a g e m e n t w e r e o u t l i n e d i n C h a p t e r I I I . W h i l e we h a v e c o n c e n t r a t e d h e r e o n t h e p r o b l e m o f s t o c h a s t i c f l u c t u a t i o n s i n t h e r e s o u r c e s t o c k , t h e f u n d a m e n t a l u n c e r t a i n t i e s i n v o l v e d when m o d e l p a r a m e t e r s a r e k n o w n o n l y i m p r e c i s e l y , o r e v e n w o r s e , when t h e c o r r e c t s t r u c t u r e o f t h e f i s h e r y i s u n c l e a r , a r e a l s o o f g r e a t i n t e r e s t . P a r a m e t e r u n c e r t a i n t y may a r i s e i n t h e s t o c k - r e c r u i t m e n t f u n c t i o n d e s c r i b i n g t h e b e h a v i o u r o f t h e f i s h s t o c k o r i n t h e f i s h e r y ' s e c o n o m i c p a r a m e t e r s ( t h e f i s h p r i c e , t h e v a r i a b l e c o s t f u n c t i o n , t h e f u t u r e c o s t o f c a p i t a l , t h e d i s c o u n t r a t e , a n d s o o n ) . O f t h e s e p o s s i b i l i t i e s , u n c e r t a i n t y i n t h e p o p u l a t i o n d y n a m i c s p a r a m e t e r s i s p e r h a p s m o s t s e r i o u s , s i n c e t h e s e p a r a m e t e r s e n t e r d i r e c t l y i n t o t h e d y n a m i c s o f t h e f i s h e r y a n d a r e i n h e r e n t l y u n c e r t a i n ( w h e r e a s t h e e c o n o m i c p a r a m e t e r s w i l l a t l e a s t b e c o m e k n o w n w i t h t i m e ) . I t s e e m s c l e a r t h a t s u c h u n c e r t a i n t i e s m u s t p l a y a n i m p o r t a n t r o l e i n d e t e r m i n i n g o p t i m a l i n v e s t m e n t s t r a t e g i e s . I n d e e d , i n i t i a l e r r o r s i n p a r a m e t e r e s t i m a t e s , c o m b i n e d w i t h s i m p l e i r r e v e r s i b l e i n v e s t m e n t s t r a t e g i e s , c o u l d l e a d t o c o n s i d e r a b l e o v e r - c a p a c i t y , p a r t i c u l a r l y i n a d e v e l o p i n g f i s h e r y . [ T h e p o t e n t i a l i m p o r t a n c e o f u n c e r t a i n t y i n a p a r a m e t e r c a n be c h e c k e d u s i n g t h e a n a l y s i s o f C h a p t e r s I I a n d I I I . I f t h e o p t i m a l p o l i c y f u n c t i o n s o b t a i n e d t h e r e a r e f a i r l y i n s e n s i t i v e t o v a r i a t i o n s i n t h e p a r a m e t e r o v e r i t s l i k e l y r a n g e , s u c h u n c e r t a i n t y c a n be f a i r l y s a f e l y i g n o r e d . H o w e v e r , o n e w o u l d n o t e x p e c t t h i s t o be t h e c a s e i n g e n e r a l . ] 143 P r e l i m i n a r y a n a l y s i s o f t h e i n v e s t m e n t p r o b l e m u n d e r p a r a m e t e r u n c e r t a i n t y , w h i l e b e y o n d t h e s c o p e o f t h i s t h e s i s , s u g g e s t s a r i c h v a r i e t y o f p o s s i b l e b e h a v i o u r a n d o f p o s s i b l e i n v e s t m e n t s t r a t e g i e s . I n p a r t i c u l a r , t h e d e v e l o p m e n t o f a d a p t i v e m a n a g e m e n t p o l i c i e s , i n w h i c h f i s h e r i e s i n v e s t m e n t r e s p o n d s t o new i n f o r m a t i o n a n d i s u s e d i n t u r n a s a t o o l t o a c q u i r e i n f o r m a t i o n , p r o m i s e s t o b e a f r u i t f u l a r e a f o r f u r t h e r r e s e a r c h . R e f e r e n c e s ACMRR W o r k i n g P a r t y o n t h e S c i e n t i f i c B a s i s o f D e t e r m i n i n g M a n a g e m e n t M e a s u r e s , R e p o r t o f t h e ACMRR W o r k i n g P a r t y o n t h e s c i e n t i f i c b a s i s o f d e t e r m i n i n g m a n a g e m e n t m e a s u r e s . FAO F i s h . R e p . 2 3 6 ( 1 9 8 0 ) . P. A n d e r s e n , S e l e c t e d a s p e c t s o f t h e b e h a v i o r o f t h e c o m p e t -i t i v e f i r m u n d e r u n c e r t a i n t y w i t h a p p l i c a t i o n t o t h e f i s h i n g f i r m a n d c o m m e n t s o n t h e c o m p e t i t i v e i n d u s t r y , P a p e r p r e s e n t e d t o t h e W o r k s h o p o n U n c e r t a i n t y a n d F i s h e r i e s E c o n o m i c s , U n i v . R h o d e I s l a n d ( 1 9 8 1 ) . P. A n d e r s e n , C o m m e r c i a l f i s h e r i e s u n d e r p r i c e u n c e r t a i n t y , J . E n v . E c o n . M g t . 9, 1 1 - 2 8 ( 1 9 8 2 ) . P. A n d e r s e n a n d J . G . S u t i n e n , A s u r v e y o f s t o c h a s t i c b i o e c o n o m i c s : m e t h o d s a n d r e s u l t s , P a p e r p r e s e n t e d t o t h e W o r k s h o p o n U n c e r t a i n t y a n d F i s h e r i e s E c o n o m i c s , U r i i v . R h o d e I s l a n d ( 1 9 8 1 ) . J . L . A r o n , H a r v e s t i n g a p r o t e c t e d p o p u l a t i o n i n a n u n c e r t a i n e n v i r o n m e n t , M a t h . B i o s c i . 4 7 , 1 9 7 - 2 0 5 ( 1 9 7 9 ) . K . J . A r r o w , O p t i m a l c a p i t a l p o l i c y w i t h i r r e v e r s i b l e i n v e s t m e n t , i n " V a l u e , C a p i t a l , a n d G r o w t h : P a p e r s i n H o n o u r o f S i r J o h n H i c k s " , e d . b y J . N . W o l f e , E d i n b u r g h U n i v e r s i t y P r e s s , E d i n b u r g h ( 1 9 6 8 ) . K . J . A r r o w a n d M. K u r z , " P u b l i c I n v e s t m e n t , t h e R a t e o f R e t u r n , a n d O p t i m a l F i s c a l P o l i c y " , J o h n s H o p k i n s P r e s s , B a l t i m o r e ( 1 9 7 0 ) . K . J . A r r o w a n d R.C. L i n d , U n c e r t a i n t y a n d t h e e v a l u a t i o n o f p u b l i c i n v e s t m e n t d e c i s i o n s , A m e r . E c o n . R e v . 6 0 , 3 6 4 - 3 7 8 ( 1 9 7 0 ) . W.J. B a u m o l , On t h e s o c i a l r a t e o f d i s c o u n t , A m e r . E c o n . R e v . 5 8 , 7 8 8 - 8 0 2 ( 1 9 6 8 ) . J . R . B e d d i n g t o n a n d R.M. M a y , H a r v e s t i n g n a t u r a l p o p u l a t i o n s i n a r a n d o m l y f l u c t u a t i n g e n v i r o n m e n t , S c i e n c e 1 9 7 , 4 6 3 - 4 6 5 ( 1 9 7 7 ) . R. B e l l m a n , " D y n a m i c P r o g r a m m i n g " , P r i n c e t o n U n i v . P r e s s , P r i n c e t o n ( 1 9 5 7 ) . R . J . H . B e v e r t o n a n d S . J . H o l t , On t h e d y n a m i c s o f e x p l o i t e d f i s h p o p u l a t i o n s , M i n i s t . A g r . F i s h . F o o d , L o n d o n . F i s h . I n v e s t . S e r . 2 ( 1 9 5 7 ) . 145 N.E. B o c k s t a e l a n d J . J . O p a l u c h , D i s c r e t e m o d e l l i n g o f b e h a v i o r a l r e s p o n s e u n d e r u n c e r t a i n t y : T h e c a s e o f t h e f i s h e r y , U n i v . R h o d e I s l a n d D e p t . R e s . E c o n . S t a f f P a p e r ( 1 9 8 1 ) . W.A. B r o c k a n d L . J . M i r m a n , O p t i m a l e c o n o m i c g r o w t h a n d u n c e r t a i n t y : T h e d i s c o u n t e d c a s e , J . E c o n . T h e o r y 1 8 , 2 2 5 - 2 4 3 ( 1 9 7 8 ) . H.F. C a m p b e l l , T h e e f f e c t o f c a p i t a l i n t e n s i t y o n t h e o p t i m a l r a t e o f e x t r a c t i o n o f a m i n e r a l d e p o s i t , C a n . J . E c o n . 1 3 , 3 4 9 - 3 5 6 ( 1 9 8 0 ) . C.W. C l a r k , " M a t h e m a t i c a l B i o e c o n o m i c s : T h e O p t i m a l M a n a g e m e n t o f R e n e w a b l e R e s o u r c e s " , W i l e y - I n t e r s c i e n c e , New Y o r k ( 1 9 7 6 a ) . C. W. C l a r k , A d e l a y e d - r e c r u i t m e n t m o d e l o f p o p u l a t i o n d y n a m i c s , w i t h a n a p p l i c a t i o n t o b a l e e n w h a l e p o p u l a t i o n , J . M a t h . B i o l o g y 3, 3 8 1 - 3 9 1 ( 1 9 7 6 b ) . C.W. C l a r k , T o w a r d s a p r e d i c t i v e m o d e l f o r t h e . e c o n o m i c r e g u l a t i o n o f c o m m e r c i a l f i s h e r i e s , C a n . J . F i s h . A q u a t . S c i . 3 7 , 1 1 1 1 - 1 1 2 9 ( 1 9 8 0 ) . C.W. C l a r k , F . H . C l a r k e a n d G. R. M u n r o , T h e o p t i m a l e x p l o i t a -t i o n o f r e n e w a b l e r e s o u r c e s t o c k s : p r o b l e m s o f i r r e v e r s i b l e i n v e s t m e n t , E c o n o m e t r i c a 4 7 , 2 5 - 4 9 ( 1 9 7 9 ) . C. W. C l a r k a n d G.P. K i r k w o o d , B i o e c o n o m i c m o d e l o f t h e G u l f o f C a r p e n t e r i a p r a w n f i s h e r y , J . F i s h . R e s . B o a r d C a n . 3 6 , 1 3 0 4 - 1 3 1 2 ( 1 9 7 9 ) . C.W. C l a r k a n d R. L a m b e r s o n , P e l a g i c w h a l i n g : a n e c o n o m i c h i s t o r y a n d a n a l y s i s , M a r i n e P o l i c y 1 0 3 - 1 2 0 ( 1 9 8 2 ) . C.W. C l a r k a n d G. R. M u n r o , T h e e c o n o m i c s o f f i s h i n g a n d m o d e r n c a p i t a l t h e o r y : a s i m p l i f i e d a p p r o a c h , J . E n v . E c o n . M g t . 2, 9 2 - 1 0 6 ( 1 9 7 5 ) . C. W. C l a r k a n d G.R. M u n r o , F i s h e r i e s a n d t h e p r o c e s s i n g s e c t o r : Some i m p l i c a t i o n s f o r m a n a g e m e n t p o l i c y , B e l l J . E c o n . 1 1 , 6 0 3 - 6 1 6 ( 1 9 8 0 ) . V. C o n r a d a n d Y. W a l l a c h , A f a s t e r SSOR a l g o r i t h m , N u m e r . M a t h . 2 7 , 3 7 1 - 3 7 2 ( 1 9 7 7 ) . D. H. 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O s i e r , "An A p p l i c a t i o n o f M a r g i n a l C o s t P r i c i n g P r i n c i p l e s t o B.C. H y d r o " , D e p a r t m e n t o f E c o n o m i c s , R e s o u r c e s P a p e r N o . 1 2 , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r ( 1 9 7 7 ) . P.H. P e a r s e ( e d . ) , S y m p o s i u m o n p o l i c i e s f o r e c o n o m i c r a t i o n a l i z a t i o n o f c o m m e r c i a l f i s h e r i e s , J . F i s h . R e s . C a n . 3 6 , 7 1 1 - 8 6 6 ( 1 9 7 9 ) . M.L. P u t e r m a n a n d M.C. S h i n , M o d i f i e d p o l i c y i t e r a t i o n a l g o r i t h m s f o r d i s c o u n t e d M a r k o v d e c i s i o n p r o b l e m s , M a n a g e m e n t S c i . 2 4 , 1 1 2 7 - 1 1 3 7 ( 1 9 7 8 ) . R.S. P i n d y c k , U n c e r t a i n t y a n d e x h a u s t i b l e r e s o u r c e m a r k e t s J . P o l i t i c a l E c o n . 8 8 , 1 2 0 3 - 1 2 2 5 ( 1 9 8 0 ) . R.S. P i n d y c k , U n c e r t a i n t y i n t h e t h e o r y o f r e n e w a b l e r e s o u r c e m a r k e t s , p r e p r i n t , M a s s . I n s t . T e c h . ( 1 9 8 2 ) . W.J. R e e d , A s t o c h a s t i c m o d e l f o r t h e e c o n o m i c m a n a g e m e n t o f a r e n e w a b l e r e s o u r c e , Math.. B i o s c i . 2 2 , 3 1 3 - 3 3 7 ( 1 9 7 4 ) . W.J. R e e d , T h e s t e a d y - s t a t e o f a s t o c h a s t i c h a r v e s t i n g m o d e l , M a t h . B i o s c i . 4 1 , 2 7 3 - 3 0 7 ( 1 9 7 8 ) . W.J. R e e d , O p t i m a l e s c a p e m e n t l e v e l s i n s t o c h a s t i c a n d d e t e r m i n i s t i c m o d e l s , J . E n v . E c o n . M g t . 6, 3 5 0 - 3 6 3 ( 1 9 7 9 ) . W.E. R i c k e r , S t o c k a n d r e c r u i t m e n t , J . F i s h . R e s . B o a r d C a n . 1 1 , 5 5 9 - 6 2 3 ( 1 9 5 4 ) . W.E. R i c k e r , " C o m p u t a t i o n a n d I n t e r p r e t a t i o n o f B i o l o g i c a l S t a t i s t i c s o f F i s h P o p u l a t i o n s " , B u l l . F i s h . R e s . B o a r d C a n . 191 ( 1 9 7 5 ) . H.E. R y d e r a n d G.M. H e a l , O p t i m a l g r o w t h w i t h i n t e r t e m p o r a l l y d e p e n d e n t p r e f e r e n c e s , R e v . E c o n . S t u d . 4 0 , 1-31 ( 1 9 7 3 ) . A. S c o t t a n d P.A. N e h e r ( e d s . ) , "The P u b l i c R e g u l a t i o n o f C o m m e r c i a l F i s h e r i e s i n C a n a d a " , E c o n o m i c C o u n c i l o f C a n a d a , O t t a w a ( 1 9 8 1 ) . W. S i l v e r t , T h e e c o n o m i c s o f o v e r - f i s h i n g , T r a n s . A m e r . F i s h . S o c . 1 0 6 , 1 2 1 - 1 3 0 ( 1 9 7 7 ) . J . B . S m i t h , An a n a l y s i s o f o p t i m a l r e p l e n i s h a b l e r e s o u r c e m a n a g e m e n t u n d e r u n c e r t a i n t y , P h . D . d i s s e r t a t i o n , U. W e s t e r n O n t a r i o , L o n d o n ( 1 9 7 8 ) . 149 V . L . S m i t h , E c o n o m i c s o f p r o d u c t i o n f r o m n a t u r a l r e s o u r c e s , A m e r . E c o n . R e v . 5 8 , 4 0 9 - 4 3 1 ( 1 9 6 8 ) . D.F. S p u l b e r , A d a p t i v e h a r v e s t i n g p o l i c i e s a n d s t a b l e e q u i l i b r i u m f o r t h e m u l t i - c o h o r t f i s h e r y u n d e r u n c e r t a i n t y , m i m e o g r a p h , D e p t . E c o n . B r o w n U n i v e r s i t y ( 1 9 7 8 ) . D.F. S p u l b e r , P u l s e f i s h i n g t h e m u l t i c o h o r t f i s h e r y u n d e r u n c e r t a i n t y , P a p e r p r e s e n t e d t o t h e W o r k s h o p o n U n c e r t a i n t y a n d F i s h e r i e s E c o n o m i c s , U n i v . R h o d e I s l a n d ( 1 9 8 1 ) . D.F. S p u l b e r , F i s h e r i e s a n d u n c e r t a i n t y , m i m e o g r a p h , D e p t . E c o n . U n i v . S o u t h e r n C a l i f o r n i a ( 1 9 8 2 ) . C . J . W a l t e r s , O p t i m a l h a r v e s t s t r a t e g i e s f o r s a l m o n i n r e l a t i o n t o e n v i r o n m e n t a l v a r i a b i l i t y a n d u n c e r t a i n p r o d u c t i o n p a r a m e t e r s , J . F i s h . R e s . B o a r d C a n . 3 2 , 1 7 7 7 - 1 7 8 4 ( 1 9 7 5 ) . C . J . W a l t e r s , p e r s o n a l c o m m u n i c a t i o n ( 1 9 8 1 ) . C . J . W a l t e r s a n d R. H i l b o r n , A d a p t i v e c o n t r o l o f f i s h i n g s y s t e m s , J . F i s h . R e s . B o a r d C a n . 3 3 , 1 4 5 - 1 5 9 ( 1 9 7 6 ) . C . J . W a l t e r s a n d R. H i l b o r n , E c o l o g i c a l o p t i m i z a t i o n a n d a d a p t i v e m a n a g e m e n t , A n n u a l R e v . E c o l . S y s t . 9, 1 5 7 - 1 8 8 ( 1 9 7 8 ) . C . J . W a l t e r s a n d D. L u d w i g , E f f e c t s o f m e a s u r e m e n t e r r o r s o n t h e a s s e s s m e n t o f s t o c k - r e c r u i t m e n t r e l a t i o n s h i p s , C a n . J . F i s h . A q u a t . S c i . 3 8 , 7 0 4 - 7 1 0 ( 1 9 8 1 ) . 1 50 A p p e n d i x A We w i s h t o e x a m i n e i n m o r e d e t a i l t h e l o n g r u n e q u i l i b r i u m v a l u e s f o r r e c r u i t m e n t , e s c a p e m e n t a n d c a p a c i t y , d e n o t e d b y R, S, K r e s p e c t i v e l y . F o r a n y g i v e n K, t h e e q u i l i b r i u m v a l u e o f S i s t h e s o l u t i o n o f : - q T K ( A 1 ) S= Max ( s ( K ) , F ( S ) e }. F o r f i x e d S, c a p a c i t y i n t h e d e l a y e d i n v e s t m e n t c a s e i s g i v e n b y K = M a x { ( l - r ) K , h ( S ) } s o t h a t t h e e q u i l i b r i u m s a t i s f i e s n+1 n ( A 2 ) K= Max { ( 1 - r ) K , h ( S ) } = h ( S ) . T h e s o l u t i o n o f ( A 1 ) a n d ( A 2 ) p r o d u c e s u n i q u e v a l u e s S a n d K, a n d h e n c e R= F ( S ) . I n t h e i n s t a n t a n e o u s i n v e s t m e n t c a s e , ( A 2 ) b e c o m e s : K= h ( R ) = h ( F ( S ) ) a n d t h e s o l u t i o n f o l l o w s s i m i l a r l y . T o r e l a t e t h e s e e q u i l i b r i a t o t h e s o l u t i o n s o f M o d i f i e d G o l d e n R u l e e q u a t i o n s , we s h a l l c o n s i d e r t h e b e h a v i o u r o f t h e v a l u e f u n c t i o n i n t h e v i c i n i t y o f t h e e q u i l i b r i u m ( R , K ) , b e g i n n i n g w i t h t h e d e l a y e d i n v e s t m e n t c a s e . A s s u m i n g t h a t t h e o p t i m a l e s c a p e m e n t S = S * ( R , K ) i s g i v e n b y h a r v e s t i n g a t f u l l c a p a c i t y , s o S * ( R , K ) = R • e x p { - q T K } , a n d a s s u m i n g h ( S ) > ( 1 - y ) ( K + d K ) , we c a n d e r i v e a n e x p r e s s i o n f o r V ( R , K ) a s f o l l o w s : K V ( R , K + d K ) = p R ( 1 - e x p { - q T ( K + d K ) } ) - c T ( K + d K ) - 6 [ h ( R - e x p { - q T ( K + d K ) } ) - ( 1 - y ) ( K + d K ) ] + a V ( F [ R - e x p { - q T ( K + d K ) } ] , h ( R • e x p { - q T ( K + d K ) } ) ) = V ( R , K ) + [ p q T S - c T + ( 1 - y ) 6 ] d K - a q T S F ' ( S ) V ( R , K ) d K R w h e r e we h a v e s i m p l i f i e d u s i n g t h e o p t i m a l i t y e q u a t i o n 151 V ( F ( S ) , K ) = 6 / o , s i n c e K = h ( S ) . H e n c e : K ( A 3 ) V ( R , K ) = p q T ( S - ( 1 / p q ) [ c - ( 1 - 7 ) 6 / T ] ) - o q T S F ' ( S ) V ( R , K ) K R A g a i n u s i n g V ( R , K ) = 6 / o , e q u a t i o n ( A 3 ) c a n be r e w r i t t e n t o g i v e : K ( A 4 ) V ( R , K ) = p ( S - x $ ) / o S F ' ( S ) R w h e r e x$= ( 1 / p q ) { c + [ ( 1 - c ) / c + y ] 6 / T } = ( 1 / p q ) { c + [ 1 - o + o y ] ( 6 / 0 T ) } . A l t e r n a t i v e l y , V ( R , K ) c a n b e d e r i v e d a s f o l l o w s : R V ( R + d R , K ) = p ( R + d R ) ( 1 - e x p { - q T K } ) - c T K - 6 [ h ( S + e x p { - q T K } d R ) - ( 1 - y ) K ] + o V ( F [ S + e x p { - q T K } d R ] , h ( S + e x p { - q T K } d R ) ) = V ( R , K ) + [ p ( 1 - e x p { - q T K } ) + o e x p { - q T K } F ' ( S ) V ( R , K ) ] d R . R w h e r e a g a i n S = R * e x p { - q T K } , a n d we h a v e u s e d V ( F ( S ) , K ) = 6 / a . K S i n c e e x p { - q T K } = S / R = S / F ( S ) , t h i s c a n be s o l v e d f o r V ( R , K ) : R ( A 5 ) V ( R , K ) = p [ F ( S ) - S ] / ( F ( S ) - c S F ' ( S ) ) R E q u a t i n g t h e e x p r e s s i o n s f o r V ( R , K ) i n ( A 4 ) a n d ( A 5 ) , a n d R r e a r r a n g i n g , we s e e t h a t t h e o p t i m a l e q u i l i b r i u m e s c a p e m e n t S s a t i s f i e s : ( A 6 ) F ' ( S ) • [ 1 - x $ / F ( S ) ] / [ 1 - x g / S ] = 1 / a R e p e a t i n g t h e a n a l y s i s f o r t h e i n s t a n t a n e o u s i n v e s t m e n t c a s e , e q u a t i o n s ( A 4 ) a n d ( A 5 ) a r e u n a l t e r e d e x c e p t t h a t t h e c o n s t a n t x§ i s c h a n g e d t o x $ * ( d e f i n e d b e l o w ) , - s o t h a t t h e o p t i m a l e s c a p e m e n t now s a t i s f i e s : ( A 6 ' ) F ' ( S ) • [ 1 - x * * / F ( S ) ] / [ 1 - x $ * / S ] = 1 / 0 w h e r e x $ * = ( l / p q ) { c +[1 - o + o y ] 6 / T } . 1 52 T h e r e f o r e t h e o p t i m a l e q u i l i b r i u m e s c a p e m e n t S i n b o t h c a s e s s a t i s f i e s a M o d i f i e d G o l d e n R u l e e q u a t i o n w h e r e t h e u n i t v a r i a b l e c o s t h a s b e e n a p p r o p r i a t e l y c h a n g e d t o i n c l u d e t h e r e n t a l c o s t o f c a p i t a l . S i n c e t h e v a l u e o f o n e u n i t o f c a p i t a l b o u g h t f o r 6 t h i s y e a r d e c r e a s e s t o 0 ( 1 - 7 ) 6 n e x t y e a r , a n a n n u a l p a y m e n t o f [ l - c ( l ~ r ) ] 6 i s r e q u i r e d t o m a i n t a i n t h e v a l u e o f t h e c a p i t a l s t o c k i n e q u i l i b r i u m . H e n c e [ ( 1 - O ) / C ] 6 a n d 7 6 a r e t h e y e a r l y c h a r g e s , p e r u n i t o f c a p a c i t y , f o r i n t e r e s t a n d d e p r e c i a t i o n , r e s p e c t i v e l y . E q u a t i o n ( A 6 ) a n d ( A 6 ' ) a r e p r e c i s e l y t h e M.G.R. e q u a t i o n s o b t a i n e d b y a s s u m i n g t h a t c a p i t a l i s p e r f e c t l y m a l l e a b l e w i t h u n i t c o s t 6/0 a n d 6 r e s p e c t i v e l y . T h i s c a n be s e e n i n t h e d e l a y e d i n v e s t m e n t c a s e a s f o l l o w s ( t h e i n s t a n t a n e o u s i n v e s t m e n t c a s e i s s i m i l a r ) : n - 1 V ( R , ,K, )= I o W ( R , S ) -61 } . n>1 n n n+1 n-1 n-1 Now la 61 = 6 I a f K - ( l - y ) K ] n>1 n+1 n>1 n+1 n n-1 n-1 =6{ (1/O) I a K - ( l / c ) K , -(1-7) l a K } n>1 n n>1 n n - 1 = 6 [ ( 1 - o + o r ) / o ] • la K - ( 6 / a ) K , n>1 n n-1 H e n c e V ( R 1 f K , ) = I a { i r ( R , S ) -6 [ ( 1 - 0+07 )/a) K } + ( 6 / o ) K , . n> 1 n n n n-1 = I 0 { r r ( R , S ) - 6 [ ( 1 ~a+ar)/a]K } n>2 n n n + J T ( R , , S 1 ) + 6 ( l - y ) K 1 . 153 I f c a p i t a l i s m a l l e a b l e , K i s p r e c i s e l y e q u a l t o E , t h e n n o p t i m a l i n s t a n t a n e o u s e f f o r t . S i n c e E = ( l / q T ) l o g ( R /S ) a n d n n n T T ( R ,S )= p ( R ,S ) - c T E we o b t a i n : n n n n n V ( R , , K , ) = ff(R,,S,) + 6 ( 1 - r ) K , n-1 + I o ( p ( R -S ) - ( 1 / q ) ( c + [ ( 1 - o + c r ) / a ] ( 6 / T ) ) l o g ( R /S )} n>2 n n n n T h e m a x i m i z a t i o n o f t h i s e x p r e s s i o n , s u b j e c t t o R = F ( S ) , n+1 n c a l l s f o r a s t e a d y - s t a t e e s c a p e m e n t g i v e n b y e q u a t i o n ( A 6 ) , n a m e l y S= S ( C l a r k , 1 9 7 6 a ) . 1 54 T a b l e I P a r a m e t e r v a l u e s u s e d i n t h e b a s e c a s e r u n s o f t h e m o d e l f o r t h e p r a w n a n d w h a l e f i s h e r i e s . T h e d a t a a r e a d a p t e d f r o m C l a r k & K i r k w o o d ( 1 9 7 9 ) a n d C l a r k & L a m b e r s o n ( 1 9 8 2 ) . ( c . d . = " c a t c h e r d a y " ) Q u a n t i t y P r a w n F i s h e r y W h a l e F i s h e r y " F i s h " p r i c e ( P ) 0.9 A $ / k g 7 0 0 0 US$/BWU V a r i a b l e C o s t ( c ) 1 6 0 0 A $ / w e e k / v e s s e l 5 0 0 0 U S $ / c . d . C a p i t a l c o s t ( 6 ) 0 . 4 7 X 1 0 6 A $ / v e s s e l 1 0 0 0 0 US$ / ( c . d . / y e a i N e t R e v e n u e i n A l t e r n a t i v e F i s h e r y (/») 0 A $ / w e e k / v e s s e l 0 U S $ / c . d . D e p r e c i a t i o n R a t e ( r ) 0.15 0.15 D i s c o u n t F a c t o r ( a ) 0.9 0.9 C a t c h a b i l i t y ( q ) 0 . 0 0 1 7 9 / w e e k / v e s s e l 1 . 3 x 1 0 - * / c . d . M a x . S e a s o n L e n g t h ( T ) 2 6 . 0 w e e k s 1.0 y e a r s N a t . M o r t a l i t y R a t e (m) 0.05 / w e e k 0.1 / y e a r M a x i m u m F e c u n d i t y ( a ) 4 2 . 0 1.15 M a x i m u m R e c r u i t m e n t ( b ) 7 . 0 X 1 0 6 k g 1 . 1 8 6 x 1 0 7 BWU 155 T a b l e I I T h e o p t i m a l v a l u e f u n c t i o n V ( R , K ) f o r t h e b a s e c a s e p r a w n f i s h e r y . R e c r u i t m e n t i s g i v e n i n m i l l i o n s o f k i l o g r a m s , c a p a c i t y i n s t a n d a r d i z e d v e s s e l s a n d " v a l u e " i n m i l l i o n s o f A u s t r a l i a n d o l l a r s . R e c r u i t m e n t 1 .0 2.5 4.0 5.5 7.0 8.5 10.0 C a p i t a l 0 3.2 3.4 3.5 3.6 3.7 3.7 3.7 2 4.0 4.3 4.6 4.8 4.9 5.1 5.2 4 4.8 5.2 5.5 5.8 6. 1 6.4 6.6 6 5.5 6.0 6.5 6.9 7.3 7.7 8.0 8 6.3 6.9 7.4 7.9 8.4 8.9 9.3 10 7.0 7.6 8.3 8.9 9.5 10.0 10.6 1 2 7.7 8.4 9.2 9.9 1 0 . 5 11.1 11.8 1 4 8.4 9. 1 9.9 10.7 1 1 . 5 1 2 . 2 12.9 1 6 9.0 9.7 1 0 . 6 1 1 . 5 12.3 13.1 1 3 . 9 18 9.5 10.3 1 1 . 3 12.2 13.1 1 4 . 0 14.8 20 10.1 10.9 11.8 12.8 13.8 14.7 1 5 . 6 22 1 0 . 5 11.4 1 2 . 3 13.4 14.4 15.4 16.3 24 1 0 . 9 11.8 12.7 13.8 1 4 . 9 1 5 . 9 1 6 . 9 26 1 1 . 3 12.2 13.1 14.2 1 5 . 3 16.4 1 7 . 5 28 1 1 . 6 12.6 1 3 . 5 1 4 . 5 15.7 16.8 17.9 156 T a b l e I I I . V a l u e s o f g (•,•,•) x F i x i n g t h e v a l u e o f i ( i = 1 , . . . , M ) , l e t x.=x , x + = x a n d i i + 1 A = x + - x . . T h e n t h e f u n c t i o n g ( x ) o f I I I . D c a n be w r i t t e n a s a n i 4 4 - r p o l y n o m i a l i n x , g ( x ) = I g ( i , n , r ) x n i r=1 x w h e r e t h e c o e f f i c i e n t s g ( i , n , r ) a r e g i v e n b y : x y = 1 y = 2 y = 3 y = 4 n=1 2 / A 3 3 / A 2 - 6 x + / A 3 - 6 x + / A 2 + 6 x 2 / A 3 3 x 2 / A 2 - 2 x 3 / A 3 n=2 1/A 2 - 3 X + / A 2 + 1 / A 3 x 2 / A 2 - 2 x + / A ~ x 3 / A 2 + x 2 / A n=4 1/A 2 - 3 X . / A 2 - 1 / A 3 x 2 / A 2 + 2 x . / A - X 3 / A 2 - X 2 / A 1 57 T a b l e I V . C o m p a r i s o n o f V a l u e F u n c t i o n s f o r  O p t i m a l D e t e r m i n i s t i c a n d S t o c h a s t i c P o l i c i e s T h e v a l u e f u n c t i o n V ( R , K ) i s e v a l u a t e d i n a s t o c h a s t i c ( f f = 0 . 5 8 ) e n v i r o n m e n t , u s i n g t w o s e t s o f p o l i c y f u n c t i o n s : ( i ) t h e o p t i m a l p o l i c i e s f o r t h e <r=0.58 s t o c h a s t i c e n v i r o n m e n t , a n d ( i i ) t h e " o p t i m a l " p o l i c i e s i f t h e f i s h e r y w e r e d e t e r m i n i s t i c . I n t h e t a b l e , r e c r u i t m e n t i s g i v e n i n m i l l i o n s o f k i l o g r a m s , c a p a c i t y K i s i n s t a n d a r d i z e d v e s s e l s , a n d v a l u e i s m e a s u r e d i n m i l l i o n s o f A u s t r a l i a n d o l l a r s . V ( R , K ) f o r t h e O p t i m a l S t o c h a s t i c P o l i c y K/R 0. 20 0.50 1 .00 2.00 3.00 4.50 7.00 2 0 . 0 0.0 4. 61 5.04 5.39 5.91 6.26 6.60 6.92 7.43 3.0 4. 75 5.21 5.60 6.12 6.58 7.10 7.74 9.81 6.0 4 . 89 5.38 5.81 6.33 6.85 7.53 8.44 11.9 9.0 5. 02 5.54 6.01 6. 54 7.08 7.90 9.04 13.7 12.0 5. 1 5 5.69 6.19 6.76 7.29 8.21 9.55 1 5 . 3 15.0 5. 27 5.84 6.36 6.97 7.50 8.48 9.99 16.6 18.0 5. 39 5.98 6.53 7.17 7.70 8.69 10.4 17.8 2 1 . 0 5. 50 6.12 6.68 7.35 7.88 8.87 10.7 18.8 V ( R , K ) f o r t h e O p t i m a l D e t e r m i n i s t i c P o l i c y K/R 0. 20 0.50 1 .00 2.00 3.00 4.50 7.00 2 0 . 0 0.0 4 . 48 • 4.90 5.23 5.75 6.11 6.45 6.76 7.25 3.0 4. 63 5.08 5.46 5.96 6.43 6.95 7.57 9.63 6.0 4. 77 5.25 5.68 6.18 6.69 7.38 8.28 11.7 9.0 4. 90 5.41 5.87 6.39 6.93 7.75 8.88 1 3 . 5 12.0 5. 03 5.57 6.06 6.62 7 . 1 5 8.06 9.40 15.1 15.0 5. 1 6 5.72 6.24 6.84 7.37 8.35 9.84 16.4 18.0 5. 28 5.87 6.41 7.05 7.58 8.57 10.2 17.6 2 1 . 0 5. 40 6.01 6.58 7.24 7.77 8.76 10.6 18.7 158 Figure 1. The optimal capacity function, h(R), and optimal escapement function, s(K), for the base case prawn fishery with instantaneous investment. Sample t r a j e c t o r i e s and the long run equilibrium (R,K) are also shown. B I O M A S S ( M I L L I O N S OF K I L O G R A M S ) 159 F i g u r e 2 . The o p t i m a l c a p a c i t y f u n c t i o n , h ( R ) , and o p t i m a l escapement f u n c t i o n , s ( K ) , f o r the base case whale f i s h e r y w i t h i n s t a n t a n e o u s i n v e s t m e n t . Sample t r a j e c t o r i e s and the l o n g run e q u i l i b r i u m (R,K) a r e a l s o shown. 160 F i g u r e 3 . The o p t i m a l p o l i c y f u n c t i o n s f o r the base case prawn f i s h e r y w i t h d e l a y e d i n v e s t m e n t . Sample t r a j e c t o r i e s and the l o n g run e q u i l i b r i u m (S,K) a r e a l s o shown. 161 F i g u r e 4 . The o p t i m a l p o l i c y f u n c t i o n s f o r the base case whale f i s h e r y w i t h d e l a y e d i n v e s t m e n t . Sample t r a j e c t o r i e s and the l o n g run e q u i l i b r i u m (S,K) a r e a l s o shown. 162 F i g u r e 5. The e f f e c t of f e c u n d i t y of the prawn s t o c k on the o p t i m a l p o l i c y f u n c t i o n s . Long run e q u i l i b r i u m p o i n t s f o r each parameter v a l u e a r e i n d i c a t e d . i g u r e 6 . T h e e f f e c t o f t h e c a r r y i n g c a p a c i t y (maximum r e c r u i t m e n t ) o f t h e p r a w n s t o c k on t h e o p t i m a l p o l i c y f u n c t i o n s . 164 F i g u r e 7. The r o l e of d e p r e c i a t i o n i n the base case prawn f i s h e r y . The z e r o - d e p r e c i a t i o n e q u i l i b r i u m c u r v e i s shown, t o g e t h e r w i t h e q u i l i b r i u m p o i n t s c o r r e s p o n d i n g t o r=0.05,0.15 and 0.20 . B I O M A S S ( M I L L I O N S O F i r K I L O G R A M S ) 165 F i g u r e 8. The r o l e of d e p r e c i a t i o n i n the whale f i s h e r y . The e q u i l i b r i u m p o i n t f o r the r=0.15 case and the e q u i l i b r i u m c u r v e f o r the r=0 case a r e shown. 166 F i g u r e 9. The e f f e c t of the d e p r e c i a t i o n r a t e on a lower f e c u n d i t y (a=14) prawn f i s h e r y w i t h o t h e r w i s e unchanged p a r a m e t e r s . 167 F i g u r e 10. The e f f e c t of the d e p r e c i a t i o n r a t e on a prawn f i s h e r y w i t h low f e c u n d i t y (a=14) and low c a p i t a l c o s t (6= A$8.32x10"). 168 F i g u r e 11. V a r i a t i o n of the o p t i m a l p o l i c y f u n c t i o n s w i t h u n i t c a p i t a l c o s t , 6, f o r the prawn f i s h e r y . 169 F i g u r e 12. V a r i a t i o n of the o p t i m a l p o l i c y f u n c t i o n s w i t h f i s h p r i c e , p, f o r the prawn f i s h e r y . 170 Figure 13. The effect of the discount factor, c, and the corresponding discount rate, on the optimal policy functions for the prawn fishery. 171 Figure 14. The optimal policy functions for a fishery with a Ricker stock-recruitment function (a=11.639, b=7.0xl0 6; see text for d e t a i l s ) . 172 F i g u r e 15. The o p t i m a l p o l i c y f u n c t i o n s , h(S) and s ( K ) , ar e shown f o r the s t o c h a s t i c (<r=0.58) base case prawn f i s h e r y . In a d d i t i o n the st e a d y s t a t e d i s t r i b u t i o n f o r t h i s o p t i m a l l y managed f i s h e r y i s a p p r o x i m a t e d by showing the e n d p o i n t s of 160 40-year sample p a t h s , b e g i n n i n g each one a t the q u a s i - e q u i l i b r i u m p o i n t , ( 4 . 3 X 1 0 6 , 7 . 7 5 ) . See t e x t f o r d e t a i l s . s(K) CO _J .. CO CO LU . ><c' Q UJ Q or cr •• cr l o 1 " ' o cr . Q_m cr PM ' h(S) IS:*. 7 i - i 1 1 1 r 3 ' B I O M A S S ( M I L L I O N S OF K I L O G R A M S ) 21 173 F i g u r e 16. O p t i m a l p o l i c y f u n c t i o n s and s i m u l a t i o n method a r e as i n F i g u r e 15, except the i n i t i a l p o i n t f o r each s i m u l a t i o n i s (2.Ox 1 0 6 , 6 . 0 ) . As e x p e c t e d , t h i s does not a f f e c t the d i s t r i b u t i o n . 174 Figure 17. Optimal policy functions and simulation method are as in Figure 15, except the i n i t i a l point for each simulation i s (6.0x10 s,9.0). Again, t h i s does not a f f e c t the steady-state d i s t r i b u t i o n . tn — i •. Ujcn to in UJ . >«• a a s« z cr <_> cr ._] cr CJ IN ' s(K) h(S) V •44^ * 11 4 • 4> 4-1. V j . »+ t 4* 4 4 1 7 4-4-4 / 1 1 1 1 r~ B I O M A S S ( M I L L I O N S OF K I L O G R A M S ) 2) 175 F i g u r e 18. O p t i m a l p o l i c y f u n c t i o n s and s i m u l a t i o n method are as i n F i g u r e 15, except the l e v e l of n o i s e has been reduced t o *=0.2; . as e x p e c t e d , the d i s t r i b u t i o n c o n c e n t r a t e s around the q u a s i - e q u i l i b r i u m p o i n t . to —i •. U J C T l to to U J . > » • o IxJ Mr- ' ' D an cr to 1"' o cr .. n cn cr CM ' h(S) ——I 1 1 1 B I O M A S S ( M I L L I O N S OF K I L O G R A M S ) IB. 21 176 Figure 19(a). The optimal policy functions for the base case stochastic prawn fishery are shown, together with 8 20-year sample paths, each beginning from the quasi-equilibrium point. 177 F i g u r e 1 9 ( b ) . The e f f e c t of s t o c h a s t i c f l u c t u a t i o n s on the o p t i m a l management of the base case prawn f i s h e r y i s shown f o r a sample 15-year outcome, governed by the p o l i c y c u r v e s i n d i c a t e d . The arrows j o i n (S,K) p o i n t s ; the p r o c e s s e s of d e p r e c i a t i o n , i n v e s t m e n t , r e c r u i t m e n t and h a r v e s t i n g o c cur s u c c e s s i v e l y between each p a i r of p o i n t s . 178 F i g u r e 20. The o p t i m a l f l e e t c a p a c i t y f u n c t i o n s , h ( S ) , a r e shown f o r the a=42 prawn f i s h e r y w i t h 4 v a l u e s of the c a p i t a l c o s t , 6=0.1175, 0.235, 0.470 and 0.705 ( A $ X 1 0 6 ) and v a r i o u s l e v e l s of u n c e r t a i n t y (a). E q u i l i b r i u m and q u a s i - e q u i l i b r i u m p o i n t s a re a l s o i n d i c a t e d . 179 F i g u r e 21. The o p t i m a l f l e e t c a p a c i t y f u n c t i o n s , h ( S ) , a r e shown f o r the a=14 prawn f i s h e r y w i t h 3 v a l u e s of the c a p i t a l c o s t , 6=0.0832, 0.235, and 0.470 ( A $ X 1 0 6 ) and u n c e r t a i n t y l e v e l g i v e n by e= 0 and 0.58 . a=o O-0.58 S i <n" J B I O M A S S ( M I L L I O N S O F K I L O G R A M S ) 180 F i g u r e 22. The j o i n t e f f e c t of the biomass growth r a t e and the l e v e l of u n c e r t a i n t y i s i n v e s t i g a t e d . O p t i m a l c a p a c i t y f u n c t i o n s a r e shown f o r a= 14, 42, 140 and 560, i n the d e t e r m i n i s t i c (*=0) and s t o c h a s t i c (^=0.58) c a s e s . o-o o-0.58 a-560 B I O M A S S i 1 r ( M I L L I O N S O F K I L O G R A M S ) 181 F i g u r e 2 3 . A s i n F i g u r e 2 2 , t h e i n t e r p l a y b e t w e e n b i o m a s s g r o w t h r a t e a n d u n c e r t a i n t y i s s h o w n , b u t f o r a l o w e r l e v e l o f u n i t c a p i t a l c o s t ( 6 = A $ 8 3 2 0 0 ) . T h e v a l u e s a= 3 . 8 2 a n d 1 4 . 0 0 , w i t h «y=0 a n d 0 . 5 8 , h a v e b e e n u s e d . c n a-0.58 182 F i g u r e 24. The d e p r e c i a t i o n r a t e and u n c e r t a i n t y : the o p t i m a l d e t e r m i n i s t i c and s t o c h a s t i c c a p a c i t y f u n c t i o n s a r e shown f o r each of 7= 0, 0.15 and 0.20 , f o r the base case prawn f i s h e r y . — o-0 a=0.58 ° 1 183 F i g u r e 25. The d e p r e c i a t i o n r a t e and u n c e r t a i n t y : the o p t i m a l d e t e r m i n i s t i c and s t o c h a s t i c c a p a c i t y f u n c t i o n s a r e shown f o r each of r= 0.05 and 0.15, f o r a f i s h e r y w i t h low f e c u n d i t y (a=14) and low c a p i t a l c o s t (6=A$83200). 184 F i g u r e 26. The d i s c o u n t r a t e and u n c e r t a i n t y : the o p t i m a l d e t e r m i n i s t i c and s t o c h a s t i c c a p a c i t y f u n c t i o n s a r e shown f o r each of o= 0.80, 0.90 and 0.99 ( w i t h r e s p e c t i v e d i s c o u n t r a t e s r=25, 11 and 1 p e r c e n t ) , i n the case of the prawn f i s h e r y . O-0.99 1 . B I O M A S S ~i r ( M I L L I O N S O F K I L O G R A M S ) 185 F i g u r e 2 7 . R i c k e r s t o c k - r e c r u i t m e n t a n d u n c e r t a i n t y : t h e o p t i m a l p o l i c y f u n c t i o n s h ( S ) a n d s ( K ) a r e shown f o r t h e p r a w n f i s h e r y w i t h R i c k e r p a r a m e t e r s a = 3 . l 7 2 a n d b = 7 . 0 x l 0 6 , a n d u n c e r t a i n t y <r= 0 a n d 0 . 5 8 . a-0 o=0.58 S i 186 F i g u r e 2 8 . T h e w h a l e f i s h e r y u n d e r u n c e r t a i n t y : t h e o p t i m a l p o l i c y f u n c t i o n s h ( S ) a n d s ( K ) a r e shown f o r e a c h o f t h e c a s e s <t= 0 , 0.1 a n d 0 . 2 . ( T h e s ( K ) c u r v e s f o r «=0 a n d *=0.1 d i f f e r n e g l i g i b l y . ) a=0 o=0.1 o=0.2 187 Figure 29. The interaction between biomass growth rate and cost of c a p i t a l in determining the role of uncertainty in f i s h e r i e s investment. Points shown represent (a,6) combinations that have been considered; and "-" symbols indicate whether investment increases or decreases with uncertainty. (The diagram i s not drawn to scale and the dividing l i n e i s approximate.) e BIOMASS GROWTH RATE 

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