Replenishable resource management under uncertainty: A reexamination of the U.S. Northern Fishery

JOURNAL

OP ENVIRONMENTAL

ECONOMICS

AND MANAGEMENT

7, 209-219 (1980)

Replenishable Resource Management under Uncertainty: A Reexamination of the U.S. Northern Fishery’ J. BARRY SMITH Department

of Economics

and lnrtitute of Applied Economic Research, Montreal, Quebec H3G IM8, Cam&

Concordia

Uniwrzity,

Received September 11, 1979 This paper partially extends the replenishable resource management literature to the case of uncertainty. Stochastic resource dynamics are defined, studied and estimated. The results of this study suggest that the misallocation of resources into the northern lobster fishery has been greater than was previously believed. While aggregate stochastic influences are small in the lobster fishery, the dynamics of whales appear to be significantly affected by stochastic influences.

INTRODUCTION

AND CONCLUSIONS

In a recent paper, Bell [2] investigated the relationship between technological externalities and common-property resources for the U.S. northern lobster fishery. Bell, citing Dow et al. [9] argued that stochastic natural influences could affect the growth of lobster. Using mean annual water temperature as a proxy for natural activity, Bell introduced harvesting into the Lotka-Volterra resource model and estimated steady-state production (extraction) and cost relationships for the fishery. The results were most interesting. In particular, Bell empirically verified the theoretical result that the long run industry supply curve (long run industry average total cost curve) was backward bending. In addition, using 1966 costs, Bell showed that socially efficient resource allocation (price = marginal cost) would involve a reduction of approximately 50% in resources allocated into the northern lobster fishery. Finally, the temperature variable played an important role in determining these results and seemed to confirm the importance of natural activity in the determination of the growth relationship. Unfortunately, Bell’s theoretical and empirical derivation of steady-state relationships was not consistent with his hypothesized model of resource regeneration with uncertain natural activity. In this paper, the stochastic replenishable resource management results of Smith [24] are used to correct some of these problems. A new model, based upon a stochastic differential equation description of the growth process for northern lobster is introduced. Asymptotic properties of the model are derived and then the model is estimated for the northern lobster fishery using data for the same period ‘The author would like to thank the members of his thesis committee: Charles Plourde, David Scheffman and Arthur Robson. As well, the suggestions of Vittorio Corbo, Franqois Bourguignon and an anonymous referee were appreciated. Any remaining errors are the responsibility of the author.

209 0095~06%/80/030209-11$02.00/O Copyrigi~t 0 1980 by Academic F’ms, Inc. All lights of repmduction in any form reserved

210

J. BARRY

SMITH

as the Bell study. The cost and revenue relationships are illustrated for 1966 in order to provide a comparison with Bell’s results. Of interest is the fact that a different pattern of results emerges. In particular, socially efficient resource allocation would involve a greater reduction in the allocation of effort into the lobster fishery than the work of Bell suggests. In addition, the role played by uncertainty in natural activity does not appear to be important in the way that Bell’s results would suggest. It is shown that, conditional upon stock size, the variability in growth induced by natural activity is very small. The reasons for the differences in results between these two studies appear to depend upon two restrictive assumptions introduced by Bell. In the first place, Bell adopts a more restrictive functional form to describe the extraction process. In this paper, Bell’s extraction assumption is tested and statistically rejected. Secondly, Bell assumes that all variables are observed at steady-state values and therefore excludes explicit dynamic considerations from his empirical analysis. In this paper, parameters of the stochastic differential equation describing the growth process are estimated. These parameter estimates are then used to quantify the steady-state cost and output relationship under uncertainty. The approach adopted here is therefore more general. Another interesting result arising from this study relates to the necessity of using stochastic resource models to gain insights into socially optimal management policies. In the case of northern lobster, a correctly specified deterministic model appears to accurately describe most of the important information about dynamic characteristics needed for optimal management of a replenishable natural resource. Thus, a partially reassuring response is offered to Zellner’s [30] worry that the ignored effects of uncertain natural activity might lead to inaccuracies in management prescriptions. Also, for the case of whales it is shown that the error of the deterministic model is of the order of 8%.

A REVIEW OF BELL’S MODEL

Bell postulates that, in the absence of extraction, interval dt follows the Lotka-Volterra equation

resource growth

over the

dN f = AN, - d$;, dt

where A is the constant birth rate of the biomass, EN, is the variable death rate, and N, is the amount of resources biomass at time t. This differential equation is plotted in Fig. 1. The globally asymptotically stable steady-state stock size is given by N* = X/E. The maximum sustainable yield stock size occurs at N = A/2c with the corresponding maximum sustainable yield given by MSY = A22/4e. Bell introduces the extraction function

Q, = k&N,,

(2)

where Q, is the landed biomass at time t, k is a scale parameter, and E, is an index of effort at time t.

STOCHASTIC

FIG.

1.

RESOURCE

MANAGEMENT

The Lotka-Volterra

growth

211

curve.

Growth, net of extraction, is then written as dN L = AN, - cN,= - kE,N,, dt

Finally, Bell introduces a random term (U,) to the right-hand side of the growth equation to measure uncertainty in natural activity. The final form of the growth model is thus: dN 2 = AN, - cN,z - kE,N, + v,N,. dt

Bell next equates the left-hand side of (4) to zero and derives the following implicit relationship between effort and steady-state stock size: 0 =A -EN,

- kE, + v,.

(5)

(5) is combined with (2) to yield:

Q, E,=Y-

kh

k= -tE,+rQ.

Finally, Bell removes the contribution of water temperature (measured in degrees Farenheit) from the error term to obtain the final equation

where V, reflects the fact that the error structure is different. Bell estimates Eq. (7) by ordinary least squares for the period 1950- 1966 with Q, and E, measured respectively as pounds of lobsters landed and numbers of traps fished. Essentially, Eq. (7) describes industry production in terms of the average

212

J. BARRY

product of effort. The estimation brackets.

Q, E,- -

SMITH

results of Bell are given in (8) with I statistics in

- 48.4 - O.O00024E, + 2.13”F,. (1.43)

(3.37)

(3.58)

(8)

Unfortunately, not all of the parameters in (7) are identified. However, it is clear that natural activity plays a very important role in the model. One would normally expect all of the parameters A, e, and k to be positive. However, it is clear that at least one of these parameters must be negative in order for the expression kh/c to be estimated as -48.4. However, this does not appear to affect the remainder of the results when Bell fixes temperature at its 1966 value and rewrites (8) as:

Q

-i = 49.4 - O.O00024E,. E*

(9)

Bell then uses (9) and additional information on the cost of effort to demonstrate that effort has been allocated in a socially suboptimal fashion into the northern lobster fishery. PROBLEMS

WITH

THE

BELL

SPECIFICATION

There are a number of major problems associated with Bell’s approach. Perhaps the best way to disentangle some of these difficulties is to return to Eq. (5) of the previous section. In particular, the steady-state implicit relationship described by Eq. (5) will not in general hold under uncertainty since U, is a random variable and it is therefore not correct to equate the random function described by (4) to zero. Nor is it in general correct to assume that (5) will hold with equality when CJ is replaced by its realized value at time t. This is because N, is also a random variable at time t with a probability distribution determined jointly by the distribution of V, and the differential equation postulated in Eq. (4). Finally, it is not valid to take an expectations operator through both sides of Eq. (5) and argue that Eq. (5) now represents an implicit relationship between N,, E,, and U, which will on average hold at the steady-state. The stochustic steady-state relationship between E,, N,, and U, is much more complicated and will, in general, involve terms in the variance of N, and U,. In what follows, a more rigorous approach is provided for the introduction of uncertainty into the growth process. The model discussed here draws heavily upon the theoretical results of Smith [24].

A STOCHASTIC

MODEL

OF RESOURCE

To begin, consider the differential

representation:

N l+h

-

N, =[XN,

-

eN,‘]h.

REGENERATION

(10)

STOCHASTIC

RESOURCE

213

MANAGEMENT

Formally, Eq. (1) from the previous discussion can be derived from Eq. (10) by dividing both sides by h and performing the limiting operation h + 0. With extraction, (10) becomes N t+h

-

N, = [AN, - EN,* - kE;N,]

h.

(11)

It will be noted that the coefficient on effort in the industry production function is not constrained to be equal to 1 as in Eq. (2) of the Bell model and therefore that a more general production (extraction) process is assumed. Consider, next, the introduction of a random term to the right-hand side of Eq. (11) to represent uncertainty in natural activity. In particular: N r+h

-

N, = [AN, - cN,* - kE;N,]

h + aN,[ B,+h - B,].

(12)

Equation (12) has the following interpretation. The change in stock size over the interval h depends upon net growth as a function of stock size at time t plus a random disturbance generated by the change in a random function B, over the interval h. The random term is scaled by the stock size at time t and a parameter u > 0. The implicit assumption is that resource uncertainty is absolutely more important the larger the stock size. On the assumption that B, is distributed as normal with mean zero and cov(B,, B,) = min[s, t] then (12) can be rewritten as N r+h

-

N, = [AN, - cN,2 - kE;N,]h

+ uNIQ,

(13)

where U, is distributed as normal with mean zero and variance h. Equation (13) describes an It6 stochastic differential equation for the stochastic resource process {N,} over small intervals h. Technical conditions relating to the circumstances under which (13) is well defined are discussed in Smith [24]. Some of these conditions will become apparent in the discussion which follows. At this point it is useful to introduce a simplifying assumption and extend the discussion of stochastic steady-state properties corresponding to (13). In particular, it is assumed for the moment that E, is constant at the value ,!?. In more compact infinitesimal notation, (13) becomes dN, = iN, - cN,2dt + ON&,

(14)

where i = (X - kE”). STOCHASTIC

STEADY-STATE

PROPERTIES

It will be recalled that because a random variable enters the right-hand side of (14), stock size at any point in time (N,) is a random variable. In addition, the (transition) probability distribution governing the possible stock realizations at any point in time will in general change over time. Intuitively, this result reflects the fact that the distribution of stock size at time t + h will depend upon the realized stock size at time t as well as the randomness in growth over the interval as expressed in Eq. (14).

214

J. BARRY

SMITH

However, it may be the case that the distribution of stock sizes evolves toward a nondegenerate distribution which is independent of time and previous stock realizations. It is natural, then, to refer to this limiting distribution as the steady-state distribution. The relationship between the concepts of deterministic and stochastic steady-states is close except that in the deterministic steady state there is only one value which the resource can take. Recall that for deterministic Lotka-Volterra model, the steady-state stock size is A/E. For the growth process described by (14), Smith [24] has shown that the steady-state distribution is a member of the Gamma family and the explicit form of the probability density function for stock sizes (N) at the steady-state is given by (15) = 0

otherwise, I(u)

= /gOOX”-‘e-*dX,

o > 0

From (15) the following moments can be derived: E[N]

^ =2-g,

(16.1)

Var[ N] = (2i - ;‘)“, 8r @N-l]

c-d-.--

- 1

r; - a2 #E[N].

(16.2)

(16.3)

These moment relations demonstrate an important feature of the stochastic model. Returning to Eqs. (14) it will be noted that if the absolute value of parameter u is ceteris paribus reduced, the weight of the random term in the growth process is correspondingly reduced. One might therefore expect that if uncertainty is not heavily weighted (or important) then the deterministic (a = 0) and stochastic specifications of the model will give similar results. Equations (16.1) and (16.2) bear out this result indicating that for small u, the steady-state distribution is closely centered around the deterministic steady-state stock size X/e.

ESTIMATION

OF THE MODEL

The sources indicated by Bell [2] were used to collect data on the landed catch (in millions of pounds) and effort (millions of traps fished) for the period 1950 to 1970. Except the inclusion of data for 1967- 1970, the sample should be very similar to that used by Bell. The data used in this study is available upon request. For the purpose of estimation, the stochastic model is reduced to a single equation in output and effort. Dividing through Eq. (13) by N, yields N l+h

-w=[h-r;V,-kE~]h+uU,. N,

(17)

STOCHASTIC

RESOURCE TABLE

Parameter

Log of likelihood

215

MANAGEMENT I

Estimate

1 Statistic

;:

2.2306 0.3961

4.540 2.579

I

0.8734 0.0415

2.291 1.989

cl

0.0478

4.447

function

32.422

Dutu Source: U.S. Department Bureau of Commercial Fisheries

of the Interior (Washington).

(1950-

1970);

Fishery

Statistics

Next, the assumption that Q, = kE:ZV, implies N, = Q,/Mp can be rewritten

(~)($-J-

1 =[A+++

of the United

States;

and therefore (17)

v,,

(18)

where V, is distributed as normal with mean 0 and variance 02h. Equation (18) is exactly the form of a nonlinear regression model. For the stochastic model, the estimate of the variance parameter (a2) is recuperated and used in subsequent analysis. It is interesting to note that (18) could be the form of a deterministic model as well. Under the deterministic assumption, the disturbance term V, would reflect random disturbances which, on average, do not affect the model. Finally, the restrictive model estimated by Bell arises from assuming that the left-hand side of (18) is identically zero. For the purpose of estimation from yearly data it was assumed that h = 1. The model was estimated using a maximum likelihood algorithm prepared by F. Bourguignon. The estimated parameters and additional summary statistics are presented in Table I. The fit appears to be quite good and the parameters are estimated with reasonable precision-in all cases significant at the 5% level. It should be noted as well, that a2 is very small and it would appear therefore that uncertainty in natural activity may not contribute heavily to growth in the resource.

TEST OF THE STRUCTURE

OF TECHNOLOGY

The restrictiveness of Bell’s assumption that the extraction parameter (a) was equal to one was tested by estimating Eq. (18) under the assumption that (Y = 1. The likelihood ratio test was used to determine whether this assumption led to a statistically significant reduction in the likelihood of the model. The likelihood ratio, distributed as x2 with one degree of freedom, was calculated as R = -2(

LLF - LLF)

= -2(27.9273

- 32.4216) = 8.9886.

The tabled x2 critical value at 1% and one degree of freedom is 6.63. Since the critical value is less than the calculated ratio, the hypothesis that (Y = 1 is rejected.

216

J. BARRY

STEADY-STATE

SMITH

CHARACTERISTICS

OF THE

FISHERY

Using the moment expressions (16.1) and (16.2) and the estimated parameters, the following characteristics of the steady-state distribution can be derived: E[ N] = 53.722 (millions of pounds),

(19.1)

Var[ N] = 0.739 (millions of pounds).

(19.2)

Under a deterministic interpretation of the model (a assumed to be 0), the steady-state stock size is given by A/e = 53.749 millions of pounds. Comparing this value with that calculated in (19. l), it is clear that a deterministic model accurately predicts the mean of the stochastic steady-state distribution. It is also useful to enquire whether the presence of uncertainty leads to any different conclusions regarding the size of sustainable yields. With a deterministic interpretation of the model, the maximum sustained yield occurs at stock size A/2e = 26.875 and is given by A2/4e = 29.973 million pounds per year. Under uncertainty, the concept of maximum sustained yield is not well defined. However, using the extraction technology implicit in Eqs. (12) and (13) it is possible to compute the maximum amount of lobster which, on average, could be landed yearly. The programming problem can be written

where E, is the expectations operator for the steady-state resource with harvesting. The solution to this problem is

distribution

of the

E* = ( 2,,.2)“a = 1.851, which in turn implies that the stochastic maximum sustained yield is equal to 26.863 millions of pounds. The variance associated with this expected yield is 0.370 millions of pounds. Once again, it is seen that the value implied by the stochastic model is very accurately predicted by the deterministic version of the model.

THE

SOCIALLY

OPTIMAL

EXTRACTION

POLICY

The introduction of a more general extraction technology in this paper leads to qualitatively similar but quantitatively much different results concerning the misallocation (P # MC) of resources into the northern lobster fishery. It is therefore useful to illustrate these results. Because of the marginal importance of the stochastic terms and further, because of the gain in computational ease, these new results will be presented in terms of a deterministic interpretation of the estimated model. Following Bell’s [2] development and using this estimate of $21.43 for the price of effort in 1966, the new cost function for 1966 can be written C(Q) = 14.29927 - (18.4839 - .61668

Q)‘/2]2’5246’5.

(22)

STOCHASTIC

RESOURCE

217

MANAGEMENT

Using the observed 1966 price of $0.754 per pound, the output level at which price equals marginal cost is given by Q* = 19.77 million pounds.

(23)

The level of effort and steady-state stock size associated with (23) are given by E* = 0.2032 million traps,

(24.1)

N* = 42.55 million pounds.

(24.2)

The actual level of effort in 1966 was 0.932 million traps. Thus, there was approximately 4.6 times too much effort (as indexed by traps) allocated into fishing in 1966. Bell estimated that there was only double the optimal amount of effort allocated towards fishing. The reason for this was that he assumed that the output elasticity of effort (a) was unity. In this paper, the output elasticity was estimated at the much smaller value of 0.3961. Thus Bell overestimates the productivity of effort in fishing and therefore the optimal amount of effort to be allocated towards fishing.

A DIGRESSION ON THE SPECIFICATION TECHNOLOGY

OF EXTRACTION

Throughout the theoretical and empirical resource literature there does not appear to be agreement regarding the properties of the industry production (extraction) function. On the one hand, Plourde [19] has assumed that the production function is concave. Others, including Spence [27], have assumed that the extraction function is characterized by increasing returns to scale. A referee for this paper has argued that most biologists would support the Bell assumption of a unitary output elasticity of effort. As well, however, the referee pointed out that Carlson et al. [4] found that the elasticity of effort was often not unitary for many fisheries when stock size was assumed fixed. The view taken in this paper has been that, at least with respect to the output elasticity of effort, the issue must be resolved empirically. Although the argument that, ceteris paribus, another trap haul should yield another lobster is appealing, it must be noted that in the aggregate, crowding by fishermen, regulation of gear and/or an uneven and unknown distribution of the resource on the ocean bed may reasonably result in an average extraction function with an output elasticity of effort less than unity. As well, this argument is equally valid when stock size is variable. Clearly the formulation of good public policy requires that more accurate estimates by made of such production characteristics. The author is presently gathering data for a comparative species study. A first indication that there may be large differences in output elasticities of effort across species is provided by the empirical results of fitting the stochastic model described in the text to a 45-year time series on whaling supplied by Dasgupta and Heal [8]. The parameter estimates are presented in Table II. Two features are noteworthy. In the first place, the estimate of the output elasticity of effort in whaling is not significantly different from unity. Secondly,

218

J. BARRY

SMITH

TABLE Parameter

Log of likelihood

II

Estimate

1 statistic

;

0.7425 0.9265

6.354 4.104

f *

13.745 0.1100 0.3321

1.911 1.723 8.943

function

- 14.244

uncertainty plays a greater role where the mean of the steady-state resource distribution is 6.247 ten thousands of whales whereas the deterministic interpretation of the model yields an 8% higher estimate of the steady-state at 6.748 ten thousands of whales. A CONCLUDING

REMARK ON THE EFFECT OF NATURAL

ACTIVITY

On the basis of the analyses conducted in this paper, it appears that random natural events may not significantly affect the growth characteristics of a resource. Therefore, optimal management prescriptions based upon deterministic models should be accurate. This conclusion is not inconsistent with any controlled experiment which might show that resources are sensitive to changes in the environment. Rather, the conclusion drawn here is that random natural events affecting a resource in its natural habitat have a very small net effect on growth. Thus, a deterministic model may provide an appropriate vehicle for deriving optimal management prescriptions.

REFERENCES 1. R. B. Ash, and M. F. Gardner, “Topics in Stochastic Processes,” Academic Press, New York (1975). 2. F. W. Bell, Technological externalities and common property resources: An empirical study of the U.S. northern lobster fishery, J. PO/. Econ. 148- 158 (1972). 3. F. Bourguignon, A particular class of continuous-time stochastic growth models, J. Econ. Theory 9 (1974). 4. E. W. Carlson, F. V. Waugh, and F. W. Bell, “Production From the Sea,” NOAA Technical Report, National Marine Fisheries Service Circular 371, Washington, D.C. (1973). 5. C. W. Clark, “Mathematical Bioeconomics: The Optimal Management of Renewable Resources,” Wiley, Toronto (1976). 6. C. Copes, The backward-bending supply curve of the fishing industry, Scot. J. Pal. Econ. 17, 69-77 (1970). 7. D. A. Cox, and H. D. Miller, “The Theory of Stochastic Processes,” Wiley, New York (1%5). 8. P. S. Dasgupta, and G. M. Heal, “Economic Theory and Exhaustible Resources,” Cambridge Univ. Press, London/New York (1979). 9. R. Dow, D. Harriman, G. Pontecorvo, and J. Storer, “The Maine Lobster Fishery,” U.S. Fish and Wildlife Service, Washington, DC. (1961). 10. W. Feller, “An Introduction to Probability Theory and Its Applications,” Vol. 2, Wiley, New York (1966). 11. A. Friedman, “Stochastic Differential Equations and Applications,” Vols. 1 and 2, Academic Press, New York (1975). 12. N. S. Goel, S. C. Maitra, and E. N. Montroll, “On the Volterra and Other Nonlinear Models of Interacting Populations,” Academic Press, New York (197 I).

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RESOURCE MANAGEMENT

219

13. H. S. Gordon, Economic theory of a common-property resource: The fishery, J. Pal. Econ. 62, 124- 142 (1954). 14. T. R. Lewis, “Optimal Resource Management Under Conditions of Uncertainty: The Case of An Ocean Fishery,” Social Science Working Paper No. 104, California Institute of Technology (1975). 15. A. J. Lotka, “Elements of Physical Biology,” Wiiams & Wilkins, Baltimore (1925). 16. P. Mandl, “Analytical Treatment of One-Dimensional Markov Processes,” Springer-Verlag, New York (1968). 17. R. C. Merton, An asymptotic theory of growth under uncertainty, Reu. Econ. Stud. 42, 375-393 (1975). 18. C. G. Plourde, A simple model of replenishable resource exploitation, Amer. Econ. Reu. 60, 518-522 (1970). 19. C. G. Plourde, Exploitation of common property replenishable resources, IV&em Econ. J. 9, 256-66 (1971). 20. M. B. Schaefer, Some considerations of population dynamics and economics in relation to the management of marine fisheries, J. Fisheries Res. Board Canud. 14, 669-681 (1957). 21. D. T. Scheffman, “A Continuous-Time Model of a Stock Market Value Maximizing Firm,” Working Paper, University of Western Ontario, London (1957). 22. A. D. Scott, The fishery: Objectives of sole ownership, J. Pot. Econ. 63, 116- 124 (1955). 23. A. D. Scott (Ed.), “Economics of Fisheries Management-A Symposium,” Institute of Animal Resource Ecology, University of British Columbia, Vancouver (1970). 24. J. B. Smith, “An halysis of Optimal Replenishable Resource Management Under Uncertainty,” unpublished Ph.D. dissertation, University of Western Ontario (1978). 25. J. B. Smith, “Long Run Replenishable Resource Management Under Uncertainty: The Case of the Ocean Fishery,” Mimeo, Concordia University (1979). 26. V. L. Smith, General equilibrium with a replenishable natural resource, Rem. Econ. Srud. Suppl. August (1974). 27. A. M. Spence, Blue whales and applied control theory, in “System Approaches and Environmental Problems” (H. W. Gottinger, Ed.), Vandenhoeck and Ruprecht, Gottinger (1975). 28. V. Volterra, “Le+ons sur la The&e de la Lutte pour la Vie” Gauthier-Villars, Paris (1931). 29. J. E. Wilen, “Common Property Resources and the Dynamics of Overexploitation: The Case of the North Pacific Fur Seal,” Paper No. 3, University of British Columbia Program in Natural Resources, Vancouver (1976). 30. A. Zellner, Management of marine resources: Some key problems requiring additional analysis, in “Economics of Fisheries Management-A Symposium” (A. D. Scott, Ed.), pp. 109-l 15, Institute of Animal Resource Ecology, University of British Columbia, Vancouver (1970).