In cboettig/pdg_control: Pretty Darn Good Control
library("methods")
library("knitr")
basename <- "manuscript"
opts_chunk$set(fig.path = paste("components/figure/", basename, "-", sep=""),
cache.path = paste("components/cache/", basename, "/", sep=""))
opts_chunk$set(cache = 2)
opts_chunk$set(tidy=FALSE, warning=FALSE, message=FALSE,
comment = NA, verbose = TRUE, echo=FALSE)
# PDF-based figures
opts_chunk$set(dev='pdf')
fig.cache <- TRUE
library("rmarkdown")
library("pdgControl")
library("reshape2")
library("plyr")
library("ggplot2")
library("data.table")
library("pander")
library("cboettigR")
library("ggthemes")
library("snowfall")
library("tidyr")
library("dplyr")
library("gtable")
library("gridExtra")
library("parallel")
options(digits=2)
theme_set(theme_tufte(base_size=12) + theme(axis.line = element_line()))
parallel = TRUE
ncpu = parallel::detectCores()
price = 10
c0 = 30
profit <- profit_harvest(price = price, c0 = c0, c1 = 0)
#c2 <- exp(seq(0, log(41), length.out = 40))-1
c2 <- seq(0, 40, length.out=100)
reduction <- 0.25
seed <- 123 # Random seed (replicable results)
delta <- 0.05 # economic discounting rate
OptTime <- 50 # stopping time
gridsize <- 50 # grid size for fish stock and harvest rate (discretized population)
sigma_g <- 0.2 # Noise in population growth
reward <- 0 # bonus for satisfying the boundary condition
z_g <- function() rlnorm(1, 0, sigma_g) # mean 1
z_m <- function() 1 # No measurement noise,
z_i <- function() 1 # No implemenation noise
f <- BevHolt # Select the state equation
pars <- c(1.5, 0.05) # parameters for the state equation
K <- (pars[1] - 1)/pars[2] # Carrying capacity (for reference
xT <- 0 # boundary conditions
x0 <- K
x_grid <- seq(0.01, 1.2 * K, length = gridsize)
h_grid <- seq(0.01, 0.8 * K, length = gridsize)
source("components/analysis.R")
Introduction
Ecosystems are dynamic and exhibit rich patterns of variability in
both time and space [@Durrett1994]. 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., @Reed1979; @Neubert2003; @Sethi2005; @Halpern2011]. However,
the policy process can often be much more sluggish
to respond to variations in ecosystem dynamics [@Walters1978; @Armsworth2010].
Moreover, stakeholders impacted by ecosystem management
may prefer some stability and not want to deal with continually changing
management recommendations [@Biais1995; @Armsworth2003;
@Patterson2007a; @Patterson2007b; @Sanchirico2008].
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.
tuna <- read.csv("components/data/tuna.csv")
grouper <- read.csv("components/data/grouper.csv")
library("scales")
ex <- rbind(cbind(tuna, species="tuna"), cbind(grouper, species="snowy grouper"))
ex_long <- gather(ex, variable, value, stock, quota)
ggplot(ex_long, aes(year,value)) +
geom_line() + geom_point() +
facet_wrap(variable~species, scales="free") +
ylab("value (mt)") + theme_bw() +
scale_x_continuous(breaks = pretty_breaks(n = 5))
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 [@ICCAT2009a]. 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 [@ICCAT2009a]. Despite the estimated population size
declining by r 100 * (1 - min(tuna$stock) / max(tuna$stock))% 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 (@Safina1998, @Sissenwine1998,
@Porch2005). 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 [@SEDAR2013]. 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 @Biais1995 and
@Patterson2007a 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
@Ludwig1980, @Feichtinger1994, @Wirl1999). 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. @Sethi2005; @Singh2006; @McGough2009),
we start from the classical model for simplicity of presentation and analysis.
With his formulation, @Reed1979 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
\begin{equation}
N_{t+1} = Z_t \frac{A (N_t - h_t)}{1 + B (N_t - h_t)}, \label{eq:state_equation}
\end{equation}
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
\begin{equation}
\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) \label{eq:objective}
\end{equation}
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
\begin{equation}
\Pi_0(N_t,h_t) = p h_t - c_0 E_t \label{eq:annual_netrevenue}
\end{equation}
where $E_t$ represents fishing effort. We assume catch is proportional
to stock size and effort expended fishing that year, $h_t = q E_t N_t$ and
constant $q>0$ is the catchability coefficient. In
Eqn. \eqref{eq:annual_netrevenue}, $p$ 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 r (pars[1]-1)/pars[2]; (specifically, $A=$ r pars[1] and
$B=$ r pars[2] in Eqn \eqref{eq:state_equation}). To characterize environmental variability, we
assume multiplicative shock $Z_t$ is distributed log-normally with
log standard deviation $\sigma_g$ = r sigma_g. In addition, we
show cases where $p=$ r price, $\delta=$ r delta and $c_0=$ r c0.
In Appendix A we illustrate that the qualitative patterns observed
here are not sensitive to the specific choices of these parameter values.
Taken together this objective function \eqref{eq:objective} and the state
equation \eqref{eq:state_equation} 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 $T =$ r OptTime using
value iteration [@Mangel1988; @Clark2000].
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, @zenodo_copy), along with scripts for replicating
the analyses presented here. To address concerns about computational reproducibility [@Boettiger2015],
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
\begin{equation}
\Pi_{1}(N_t,h_t, h_{t-1}) = \Pi_0 - c_1 | h_t - h_{t-1} | \,.
\label{eq:Pi_1}
\end{equation}
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:
\begin{equation}
\Pi_{2}(N_t,h_t, h_{t-1}) = \Pi_0 - c_2 ( h_t - h_{t-1})^2 \,.
\label{eq:Pi_2}
\end{equation}
Finally, we assume instead that there is a fixed cost associated with
any adjustment of policy, regardless of how large that adjustment might
be
\begin{equation}
\Pi_{3}(N_t,h_t, h_{t-1}) = \Pi_0 - c_3 (1-\mathbf{I}(h_t, h_{t-1})) \,,
\label{eq:Pi_3}
\end{equation}
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. \eqref{eq:objective}.
We note that the fixed cost has conceptual analogues to the set-up cost of
@Reed1974 and @Spulber1982 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. \eqref{eq:state_equation}, 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 Appendix A, 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: $\Pi_4 (N_t,h_t) = p h_t - c_0 E_t-c_4E_t^2$. 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.
```r{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.", dependson="quadcosts", fig.width=5, fig.height=3, cache = fig.cache, dev=c("pdf", "svg", "png")}
relabel <- c(L1 = substitute(paste(Pi[1])),
L2 = substitute(paste(Pi[2])),
fixed = substitute(paste(Pi[3])))
ggplot(fees, aes(c2, (npv0-value)/npv0, lty=variable, color=variable)) +
geom_line() +
geom_hline(aes(yintercept=reduction), linetype=4) +
xlab("Penalty coefficient") +
ylab("Reduction in net present value") +
scale_linetype_discrete(labels=relabel, breaks = c('L1', 'L2', 'fixed'), name = 'cost model') +
scale_color_discrete(labels=relabel, breaks = c('L1', 'L2', 'fixed'), name = 'cost model') +
annotate("text", mean(c2), reduction+.02, label = "25% reduction in NPV")
write.csv(fees, "components/data/figure2.csv")
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 `r 100*(1-reduction)`%
of its unconstrained value when optimally managed. To demonstrate that the results
shown here are independent of this choice of calibrating to a `r 100*(1-reduction)`%
reduction, we conduct the same analysis with a 10% and 30% reduction rate as well.
The corresponding figures can be found in Appendix A.
Results
=======
Effect of policy adjustment costs on optimal quotas and stock sizes
-------------------------------------------------------------------
```r
labeller <- function(variable,value){
return(relabel[paste(value)])
}
dt3 <- dt %>%
filter(replicate=="rep_17" & time < OptTime) %>%
select(time, harvest, harvest_alt, penalty_fn) %>%
gather(variable, value, -time, -penalty_fn)
dt3$penalty_fn <- factor(dt3$penalty_fn, levels=c("L1", "L2", "fixed"))
ggplot(dt3, aes(time, value, col=variable, lty=variable)) +
geom_line(lwd=1) +
facet_grid(penalty_fn~., labeller = labeller) +
labs(x="time", y="harvest", title = "Example Harvest Dynamics") +
scale_color_discrete(labels=c("Adjustment penalty", "No penalty"), name="") +
scale_linetype_discrete(labels=c("Adjustment penalty", "No penalty"), name="") +
theme_bw()
write.csv(dt3, "components/data/figure3.csv")
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 Appendix A. 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 r 100*reduction% 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 r reduction *100% of $NPV_0({\bf h_0^*})$. To demonstrate this,
we solved for the optimal policy under each of the three adjustment
cost scenarios for r length(c2) different $c_i$ coefficients. For each
resulting policy, we then simulated r length(reps) 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.
stats_df %>%
filter(variable != "cross.correlation") %>%
separate(variable, c("measurement", "statistic"), sep="\\.") %>%
filter(measurement == 'harvest') %>%
spread(statistic, value) ->
f4df
f4df$penalty_fn <- factor(f4df$penalty_fn, levels = c("L1", "L2", "fixed"))
p4 <- ggplot(f4df, aes(x = penalty_fraction, fill=penalty_fn, col=penalty_fn)) +
coord_cartesian(xlim = c(0, .3)) +
scale_color_discrete(labels=relabel, name="cost model") +
scale_fill_discrete(labels=relabel, name="cost model")
p4a <- p4 +
stat_summary(aes(y=autocorrelation),
fun.y = mean,
fun.ymin = function(x) mean(x) - sd(x),
fun.ymax = function(x) mean(x) + sd(x),
geom = "ribbon", alpha=0.3, colour=NA) +
stat_summary(aes(y=autocorrelation), fun.y = mean, geom = "line") +
xlab("")
p4b <- p4 +
stat_summary(aes(y=variance),
fun.y = mean,
fun.ymin = function(x) mean(x) - sd(x),
fun.ymax = function(x) mean(x) + sd(x),
geom = "ribbon", alpha=0.3, colour=NA) +
stat_summary(aes(y=variance), fun.y = mean, geom = "line") +
xlab(expression(paste("Penalty size (as fraction of ", NPV[0], ")")))
legend <- gtable_filter(ggplot_gtable(ggplot_build(p4a)), "guide-box")
grid.arrange(do.call(arrangeGrob,
lapply(list(p4a, p4b),
function(x) x + theme_bw() + theme(legend.position="none"))),
legend,
ncol = 2,
widths = c(5,2))
write.csv(f4df, "components/data/figure4.csv")
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 @Ludwig1980 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 results shown in Appendix A provide
further evidence that these patterns are independent of the exact reduction shown by providing
results analogous to Figure 3 (A8, A9) and Figure 5 (A10, A11), and Figure 6 (A6).
fraction_no_shift <- function(x){
noshift <- x[2:length(x)] - x[1:length(x)-1] == 0
strictly_positive <- (x > 0.01 )[-1] # drop first point,
who <- noshift & strictly_positive # True whenever consecutive pairs did not shift and the second in the pair was nonzero harvest
sum(who)/length(who)
}
shifts <- dt[, fraction_no_shift(harvest), by=c("penalty_fn", "replicate")]
ave <- shifts[, mean(V1), by=penalty_fn]
no_shifts <- ave$V1
names(no_shifts) <- ave$penalty_fn
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 r no_shifts[["L2"]] * 100% of the time for quadratic costs $\Pi_2$, compared with
r no_shifts[["L1"]] * 100% with linear costs $\Pi_1$ and r no_shifts[["fixed"]]*100%
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
profits <- dt[, sum(profit_fishing), by=c('penalty_fn', 'replicate') ]
costs <- dt[, sum(policy_cost), by=c('penalty_fn', 'replicate') ]
reed_profits <- dt[, sum(profit_fishing_alt), by=c('penalty_fn', 'replicate') ]
reed_costs <- dt[, sum(policy_cost_alt), by=c('penalty_fn', 'replicate') ]
setnames(profits, "V1", "profits")
setnames(reed_profits, "V1", "profits")
Reed <- cbind(reed_profits, costs = reed_costs$V1, Assumption = "No adjustment penalty")
Adj <- cbind(profits, costs = costs$V1, Assumption = "Adjustment penalty")
hist_dat <- melt(rbind(Adj, Reed), id=c("penalty_fn", "replicate", "Assumption"))
````
```r
fig5_caption <- paste("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 Appendix A. 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.")
fig5_dat <- hist_dat %>%
filter(penalty_fn=="L1") %>%
select(-penalty_fn) %>%
unite(dist, Assumption, variable, sep = " : ")
labels <- c(expression(paste("Costs under ", Pi[1])),
expression(paste("Revenue under ", Pi[1])),
expression(paste("Cost under ", Pi[0])),
expression(paste("Revenue under ", Pi[0])))
ggplot(fig5_dat) +
geom_density(aes(value, fill=dist, color=dist, lty=dist), alpha=0.5, size=1) +
scale_fill_discrete(name="", labels=labels) +
scale_color_discrete(labels=labels, name="") +
scale_linetype_discrete(labels=labels, name="") +
xlab("Realized NPV") +
ylab("Probability density of revenue/cost")
write.csv(hist_dat, "components/data/figure5.csv")
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 r length(reps) 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.
who <- c("penalty_fn", "ignore_fraction", "assume_fraction", "reduction")
table1 <- arrange(error_df[who], reduction)
names(table1) = c("penalty.fn", "ignoring", "assuming", "reduction")
table1_long <- melt(table1, id = c('penalty.fn', 'reduction'))
table1_long$reduction <- as.factor(table1_long$reduction)
table1_long <- subset(table1_long, reduction != "0.3")
table1_long$penalty.fn <- factor(table1_long$penalty.fn, levels = c("L1", "L2", "fixed"))
ggplot(filter(table1_long, reduction == 0.25),
aes(penalty.fn, value, fill = variable)) +
geom_bar(stat="identity", position="dodge") +
scale_fill_discrete(name="error type") +
scale_x_discrete(labels = c('L1' = substitute(paste(Pi[1])),
'L2' = substitute(paste(Pi[2])),
'fixed' = substitute(paste(Pi[3])))) +
xlab("Cost model") +
ylab(expression(NPV[0]))
write.csv(table1_long, "components/data/figure6.csv")
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 r reduction*100% 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 r (1 -reduction)*100% 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 r (1 - reduction) * 100% 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 r (1-reduction)*100% 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 Appendix A 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 Appendix A.
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 [@Reed1979; @Clark2010].
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 [@Feichtinger1994; @Wirl1999] 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 [@Brown1974;
@Lewis1981; @McGough2009]; (see Appendix A 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 [@Brown1974; @Lewis1981; @Feichtinger1994; @Wirl1999; @McGough2009],
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 Appendix A
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 [@Homans1997],
or models that derive policy recommendations endogenously by modeling strategic
interactions between different stakeholders [@Kaitala1993; @Laukkanen2003].
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 @Reed1979 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 @Sethi2005; @Singh2006; @McGough2009).
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 [@McFadden1975; @McFadden1976]. 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 @Reed1974) 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 [@Scheffer2009]. 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 [@Boettiger2013].
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.
References
cboettig/pdg_control documentation built on May 13, 2019, 2:10 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
library("methods") library("knitr") basename <- "manuscript" opts_chunk$set(fig.path = paste("components/figure/", basename, "-", sep=""), cache.path = paste("components/cache/", basename, "/", sep="")) opts_chunk$set(cache = 2) opts_chunk$set(tidy=FALSE, warning=FALSE, message=FALSE, comment = NA, verbose = TRUE, echo=FALSE) # PDF-based figures opts_chunk$set(dev='pdf') fig.cache <- TRUE library("rmarkdown") library("pdgControl") library("reshape2") library("plyr") library("ggplot2") library("data.table") library("pander") library("cboettigR") library("ggthemes") library("snowfall") library("tidyr") library("dplyr") library("gtable") library("gridExtra") library("parallel") options(digits=2) theme_set(theme_tufte(base_size=12) + theme(axis.line = element_line()))
parallel = TRUE ncpu = parallel::detectCores() price = 10 c0 = 30 profit <- profit_harvest(price = price, c0 = c0, c1 = 0) #c2 <- exp(seq(0, log(41), length.out = 40))-1 c2 <- seq(0, 40, length.out=100) reduction <- 0.25 seed <- 123 # Random seed (replicable results) delta <- 0.05 # economic discounting rate OptTime <- 50 # stopping time gridsize <- 50 # grid size for fish stock and harvest rate (discretized population) sigma_g <- 0.2 # Noise in population growth reward <- 0 # bonus for satisfying the boundary condition z_g <- function() rlnorm(1, 0, sigma_g) # mean 1 z_m <- function() 1 # No measurement noise, z_i <- function() 1 # No implemenation noise f <- BevHolt # Select the state equation pars <- c(1.5, 0.05) # parameters for the state equation K <- (pars[1] - 1)/pars[2] # Carrying capacity (for reference xT <- 0 # boundary conditions x0 <- K x_grid <- seq(0.01, 1.2 * K, length = gridsize) h_grid <- seq(0.01, 0.8 * K, length = gridsize)
source("components/analysis.R")
Introduction
Ecosystems are dynamic and exhibit rich patterns of variability in both time and space [@Durrett1994]. 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., @Reed1979; @Neubert2003; @Sethi2005; @Halpern2011]. However, the policy process can often be much more sluggish to respond to variations in ecosystem dynamics [@Walters1978; @Armsworth2010]. Moreover, stakeholders impacted by ecosystem management may prefer some stability and not want to deal with continually changing management recommendations [@Biais1995; @Armsworth2003; @Patterson2007a; @Patterson2007b; @Sanchirico2008]. 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.
tuna <- read.csv("components/data/tuna.csv") grouper <- read.csv("components/data/grouper.csv") library("scales") ex <- rbind(cbind(tuna, species="tuna"), cbind(grouper, species="snowy grouper")) ex_long <- gather(ex, variable, value, stock, quota) ggplot(ex_long, aes(year,value)) + geom_line() + geom_point() + facet_wrap(variable~species, scales="free") + ylab("value (mt)") + theme_bw() + scale_x_continuous(breaks = pretty_breaks(n = 5))
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 [@ICCAT2009a]. 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 [@ICCAT2009a]. Despite the estimated population size
declining by r 100 * (1 - min(tuna$stock) / max(tuna$stock))% 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 (@Safina1998, @Sissenwine1998,
@Porch2005). 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 [@SEDAR2013]. 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 @Biais1995 and @Patterson2007a 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 @Ludwig1980, @Feichtinger1994, @Wirl1999). 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. @Sethi2005; @Singh2006; @McGough2009), we start from the classical model for simplicity of presentation and analysis.
With his formulation, @Reed1979 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
\begin{equation} N_{t+1} = Z_t \frac{A (N_t - h_t)}{1 + B (N_t - h_t)}, \label{eq:state_equation} \end{equation}
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
\begin{equation} \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) \label{eq:objective} \end{equation}
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
\begin{equation} \Pi_0(N_t,h_t) = p h_t - c_0 E_t \label{eq:annual_netrevenue} \end{equation}
where $E_t$ represents fishing effort. We assume catch is proportional to stock size and effort expended fishing that year, $h_t = q E_t N_t$ and constant $q>0$ is the catchability coefficient. In Eqn. \eqref{eq:annual_netrevenue}, $p$ 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 r (pars[1]-1)/pars[2]; (specifically, $A=$ r pars[1] and
$B=$ r pars[2] in Eqn \eqref{eq:state_equation}). To characterize environmental variability, we
assume multiplicative shock $Z_t$ is distributed log-normally with
log standard deviation $\sigma_g$ = r sigma_g. In addition, we
show cases where $p=$ r price, $\delta=$ r delta and $c_0=$ r c0.
In Appendix A we illustrate that the qualitative patterns observed
here are not sensitive to the specific choices of these parameter values.
Taken together this objective function \eqref{eq:objective} and the state
equation \eqref{eq:state_equation} 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 $T =$ r OptTime using
value iteration [@Mangel1988; @Clark2000].
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, @zenodo_copy), along with scripts for replicating the analyses presented here. To address concerns about computational reproducibility [@Boettiger2015], 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
\begin{equation} \Pi_{1}(N_t,h_t, h_{t-1}) = \Pi_0 - c_1 | h_t - h_{t-1} | \,. \label{eq:Pi_1} \end{equation}
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:
\begin{equation} \Pi_{2}(N_t,h_t, h_{t-1}) = \Pi_0 - c_2 ( h_t - h_{t-1})^2 \,. \label{eq:Pi_2} \end{equation}
Finally, we assume instead that there is a fixed cost associated with any adjustment of policy, regardless of how large that adjustment might be
\begin{equation} \Pi_{3}(N_t,h_t, h_{t-1}) = \Pi_0 - c_3 (1-\mathbf{I}(h_t, h_{t-1})) \,, \label{eq:Pi_3} \end{equation}
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. \eqref{eq:objective}. We note that the fixed cost has conceptual analogues to the set-up cost of @Reed1974 and @Spulber1982 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. \eqref{eq:state_equation}, 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 Appendix A, 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: $\Pi_4 (N_t,h_t) = p h_t - c_0 E_t-c_4E_t^2$. 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.
```r{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.", dependson="quadcosts", fig.width=5, fig.height=3, cache = fig.cache, dev=c("pdf", "svg", "png")}
relabel <- c(L1 = substitute(paste(Pi[1])), L2 = substitute(paste(Pi[2])), fixed = substitute(paste(Pi[3])))
ggplot(fees, aes(c2, (npv0-value)/npv0, lty=variable, color=variable)) + geom_line() + geom_hline(aes(yintercept=reduction), linetype=4) + xlab("Penalty coefficient") + ylab("Reduction in net present value") + scale_linetype_discrete(labels=relabel, breaks = c('L1', 'L2', 'fixed'), name = 'cost model') + scale_color_discrete(labels=relabel, breaks = c('L1', 'L2', 'fixed'), name = 'cost model') + annotate("text", mean(c2), reduction+.02, label = "25% reduction in NPV")
write.csv(fees, "components/data/figure2.csv")
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 `r 100*(1-reduction)`% of its unconstrained value when optimally managed. To demonstrate that the results shown here are independent of this choice of calibrating to a `r 100*(1-reduction)`% reduction, we conduct the same analysis with a 10% and 30% reduction rate as well. The corresponding figures can be found in Appendix A. Results ======= Effect of policy adjustment costs on optimal quotas and stock sizes ------------------------------------------------------------------- ```r labeller <- function(variable,value){ return(relabel[paste(value)]) } dt3 <- dt %>% filter(replicate=="rep_17" & time < OptTime) %>% select(time, harvest, harvest_alt, penalty_fn) %>% gather(variable, value, -time, -penalty_fn) dt3$penalty_fn <- factor(dt3$penalty_fn, levels=c("L1", "L2", "fixed")) ggplot(dt3, aes(time, value, col=variable, lty=variable)) + geom_line(lwd=1) + facet_grid(penalty_fn~., labeller = labeller) + labs(x="time", y="harvest", title = "Example Harvest Dynamics") + scale_color_discrete(labels=c("Adjustment penalty", "No penalty"), name="") + scale_linetype_discrete(labels=c("Adjustment penalty", "No penalty"), name="") + theme_bw() write.csv(dt3, "components/data/figure3.csv")
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 Appendix A. 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 r 100*reduction% 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 r reduction *100% of $NPV_0({\bf h_0^*})$. To demonstrate this,
we solved for the optimal policy under each of the three adjustment
cost scenarios for r length(c2) different $c_i$ coefficients. For each
resulting policy, we then simulated r length(reps) 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.
stats_df %>% filter(variable != "cross.correlation") %>% separate(variable, c("measurement", "statistic"), sep="\\.") %>% filter(measurement == 'harvest') %>% spread(statistic, value) -> f4df f4df$penalty_fn <- factor(f4df$penalty_fn, levels = c("L1", "L2", "fixed")) p4 <- ggplot(f4df, aes(x = penalty_fraction, fill=penalty_fn, col=penalty_fn)) + coord_cartesian(xlim = c(0, .3)) + scale_color_discrete(labels=relabel, name="cost model") + scale_fill_discrete(labels=relabel, name="cost model") p4a <- p4 + stat_summary(aes(y=autocorrelation), fun.y = mean, fun.ymin = function(x) mean(x) - sd(x), fun.ymax = function(x) mean(x) + sd(x), geom = "ribbon", alpha=0.3, colour=NA) + stat_summary(aes(y=autocorrelation), fun.y = mean, geom = "line") + xlab("") p4b <- p4 + stat_summary(aes(y=variance), fun.y = mean, fun.ymin = function(x) mean(x) - sd(x), fun.ymax = function(x) mean(x) + sd(x), geom = "ribbon", alpha=0.3, colour=NA) + stat_summary(aes(y=variance), fun.y = mean, geom = "line") + xlab(expression(paste("Penalty size (as fraction of ", NPV[0], ")"))) legend <- gtable_filter(ggplot_gtable(ggplot_build(p4a)), "guide-box") grid.arrange(do.call(arrangeGrob, lapply(list(p4a, p4b), function(x) x + theme_bw() + theme(legend.position="none"))), legend, ncol = 2, widths = c(5,2)) write.csv(f4df, "components/data/figure4.csv")
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 @Ludwig1980 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 results shown in Appendix A provide further evidence that these patterns are independent of the exact reduction shown by providing results analogous to Figure 3 (A8, A9) and Figure 5 (A10, A11), and Figure 6 (A6).
fraction_no_shift <- function(x){ noshift <- x[2:length(x)] - x[1:length(x)-1] == 0 strictly_positive <- (x > 0.01 )[-1] # drop first point, who <- noshift & strictly_positive # True whenever consecutive pairs did not shift and the second in the pair was nonzero harvest sum(who)/length(who) } shifts <- dt[, fraction_no_shift(harvest), by=c("penalty_fn", "replicate")] ave <- shifts[, mean(V1), by=penalty_fn] no_shifts <- ave$V1 names(no_shifts) <- ave$penalty_fn
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 r no_shifts[["L2"]] * 100% of the time for quadratic costs $\Pi_2$, compared with
r no_shifts[["L1"]] * 100% with linear costs $\Pi_1$ and r no_shifts[["fixed"]]*100%
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
profits <- dt[, sum(profit_fishing), by=c('penalty_fn', 'replicate') ] costs <- dt[, sum(policy_cost), by=c('penalty_fn', 'replicate') ] reed_profits <- dt[, sum(profit_fishing_alt), by=c('penalty_fn', 'replicate') ] reed_costs <- dt[, sum(policy_cost_alt), by=c('penalty_fn', 'replicate') ] setnames(profits, "V1", "profits") setnames(reed_profits, "V1", "profits") Reed <- cbind(reed_profits, costs = reed_costs$V1, Assumption = "No adjustment penalty") Adj <- cbind(profits, costs = costs$V1, Assumption = "Adjustment penalty") hist_dat <- melt(rbind(Adj, Reed), id=c("penalty_fn", "replicate", "Assumption")) ```` ```r fig5_caption <- paste("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 Appendix A. 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.")
fig5_dat <- hist_dat %>% filter(penalty_fn=="L1") %>% select(-penalty_fn) %>% unite(dist, Assumption, variable, sep = " : ") labels <- c(expression(paste("Costs under ", Pi[1])), expression(paste("Revenue under ", Pi[1])), expression(paste("Cost under ", Pi[0])), expression(paste("Revenue under ", Pi[0]))) ggplot(fig5_dat) + geom_density(aes(value, fill=dist, color=dist, lty=dist), alpha=0.5, size=1) + scale_fill_discrete(name="", labels=labels) + scale_color_discrete(labels=labels, name="") + scale_linetype_discrete(labels=labels, name="") + xlab("Realized NPV") + ylab("Probability density of revenue/cost") write.csv(hist_dat, "components/data/figure5.csv")
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 r length(reps) 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.
who <- c("penalty_fn", "ignore_fraction", "assume_fraction", "reduction") table1 <- arrange(error_df[who], reduction) names(table1) = c("penalty.fn", "ignoring", "assuming", "reduction") table1_long <- melt(table1, id = c('penalty.fn', 'reduction')) table1_long$reduction <- as.factor(table1_long$reduction) table1_long <- subset(table1_long, reduction != "0.3") table1_long$penalty.fn <- factor(table1_long$penalty.fn, levels = c("L1", "L2", "fixed")) ggplot(filter(table1_long, reduction == 0.25), aes(penalty.fn, value, fill = variable)) + geom_bar(stat="identity", position="dodge") + scale_fill_discrete(name="error type") + scale_x_discrete(labels = c('L1' = substitute(paste(Pi[1])), 'L2' = substitute(paste(Pi[2])), 'fixed' = substitute(paste(Pi[3])))) + xlab("Cost model") + ylab(expression(NPV[0])) write.csv(table1_long, "components/data/figure6.csv")
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 r reduction*100% 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 r (1 -reduction)*100% 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 r (1 - reduction) * 100% 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 r (1-reduction)*100% 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 Appendix A 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 Appendix A.
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 [@Reed1979; @Clark2010]. 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 [@Feichtinger1994; @Wirl1999] 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 [@Brown1974; @Lewis1981; @McGough2009]; (see Appendix A 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 [@Brown1974; @Lewis1981; @Feichtinger1994; @Wirl1999; @McGough2009], 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 Appendix A 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 [@Homans1997], or models that derive policy recommendations endogenously by modeling strategic interactions between different stakeholders [@Kaitala1993; @Laukkanen2003]. 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 @Reed1979 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 @Sethi2005; @Singh2006; @McGough2009).
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 [@McFadden1975; @McFadden1976]. 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 @Reed1974) 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 [@Scheffer2009]. 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 [@Boettiger2013].
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.
References
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.