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%              submited 13 aug  1996
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\markboth{F. B. Hanson and D. Ryan, Harvesting with Price Dynamics}{F. B. Hanson and D. Ryan, Harvesting with Price Dynamics}
%\markright{HARVESTING WITH PRICE DYNAMICS}

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\begin{document}
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\begin{flushleft}{
{\Large\bf Optimal Harvesting with Both Population and Price 
Dynamics}
~\\*[0.5in]
{{Floyd B. Hanson}
\footnote{Research supported in part by the National Science 
Foundation under Grants DMS 91-02343, 93-01107 and 96-26692 at the 
University of Illinois at Chicago.  This paper is to appear in {\bf Mathematical
Biosciences}, vol. ?, pp. 1-18?, 1998.}}\\
{\small\it 
Department of Mathematics, Statistics, and Computer Science (M/C 249)\\
University of Illinois at Chicago\\
851 S. Morgan\\
Chicago, IL 60607-7045, USA\\*[0.1in]
Telephone: 312-413-2142\\
Fax: 312-996-1491\\
Email:  hanson@uic.edu
}\\*[0.125in]
{and}\\*[0.125in]
{Dennis Ryan}\\
{\small\it 
Division of Science and Mathematics\\
McKendree College\\
Lebanon, IL 62254, USA\\*[0.1in]
Telephone:  618-537-6937\\
Fax:  618-537-6259\\
Email:  dryan@a1.mckendree.edu
}\\*[0.2in]
{\it Keywords:} {price uncertainty, stochastic growth, optimal harvesting, 
bioeconomics}\\*[0.2in]
{\it Running Title:} {Harvesting with Price Dynamics}
}\end{flushleft}
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\begin{abstract}
We consider the effects of large inflationary price fluctuations
on the computed optimal harvest strategy for a randomized Schaefer model.  
Both prices and population sizes are assumed random with both background 
(Wiener) and jump (Poisson) components.  
Population fluctuations are assumed to be density independent, 
i.e., relative changes are independent of population size.  
Stochastic dynamic programming is employed to find the
optimal harvesting effort and economic return for a realistic set of
bioeconomic data for Pacific halibut. It is found that inflationary effects
have a pronounced influence on the optimal return, even in a
hazardous or disastrous environment.  However, optimal harvesting effort
levels are much less sensitive to inflationary effects.  
\end{abstract}

\section{Introduction}    %SECTION 1
%
Bioeconomic resource models incorporating random fluctuations in either
population size or model parameters have been the subject of much interest.
Reed~\cite{1,2} considered optimal harvest and escapement policies in the
presence of general discrete-time multiplicative noise for a variety of
assumptions. Gleit~\cite{3} gave an exact solution for the optimal present
value, and corresponding linear optimal control, of an exponentially growing
resource subject to Gaussian white noise fluctuations. Ludwig~\cite{4,5}
solved by perturbation methods the more general control problem for populations 
with Ricker type growth influenced by lognormal noise, while Ludwig and Varah 
studied these problems numerically in~\cite{6}. Ryan and Hanson~\cite{7} solved
exactly the optimal harvest problem for constant effort and exponential
growth in the presence of large fluctuations modeled by Poisson processes and
in~\cite{8} numerically constructed the optimal feedback control for logistic
growth with the same type of noise. See also~\cite{7,8} for a more extensive
bibliography.

In the above models per unit prices are either constant or exogenous and
deterministic. However, random price fluctuations are a realistic effect and
have been incorporated in standard resource models. Andersen~\cite{9} has
studied continuous time optimal harvest models with logistic growth when
prices follow a general probability distribution. Lewis~\cite{10,11} has
examined similar models in discrete time when both prices and population size
are allowed to be random. Pindyck~\cite{12} has studied the economic
consequences of uncertainty in population size as well as unit price for a
variety of continuous-time harvest models, using Gaussian distributed Wiener 
processes to model fluctuations. 
Clark~\cite{13} discusses a discrete-time model with randomly
varying seasonal prices. Ryan~\cite{14} considered a model in which the unit
price changes suddenly at a random time. Comprehensive introductions to these
problems are given by Mangel~\cite{15} and Andersen and Sutinen~\cite{16}.

In the present paper, we explore the effects of price fluctuations on the
computed optimal harvest strategy for a randomized Schaefer type model. In
our model unperturbed prices consist of an inflation adjusted constant price
term plus a supply/demand term. Random price variation is incorporated into
the model through a multiplicative random process that includes both small
continuous-time fluctuations and the possibility of occasional, large random
changes. Since our primary interest is to model, and study numerically, the
effects of randomness on supply/demand factors, we ignore both random and
inflationary effects on the postulated cost function. Technically, random and
inflationary effects of the type hypothesized for the supply/demand function
are easy to incorporate into the model. However, random fluctuations in cost
are likely to be much more complicated and require a more complicated general
model. Thus, we restrict our attention to serially uncorrelated exogenous
random price and population fluctuations. 
Our analytical emphasis is primarily numerical.

The presence of such fluctuations is well documented in fisheries. 
Figure~\ref{FIG1} shows price versus year and 
Figure~\ref{FIG2} shows catch versus year for the Pacific halibut 
({\it Hippoglossus hippoglossus}) system.  
See the 1984 and 1985 International Pacific Halibut Commission (IPHC)
Annual Reports~\cite{17a,17}. The price data
show low level fluctuations followed by a precipitous decline followed by
moderate fluctuations over a short time period. 
%
\begin{figure}[htb]
\begin{center}
~\\*[-10pt]
\hspace*{-10pt}
\epsfysize300pt
\epsffile{Fig1YPBig.eps}
\caption{\label{FIG1}
Pacific halibut prices in U.S.\ dollars per kilogram for each year
from 1935 to 1985. The source of the data in the table in Appendix~II in the
IHPC 1984 and 1985 Annual Reports \cite{17a,17}.
}
\end{center}
\end{figure}
%
%
\begin{figure}[htb]
\begin{center}
~\\*[-10pt]
\hspace*{-20pt}
\epsfysize300pt
\epsffile{Fig2YHBig.eps}
\caption{\label{FIG2}
U.S.-Canadian catch in millions of kilograms for each year from
1935 to 1985. The source is the same as Figure~1.
}
\end{center}
\end{figure}
%
The price versus catch data as shown in
Figure~\ref{FIG3} reflect similar but more pronounced fluctuation with a 
prominent
trough in the highly inflationary time around 1979. In general, random catch
and recruitment fluctuations are particularly well documented. See~\cite{8}
for a more detailed survey.
%
\begin{figure}[htb]
\begin{center}
~\\*[-10pt]
\hspace*{-20pt}
\epsfysize300pt
\epsffile{Fig3HPFitBig.eps}
\caption{\label{FIG3}
Pacific halibut price in U.S.\ dollars per kilogram versus catch in
millions of kilograms for the years from 1935 to 1985. The source is the same
as Figure~1.  Also, the linear regression for the price times catch as a
function of catch from 1980 to 1985 is displayed as the
smooth hyperbolic price curve.
}
\end{center}
\end{figure}
%

In the present paper we explore the effects of both random population
fluctuations as well as random price fluctuations on the computed optimal
harvest strategy. Our model is new in that it simultaneously incorporates the
possibility of large fluctuations in both resource size and per unit prices
while maintaining the general structure employed in the previously discussed
work. Section~2 briefly develops the deterministic model. The stochastic
model is presented in Section~3. Numerical methods and results are discussed 
in Section~4.

\section{Deterministic Model}      %SECTION 2
%
A frequently employed model for the harvesting of a renewable resource of
size $N(s)$ at time~$s$ is the differential equation 
\begin{equation}\label{2.1}     %EQUATION 2.1
dN(s) = \left[r_1 N(1 - N/K) - H(s)\right]\, ds, \qquad s > 0, \quad N(0)
=x. 
\end{equation}
Here, $r_1$ and $K$ are the population's intrinsic growth rate and carrying
capacity, respectively. The harvest term is assumed to be given by 
{\it catch per unit effort hypothesis}~\cite{18} 
\begin{equation}\label{2.15}    %EQUATION 2.15
H(s) = q\cdot E\cdot N(s),
\end{equation}
where $q$ is the catchability coefficient.  The effort $E = E(N(s),s)$, 
in feedback control form here, is a measure of harvesting effort 
and is assumed to satisfy the condition 
$$E_{\min}\le E\le E_{\max} < \infty.$$

The value of the harvest is given by the discounted present value of future
resources
\begin{equation}\label{2.2} %EQUATION 2.2
v(x; E) = \int_0^T e^{-\delta s} [pqEN(s)-c(E)]\,ds,
\end{equation}
with $T$ the time horizon, $\delta$ the discount rate, $p$ the price of a unit
of harvested biomass, and $c(E)$ the cost of a unit of effort when the
population size is~$N$.  The instantaneous net return or profit is given by
$$R(s) = p\cdot H(s) - c(E) = p\cdot q\cdot E\cdot N(s) - c(E)$$
at time $s$.
If it is assumed that the goal of the harvest is to find the effort 
level $E^*$ that maximizes the total profit, then we must compute 
\begin{eqnarray}\label{2.3}  %EQUATION 2.3
v^*(x) \equiv v \left(x; E^*\right) = \max_E[v(x;E)], 
\end{eqnarray}
subject to the dynamical constraint in Eq.~(\ref{2.1}).  This is a problem in 
optimal control theory and can be
studied using Pontryagin's maximum principle~(see Clark \cite{18} on the
maximum principle in the context of fishery bioeconomics).  However, that
method does not readily extend to the stochastic case. A more efficient form
for computing optimal controls in the presence of random fluctuations is the
Bellman equation of dynamic programming (see Bryson and Ho \cite{23}, for
instance). Thus, we consider the current value form of Eq.~(\ref{2.2}) given by
\begin{equation}\label{2.4}    %EQUATION 2.4
V(x,t;E) = \int_t^T e^{-\delta(s-t)} [pqEN(s) - c(E)]\,ds,
\end{equation}
and apply the principle of optimality to derive an equation for 
$$V^*(x,t) = \max_E[V(x,t; E)],$$ 
(see \cite{8} for a simple, formal derivation),
\begin{equation}    %EQUATION 2.5
V_t^*(x,t)+ r_1 x(1 - x/K) V_x^*(x,t) - \delta V^*(x,t) + S^*(x,t) =0,  
\end{equation}   
where
\begin{equation}\label{2.6}   %EQUATION 2.6
S^*(x,t) = \max_E\left[\left(p - V_x^*(x,t)\right)q E x - c(E)\right].
\end{equation}

Let $E_R(x,t)$ be the regular solution of $c'(E) = (p - V_x^*(x,t))qx$ 
corresponding to the unconstrained maximum in Eq.~(\ref{2.6}).  For instance,
in the case of quadratic costs with
$c(E)=c_1\cdot E+c_2\cdot E^2$,
we obtain
$$E_R(x,t)=\frac{(p - V_x^*(x,t))qx-c_1}{2\cdot c_2}.$$
Upon imposing the constraints on the harvesting effort~$E$, we get 
the {\it bang-regular-bang} control 
\begin{eqnarray}\label{2.7}   %EQUATION 2.7
E^*(x,t) =  \left\{\begin{array}{ll}
E_{\max}, & E_{\max} \le E_R(x,t)\\
E_R(x,t), &E_{\min} \le E_R(x,t) \le E_{\max} \\    
E_{\min}, &E_R(x,t) \le E_{\min} 
\end{array}\right. .
\end{eqnarray}

The full problem is determined by imposing the final boundary condition 
\begin{eqnarray}\label{2.8}     %EQUATION 2.8 
V^*(x,T) = 0 ,   
\end{eqnarray}
and natural boundary condition 
\begin{eqnarray}\label{2.9}     %EQUATION 2.9
V^* (0,t) = -\frac{(c_1+c_2E_{\min})E_{\min}}{\delta}
\left(1-e^{-\delta(T-t)}\right),
\end{eqnarray}
in the case of increasing, convex quadratic costs, i.e., $c_1>0$ and $c_2>0$,
provided that the discount rate $\delta$ is positive.  Note that if $x=0$,
then $N(t)=0$ by the vanishing of the right hand side of (\ref{2.1}) with
(\ref{2.15}) at extinction and the optimal
harvesting effort must be at the minimum $E_{\min}$ due to the negativity
of the cost function.

See \cite{8} for details. This is the form of the problem most easily 
generalized to the stochastic model discussed in Section~3.

\section{Stochastic Model}     %SECTION 3
%
An It\^{o} stochastic differential equation describing the growth and 
harvesting of a
model resource population subjected to large random changes in size is the
randomized Schaefer model~\cite{8}
\begin{eqnarray}\label{3.1}    %EQUATION 3.1
dN(s) &=& [r_1 N(1-N/K)-H(s)]\,ds+\sigma_1 N\, dW_1(s) +N\sum_{j=1}^n a_j\,dZ_j
(s,f_j),\nonumber\\
&~&\\\nonumber
N(t) &=& x,
\end{eqnarray}
where $r_1$, $K$ and $H(s)=qEN(s)$ are as described in Section~2.

There are two random components in Eq.~(\ref{3.1}).  Large rare
finite amplitude fluctuations are characterized by the density independent,
compound Poisson process 
$$N\sum_{j=1}^n a_j\,dZ_j(s;f_j).$$ 
Here, the relative
jump amplitude $a_j>-1$, the jump rate $f_j>0$, the Wiener noise
coefficient $\sigma_1\ge 0$, and the $j$th incremental Poisson process
$dZ_j(s ;f_j)$ has infinitesimal mean and variance $f_j\,ds$ drawn
from the set $\{dZ_1,\ldots,dZ_n\}$ of independent Poisson processes.
The independent density amplitude factor~$N$ is chosen so that any
variation is proportional to current population size as measured
by the~$a_j$, i.e., relative changes are independent of density~$N$.
Background fluctuations are modeled by the normalized, Gaussian
distributed Wiener process~$dW_1$ with zero mean and infinitesimal
variance $ds$.  Thus, in the model~(\ref{3.1}) the population size
is known at a time $s>0$ with future population size determined
from the deterministic component, the serially uncorrelated
fluctuations occurring from $dW_1$, and the random increases
occurring at the times of events of the $dZ_j$ with frequency~$f_j$.
Large scale effects, typically brought about by very complex and
poorly understood mechanisms, are thus estimated in~(\ref{3.1}) as
a lumped sum of density independent terms.  Such large fluctuations
are commonly observed in marine recruitment data and are well
documented in the fisheries literature \cite{19,20,21,CJFAS87}.
An analysis of the moments to the process defined by~(\ref{3.1})
in the absence of harvesting is given in \cite{22}.

In the economic component of the model we assume management is neutral to
risk and that prices are random and given by the supply/demand relation
\begin{equation}\label{3.2}    %EQUATION 3.2
P(s)=\left(\frac{p_0}{H(s)}+p_1\right) Y(s),
\end{equation}
where again $H(s)=qEN(s)$ is the amount of harvested biomass, 
$P(s)\cdot H(s)$ is the gross return on harvest $H(s)$,
$p_1$ is a constant price per unit harvested biomass coefficient, 
and $p_0$ is the supply/demand coefficient component of the price.  
In~(\ref{3.2}), $Y(s)$ is a fluctuating inflationary factor satisfying the 
It\^{o} stochastic differential equation
\begin{equation}\label{3.3}    %EQUATION 3.3
dY(s)=r_2 Y\,ds+\sigma_2 Y\,dW_2 (s)+ Y\sum_{j=1}^m b_j \,dQ_j(s;g_j),\qquad
Y(t)=y,
\end{equation}
with relative jump amplitude $b_j>-1$, jump rate $g_j>0$, 
and Wiener coefficient $\sigma_2\ge 0$.
The $dQ_j$ for $j=1,\ldots,m$, and
$dW_2$ are, respectively, incremental Poisson processes and a Wiener
process, as in~(\ref{3.1}).  Equation~(\ref{3.2}) describes prices inflated at 
an annual rate $r_2$, subjected to the rapid background perturbations of 
$dW_2$ as well as the occasional random jump increases or decreases 
brought about by the $dQ_j$. Such a characterization for the price $P$ 
is plausible from Figure~\ref{FIG3}, which shows price
versus catch for the Pacific halibut fishery from 1935 to 1985.  The
estimated mean price is hyperbolic in nature with both types of fluctuations
evident about the mean.  In particular, in the absence of discernible
correlations in the fluctuations the data suggest the multiplier~$Y$ to be
independent of~$P$.

Computation of the optimal exploitation policy in the stochastic case is
much more complicated than that for the deterministic 
model~(\ref{2.1})--(\ref{2.4}). 
Equation~(\ref{2.4}) must be modified to account for the random terms 
describing fluctuations in $N$ and $P$ and given by~(\ref{3.1})--(\ref{3.3}).  
Corresponding to~(\ref{2.3})--(\ref{2.4}), we seek a policy $E^*(x,y,t)$ that 
maximizes the expected discounted current value
\begin{eqnarray}\label{3.4}   %EQUATION 3.4
V(x,y,t) ={\rm Mean} \left[\int_t^T e^{-\delta(s-t)}\left[(p_0+p_1
qEN)Y(s)-c(E)\right]\,ds\mid N(t)=x, \; Y(t)=y\right],
\end{eqnarray}
where 
$${\rm Mean}=
{\raisebox{-1ex}[1em][1.5em]{${\displaystyle\rm Mean}
\atop{\scriptstyle\bf\{dZ,dQ,dW\}}$}}$$ 
denotes the mean or expectation 
taken over the vector processes ${\bf dZ}=[dZ_1,\ldots,dZ_n]$, ${\bf
dQ}=[dQ_1,\ldots,dQ_n]$, and ${\bf dW}=[dW_1,dW_2]$.  
Here, 
\begin{equation}\label{3.45}   %EQUATION 3.45
P(s)\cdot H(s)-c(E) = (p_0+p_1\cdot H(s))\cdot Y(s) -c(E),
\end{equation}
is the net return on harvested biomass $H(s)$ at time $s$.  
We further specialize effort costs to the quadratic form 
\begin{equation}\label{3.46}   %EQUATION 3.46
c(E)=c_1 E+c_2 E^2.  
\end{equation}
The additional quadratic
cost term $c_2 E^2$, appropriately scaled, may be viewed as a perturbation 
on the more typically employed linear costs as well as a technique to avoid 
difficulties
inherent in the computation of singular controls \cite{18,23}.  We assume
that both $c_1$ and $c_2$ are positive, so that costs are an increasing 
function and costs grow faster than a linear function of effort. Such a
cost function is, however, relatively common and has been employed in
fisheries studies by a number of authors \cite{10,11,24,25,26}.

Note that we do not include inflationary effects in the costs since our main
focus in on price dynamics, but since we have modeled the price dynamics
including inflationary effects, the discount rate $\delta$ in Eq. (\ref{3.4})
must be considered the nominal discount rate rather than the real or
inflation corrected discount rate.  Also, we have selected a finite horizon 
$T$ rather than an infinite horizon, since we believe that the finite
horizon case embodies more realism, particularly when motivated by fisheries
problems where the fishing season can be rather short and the environment
dynamic.  In this current paper, we are concerned with the dynamic problem
and not the equilibrium solutions such as those associated with the infinite
horizon case.

Since $N$ and $Y$ are stochastic processes and $E$ is a function of the
state as well as time, the easiest approach to the calculation of $(E^*, V^*)$
is via the Bellman equation of continuous-time dynamic programming 
\cite{8,15,27}.  Since $N$ and $Y$ involve discontinuous processes, the
Bellman equation will involve functional delay terms in both $x$ and $y$ as
well as a second-order derivative term arising from the Wiener processes
in~(\ref{3.1}) and~(\ref{3.3}).  Thus, $E^*$ and $V^*$ satisfy
\begin{eqnarray}\label{3.5}   %EQUATION 3.5
0 &=& V^*_t+r_1 x(1-x/K)V^*_x+\frac{\sigma_1^2 x^2}{2} V_{xx}^* - \delta V^* 
+ \sum_j f_j\left[V^*\left((1+a_j) x,y,t\right)-V^*(x,y,t)\right] 
\nonumber\\&~&\\
&+& r_2yV^*_y + \frac{\sigma_2^2 y^2}{2}V_{yy}^*+\sum_j 
g_j\left[V^*(x,(1+b_j)y,t)- V^*(x,y,t)\right]+S^*(x,y,t),\nonumber
\end{eqnarray}
where $S^*$ is the control switching term containing the argument of the
maximum in~(\ref{3.5}),
\begin{equation}\label{3.6}    %EQUATION 3.6
S^*(x,y,t) =\max_E\left[p_0\cdot y+\left(p_1\cdot y
- V^*_x(x,y,t)\right)q E x
-\left(c_1 E +c_2 E^2\right)\right],
\end{equation}
with unconstrained, regular control given as
\begin{equation}\label{3.65}    %EQUATION 3.65
E_R(x,y,t)=\frac{(p_1\cdot y - V_x^*(x,y,t))\cdot q\cdot x-c_1}{2\cdot c_2},
\end{equation}
determined from the argument of the maximum in~(\ref{3.6}), and with
the constrained, optimal control given as
\begin{eqnarray}\label{3.66}    %EQUATION 3.66
E^*(x,y,t) =  \left\{\begin{array}{ll}
E_{\max}, & E_{\max} \le E_R(x,y,t)\\
E_R(x,y,t), &E_{\min} \le E_R(x,y,t) \le E_{\max}\\
E_{\min}, &E_R(x,y,t) \le E_{\min}
\end{array}\right., 
\end{eqnarray}
similar to the form in Eq. (\ref{2.7}) in the one-dimensional, 
deterministic case, but with a two-dimensional dependence.

Equation~(\ref{3.5}) is augmented by the side conditions
\begin{eqnarray}   %EQUATION 3.7  3.8
V^*(x,y,T) &=& 0,\label{3.7}\\
V^*(0,0,t) &=& -\frac{(c_1+c_2E_{\min})E_{\min}}{\delta}
\left(1-e^{-\delta(T-t)}\right),
\label{3.8}
\end{eqnarray}
with the same reasoning given for (\ref{2.8}) and (\ref{2.9}) in the 
deterministic case, but here with $Y(s)\equiv 0$ when $y=Y(t)=0$.
The most appropriate method of solution for~(\ref{3.5})--(\ref{3.8}) appears to 
be numerical.  Perturbative techniques might be suggested for $\sigma_1$ and
$\sigma_2$ small, but these are not likely to be effective because of the
$O(1)$ nature of the functional terms in~(\ref{3.5}).  The numerical procedures 
are discussed more fully in Section~4.

\section{Numerical Results}   %SECTION 4

%
In this section we examine the results of the numerical solution of 
Eq.~(\ref{3.5}) for certain values of the parameters. The numerical solution 
of~(\ref{3.5}) is also outlined.

Our parameter values are based on Pacific halibut data over a number of years
and come from a variety of sources. We use estimates of $r_1 = 0.71$/year and
$K = 80.5 \times 10^6\,{\rm Kg}$ (see Clark \cite{18}). The price and cost data
were taken from the 1984 and 1985 IPHC Annual Reports~\cite{17a,17} for the 
period 1980 to
1985 to allow some temporal perspective and avoid the anomalous inflationary
period of the late 1970's. Linear regression was used to 
fit value versus catch, $\widetilde{V}=p_0+p_1\cdot H$, for 1980 to 1985. 
Although we use only six data points, 
$p_0 = \$8.46/$year and $p_1 = \$1.59/$Kg with 78\% 
of the variance explained.  The results of the linear regression for the 
value is displayed in Figure~\ref{FIG3} as the smooth hyperbolic curve of the 
price versus catch 
$$\widetilde{P}=\widetilde{V}/H = p_0/H+p_1,$$
fitting only the higher price fluctuations in 
the original IPHC price versus catch data from 1935 to 1985.

Other parameter values are
taken to be: 
$r_2=0.01/$year, 
$T = 10$ years, $\delta= 0.06/$year, 
$c_1 = \$96\times 10^{-6}/(\mbox{skate-year}) / {\rm year}$
(a standard skate is a 550 meter ground line with 100 hooks. 
Note that in the IPHC data \cite{17a,17} the annual effort is given in units
of skates with year dimensions implicit, thus year dimensions have been 
explicitly added here to effort, cost and catchability to preserve dimensional
correctness), 
$c_2 = \$0.10\times 10^{-6}/(\mbox{skate-year})^2/{\rm year}$,
$q = 3.30\times 10^{-6}/(\mbox{skate-year})/{\rm year}$, 
$E_{\min} = 0$, and $E_{\max} = r_1/q =0.2152 \times 10^6$ skate-years. 
Since our primary focus is on discontinuous effects, 
we take $\sigma_1 = 0$ and $\sigma_2 = 0$, removing the continuous background 
noise. 
We further lump the additive effects of the jumps by taking 
$f_1 = 0.5$ with $f_j = 0$ for $j \ge 2$,
$a_1 = -0.5$ with $a_j = 0$ for $j \ge 2$,
$g_1 = 0.5$ with $g_j = 0$ for $j \ge 2$, and
$b_1 = 0.5$ with $b_j = 0$ for $j \ge 2$.
%

The numerical solution of~(\ref{3.5}) has been obtained by employing
a hybrid extrapolated predictor-corrector and Crank-Nicolson finite difference 
method modified to account for
\begin{enumerate}
\item functional terms that appear due to the Poisson processes used to 
characterize the large fluctuations in both population and inflation rate,
and \item the maximization embodied in the switching term~(\ref{3.6}).
\end{enumerate}
We discretize using $x_i=(i-1)\Delta x$, $i=1, \cdots, N_x$ for the population,
$y_j=(j-1)\Delta y$, $j=1, \cdots, N_y$ for the inflationary factor, 
and $t_k=T-(k-1)\Delta t$, $k=1, \cdots, N_t$ for the time, where 
$\Delta x=K/(N_x-1)$, $\Delta y=e^{r_2T}/(N_y-1)$, and $\Delta t=T/(N_t-1)$.
The dependent variable $V^*(x_i,y_j,t_k)$ is represented by the discrete
variable $V_{i,j,k}$.  Second order central finite differences are used for 
spatial derivatives such that $V_x^*(x_i,y_j,t_k)$ is approximated by 
{\it DVX}$_{i,j,k}=\frac{1}{2}(V_{i+1,j,k}-V_{i-1,j,k})/\Delta x$
and $V_y^*(x_i,y_j,t_k)$ is approximated by
{\it DVY}$_{i,j,k}=\frac{1}{2}(V_{i,j+1,k}-V_{i,j-1,k})/\Delta y$,
with appropriate forms for the boundaries.
The second derivatives terms $V_{xx}^*(x_i,y_j,t_k)$ and $V_{yy}^*(x_i,y_j,t_k)$
are discretized by the central difference formulas 
{\it DDVX}$_{i,j,k}=(V_{i+1,j,k}-2V_{i,j,k}+V_{i-1,j,k})/(\Delta x)^2$ and
{\it DDVY}$_{i,j,k}=(V_{i,j+1,k}-2V_{i,j,k}+V_{i,j-1,k})/(\Delta y)^2$, 
respectively.
The backward time derivative $V_t^*(x_i,y_j,t_{k+0.5})$ is approximated by
{\it DVT}$_{i,j,k}=-(V_{i,j,k+1}-V_{i,j,k})/\Delta t$, which is also a 
second order central
finite difference, but  about the half time steps of the Crank-Nicolson method.
The functional terms $V^*((1+a_l)x_i,y_j,t_k)$ and $V^*(x_i,(1+b_l)y_j,t_k)$ are
approximated, respectively, by linear interpolation between the two nearest 
nodal values $V_{\hat{i},\hat{j},k}$ consistent with order of the errors in the 
second order central finite differences used for the spatial derivatives.
We denote the linear interpolation of 
$V^*((1+a_l)x_i,y_j,t_k)$ by {\it ZV}$_{i,j,k,l}$ and 
$V^*(x_i,(1+b_l)y_j,t_k)$ by {\it QV}$_{i,j,k,l}$.  

\newcommand{\VE}{\mbox{\it VE}}
\newcommand{\SE}{\mbox{\it SE}}
\newcommand{\ZVE}{\mbox{\it ZVE}}
\newcommand{\QVE}{\mbox{\it QVE}}
\newcommand{\DVXE}{\mbox{\it DVXE}}
\newcommand{\DVYE}{\mbox{\it DVYE}}
\newcommand{\DDVXE}{\mbox{\it DDVXE}}
\newcommand{\DDVYE}{\mbox{\it DDVYE}}
\newcommand{\VM}{\mbox{\it VM}}
\newcommand{\EM}{\mbox{\it EM}}
\newcommand{\ERM}{\mbox{\it ERM}}
\newcommand{\SM}{\mbox{\it SM}}
\newcommand{\ZVM}{\mbox{\it ZVM}}
\newcommand{\QVM}{\mbox{\it QVM}}
\newcommand{\DVXM}{\mbox{\it DVXM}}
\newcommand{\DVYM}{\mbox{\it DVYM}}
\newcommand{\DDVXM}{\mbox{\it DDVXM}}
\newcommand{\DDVYM}{\mbox{\it DDVYM}}
The Crank-Nicolson average for
the midpoint time-step $V_{i,j,k+0.5}$ is denoted by 
$\VM_{i,j,k}\equiv 0.5(V_{i,j,k}+V_{i,j,k+1})$, 
and the accelerating extrapolated starting value by 
$\VE_{i,j,k}\equiv 0.5(3V_{i,j,k}-V_{i,j,k-1})$ provided $k\leq 2$, 
with corresponding notations for the spatial derivatives, 

Thus, the discrete extrapolated, predictor approximation 
corresponding to the Bellman equation
(\ref{3.5}) is
\begin{eqnarray}\label{4.01}    %EQUATION 4.01
V_{i,j,k+1}^{(p)}&=&V_{i,j,k}^{(c,*)}
+\Delta t\left[r_1x_i(1-x_i/K)\DVXE_{i,j,k}
+\frac{1}{2}\sigma_1^2x_i^2\DDVXE_{i,j,k} - \delta \VE_{i,j,k}
\right.
\nonumber\\
&+&\Sigma_l f_l (\ZVE_{i,j,k,l}-\VE_{i,j,k})
+r_2y_j\DVYE_{i,j,k} 
\\\nonumber
&+& \left. 
\frac{1}{2}\sigma_2^2y_j^2\DDVYE_{i,j,k}
+\Sigma_l g_l (\QVE_{i,j,k,l}-\VE_{i,j,k}) 
+\SE_{i,j,k}\right],
\end{eqnarray}
where $V_{i,j,k}^{(c,*)}$ is the final correction from the $k$th backward time
step, 
$\DVXE_{i,j,k}=0.5(\VE_{i+1,j,k}-\VE_{i-1,j,k})/\Delta x$,
for example, and 
$\VE_{i,j,k}\equiv 0.5(3V_{i,j,k}^{(c,*)}-V_{i,j,k-1}^{(c,*)})$ 
provided $k\leq 2$ so that corrections are available on at least two starting
time steps.  In the predictor evaluation step, $\DVXM$, $\DVYM$, $\DDVXM$, 
$\DDVYM$, $\ZVM$, and $\QVM$ are evaluated using the discrete values
$$\VM_{i,j,k}^{(p)}=0.5(V_{i,j,k}^{(c,*)}+V_{i,j,k+1}^{(p)}).$$
From (\ref{3.65}), it follows that the regular control $E_R(x_i,y_j,t_{k+0.5})$
at the predictor step is given approximately by 
\begin{eqnarray}\label{4.02}    %EQUATION 4.02
\ERM_{i,j,k}^{(p)}=(p_1Y_j-\DVXM_{i,j,k}^{(p)}\cdot q\cdot x_i-c_1)/(2c_2).
\end{eqnarray}
The predicted, constrained, optimal control $\EM_{i,j,k}^{(p)}$ is computed 
using composite formula (\ref{3.66}) with $\ERM_{i,j,k}^{(p)}$ substituted for
$E_R(x,y,t)$ on the right hand side and the maximized control switching term 
is computed from the argument of the maximum in (\ref{3.6}) by substituting 
the optimal $\EM_{i,j,k}^{(p)}$ for $E$.

Consequently, the $(L+1)$th correction to the discretized Bellman equation
is given by
\begin{eqnarray}\label{4.03}    %EQUATION 4.03
V_{i,j,k+1}^{(c,L+1)}&=&V_{i,j,k}^{(c,*)}
+\Delta t\left[r_1x_i(1-x_i/K) \DVXM_{i,j,k}^{(c,L)}
+\frac{1}{2}\sigma_1^2x_i^2\DDVXM_{i,j,k}^{(c,L)} - \delta \VM_{i,j,k}^{(c,L)}
\right.
\nonumber\\
&+&\Sigma_l f_l (\ZVM_{i,j,k,l}^{(c,L)}-\VM_{i,j,k}^{(c,L)})
+r_2y_j\DVYM_{i,j,k}^{(c,L)} 
\\\nonumber
&+& \left. \frac{1}{2}\sigma_2^2y_j^2\DDVYM_{i,j,k}^{(c,L)}
+\Sigma_l g_l (\QVM_{i,j,k,l}^{(c,L)}-\VM_{i,j,k}^{(c,L)}) 
+\SM_{i,j,k}^{(c,L)}\right],
\end{eqnarray}
for $L=0$ to $L^*$, where $\VM_{i,j,k}^{(c,0)}=\VM_{i,j,k}^{(p)}$, i.e., the
prediction is the $0$th correction.  The subsequent correction evaluation
step is again the Crank-Nicolson average
$$\VM_{i,j,k}^{(c,L)}=0.5(V_{i,j,k}^{(c,*)}+V_{i,j,k+1}^{(c,L)}),$$
which is used to calculate $(L+1)$th corrections for all differenced 
derivatives and functions terms as well as $\ERM_{i,j,k}^{(c,L+1)}$,
$\EM_{i,j,k}^{(c,L+1)}$ and $\SM_{i,j,k}^{(c,L+1)}$.

Corrections are continued until a relative stopping criterion,
$$
|V_{i,j,k+1}^{(c,L+1)}-V_{i,j,k+1}^{(c,L)}|<\varepsilon|V_{i,j,k+1}^{(c,L)}|
$$ 
is satisfied for all $\{i,j\}$ at
fixed discrete time $k+1$ and some relative tolerance $\varepsilon>0$ 
with the stopped correction counter denoted by 
$L_k^*=L+1$ .  The final correction value that is used in the next time 
step is more concisely denoted by $V_{i,j,k}^{(c,*)}=V_{i,j,k}^{(c,L_k^*)}$.
Typically, only a few corrections are needed for reasonable accuracy, beyond
the starting, final value at $k=1$.  

The convergence of the corrections is not a simple matter and convergence 
difficulties increase with the dimension of the state space, since the
convergence of the discretized stochastic dynamic programming procedure
critically depends on the mesh ratio of $\Delta t$ compared to some metric of 
$\Delta x$ and $\Delta y$.  For more information on the approximate 
quasi-deterministic convergence criteria used, comparison to other methods,
and additional references the reader is
referred to the survey chapter of Hanson \cite{CADS96}.

%
Figure~\ref{FIG4} shows the optimal current value $V^*(K,y,t)$ 
in million dollar units using optimal effort $qE^*(K,y,t)/r_1$ 
versus a scaled price factor $y\cdot\exp(-r_2 T)$, i.e., with the 
deterministic inflationary part $\exp(+r_2 T)$ at the final time scaled out. 
%
\begin{figure}[htb]
\begin{center}
~\\*[-10pt]
\hspace*{-20pt}
\epsfysize300pt
\epsffile{Fig4YVTBig.eps}
\caption{\label{FIG4}
Optimal current value, $V^*(K,y,t)$, in millions of U.S.\ dollars
versus the scaled price factor, $y\cdot\exp(-r_2\cdot T)$,
with time parameter $t = 0.0$, 2.0, 4.0, 6.0, 8.0, 10.0 for each curve ordered
from top to bottom, respectively,
and with population size fixed at carrying capacity $x = K$.
}
\end{center}
\end{figure}
%
The figure is intended to show the effects on the optimal current value
due to the inflationary factor.
The curves indexed by time-to-go, $T - t = 0$, 2, 4, 6, 8, 10 
starting at the bottom along the abscissa at $T - t = 0$ (i.e., the final
time $t=T$) to the uppermost curve at $T - t = 10$ (i.e., initial time $t=0$).  
As expected, the optimal current value increases as a function of increasing
scaled inflation factor with a nearly constant slope for fixed $T-t$, 
except for the zero final current value at $t=T$.
From the curves $T-t = 2$ to 4, 4 to 6, 6 to 8
and 8 to 10 the current value shows a substantial increase of about
2.9, 2.2, 2.0 and 1.9 times, respectively.
The optimal current value as a function of
population size (not pictured) is relatively flat but shows similar large
increases when indexed over the indicated intervals. The displayed curves are
essentially linear with slope approximately $p_1qE(K,y,t)N(K,t)$ for the
relatively short horizon $T = 10$.  Thus, even when influenced by density
independent disasters, the optimal current value is extremely sensitive to the
stochastic inflationary factor with the rate of increase increasing for
longer times.
As a word of caution in interpreting Fig.~\ref{FIG4}, we note that
the scaled inflationary factor $y\cdot\exp(-r_2 T)$ just indicates a rough,
average scaling of $y$ for this model and does not indicate an exponential
growth in the price of halibut since we took only price parameters from
the halibut fishery and not inflation parameters.

Since the price $P$ is time dependent, the instantaneous return or value will
be altered by both changes in the stock level $N(t)$ and the price level
$P(t)$.  
In order to motivate this and our more general numerical results,
we examine the simplifying quasi-deterministic approximation~\cite{22},
bearing in mind that the results displayed in Figs.~\ref{FIG4} 
through~\ref{FIG6} are for
the stochastic problem with random Poisson jumps in price and population. 
The rate of change in the inflationary factor can be approximated
by the quasi-deterministic approximation
to Eq. (\ref{3.3}), where $Y_{QD}$ satisfies 
\begin{eqnarray}\label{4.2}    %EQUATION 4.2
dY_{QD}(t) = {\rm Mean}\left[ dY(t) \mid Y(t) = Y_{QD}(t) \right]
     = \left(r_2+\sum_j b_j g_j\right)Y_{QD}(t)\, dt ,
\end{eqnarray}
i.e., approximated by the exponential growth:
\begin{eqnarray}\label{4.3}    %EQUATION 4.3
Y_{QD}(t) = y(0)\exp\left(\left(r_2+\sum_j b_j g_j\right)t\right).
\end{eqnarray}
Similarly, the stock level has the quasi-deterministic approximation, 
$N_{QD}(t)$, assuming in (\ref{3.1}) that the effort $E$ is constant 
and that
$$H_{QD}(t) = q\cdot E\cdot N_{QD}(t),$$
for simplicity,
\begin{eqnarray}\label{4.4}    %EQUATION 4.4
dN_{QD}(t) &=& {\rm Mean} \left[ dN(t) \mid N(t) = N_{QD}(t) \right]
\nonumber\\&~&\\\nonumber
&=& \left(r_1\cdot(1-N_{QD}(t)/K)- q\cdot E + 
\sum_j a_j f_j\right)N_{QD}(t)\, dt ,
\end{eqnarray}
i.e., a ``Schaefer'' model modified by the linear, mean jump contribution.
The quasi-deter\-ministic price, from Eq. (\ref{3.2}), is then
$$P_{QD}(t)=\left(p_0/H_{QD}(t)+p_1\right)\cdot Y_{QD}(t).$$
Consequently, the approximate instantaneous return, 
$$R_{QD}(t) \equiv P_{QD}(t)\cdot H_{QD}(t) - c(E),$$
has a marginal rate of increase that decomposes into
\begin{eqnarray}\label{4.5}    %EQUATION 4.5
\frac{dR_{QD}}{dt}(t) = p_1\cdot q\cdot E\cdot \frac{dN_{QD}}{dt}(t)
+\left(p_0+p_1\cdot q\cdot E\cdot N_{QD}(t)\right) \cdot \frac{dY_{QD}}{dt}(t).
\end{eqnarray}
Thus the approximate immediate return changes with the changes in stock
level, but also with the average approximate inflationary jump 
rate, which will be more rapid for longer times from (\ref{4.3}).
Note that this approximate result ignores changes in the harvesting effort.

The wide separation in the curves may be accounted for by noting that
starting the discounting at $s = t$ makes the Bellman equation autonomous.
Since the Wiener and Poisson processes are stationary each separate curve
represents the expected addition to $V^*$ starting from the previous time. In
other words $y$ can be thought of as a restarted inflation rate at any time.
Following a jump in price this increment is positive and augments the
marginal increase in the expected value of current yield revenues.

Figure~\ref{FIG5} shows the scaled optimal feedback effort $qE^*(K,y,t)/r_1$ 
versus $y\cdot\exp(-r_2T)$ for time-to-go $T - t = 0$, 2, 4, 6, 8, 10.
This shows that optimal effort is not very sensitive to the lumped effects of
the inflation factor and the $dQ_j$. While the optimal current values differ
by over 380\% between $T - t = 2$ and $T - t = 0$ at $x = K$, the optimal
effort needed to obtain these levels shows a relative difference of about
$-0.13\%$ for the same values, i.e., a slight decrease. This is consistent
over the full range of time-to-go values. Thus stochastic inflationary
effects dramatically change optimal return while leaving effort levels
relatively untouched. This is reasonable since for $x$ near $K$ for
increasing times the rapid inflationary increase in $P$ is offset by a rapid
increase in the shadow price $V_x^*$. Away from the constraints this tends to
keep effort levels slowly changing.
%
\begin{figure}[htb]
\begin{center}
~\\*[-10pt]
\hspace*{-20pt}
\epsfysize300pt
\epsffile{Fig5YETBig.eps}
\caption{\label{FIG5}
Optimal feedback effort, $q\cdot E^*/r_1(K,y,t)$,
in dimensionless form
versus the scaled price inflation factor, $y\cdot\exp(-r_2\cdot T)$,
with time parameter covering $t = 0.0$, 2.0, 4.0, 6.0, 8.0, 10.0 for each curve
closely spaced from the bottom to top, respectively,
and with population size fixed at carrying capacity $x = K$.
}
\end{center}
\end{figure}
%

Figure~\ref{FIG6} displays the sensitivity of the optimal current value, 
$V^*(K,y,0)$, to the inflation price factor rate $r_2$. 
The curves are parameterized by the scaled inflation price factor 
$y\cdot\exp(-r_2T)$ ranging from 1.0 for the topmost curve
to 0.2 at the bottom in steps of 0.2. The convexity of the curves is most
upward for $r_2$ near 0.2 and the scaled price factor near 1.0 suggesting
that return is sensitive to the starting inflation rate. For $y$ near zero the
expected return will stay positive as long as the shadow price, $V_x^*$, is 
less than $(p_0 + p_1qEx)y$, in which case we see the expected present value
approximately proportional to~$y$. For the parameter values chosen, the
combined effects of the random jumps $dQ_j$ in the price factor 
and the inflationary drift $r_2\cdot Y$ overcome the effect of the occasional 
disaster according to the random population jumps $dZ_j$ that might drive the 
population to near zero.  These combined 
effects are more pronounced for higher inflation levels with higher 
variability as the inflation rate increases.
%
\begin{figure}[htb]
\begin{center}
~\\*[-10pt]
\hspace*{-20pt}
\epsfysize300pt
\epsffile{Fig6R2VYBig.eps}
\caption{\label{FIG6}
Sensitivity of optimal current value, $V^*(K,y,0)$, to the inflation
price factor rate $r_2$, with curves parameterized by
the scaled inflation price factor, $y\cdot\exp(-r_2\cdot T)$,
ranging from 1.0 at the top to 0.2 at the bottom in steps of 0.2,
with time fixed at the initial value $t=0.0$,
and with population size fixed carrying capacity $x = K$.
}
\end{center}
\end{figure}
%

\section{Summary}     %SECTION 5
%
We have examined the effects of random price fluctuations on the computed
optimal harvest strategy and return for a randomized Schaefer model with
density independent disasters. Model population and economic parameters were 
taken from \cite{18} and the 1984 and 1985 IPHC Annual Reports \cite{17a,17}. 

We have found that random price fluctuations that include large inflationary
increases against a background of continuous inflationary growth strongly
affect optimal return but have a much less significant impact on optimal
effort. Random inflationary effects, even in the presence of a hazardous
environment, are therefore much more likely to play a role in determining
optimal return than in scheduling effort.

\newpage
%\newpage
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%\newpage
%\noindent{\bf FIGURE CAPTIONS}

%\bigskip
%\noindent
%Figure 1.  Pacific halibut prices in U.S.\ dollars per kilogram for each year
%from 1935 to 1985. The source of the data in the table in Appendix~II in the
%IHPC 1984 and 1985 Annual Reports \cite{17a,17}.

%\bigskip
%\noindent
%Figure 2.  U.S.-Canadian catch in millions of kilograms for each year from
%1935 to 1985. The source is the same as Figure~1.

%\bigskip
%\noindent
%Figure 3.  Pacific halibut price in U.S.\ dollars per kilogram versus catch in
%millions of kilograms for the years from 1935 to 1985. The source is the same
%as Figure~1.  Also, the linear regression for the price times catch as a
%function of catch from 1980 to 1985 is displayed as the
%smooth hyperbolic price curve.

%\bigskip
%\noindent 
%Figure 4.  Optimal current value, $V^*(K,y,t)$, in millions of U.S.\ dollars
%versus the scaled price factor, $y\cdot\exp(-r_2\cdot T)$, 
%with time parameter $t = 0.0$, 2.0, 4.0, 6.0, 8.0, 10.0 for each curve ordered
%from top to bottom, respectively, 
%and with population size fixed at carrying capacity $x = K$.

%\bigskip
%\noindent
%Figure 5.  Optimal feedback effort, $q\cdot E^*/r_1(K,y,t)$, 
%in dimensionless form 
%versus the scaled price inflation factor, $y\cdot\exp(-r_2\cdot T)$,
%with time parameter covering $t = 0.0$, 2.0, 4.0, 6.0, 8.0, 10.0 for each curve
%closely spaced from the bottom to top, respectively,
%and with population size fixed at carrying capacity $x = K$.

%\bigskip
%\noindent 
%Figure 6.  Sensitivity of optimal current value, $V^*(K,y,0)$, to the inflation
%price factor rate $r_2$, with curves parameterized by 
%the scaled inflation price factor, $y\cdot\exp(-r_2\cdot T)$, 
%ranging from 1.0 at the top to 0.2 at the bottom in steps of 0.2,
%with time fixed at the initial value $t=0.0$, 
%and with population size fixed carrying capacity $x = K$.
%
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