# Optimal management of a stochastically varying population when policy
adjustment is costly

Carl Boettiger  cboettig (at) berkeley.edu Michael Bode  James N. Sanchirico
Jacob LaRiviere  Alan Hastings  Paul R. Armsworth  Department of Environmental
Science, Policy and Management, University of California, Berkeley, 130
Mulford Hall #3114, Berkeley, CA 94720-3114, USA  Department of Environmental
Science and Policy, University of California, Davis  School of Botany,
University of Melbourne, Australia  Department of Ecology and Evolutionary
Biology, University of Tennessee, Knoxville  Department of Economics,
University of Tennessee, Knoxville

###### Abstract

Ecological systems are dynamic and policies to manage them need to respond to
that variation. However, policy adjustments will sometimes be costly, which
means that fine-tuning a policy to track variability in the environment very
tightly will only sometimes be worthwhile. We use a classic fisheries
management question – how to manage a stochastically varying population using
annually varying quotas in order to maximize profit – to examine how costs of
policy adjustment change optimal management recommendations. Costs of policy
adjustment (here changes in fishing quotas through time) could take different
forms. For example, these costs may respond to the size of the change being
implemented, or there could be a fixed cost any time a quota change is made.
We show how different forms of policy costs have contrasting implications for
optimal policies. Though it is frequently assumed that costs to adjusting
policies will dampen variation in the policy, we show that certain cost
structures can actually increase variation through time. We further show that
failing to account for adjustment costs has a consistently worse economic
impact than would assuming these costs are present when they are not.

###### keywords:

optimal control , fisheries , ecological management , adjustment costs ,

## Introduction

Ecosystems are dynamic and exhibit rich patterns of variability in both time
and space (Durrett and Levin 1994). In designing management policies for
ecosystems, managers need to decide how much of that variation to respond to.
Managers could try to track variations in ecosystem dynamics very closely by
setting policies that are extremely responsive to the environment. Many
theoretical studies that seek to identify optimal policies for exploited
populations and communities adopt this approach and largely ignore the
challenges that would be involved in implementing such recommendations (e.g.,
Reed 1979, Neubert 2003, Sethi et al. 2005, Halpern et al. 2011). However, the
policy process can often be much more sluggish to respond to variations in
ecosystem dynamics (Walters 1978, Armsworth et al. 2010). Moreover,
stakeholders impacted by ecosystem management may prefer some stability and
not want to deal with continually changing management recommendations (Biais
1995, Armsworth and Roughgarden 2003, Patterson 2007, Patterson and Resimont
2007, Sanchirico et al. 2008). In other words, whatever gains are available
from fine-tuning a policy prescription to more closely reflect environmental
variation should be traded off against potential costs associated with the
more responsive approach to management this would require.

Figure 1: Estimated stock abundances (top panels) show much greater variation
than the corresponding harvest policies (bottom). Left panels: ICCAT Stock
assessments and Total Allowable Catch (TAC, before adjustments for national
underages and overages) for bluefin tuna, 1982 \- 2007. Right panels: SEDAR
stock assessments and annual quotas for south Atlantic snowy grouper,
1994-2012.

To illustrate this tradeoff, we use an example from fisheries management.
Figure 1, top-left panel, shows a time series for the estimated population
size, represented as spawning stock biomass, of west Atlantic bluefin tuna
(_Thunnus thynnus_ , henceforth bluefin) from a recent stock assessment
(International Commission for the Conservation of Atlantic Tunas (ICCAT)
2009). The bottom-left panel shows the catch quota (Total Allowable Catch or
TAC, before being adjusted for overages or underages in the catch of each
country in the previous year) for the stock as set by the relevant management
agency (International Commission for the Conservation of Atlantic Tunas
(ICCAT) 2009). Despite the estimated population size declining by 53.3%
between 1982 and 1991, the quota was not changed during this period. Instead,
the quota only changed occasionally and in between times was left unaltered.
Fishery management decisions regarding this species can be highly contentious
(Safina (1998), Sissenwine et al. (1998), Porch (2005)). Moreover, the stock
is fished by fleets from many nations with quotas being set by a multilaterial
management agency through a process of negotiation. As such, we might
reasonably anticipate that for this species there could be substantial
transaction costs involved in reaching agreement over any quota change, or
constraints on the frequency with which the those changes could occur. Either
could contribute to the observed quota stability.

Such patterns of variable stocks managed under far less variable quotas are
extremely common, as in the instance of the management of south Atlantic snowy
grouper (_Epinephelus niveatus_) shown in the righthand panels (SEDAR 2013).
Though the stock assessment shows similarly frequent variation between 1994
and 2012, the quota has only been adjusted twice during this period. In
contrast to the adjustment in the bluefin quotas, both adjustments in snowy
grouper quotas were phased in through steps over several years: 1994-96 and
2005-08. More generally, reviews by Biais (1995) and Patterson (2007) document
many cases where changes in catch quotas that a management agency set were
more modest than changes that would be recommended just by considering
variations in stock abundance.

We use a classic fisheries management question to examine how accounting for
costs of policy adjustment can change optimal policies (see also Ludwig
(1980), Feichtinger et al. (1994), Wirl (1999)). We focus on how harvest
quotas for a stochastically varying fish population can be chosen to maximize
the net present value of a fishery. Our formulation and solution method
largely follow Reed’s classic treatment on this question, a treatment repeated
widely in bioeconomic textbooks. We note that while later work has extended
this treatment to deal with a variety of other issues (e.g. Sethi et al.
(2005); Singh et al. (2006); McGough et al. (2009)), we start from the
classical model for simplicity of presentation and analysis.

With his formulation, Reed (1979) showed that a constant escapement policy
could be optimal under certain conditions. Such a policy involves choosing
annual quotas that are perfectly responsive to recruitment variation in a fish
stock. In poor recruitment years, the quota is set to zero and no fishing is
allowed. Any time a recruitment pulse exceeds the escapement threshold, a
quota is set that allows the fishery to exactly compensate through harvesting,
thereby maintaining the optimal escapement level. However, in that analysis,
Reed did not account for any costs of policy adjustment, which for such a
responsive management strategy potentially could be large.

As the examples in Figure 1 make clear, management policies will rarely be as
responsive as this constant escapement policy assumes. In this paper, we
consider a case where managers seek to balance the benefits in terms of
increased profits from fishing from more finely tracking recruitment
variations with the growing costs associated with adjusting policies
frequently to do so. The policy adjustment costs involved could reflect pure
administrative transaction costs or preferences held by fishermen, fish
processing plants or other stakeholders, for less variable quotas. In seeking
to account for these policy adjustment costs, we recognize that we do not know
just what functional form they should take and that it will require
substantial empirical work to estimate that. Therefore, we investigate three
candidate functional forms that represent qualitatively different assumptions
about how these costs operate to determine whether the results we obtain are
sensitive to such differences.

## Methods

### Fish population dynamics (state equation)

We will assume Beverton-Holt dynamics with multiplicative environmental noise

$N_{t+1}=Z_{t}\frac{A(N_{t}-h_{t})}{1+B(N_{t}-h_{t})},$ (1)

where $N_{t}$ the stock size, $h_{t}$ the harvested level, $Z_{t}$ gives the
stochastic shocks, which we assume are log-normally distributed and $A$ and
$B$ are positive constants.

We assume managers set an annual quota for harvesting $h_{t}$ (the control
variable) after observing the stock size that year $N_{t}$ (the state
variable), but while still being uncertain about future environmental
conditions and stock sizes. Through time, this gives a time path of management
actions ${\bf h}=(h_{1},h_{2},\dots)$ that depend on the stock sizes that were
observed (a state dependent control rule).

We assume that managers choose annual quotas to maximize the expected net
present value (NPV) of the fishery. We take as a base case the situation where
there are no costs associated with policy adjustment and the managers’
objective is

$\max_{{\bf h}}\mathbf{E}(NPV_{0})=\max_{{\bf
h}}\sum_{0}^{T_{\mathrm{max}}}\mathbf{E}\left(\displaystyle\frac{\Pi_{0}(N_{t},h_{t})}{(1+\delta)^{t+1}}\right)$
(2)

where $\mathbf{E}$ is the expectation operator, $\delta>0$ is the discount
rate and $\Pi_{0}$ is the net revenue from operating the fishery in a given
year. In this base case, we assume the annual dockside (that is, before
accounting for any adjustment costs that may be incurred) net revenue from
fishing is

$\Pi_{0}(N_{t},h_{t})=ph_{t}-c_{0}E_{t}$ (3)

where $E_{t}$ represents fishing effort. We assume catch is proportional to
stock size and effort expended fishing that year, $h_{t}=qE_{t}N_{t}$ and
constant $q>0$ is the catchability coefficient. In Eqn. (3), $p$ is the price
per unit harvest and $c_{0}$ the cost of fishing per unit effort and, to
simplify the presentation of results, we assume that these are constants with
$p>0$ and $c_{0}>0$.

To simplify presentation of the results, we will illustrate cases where the
growth parameters are chosen so that the equilibrium biomass for the
equivalent deterministic model without harvesting is 10; (specifically, $A=$
1.5 and $B=$ 0.05 in Eqn (1)). To characterize environmental variability, we
assume multiplicative shock $Z_{t}$ is distributed log-normally with log
standard deviation $\sigma_{g}$ = 0.2. In addition, we show cases where $p=$
10, $\delta=$ 0.05 and $c_{0}=$ 30\. In the supplement we illustrate that the
qualitative patterns observed here are not sensitive to the specific choices
of these parameter values.

Taken together this objective function (2) and the state equation (1) define a
stochastic dynamic programming problem that we solve using backwards recursion
via Bellman’s equation. We denote the resulting state-dependent, optimal
control as $\mathbf{h}_{0}^{\ast}$. We solve this problem on a finite time
horizon of $T=$ 50 using value iteration (Mangel and Clark 1988, Clark and
Mangel 2000).

R code for implementing the dynamic programming algorithm in this context is
provided as a supplementary R package to the paper,
(github.com/cboettig/pdg_control, Boettiger et al. (2015)), along with scripts
for replicating the analyses presented here. To address concerns about
computational reproducibility (Boettiger 2015), a Docker container image of
the software environment is also provided.

### Costs of policy adjustment

We compare this base case to three alternative problem formulations, each
reflecting different plausible functional forms that costs of policy
adjustment could take. In each, we assume managers can adjust the quota set in
the fishery $h_{t}$ in a given year and that any policy adjustment costs are
associated with changes to this control variable. In each case, we assume
there is no cost in initially setting the harvest policy at time 0.

First we assume that policy adjustment costs are directly proportional to the
magnitude of the change in policy being proposed, such that larger changes to
annual harvesting quotas incur greater policy adjustment penalties.
Specifically, we replace $\Pi_{0}$ in the $NPV_{0}$ equation with

$\Pi_{1}(N_{t},h_{t},h_{t-1})=\Pi_{0}-c_{1}|h_{t}-h_{t-1}|\,.$ (4)

Next we continue to assume that policy adjustment costs depend on the
magnitude of the change in policy being proposed. However, we consider a case
where this dependence is nonlinear with big changes in policy being
disproportionately expensive:

$\Pi_{2}(N_{t},h_{t},h_{t-1})=\Pi_{0}-c_{2}(h_{t}-h_{t-1})^{2}\,.$ (5)

Finally, we assume instead that there is a fixed cost associated with any
adjustment of policy, regardless of how large that adjustment might be

$\Pi_{3}(N_{t},h_{t},h_{t-1})=\Pi_{0}-c_{3}(1-\mathbf{I}(h_{t},h_{t-1}))\,,$
(6)

where indicator function $\mathbf{I}(h_{t},h_{t-1})$ takes the value 1 if
$h_{t}=h_{t-1}$ and zero otherwise. For each $\Pi_{i}$, we can then define a
new objective function $NPV_{i}$ similar to that in Eqn. (2). We note that the
fixed cost has conceptual analogues to the set-up cost of Reed (1974) and
Spulber (1982) but here the fee is associated with a change in harvesting
($h_{t}\neq h_{t-1}$) and not with harvesting _per se_ ($h_{t}>0$) as in those
studies.

Taken together with the state equation Eqn. (1), each of these new objective
functions defines a different stochastic dynamic programming problem. Again,
we solve them numerically using backwards recursion. To include costs of
policy adjustment, we expand the state space to include both the current stock
size and the management action taken on the previous time step
$(N_{t},h_{t-1})$. For each new objective function $\max NPV_{i}$, we denote
the corresponding optimal control policy as the vector $\mathbf{h}_{i}^{*}$.

Note that this problem is much larger computationally than the classic
formulation of this stochastic dynamic programming problem for a single stock.
Whereas the classic problem considers each of $Q$ possible discrete quotas for
$S$ possible stock values at each time $t$, for a search space of size
$S\times Q$ per timestep; this formulation must also consider all possible
values of quota in the _previous_ time step: since the costs that will follow
depend on whether and by how much the harvest policy will change. This creates
a total of $S\times Q\times Q$ configurations.

In the Supplementary Material, we compare our results with more conventional
fisheries economic formulation in which additional costs are applied to the
control variables themselves, as opposed to adjustments to the controls:
$\Pi_{4}(N_{t},h_{t})=ph_{t}-c_{0}E_{t}-c_{4}E_{t}^{2}$. Additional costs of
this form tend to have a smoothing effect on optimal quotas, but also change
the long-term average stock size or quota size that is optimal.

### Comparing apples to apples

Each policy adjustment cost function is characterized in terms of a cost
coefficient $c_{i}$. However, $c_{i}$ takes different units for each
functional form. Therefore, if seeking to compare the relative effect of each
penalty function on optimal management, it is unclear what parameter values
should be used. To address this issue, we calibrate choices of $c_{i}$ so that
each has a comparable impact on the optimal fishery $NPV$. Fortunately, the
optimal control framework provides a completely natural way to make this
comparison: in terms of the _economic value_ (measured by $NPV$) of the stock
in each case. We will calibrate our realized economic value under each policy
relative to economic value of the stock when adjustments to the policy are
free ($NPV_{0}$).

Figure 2 shows the calibration graphically. Each curve plots the change in
maximum expected $NPV$ for a given cost structure as the $c_{i}$ coefficient
is increased. In each case, the figure shows maximum expected NPV with policy
adjustment costs as a proportion of the maximum expected NPV available in the
basic problem $NPV_{0}({\bf h_{0}^{*}})$ without policy adjustment costs.

Figure 2: Expected net present value by functional form of penalty. Horizontal
axis shows the coefficient $c_{i}$ governing the magnitude of the policy cost,
while vertical axis shows fraction of the maximum expected net present value
dissipated by the policy cost, $\tfrac{NPV_{0}-NPV_{i}}{NPV_{0}}$. The
horizontal line indicates a value of the stock that is reduced by 25% from the
maximum expected value in the absence of policy adjustment costs,
$NPV_{0}({\bf h_{0}^{*}})$. Selecting the coefficient $c_{i}$ corresponding to
this value in each functional form allows us to make consistent comparisons
across the different functional forms of policy costs.

To compare the impact of penalty functions on optimal management across the
different functional forms, we select penalty cost coefficients that induce
the same reduction in maximum expected NPV. For example, the dashed vertical
line in Figure 2 maps the needed cost coefficient $c_{i}$, for each penalty
function, such that the fishery is worth 75% of its unconstrained value when
optimally managed. To demonstrate that the results shown here are independent
of this choice of calibrating to a 75% reduction, we conduct the same analysis
with a 10% and 30% reduction rate as well. The corresponding figures can be
found in the Supplemental Materials.

## Results

### Effect of policy adjustment costs on optimal quotas and stock sizes

Figure 3: Example realization of optimal harvesting strategy under the
different functional forms of adjustment costs. When costs vary linearly with
the size of the adjustment ($\Pi_{1}$), periods with no adjustment are more
common than if adjustments were free. When costs vary quadratically with
adjustment size ($\Pi_{2}$), small adjustments are relatively cheap and thus
the resulting optimal policy always changes, but by a smaller amount than if
adjustments were free. When adjustments incur a fixed fee independent of size
($\Pi_{3}$), the optimal strategy either remains unchanged or overshoots the
cost-free optimum.

Figure 3 illustrates how the different forms of adjustment cost can impact the
dynamics of the optimal harvest policy. Corresponding stock sizes can be seen
in the supplement. Each panel is generated against the same sequence of
environmental variability so that they can be compared directly. The harvest
policy chosen shows a systematic deviations from the adjustment cost free
optimum, depending on the structure of the cost function.
In each case, the optimal solution without any adjustment cost is shown by the
dashed grey line, with the policy induced by optimization under the given cost
structure (equivalent to a 25% reduction in maximum expected NPV) overlaid in
solid blue.

The first panel shows a typical pattern resulting from linear adjustment costs
($\Pi_{1}$). The optimal policy tends to avoid very small policy adjustments,
resulting in periods of a constant policy followed by sudden bursts of
adjustment. This results in a relatively step-like policy pattern. In
contrast, the second formulation (quadratic costs, $\Pi_{2}$)
disproportionately penalizes large policy adjustments. The corresponding
optimal policy is typified by the middle panel, responding to each of the
fluctuations in stock, made by the cost-free policy, but with smaller
magnitude response than the equivalent cost-free optimal solution. This
results in a smoother $h_{t}$ curve, one that undershoots the larger
oscillations seen in the cost-free optimum in favor of a policy that changes
incrementally each year. Finally, the optimal policy for the third, ‘fixed
fee’ formulation ($\Pi_{3}$) only makes large adjustments, as one might
expect, because the magnitude of the adjustment made is not reflected in the
resulting cost.

These patterns are consistent across stochastic replicates over a range of
penalty magnitudes. The comparisons shown in Figure 3 are for one realization
(i.e. a particular sequence of random number draws representing environmental
variability) and are made for one particular magnitude of policy adjustment
costs, which have been calibrated to equal to 25% of $NPV_{0}({\bf
h_{0}^{*}})$. To demonstrate this, we solved for the optimal policy under each
of the three adjustment cost scenarios for 100 different $c_{i}$ coefficients.
For each resulting policy, we then simulated 500 stochastic replicates of the
stock dynamics managed under that policy. To summarize the patterns shown in
Figure 3 and characterize the impact of increasingly large fraction of the
economic value (NPV) being consumed by policy adjustment costs, we examine the
response of several summary statistics to increasing adjustment penalties (as
a fraction of the adjustment-free value) in Figure 4. We show the impact of
policy adjustment costs on the variance & autocorrelation of the harvests
through time.

Figure 4: Variance and autocorrelation of harvest dynamics, as a function of
the penalty coefficient. Adjustment penalty is calculated as percentage of the
maximum expected NPV in the basic case with no policy adjustment costs.

Despite summarizing across such a large ensemble, Figure 4 confirms the
patterns we observed in the individual realizations shown in Figure 3. For
example, when small policy adjustments cost little but large adjustments are
expensive ($\Pi_{2}$), we see the smoothing signal that we might have expected
(see also Ludwig (1980) for this particular case). As cost penalties increase
in severity (moving right along the horizontal axis, larger fractions of the
economic value are consumed by adjustment costs), the variance in quotas
through time decreases and the autocorrelation of quotas through time
increases. When adjustment costs are zero, all policies are slightly
negatively autocorrelated: this reflects the fact that a high harvest year is
usually followed by a low harvest year that allows the stocks to recover.
However, as adjustment costs increase, the quadratic policy tends to
undershoot when stocks jump to very high or low levels, adjusting to
compensate over multiple years, as seen in Figure 3. As a result,
autocorrelation increases.

Interestingly, as suggested by the realization in Fig. 3, including a fixed
cost ($\Pi_{1}$) of policy adjustment increases the variation in harvests
through time. This is the opposite of a smoothing effect.

Finally, the case where policy adjustment costs scale linearly with the size
of the adjustment ($\Pi_{1}$) appear to be something of a middle of the road
strategy, in that increasing the severity of policy adjustment costs. To
reveal the particular impact of policy adjustment costs of this type requires
a more targeted summary statistic. Specifically, for each run we calculated
the frequency with which the optimal policy involved maintaining a positive
quota across multiple time steps unaltered. This type of policy is arguably
the most commonly observed behavior in TAC management, but is one that is very
rarely observed to be part of the optimal management strategy in the basic
model without policy adjustment costs (Eqn. 3) or when optimizing against
$\Pi_{2}$ or $\Pi_{3}$.

Figure 4 shows these qualitative patterns still hold at when the models are
calibrated to a range of different reductions in NPV, from 0% to 30%, though
naturally the magnitude of the differences is greatest when the penalty is
larger. Note that the supplementary material provide further evidence that
these patterns are independent of the exact reduction shown by providing
results analogous to Figure 3 (S8, S9) and Figure 5 (S10, S11), and Figure 6
(S6).

While still not common for the particular parameter combinations we examine,
we find that positive unaltered quotas through time are much more likely to
occur when optimizing against $\Pi_{1}$ where policy adjustment costs scale
linearly with the size of proposed policy changes: over the 100 replicates
time series the harvest policy is strictly positive and identical in
consecutive intervals only 9.88% of the time for quadratic costs $\Pi_{2}$,
compared with 20.78% with linear costs $\Pi_{1}$ and 0.37% for fixed costs
$\Pi_{3}$. Moreover, these occurrences increase in frequency as the severity
of these costs ($c_{i}$) increases. Note that the restriction of only positive
quotas lets us distinguish between cases that are constant due purely to
adjustment costs from cases that are constant purely due to boundary effects.

### Consequences of policy adjustment costs

Figure 5: Distribution of revenues from fishing and costs paid to adjustment.
Plot shows distributions for the linear costs structure ($\Pi_{1}$) only,
alternate penalties can be found in the supplementary materials. Dockside
revenues are always higher than adjustment costs. When those costs are not
accounted for in the policy (‘no adjustment costs’), it is possible to obtain
only marginally higher revenues, but pay higher adjustment costs.

Next we examine the consequences either of ignoring policy adjustment costs
when they are present or assuming they are present when they are not.

To do so, we will simulate managing a fish stock when adjustments are costly.
We will perform 500 replicate simulations of this for each of the three cost
structures, $\Pi_{1}$, $\Pi_{2}$, & $\Pi_{3}$. For each replicate, we will use
two different management policies: one which ignores the adjustment costs,
(that is, the Reed optimum policy, $\Pi_{0}$), and a second which accounts for
the adjustment cost in place.

By comparing the NPV of this second scenario against the first, we can
quantify the impact of ignoring costs when they are present.

Figure 5 shows that the difference in dockside revenue (labeled “profits” in
the figure) between these two scenarios (green vs purple distributions) is
very small, while a larger difference exists between the costs paid for making
policy adjustments (red vs blue). This is predicted by the optimization: the
first scenario must outperform (or at least equal) the NPV of the second
scenario in aggregate, and as it cannot do so through higher profits (these
having already been maximized by the cost-free strategy), it must do so
through lower costs. Interestingly, for this set of parameters very little is
lost in revenue. This suggests that the impact of assuming costs are present
when they are not is rather small, as the revenue is nonetheless near optimal.
This also suggests that the converse error – ignoring costs when they are
present – is more severe.

Figure 6: Relative cost of ignoring adjustment costs when they are present
(‘ignoring’, red) vs assuming the adjustment costs when they are absent
(‘assuming’, blue). All values are relative to the cost free adjustment
optimum, $NPV_{0}$. In each case, it is less costly to assume adjustment costs
are present when they are not than it is to ignore them when they are
present.Supplementary Figure S6 shows how the magnitude of this effect changes
with the size of the reduction to $NPV$.

We illustrate the differences between these two errors directly in Figure 6.
Subtracting revenue from adjustment costs and considering only the mean across
the replicates, we can derive the expected $NPV$ in each case. To standardize
against a common baseline, we will express this value as a fraction of the
cost free optimum, $NPV_{0}$. Note that we have calibrated the optimal
solutions to anticipate that 25% of their $NPV$ would go into adjustment
costs. When these adjustment costs are not actually present (blue bars, Fig
6), we thus expect a $NPV$ higher than the anticipated 75% but not quite as
high as the optimal policy for that scenario (100%). When adjustment costs are
present but the policy assumes that they are not (red bars), we would expect
the $NPV$ to be less than 75% of $NPV_{0}$ (since that’s as good as the
optimal solution for this scenario can do).

Figure 6 shows that across each penalty form $\Pi_{1}$, $\Pi_{2}$ & $\Pi_{3}$,
the economic impact of ignoring adjustment costs when they are present is
higher than accounting for them when they are absent. This holds even after
accounting for effect described above: the red bars are consistently well
below the 75% of $NPV_{0}$, while the blue bars are barely below $NPV_{0}$ (a
value of 1 on the vertical axis). We also see that the impacts on net present
value of the fishery of incorrectly assuming or ignoring policy adjustment
costs are most severe when these costs are assumed to scale quadratically with
the size of the policy change being implemented ($\Pi_{2}$). In the
Supplementary Materials we show that pattern holds across different severity
of adjustment costs (robust to choice of $c_{i}$), though the larger the
adjustment cost, the greater the difference between these two errors.

Note that this analysis does not consider the possibility that an adjustment
cost is present, but of a different functional form than the policy assumes.
The consequences of managing under the wrong functional form can be as or more
severe than simply ignoring adjustment costs in the optimization altogether.
We illustrate several of these mismatch errors in the Supplementary Material.

## Discussion

Policymakers managing ecological systems that vary in space and time must
evaluate how much of that variation to reflect in management recommendations.
Finer tuning a policy to respond to frequent variations in ecological dynamics
may incur increased transaction costs associated with constantly revisiting
past policy decisions. A more pragmatic approach would be one that balances
benefits from responding to frequent variations in ecological conditions with
the increased transaction costs involved. As a first step towards exploring
these ideas, we revisited a classic problem from bioeconomics concerning the
optimal management of a fish stock subject to stochastically varying
recruitment (Reed 1979, Clark 2010). We examined how optimal policy
recommendations – here annual catch quotas – changed when accounting for costs
associated with policy adjustment and what the implications of following these
policy recommendations would be for the exploited population. We also compared
how the value of the fishery is affected by managers either under- or
overestimating the importance of policy adjustment costs of this type.

Estimating policy adjustment costs and how they respond to the size of
proposed policy changes would be empirically challenging. Recognizing this
fact, we compared different plausible forms that these costs might take. We
compared two cost structures where we assumed the magnitude of policy
adjustment costs increased with the magnitude of the change in policy being
proposed with an alternative formulation in which we assumed there was a fixed
cost associated with making any change to current policy. While each
formulation provides only very phenomenological representations of the policy-
setting environment, we believe that each enables us to explore meaningful
differences in how policy adjustment costs might operate. That being said, in
the real world we might expect the different types of adjustment cost to
operate in combination.

The biggest differences between the representations of policy adjustment cost
that we consider are between those that have a smoothing effect on annual
quotas and those that do not. The few past studies that incorporate policy
adjustment costs in models of fisheries or other environmental management
contexts (Feichtinger et al. 1994, Wirl 1999) have assumed costs of policy
adjustment increased as a quadratic function of the magnitude of the policy
change being proposed (analogous to $\Pi_{2}$ formulation). In effect, this
means that small changes to annual quotas incur little extra cost, but large
changes to annual quotas become disproportionately expensive to make.
Including costs of this form smooths inter-annual variation in the recommended
catch quotas. Also, the catches and remaining stock sizes corresponding to
optimal management become more autocorrelated in time, because it takes the
fishery multiple timesteps to harvest down peaks in abundance that follow
large recruitment pulses. The effects of smoothing here are similar to those
predicted when assuming the cost per unit effort is increasing in the amount
of effort expended (Brown 1974, Lewis 1981, McGough et al. 2009); (see
Supporting Information for a summary of the relevant results) as opposed to
associating extra costs with changes to policy _per se_. Smoothing effects of
this type are what one might have expected when including policy adjustment
costs.

Standing in sharp contrast to these smoothing predictions is our finding that
including policy adjustment costs can actually _increase_ the variability of
quotas through time if there is a fixed cost associated with making any
changes to current policy (e.g. costs of running relevant stakeholder meetings
and public consultations on proposed policy changes, $\Pi_{3}$). With this
formulation, as stock sizes vary in response to recruitment, the fishery
manager must balance the cost of brokering a policy change with the cost in
forgone fisheries revenue from not responding to favorable recruitment pulses.
The optimal policy involves ignoring small variations in recruitment, but then
assigns a larger quota when particularly strong recruitment years arise than
would have been the case in the absence of policy adjustment costs. The other
functional form we consider, in which the costs of policy adjustment scale
linearly with the size of the adjustment ($\Pi_{1}$), has less obvious effects
on optimal policies. Interestingly, it is this structure that most frequently
produces stretches of strictly positive but unchanging quotas of the type most
commonly encountered in real world applications.

That the different representations of policy adjustment costs result in such
different dynamics suggests that researchers constructing fisheries economics
models should proceed cautiously when choosing how to represent these costs.
However, in our own experience, we have found that while modeling studies
sometimes mention policy adjustment costs when motivating model assumptions
(Brown 1974, Lewis 1981, Feichtinger et al. 1994, Wirl 1999, McGough et al.
2009), they rarely provide much justification for the choice of functional
form used or test the sensitivity of any conclusions drawn to alternative
specifications. Indeed, we were surprised to find such clear differences
between the functional forms, because we anticipated that the limiting case of
increasing policy adjustment costs within each functional form should be the
same, namely a constant annual quota that does not change through time.

We also compared the efficiency costs that would result from failing to
account for policy adjustment costs, if they are present, with those involved
in assuming them when in fact they are absent. The results of this comparison
were not sensitive to the particular form of policy adjustment costs assumed.
Instead, we always found the efficiency costs of ignoring policy adjustment
costs when they were present to be much larger than the efficiency costs of
assuming policy adjustment costs applied when they were in fact absent. Policy
adjustment costs affect the overall value of the fishery in two ways here.
First, there is the direct cost associated with each quota change. Second
there is a cost in foregone revenue from missed catches when not following
what would be the optimal policy if quotas were free to track recruitment
variability. Our finding that it is more costly to ignore policy adjustment
costs when they are present arises because the first, more direct, cost
contribution here is the larger.

Assuming an adjustment cost exists when in fact it is absent is not the same
as using the wrong functional form when it is present. In the supplement we
show that choice of the functional form makes a significant difference. We
show it can be better to ignore policy costs than to derive a policy based on
assuming the wrong functional form. This result further underscores the
importance of modelers using caution when seeking to account for these costs.
Assuming an arbitrary form in order to capture the influence of adjustment
costs is thus unlikely to be instructive.

As with any modeling analysis, our formulation makes many assumptions. For
example, we assume that the fishery in question is being optimally managed by
a policymaker who acts as the “sole owner” of the stock. This approach is
different to models that assume fisheries are not well managed, e.g. by
assuming regulated open access conditions (Homans and Wilen 1997), or models
that derive policy recommendations endogenously by modeling strategic
interactions between different stakeholders (Kaitala 1993, Laukkanen 2003). As
such, we anticipate that our approach will be more relevant to some fisheries,
particularly domestic fisheries in developed countries that are subject to
relatively strong regulatory regimes, than to others (e.g., artisanal
fisheries that are subject to weaker regulation). To focus on the effects of
policy adjustment costs, we focused narrowly on the basic model specification
of Reed (1979) and examined how the predictions of this classic problem were
changed by introducing policy adjustment costs of different forms. However,
there have been many elaborations on Reed’s basic approach that increase the
realism of the optimization models involved by relaxing other assumptions (see
for example Sethi et al. (2005); Singh et al. (2006); McGough et al. (2009)).

One obvious research avenue suggested by our models is that of empirically
estimating costs of policy adjustment. A direct estimation approach could
quantify some sources of policy adjustment costs, e.g. costs to processing
plants that arise from having more variable catches. However, other more
intangible sources of policy adjustment costs, e.g. preferences of
policymakers or different stakeholders for less variable quotas, might be
missed. An alternative, more holistic, approach would be to apply revealed
preference methods to fisheries management agencies themselves, in the
tradition of (McFadden 1975, 1976). Such an analysis would involve comparing
quotas that were set relative to stock sizes as they were estimated at the
time each management decision was taken to try to infer what objective
managers were maximizing.

A worthwhile modeling extension suggested by our results would be to examine
the implications of policy adjustment costs for risks of stock collapse. This
includes models with larger stochasticity as well as model formulations that
can permit alternative stable states. Examining risks of stock collapse is not
possible with the Reed model formulation that we followed here (both cases are
inconsistent with the assumptions of Reed (1974)) but would be worthwhile in
light of our finding that the variance and autocorrelation in stock sizes
through time can be affected when accounting for costs of policy adjustment.
Both are properties known in other modeling settings to be associated with
changes to the risk of extinction or of transitions between alternative stable
states (Scheffer et al. 2009). This highlights the risk that changes in
management policies can either mimic or mask such possible early warning signs
of sudden transitions, and points to another way in which more context is
needed before such approaches can be useful to management (Boettiger et al.
2013).

When facing highly variable ecological systems, how often should natural
resource managers respond? A highly interventionist strategy would track
ecosystem variation very closely. Alternatively, a manager might choose only
to take action or change policy only when conditions look very different to
those previously experienced. We took a modeling approach to begin to explore
these ideas. Specifically, we focused on a well-known problem from fisheries
management and examined how optimal management recommendations changed when we
accounted for costs associated with frequently changing management decisions.
While we focused on a fisheries context, our findings would be relevant to
many other settings where natural resource managers revisit management
decisions through time in light of ecological variability, including game
management, management of instream flow rates, fire management, habitat
restoration, and managing for endangered species. The analyses that we present
can also be thought of as providing a temporal counterpart to discussions
about the spatial scale over which ecosystem management.

## Acknowledgements

This work was assisted through participation in “Pretty Darn Good Control”
Investigative Workshop at the National Institute for Mathematical and
Biological Synthesis, sponsored by the National Science Foundation through NSF
Award #DBI-1300426, with additional support from The University of Tennessee,
Knoxville. The authors acknowledge helpful discussions and input from working
group co-organizers Megan Donahue and Carl Toews, and participants Marie-Josee
Fortin, Dan Ryan, Frank Doyle, Claire Paris, Iadine Chades and Mandy
Karnauskas, as well as the two anonymous reviewers whose feedback strengthened
this paper. We also acknowledge the support of NSF Grant DBI-1306697 to CB and
ARC DECRA (DE130100572) & the ARC Centre of Excellence for Environmental
Decisions to MB.

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Astronomy Data Centre (CADC/NRC/CSA). This research has made use of the
NASA/IPAC Extragalactic Database (NED), which is funded by the National
Aeronautics and Space Administration and operated by the California Institute
of Technology. We acknowledge the usage of the HyperLeda database
(http://leda.univ-lyon1.fr). This research made use of NumPy (Harris et al.,
2020), SciPy (Virtanen et al., 2020), SymPy (Meurer et al., 2017), and
Astropy,151515http://www.astropy.org a community-developed core Python package
for Astronomy (Astropy Collaboration et al., 2013, 2018). This research was
(partially) based on data from the MILES project. This research has made use
of the integral-field spectroscopy data-reduction tool P3D, which is provided
by the Leibniz-Institut für Astrophysik Potsdam (AIP).

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##  Appendix A Could the JWST be used to find the black hole masses of relic
progenitors at $z\gtrsim 1$?

We have found that the mass of the black hole sitting at the centre of NGC
1277 is $M_{\rm BH}=\left(4.70\pm 0.11\right)\times 10^{9}\,{\rm M}_{\odot}$.
This very precise determination was made using the NIFS data from Walsh et al.
(2016, our cA1 fit for NGC 1277). Surprisingly, if we use the GCMS data only
we obtain nearly the same result with a relatively small one-sigma uncertainty
of 20% ($M_{\rm BH}=\left(4.9\pm 0.9\right)\times 10^{9}\,{\rm M}_{\odot}$; cC
fit). The results are broadly consistent for all the fits made assuming a
cylindrically-aligned velocity ellipsoid, irrespective of whether we use the
data from NIFS, from GCMS, or both datasets.

One might wonder whether these concordant black hole mass values between fits
resorting to datasets with a very different angular resolution is merely a
coincidence. The fact that by using adaptive optics or the JWST we can now
obtain integral field data of the progenitor of a massive ETG at $z\gtrsim 1$
with a physical resolution comparable to that given by the GCMS for NGC 1277
makes it worth to study this issue a bit further.

A possible explanation for the GCMS data alone being able to constrain the
black hole mass is that some high angular resolution information is introduced
through MGE parametrisations of HST images. In order to simulate the MGEs of a
galaxy at high redshift seen through the JWST, we first need to degrade the
HST images.

In a standard cosmology, at $z=1.2$ one arcsecond corresponds to roughly 8.4
kpc (compared to 332 pc at the distance of NGC 1277). To reproduce how NGC
1277 would look at $z=1.2$ seen through the JWST, we degraded the HST F475W,
F625W, and F850LP images by reducing their side by a factor 16 (that is, by
coadding the flux from a box of 16 pixels by 16 pixels to generate a single
pixel). The factor 16 comes from the difference in angular size ($8400\,{\rm
pc/332}\,{\rm pc}\approx 25$) and the $0\aas@@fstack{\prime\prime}05$ versus
$0\aas@@fstack{\prime\prime}031$ difference in pixel size between the HST ACS
and the JWST Near Infrared Camera (NIRCam; Rieke et al. 2005). Then, we
convolved the images with a JWST PSF simulated by WebbPSF (Perrin et al.
2014). Given the wavelength shift for a $z=1.2$ object, we convolved the F625W
image with a F140M PSF. The mass map is built from the F475W and the F850LP
filters (Sect. 3.2) that, given the redshift, would roughly correspond to the
F115M and F182M filters, respectively. However, the PSF mismatch between these
two filters is rather large and introduces artefacts in the colour map used to
compute the mass map. This is why, we convolved the two filters with the F182M
PSF, so we simulated a PSF matching. Effectively, the resampling and
convolution corresponds to a degradation of the width of the core of the PSF
of a factor of $\sim 15$.

We produced a mask based on the degraded images, hence missing small stars and
the circumnuclear dust ring that are obvious in the original HST images. The
degraded F625W image and the mass map were parametrised with the MGEs listed
in Tables 5 and 6, respectively.

Table 5: Parameters of the MGE decompositions of the degraded surface
brightness map of NGC 1277

${\rm log}\,I_{0}\,({\rm log}\,({\rm L}_{\odot}\,{\rm pc^{-2}}))$ | $\sigma_{\star}\,({\rm arcsec})$ | $q^{\prime}_{\star}$
---|---|---
4.3 | 0.356 | 0.650
3.6 | 1.129 | 0.801
3.2 | 2.517 | 0.500
2.8 | 5.547 | 0.500
2.0 | 12.816 | 0.512
1.4 | 12.816 | 1.000
161616$I_{0}$ stands for the central surface brightness of the projected
Gaussian components, $\sigma_{\star}$ denotes their dispersion, and
$q^{\prime}_{\star}$ their projected axial ratio.

Table 6: Parameters of the MGE decompositions of the degraded surface mass
density map of NGC 1277

${\rm log}\,\Sigma_{0}\,({\rm log}\,({\rm M}_{\odot}\,{\rm pc^{-2}}))$ | $\sigma_{\star}\,({\rm arcsec})$ | $q^{\prime}_{\star}$
---|---|---
4.8 | 0.731 | 0.579
4.1 | 2.189 | 0.600
2.7 | 5.542 | 0.558
3.4 | 5.858 | 0.450
2.7 | 13.023 | 0.600
171717Same as Table 5, but with $\Sigma_{0}$ indicating the projected central
surface mass density of the Gaussian components.

Figure 22:  Major- (continuous line and coordinate $x$) and minor-axis (dashed
line and coordinate $y$) surface brightness profiles of NGC 1277. Black lines
come from our original MGE parametrisation (Fig. 8 and Sect. 3.2) and grey
ones were obtained from HST data that were degraded to a resolution
corresponding to that of a galaxy at $z=1$. Figure 23:  As Fig. 22, but for
the surface density profiles. As in Fig. 9, a Salpeter IMF was assumed. Figure
24:  Same as Fig. 11, but this time we also fitted $M_{\rm BH}$ and we used
MGE parametrisations obtained from degraded HST images so to simulate how NGC
1277 would look like at $z=1$ (fit cCd).

The total luminosity of NGC 1277 as obtained from the MGE decomposition of the
degraded image is $L=2.1\times 10^{10}\,L_{\odot r}$ (the same that was found
for the original image). The total mass obtained from the degraded mass map is
$M=1.1\times 10^{11}\,{\rm M}_{\odot}$ (versus $M=1.3\times 10^{11}\,{\rm
M}_{\odot}$ for the original map before applying the correction for the
mismatch parameter).

In Figs. 22 and 23 we show the surface brightness and surface density profiles
obtained from the degraded data and we compare them to those obtained from the
original HST data (Figs. 8 and 9). Unsurprisingly, the density peaks within
the central arcsecond are washed out by the image degradation.

Table 7: JAM model fit parameters for NGC 1277 with the MGE parametrisation
obtained from degraded HST data simulating JWST observations of a galaxy at a
redshift $z=1.2$

|  | Fitting approach | $q_{\star\,{\rm min}}$ | $\beta_{z}$ | $\beta_{z,{\rm out}}$ | $\alpha$ | $M_{\rm BH}$ | $f_{\rm DM}(6\,{\rm kpc})$
---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  | $\left(10^{9}\,{\rm M}_{\odot}\right)$ | $f_{\rm DM}(5\,{R_{\rm e}})$
Cylindric | Single $\beta_{z}$ | cCd GCMS all free | $0.29\pm 0.05$ | $0.14\pm 0.08$ | $-$ | $1.60\pm 0.04$ | $11.7\pm 1.1$ | $<0.02$

181818Fit cCd is shown in Fig. 24. The errors and upper limits correspond to
formal one sigma uncertainties.

We produced Jeans models of NGC 1277 as in Sect. 4, but this time using the
MGEs obtained from degraded data. Since we want to determine the black hole
mass, we did a fit akin cC in Table 3, that is with a free $M_{\rm BH}$, a
cylindrical velocity ellipsoid, and no variation in $\beta_{z}$ with radius.
We call this model cCd (‘d’ stands for degraded). The fit is shown in Fig. 24
and the fitted parameters are listed in Table 7.

Once again, we find a negligible dark matter fraction within 6 kpc. This
stresses the robustness of our dark matter halo caracterisation. Perhaps the
most surprising result is the detection of an extremely massive black hole,
with $M_{\rm BH}=(1.17\pm 0.11)\times 10^{10}\,{\rm M}_{\odot}$. This mass is
about two times larger than that found in Sect. 4.1. The mismatch parameter
that we find here, $\alpha\approx 1.6$, is also significantly larger than in
Sect. 4.1 ($\alpha\approx 1.2$) to compensate for the lower central surface
mass density obtained from the degraded data when compared to the original HST
data.

Why is the black hole mass so large when using low angular resolution data?
Accounting for the mismatch parameter, the stellar mass within the central kpc
of NGC 1277 is $M_{\star}(1\,{\rm kpc})=5.8\times 10^{10}\,{\rm M}_{\odot}$
for the original MGE parametrisation. But this goes down to only
$M_{\star}(1\,{\rm kpc})=4.2\times 10^{10}\,{\rm M}_{\odot}$ when using the
degraded data. The central $V_{\rm rms}$ peak is the consequence of both a
black hole and a peaked stellar distribution. If the peak in the stellar
distribution is smoothened or erased, the algorithm assigns a larger mass to
the black hole to compensate for the loss of the stellar cusp. Hence, black
hole mass estimates obtained from insufficiently sharp data can only yield
upper boundaries to the actual values.

Our test underlines the importance of properly characterising the central
stellar distribution to be able to give more than an upper limit to the mass
of the central black hole. Using images with the JWST resolution concomitantly
with a deconvolution method such as the Lucy-Richardson algorithm (Richardson
1972; Lucy 1974) it should be possible to recover information about the
central stellar peak. Hence, such techniques should allow obtaining reasonable
black hole mass estimates for galaxies with a central velocity dispersion
cusp.