Abstract
Abstract. We analyze the eect of environmental uncertainties on optimal shery management in a bio-economic shery model. Unlike most of the literature on resource economics, but in line with with ecological models, we allow the different biological processes of survival and recruitment to be aected dierently by environmental uncertainties. We show that the overall eect of uncertainty on the optimal size of a sh stock is ambiguous, depending on the prudence of the value function. For the case of a risk-neutral shery manager, the overall eect depends on the relative magnitude of two opposing eects, the`convex-cost eect' and the`gambling eect'. We apply the analysis to the Baltic cod and the North Sea herring sheries, concluding that for risk neutral agents the net eect of environmental uncertainties on the optimal size of these sh stocks is negative, albeit small in absolute value. Under risk aversion, the eect on optimal stock size is positive for suciently high coecients of constant relative risk aversion.
Introduction
Environmental uncertainties have important eects on the development of sh stocks (Hilborn and Walters 1992) . Accordingly, stochastic uctuations in environmental variables such as temperature, salinity, or oxygen concentration, have to be taken into account when discussing optimal shery management. For riskneutral agents, Reed (1979) shows that optimal management of a sh stock is characterized by a constant escapement policy, i.e. it ensures that a constant proportion of the stock remains in the sea after shing.
Introduction
1 Reed (1979) also shows that the optimal constant escapement level in a stochastic environment is equal to, or larger than, the optimal escapement level in a deterministic setting where the unit harvesting cost function fullls a number of regularity assumptions. Various articles have rened Reed's seminal work by adding multiple uncertainties (Clark and Kirkwood 1986, Sethi et al. 2005) , costly capital adjustments (Singh et al. 2006 ), choice of regulatory instrument (Weitzman 2002) , spatial structure of the resource, (Costello and Polasky 2008) and management with environmental prediction (Costello et al. 2001) . All these studies assume risk-neutral decision-makers.
2
One thing that most of these models have in common is that environmental stochasticity is modeled by an i.i.d. random variable z t multiplied by the average stock-growth function f (x t ) of the resource stock x t at time t, i.e. x t+1 = z t ·f (x t ).
3
The stock-growth function combines the dierent biological processes of recruitment (young sh entering the harvested stock), survival, and growth in terms of weight. Multiplying the average stock-growth function by one random variable implies that all biological processes are equally aected by environmental uctuations. From a biological point of view, however, it seems more plausible to assume that the respective processes would be inuenced dierently by uctuations in the environmental conditions. In most ecological stock-assessment models of marine sh populations the reproduction process is considered to be more sensitive to environmental uctuations than the survival of adult sh, which is usually assumed to be constant. 4 Accordingly, we split the stock-growth function into two func-tions describing the processes of recruitment and growth on the one hand and of adults surviving natural mortality on the other. The recruitment and growth are assumed to be stochastic, whereas natural mortality is assumed to be xed.
3
We examine the eect of uncertainty on the optimal size of a sh stock under both risk neutrality and risk aversion. Whether or not optimal escapement increases with uncertainty is connected with the prudence (Kimball 1990 ) of the value function. If the value function of the shery considered exhibits positive prudence, the optimal policy will involve precautionary savings in the natural capital stock and hence higher stock size under uncertainty than in the deterministic case. If the value function exhibits negative prudence, the optimal stock size under uncertainty will be lower than in the deterministic setting. As the value function depends on (i) the biomass growth function, (ii) the prot function, and (iii) the representative sherman's utility function, all three have an inuence on whether optimal escapement increases or decreases with environmental uncertainty.
3
Considering risk-neutral stakeholders rst, we show that the overall eect of uncertainty on the optimal size of a sh stock is ambiguous, depending on the relative magnitude of two opposing eects, the`convex-cost eect' and the`gambling eect'. The`convex-cost eect' reduces optimal escapement under uncertainty over and against the deterministic case. Because harvesting costs are convex in the sh stock, expected harvesting costs are larger when sh stock growth is uncertain than they are at an expected stock level. The`gambling eect', on the other hand, increases optimal escapement under uncertainty. It comes about because uncertainty is multiplicatively connected to stock growth, so the distribution of the next period's resource rents is positively skewed. This eect induces a risk-neutral shery manager to`bet' on favorable environmental conditions.
3
For the case of a risk-averse representative sherman, we show for a special case that optimal escapement increases with uncertainty if the coecient of relative risk aversion is large enough.
3
In quantitative terms, we apply the model to Eastern Baltic cod and North Sea herring sheries and conclude that under risk neutrality the net eect of envi-ronmental uncertainties on the optimal size of these sh stocks is negative, albeit small in absolute value. Under risk aversion we observe a positive eect of uncertainty on optimal stock size for suciently high coecients of constant relative risk aversion.
3
The paper is structured as follows: In the next section we set up the model and solve the corresponding optimization problem. In Section 2 we consider optimal harvesting under risk neutrality, identifying the convex-cost eect and the gambling eect. The case of a risk-averse representative sherman is discussed analytically in Section 3. We then apply our model to sheries for Baltic cod (Section 4) and North Sea herring (Section 5), considering both risk neutrality and risk aversion. Section 6 summarizes and discusses the results.
A Fishery Model with Environmental Uncertainty
We consider a simple biomass model with stochastic recruitment in discrete time.
A Fishery Model with Environmental Uncertainty
The growth of biomass x t from time step t to t + 1 is described by the equation
A Fishery Model with Environmental Uncertainty
where s t denotes the escapement, i.e. the biomass that remains in the ecosystem after harvest h t so that s t = x t − h t . The period between t and t + 1 is divided into two parts. Harvesting takes place in the rst part. In the second, the remaining sh biomass, i.e. the escapement, reproduces and grows in weight. A fraction of the sh dies by natural causes. The term g(s) represents the survival of adult sh.
A Fishery Model with Environmental Uncertainty
The reproduction and growth process is represented as the stock-recruitment relationship r(s). We assume that g(s) and r(s) are dierentiable, (strictly) concave, and non-decreasing. Only the stock-recruitment process is assumed to be sensitive to uctuations in environmental conditions; the natural mortality of adult sh is assumed to be xed. The assumption of a xed mortality rate is customary in biological stock assessment models (Hilborn and Walters 1992, ICES 2011a,b) . Environmental uncertainty aecting recruitment is represented by the random variable z t . It is independent and identically distributed over time with an expected value equal to one, E[z t ] = 1.
A Fishery Model with Environmental Uncertainty
The price per unit of sh p is constant, i.e. the shery is small compared to the overall market. We assume a generalized Schaefer production function for the instantaneous harvest rateh in the shing season,h = q(x) K t . We use q(x)
A Fishery Model with Environmental Uncertainty
to denote the catch per unit of eort, K t to denote the eort of harvesting sh (which is assumed to be constant throughout the shing season), andx to denote the current stock size. Accordingly,x = x t at the beginning of year t's shing season andx = s t at the end (Reed 1979 , Clark 1990 . Harvesting costs C t are proportional to eort K t , with ζ as the costs per unit eort C t = ζ K t . Thus we obtain a unit cost function for harvesting sh, c(x) = ζ/q(x). We assume that, in general, unit harvesting costs are weakly decreasing and weakly convex in the stock size, i.e. c (x) ≤ 0 and c (x) ≥ 0. This means that the catch per unit of eort q(x) is non-decreasing with population abundance. Convexity also implies that the increase of the unit harvesting costs induced by a one-unit decrease in stock is greater for lower stock abundance than for higher stock abundance. A common specication of the harvesting cost function is c(x) = cx −χ with χ > 0.
A Fishery Model with Environmental Uncertainty
In that special casex c(x) is strictly concave (for 0 < χ < 1), constant (for χ = 1), or strictly convex (for χ > 1). For χ = 0, catch per unit eort is independent of stock abundance, the unit harvesting cost is constant with C = ζ, andx c(x) is linear in stock. In empirical terms the most relevant case is χ ∈ (0, 1). In a study of 297 sheries, Harley et al. (2001) nd typical values of χ to be between 0.64 and 0.75.
A Fishery Model with Environmental Uncertainty
During the harvesting season, each ton of sh caught reduces the stock by one ton. Therefore the aggregate annual prot Π t is obtained by integrating the ow of prots over the whole shing season Π t = xt st (p − c(x)) dx. The shery manager aims to maximize the well-being of the representative sherman earning his income from shing prots:
A Fishery Model with Environmental Uncertainty
Here the operator E denotes the expectation over the probability distribution of the random process {z t } and ρ ∈ (0, 1) is the discount factor. Fishermen are typically averse to uctuations in income. We nd this reected in the management plans for Baltic cod which contain rules to limit uctuations in total allowable catches from year to year. 6 We take this eect into account by assuming that instantaneous utility u(Π t ) derived from shing income is increasing and weakly concave, u (Π t ) > 0 and u (Π t ) ≤ 0. 7 For a risk-neutral sherman, the instantaneous utility function is linear, u(Π t ) ≡ Π t . For a risk-averse sherman, the instantaneous utility function is strictly concave, u (Π t ) < 0.
A Fishery Model with Environmental Uncertainty
Using J(x) to denote the value function associated with the stochastic optimization problem (2), the Bellman equation reads
A Fishery Model with Environmental Uncertainty
In the following, we use S * (x) to denote the optimal feedback policy obtained as a solution of (3) for the stochastic case andS * (x) to denote the optimal feedback policy for the corresponding deterministic model, where z t ≡ 1 for all t. The question we are asking in this paper is whether, for a given stock size x, the solution S * (x) for the stochastic problem is larger than, equal to, or smaller than the solutionS(x) for the deterministic model.
A Fishery Model with Environmental Uncertainty
To adress this question, we consider the rst-order condition for optimal es-
A Fishery Model with Environmental Uncertainty
This condition states that for the optimal escapement level s * at a given stock size x the current marginal prots of the last unit of sh harvested (left-hand side, LHS) equal the discounted expected marginal prots of an additional unit of sh that escapes shing (right-hand side, RHS). Uncertainty only makes a dierence to the RHS of this equation. As the LHS is monotonically increasing in s, the optimal escapement level at a given stock size x will be higher, the higher the RHS of (4) is. Thus, the optimal escapement level will be higher (lower) under uncertainty than with the deterministic setting if the RHS of (4) is higher (lower) when z is stochastic than in the deterministic case z ≡ 1. This, in turn, depends on the curvature of the derivative of the value function with respect to the escapement level, dJ(·)/ds, in z. So the question whether or not optimal escapement increases with uncertainty is connected to the prudence (Kimball 1990 ) of the value function
A Fishery Model with Environmental Uncertainty
. If the value function of the shery in question exhibits positive prudence, the optimal policy will involve precautionary savings in the natural capital stock and hence higher stock size under uncertainty than in the deterministic case. If by contrast the value function exhibits negative prudence, the optimal stock size under uncertainty will be lower than in the deterministic setting.
A Fishery Model with Environmental Uncertainty
As the value function depends on (i) the biomass growth function, (ii) the prot function, and (iii) the representative sherman's utility function, all three have an inuence on whether optimal escapement increases or decreases with environmental uncertainty. For the detailed analysis of the combined eect we proceed in three stages. In Section 2 we study how optimal harvesting is aected by risk when shermen are risk-neutral. Section 3 derives analytical results for risk-averse shermen, but it requires relatively restrictive assumptions on biomass growth function, harvesting technology, and preferences to derive a closed-form expression for the value function. The sections on 4 and 5 are applications of our analysis to the Baltic cod and North Sea herring sheries.
Optimal Harvesting under Risk Neutrality
For the risk-neutral case u(Π t ) ≡ Π t , both the LHS and the RHS of (4) are independent of the current stock size x. Thus, the solution of this stochastic optimization problem is state-independent (see Appendix A). Optimal feedback policy S * (x) is the most rapid approach to the constant optimal escapement level s * ,
Optimal Harvesting under Risk Neutrality
Optimal escapement level s * is determined by the following condition (see Appendix B):
Optimal Harvesting under Risk Neutrality
This condition states that for the optimal escapement level s * current marginal prots from the last unit of sh harvested equal the discounted expected marginal prots from an additional unit that escapes shing. The expected marginal prot on the RHS of (6) can be divided into two eects. The rst term on the RHS represents the expected marginal prots from the additional surviving adults. The second term on the RHS stands for the expected marginal prots from additional recruits.
Optimal Harvesting under Risk Neutrality
In the risk-neutral case, the curvature properties of the marginal cost function are essential to determine the prudence of the value function. To compare the optimal escapement level in the stochastic case with the optimal escapement levels * in the deterministic case (i.e., for z t ≡ 1 in Equation (1)), we consider the following equivalent to condition (6) in the deterministic setting:
Optimal Harvesting under Risk Neutrality
We obtain a higher (lower) optimal escapement level when the next period's expected marginal costs are lower (higher) than the marginal costs at the expected next period's stock level (which coincides with the deterministic case as we have
Optimal Harvesting under Risk Neutrality
We consider the eects for additional surviving adults and for additional recruitment separately. The rst term on the RHS of condition (6) in the stochastic case is smaller than the rst term on the RHS of condition (7) in the deterministic case. This is due to the convexity of the marginal harvesting cost function. We refer to this eect as the`convex-cost eect'. The second term on the RHS of condition (6) will be larger than the second term on the RHS of condition (7) if functionx c(x) is concave. We refer to this eect as the`gambling eect'.
Convex-cost eect
If marginal harvesting costs are convex in the sh stock, they will also be convex in the random variable. Expectation E c g(s * ) + z t r(s * ) is over a convex function of the random variable, so expected marginal harvesting costs are greater than the marginal costs at expected stock growth c g(s * ) + r(s * ) as E[z] = 1. The convex marginal cost function implies that the increase in marginal harvesting costs for a stock growth below the mean is greater than the decrease in marginal harvesting costs for a stock growth above the mean. Accordingly, the expected marginal harvesting costs are greater under uncertainty than in the deterministic case. We refer to this eect, which tends to reduce the optimal escapement level, as the convex-cost eect.
Convex-cost eect
Intuitively, the eect of convex marginal harvesting costs is similar to the eect of risk aversion, so it is optimal to invest less if the asset is risky. Accordingly, it is also intuitive that under uncertainty the convex-cost eect will reduce optimal escapement over and against the deterministic model. More precisely, the convexity of marginal harvesting costs reduces the prudence of the value function. If the convex-cost eect were the only eect present, the prudence of the value function would be unambiguously negative, and the optimal escapement level would increase with environmental uncertainty.
Gambling eect
If the functionx c(x) is convex, the second term on the RHS of condition (6) will be lower than the second term on the RHS of (7). The prudence of the value function would be unambiguously negative, so the optimal escapement level under uncertainty would be unambiguously lower than in the deterministic setting.
Gambling eect
As set out earlier, the more relevant case in empirical terms is wherex c(x) is concave inx. Here the expression z t c g(s * ) + z t r(s * ) is a concave function in the random variable z t . The expected marginal costs E z t c g(s * )+z t r(s * ) will then be smaller than the marginal costs at expected stock growth c g(s
Gambling eect
(which coincides with the deterministic case).
Gambling eect
The economic intuition for this eect is as follows: As uncertainty is multiplicatively connected with stock growth, the distribution of next period's resource rents is positively skewed and has a fat tail at high rents. Under favorable environmental conditions, a marginal increase in current escapement will result in a strong marginal increase in the sh stock, producing both a large harvest and low marginal harvesting costs in the next period. Under adverse environmental conditions, marginal harvesting costs in the next period will be high because of the low sh stock, but this eect is dampened by the fact that the harvest is small.
Gambling eect
So the expected marginal increase in harvesting costs with a marginal increase in escapement is lower than the marginal increase in harvesting costs under expected stock growth. In other words, a risk-neutral shery manager will tend to bet on favorable environmental conditions. This is why we call this eect the gambling eect.
Gambling eect
The gambling eect tends to increase the prudence of the value function, i.e. to increase the optimal escapement level under environmental uncertainty.
9
Since the convex-cost eect and the gambling eect work in opposite directions, the overall result is ambiguous. The optimal escapement level could be either higher or lower than in the deterministic case. 10 2.3 A special case: when survival is proportional to recruitment
9
In our model, a special case arises if survival is proportional to recruitment, i.e. if there exists some κ ∈ [0, 1) such that (1 − κ) g(s) = κ r(s). In this case, biomass growth (equation 1) can be written as
9
wherez = κ + (1 − κ) z is a random variable with mean 1 and
9
is the expected biomass growth function. This special case deserves attention, as the model for it is equivalent to the model studied by Reed (1979) . In the latter case the optimality condition (6) simplies to
9
If the functionx c(x) is concave inx, the optimal escapement level in the stochastic model will be unambiguously higher than in the deterministic case. In other words the gambling eect will outweighs the convex-cost eect.
11
If we further specify the unit cost function c(x) = c x −χ with χ ∈ (0, 1) and assume a log-normal distribution of environmental stochasticityz with mean µz = 1 and standard deviation σz, condition (9) can be written as follows (see Appendix C):
11
The left hand-side of this equation increases with the degree of uncertainty, as measured by the variance σ 2 z . Accordingly, the optimal escapement level increases monotonically with uncertainty. In quantitative terms, the inuence of uncertainty on the optimal escapement level will typically be small. It is maximal for χ = 1/2, as then the exponent of the factor (1 + σ 2 z ) χ (1−χ)/2 on the RHS of (10) reaches its maximum for all values of χ ∈ (0, 1). But even in this case and for an unrealistically high degree of uncertainty σ 2 z = 1, this factor changes the RHS of (10) by less than ten percent, as (1 + σ 2 z ) χ (1−χ)/2 = 2 1/8 < 1.10.
Optimal Harvesting under Risk Aversion
In the case of risk aversion, it is in general not possible to solve the Bellman equation (3) analytically. For special cases an analytical solution is however feasible.
12
To study the case of a risk-averse representative sherman analytically, we thus have to further specify the model.
12
First, we neglect harvesting costs in this section, i.e. we assume c(x) ≡ 0. This not only simplies the analysis, it also enable us to better isolate the eect of risk aversion. The point is that in the absence of harvesting costs, both the convex-cost eect and the gambling eect vanish, and optimal escapement for the risk-neutral case would be independent of uncertainty.
12
Second, we focus on the case where survival is proportional to recruitment and the biomass growth function is given by (8). We furthermore assume that the biomass growth function f (s) has the functional form
12
with positive constants α, β, and φ. A special case of this biomass growth function is the Beverton-Holt function f (s) = (s/α)/(1 + (s/β)) obtained by setting φ = 2.
12
Third, we assume an instantaneous utility function with constant relative risk
12
with ν > 0. This model is analytically solvable for the special case φ = ϑ. In Appendix D we show that the value function is
12
with some constants ψ 1 > 0 and ψ 2 , and that the optimal escapement rule is
12
It is obvious that if ϑ < 1 (ϑ > 1), optimal escapement will be lower (higher), the higher the uncertainty is. ϑ < 1 means not only that risk aversion is relatively low but also that the curvature of the biomass growth function is relatively high.
13
We can again connect this result to the prudence of the value function J(z f (s)).
13
In this case, the function inside the expectation operator on the RHS of (4) is dJ(z f (s))/ds = ψ 1 z 1−ϑ α s 1−ϑ . It is concave (convex) in ϑ if ϑ < 1 (ϑ > 1). Thus, a value function that exhibits negative (positive) prudence will give rise to a lower (higher) optimal escapement for ϑ < 1 (ϑ > 1).
Quantitative Example I: Baltic Cod Fishery
Our rst quantitative example is the Baltic cod shery. The water in the Baltic Sea is brackish, making it a marginal area for cod. The sh population depends on uctuating fresh water inows from the North Sea to increase the salinity level to a degree where their eggs can hatch (Rockmann et al. 2007 ). Accordingly, the recruitment process for Baltic cod is highly uncertain and represents a useful example for the stochastic recruitment model.
Quantitative Example I: Baltic Cod Fishery
To estimate the biological growth function we use stock assessment data (years
1966-2009) from the International Council of the Exploration of the Sea (ICES 2011
). We combine the data for total stock biomass in year t, T SB t , total harvest H t (as the sum of ocial landings and discards) and natural mortality (assumed to be xed at M = 0.2, as in the ICES stock assessments) to obtain the escapement S t and the recruitment biomass R t as follows
1966-2009) from the International Council of the Exploration of the Sea (ICES 2011
The recruitment variable R t thus encompasses both the reproduction process and growth in weight. It is calculated as the total stock biomass at the beginning of period t + 1 minus the fraction of period t's escapement that survives natural mortality. We assume that recruitment follows a stochastic Beverton-Holt (1957) stock-recruitment function r(s t ) = z t α 1 s t /(1 + α 2 s t ). 14 Using the LevenbergMarquardt algorithm for nonlinear least squares, we estimated the equation
1966-2009) from the International Council of the Exploration of the Sea (ICES 2011
assuming that ε t is an independent and identically normally distributed random variable with zero mean. 15 We obtain estimatesα 1 = 1.189 with a standard error of 0.111,α 2 = 1.525 million tons with a standard error of 0.496 million tons, and an estimateσ 2 = 0.083 million tons for the standard deviation of z t = exp(ε t ).
1966-2009) from the International Council of the Exploration of the Sea (ICES 2011
Because of the logarithmic specication of (15), estimateα 1 is biased. For our numerical computations, we use the adjusted value α 1 =α 1 exp(−0.5σ 2 ) = 1.140.
1966-2009) from the International Council of the Exploration of the Sea (ICES 2011
To estimate the parameters of the harvesting function, we specify the cost function C = ζK t and q(x) = q 0 x χ . With this specication, total shing eort in year t is 1990 ). We use data on eort as days at sea for the Danish eet from (Fiskeridirektoratet 2010) and ( Our rst step in the quantitative analysis for Baltic cod is to consider the optimal escapement levels for a risk-neural representative sherman. To determine the optimal escapement levels in the deterministic and stochastic recruitment models, we solve conditions (6) and (7) using these specications numerically. For our sensitivity analysis, we use random samples of 1000 sets of parameter values for α 1 , α 2 , q 0 and ζ, assuming that the parameter values are independently normally distributed with means and standard deviations as obtained from the estimations (or variance-covariance matrix from the estimation for α 1 and α 2 ). For each parameter set, we compute the optimal escapement and determine the standard deviation of the sample of optimal escapement levels thus obtained. we assume an instantaneous utility function with constant relative risk aversion
1966-2009) from the International Council of the Exploration of the Sea (ICES 2011
and use the same biomass growth function, marginal cost function, and discount rate as before. We solve the stochastic optimization problem (2) numerically for dierent risk-aversion coecients ϑ by numerically computing the value function J(x). To do so, we use the collocation method (Miranda and Fackler 2002) , where the value function J(x) is approximated by a nite linear combination of Chebychev polynomials.
16
The results are shown in Figure 1 .
Figure 1 about here
The upper panel shows the optimal feedback policies S * (x) under uncertainty (σ 2 z = 0.0834) andS * (x) in the deterministic case for three dierent coecients of risk aversion: the risk-neural case ϑ = 0, slight risk aversion ϑ = 0.1, and stronger risk aversion ϑ = 0.5. For the risk-neutral case, the optimal policy is the most rapid approach to constant escapement (cf. section 2). The higher the risk aversion, the smoother the optimal policy becomes: escapement is relatively lower (harvest is higher) at relatively low stock sizes, and relatively higher (harvest is lower) at higher stock sizes.
Figure 1 about here
For all three coecients of risk aversion, there is hardly any dierence between the optimal policies under uncertainty and in the deterministic setting. The lower panel in Figure 1 shows the dierence S * (x) −S * (x) for three dierent stock levels, x ∈ {0.15, 0.50, 1.8} million tons. Note that the scale of the y axis in the lower panel is in thousands of tons, while in the upper panel it is in millions of tons.
Figure 1 about here
This shows that the dierence is well below 1 percent of the optimal escapement.
Figure 1 about here
Although the overall eect is small, the eect of increasing risk aversion is unambiguous. The higher the coecient of risk aversion is, the higher is the dierence in optimal escapement under uncertainty and in the deterministic case. The two lower stock sizes considered (x = 0.15 and x = 0.50 million tons) are smaller than the optimal escapement s * in the risk-neutral case. For these stock sizes, the optimal escapement for ϑ → 0 is the same with and without uncertainty, as it simply equals the current stock size. The higher stock size x = 1.8 million tons is above s * . In this case, the optimal escapement for a risk-neutral representative sherman is lower under uncertainty than in the deterministic case (see above).
Figure 1 about here
With increasing risk aversion, the dierence also becomes positive for this stock size .
North Sea Herring Fishery
Our second case study is the North Sea herring shery. Here we use the same functional specications as for the Baltic cod shery. For North Sea herring, we use the price and cost function from Nostbakken (2008) , where p = 2.465 NOK/ kg, ζ = 1, 189, 565 NOK/per vessel-year and a catchability per vessel-year of q 0 = 0.0011. We again normalize the price to unity and obtain ζ/(p * q 0 ) = 0.439
North Sea Herring Fishery
Euros/kg and χ = 1 as parameters of the cost function.
North Sea Herring Fishery
To estimate the parameters of the biological growth function, we use ICES (1998; 2007) data for the period 1947-2005 to calculate the escapement S t as the product of the total biomass X t and e −F , where F is the mean shing mortality rate for age classes from 2 to 6. We again assume a deterministic natural mortality with a rate of M = 0.16 (as in the ICES stock assessments for herring) and a BevertonHolt function for the stock-recruitment relationship. Using the same model (15) and regression method as for cod, we obtain estimatesα 1 = 2.048 (standard error 0.266),α 2 = 0.956 million tons (standard error 0.204 million tons), andσ 2 = 0.104 million tons for the standard deviation of z t = exp(ε t ). Again, we use the adjusted value α 1 =α 1 exp(−0.5σ 2 ) = 1.9445 for our numerical analysis.
17
For the risk-neutral case we compute an optimal escapement level s * = 2.769 million tons (with a standard deviation of 0.302 million tons) in the stochastic model. In the deterministic model the optimal escapement level iss * = 2.780 million tons.Thus we again observe a slightly lower optimal escapement level in the stochastic case. The dierence is larger than with the Baltic cod shery but at 10 356 tons (with a standard deviation of 4 784 tons) still quite small.
17
In Figure 2 we show the optimal policies for risk-averse shermen.
Figure 2 about here
The results are similar to those obtained for Baltic cod. The dierences in optimal escapements between the stochastic and deterministic cases are small in absolute value. The unambiguous eect of risk aversion is that dierence increases and, for suciently high degrees of risk aversion, optimal escapement is higher under uncertainty than in the deterministic case.
Conclusion
In this paper we studied the eects of environmental uncertainties on optimal shery management for both risk-neutral and risk-averse shermen. To account for natural mortality and recruitment, we split the stock growth function of the sh stock into two processes. Following the biological approach taken in stock assessment models, we assume that natural mortality is xed at a given value. The recruitment process, by contrast, depends on stochastically uctuating environmental conditions.
Conclusion
We have demonstrated that the optimal escapement level can be higher or lower than in the deterministic setting, depending on the prudence of the value function.
Conclusion
This in turn depends on (i) the biomass growth function, (ii) the prot function and (iii) the representative sherman's utility function. Positive prudence gives rise to higher optimal escapement, whereas negative prudence results in a lower optimal escapement level.
Conclusion
For risk-neutral sherman we showed that whether the question or not the optimal escapement increases with uncertainty is inuenced by two counteracting cost eects: the convex-cost eect and the gambling eect. The convex-cost eect results from higher expected marginal costs due to the convexity of the cost function and tends to reduce optimal escapement. Intuitively, the convex-cost eect means that optimal investment is smaller when the asset is risky. The gambling eect results from lower expected marginal costs under uncertainty because the harvestable biomass increases more strongly under favorable environmental conditions than under adverse environmental conditions. The gambling eect thus tends to increase optimal escapement. Under risk aversion, the eect of uncertainty on optimal escapement is still ambiguous, depending on the stock growth function parameter and the coecient of constant relative risk aversion.
Conclusion
To quantify the eect of uncertainty both under risk neutrality and risk aversion, we applied the model to the Baltic cod and the North Sea herring sheries.
Conclusion
Under risk neutrality we observed in both sheries lower optimal escapement in the stochastic environment than in the deterministic setting, but the dierence is small in absolute value. In the setting with risk-averse shermen we found higher optimal escapement levels for suciently high coecients of constant relative risk aversion. Again, the dierence is very small, and well below one percent of optimal 8 Because the optimization problem (2) is autonomous, the value function J(x) does not depend on time.
Conclusion
9 It may appear contradictory that the`gambling eect' leads to a higher degree of prudence.
Conclusion
However, the term prudence has been coined because of its eect a higher degree of prudence induces higher precautionary savings and not because of a particular motivation why an agent saves more under uncertainty.
Conclusion
10 Under constant unit harvesting costs both the gambling and the convex-cost eect vanish and the optimal escapement level in the stochastic and deterministic model are the same.
Conclusion
11 Reed's (1979) intuitive explanation for a higher optimal escapement level is that the marginal average annual harvesting cost in the stochastic model resulting from an increase in the escapement level [...] is, because of the averaging process, less than the corresponding marginal cost in the deterministic model.
Conclusion
12 To our knowledge, no analytically solvable model has been available for the discrete-time model so far. For a similar problem in continuous time, Pindyck (1984) provides three examples of analytically solvable models.
Conclusion
13 If we again assume that z is log-normally distributed with variance σ
Conclusion
(1−ϑ)/2 (see appendix C).
Conclusion
14 We also estimated the more general growth function of r(s t ) = α s
Conclusion
found the parameters to be not signicant.
15
A Durbin-Watson test shows no autocorrelation in the error terms (DW = 1.718, p = 0.312). 
B Optimal Escapement Level
We now consider the optimal escapement level in the stochastic growth model. By dierentiating equation (18) with respect to s T −(n−1) we get the following condition: π (s * ) = ρ E π g(s * ) + z t r(s * ) g (s * ) + z t r (s * ) .
B Optimal Escapement Level
Substituting p − c(v) for π (v) and rearranging we get p − c(s * ) = ρ g (s * ) p − E c g(s * ) + z t r(s * ) + ρ r (s * ) p − E z t c g(s * ) + z r(s * ) .
B Optimal Escapement Level
C Increasing uncertainty when survival is proportional to recruitment
B Optimal Escapement Level
Inserting the cost function c(x) = ζx −χ in equation (9) and rearranging we get p − c(s
B Optimal Escapement Level
Withz as a lognormally-distributed random variable the expectation can be cal- 
B Optimal Escapement Level
The Bellman-equation reads
B Optimal Escapement Level
Using the guess (13) again, we obtain
B Optimal Escapement Level
Using (26) in (25), we obtain (14). 

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