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Structural Behavioral Models for Rights-Based FisheriesI

Matthew N. Reimer∗

Department of Agricultural & Resource Economics, University of California, Davis, One Shields Ave,

Davis, CA 95616 USA

Joshua K. Abbott

School of Sustainability, Global Institute of Sustainability, and Center for Environmental Economics and

Sustainability Policy, Arizona State University, P.O. Box 875502, Tempe, AZ 85287 USA

Alan Haynie

Resource Ecology and Fisheries Management Division, Alaska Fisheries Science Center, National Marine

Fisheries Service, NOAA, 7600 Sand Point Way NE, Bldg. 4, Seattle, WA 98115 USA

Abstract

Rights-based management is prevalent in many ﬁsheries, yet spatiotemporal models of ﬁsh-

ing behavior do not reﬂect such institutional settings. We develop a model of spatiotemporal

ﬁshing behavior that incorporates the dynamic and general equilibrium elements of catch-

share ﬁsheries. We propose an estimation strategy that is able to recover structural behav-

ioral parameters through a nested ﬁxed-point maximum likelihood procedure. We illustrate

our modeling approach through a Monte Carlo analysis and demonstrate its importance for

predicting out-of-sample counterfactual policies.

Keywords: structural econometrics, rights-based ﬁsheries, discrete choice models

IWe thank several individuals for helpful comments, including Corbett Grainger, Ling Huang, Linda

Nøstbakken, Kathleen Segerson, and Brett Watson; seminar participants at Arizona State University, Uni-

versity of Alaska Anchorage, University of Connecticut, and the Norwegian School of Economics; and session

participants at the IIFET 2018, NAAFE 2019, AERE 2019, and ASSA 2020 meetings. Funding for this

research was provided by the North Paciﬁc Research Board (NPRB-1607). The scientiﬁc results and con-

clusions, as well as any views or opinions expressed herein, are those of the authors and do not necessarily

reﬂect those of the authors’ aﬃliate or funding organizations. We declare no conﬂicts of interest and that

we have no relevant or material ﬁnancial interests that relate to the research in this paper.

∗Corresponding author

Email addresses: mnreimer@ucdavis.edu (Matthew N. Reimer), Joshua.K.Abbott@asu.edu (Joshua

K. Abbott), Alan.Haynie@noaa.gov (Alan Haynie)

1

1. Introduction

Economists are often called upon to inform policy makers of the potential consequences of

proposed environmental and natural resource regulations. For economists to oﬀer reliable ad-

vice, their models must adequately capture individual decision-making processes, contextual

variables, and institutional settings to provide externally valid predictions across the range

of policy scenarios of interest to decision-makers (Lucas,1976;Heckman,2010). If the range

of these counterfactuals deviates markedly from in-sample conditions, then purely empirical,

reduced-form descriptions of behavior will likely be unsatisfactory. Instead, structural mod-

els that explicitly model individuals’ decision-making process in terms of objective-seeking

(e.g., proﬁt or utility-maximizing) behavior under the salient economic, environmental, and

institutional constraints are needed (Wolpin,2007;Keane,2010).

In this paper, we develop a structural approach for estimating individual commercial-

ﬁshing behavior under rights-based management institutions that provides consistent esti-

mates of behavioral parameters and is capable of predicting out-of-sample counterfactual

policies. Despite the prevalence of rights-based management in today’s developed-world

ﬁsheries, most empirical models of commercial ﬁshing behavior—those intended to inform

management decision making—do not explicitly reﬂect the incentives and constraints un-

derlying rights-based institutions. Instead, they reﬂect the implicit theoretical assumptions

of regulated open or limited access ﬁsheries. As such, even if these models are calibrated on

behavior under rights-based management, they do not capture the theoretical mechanisms

by which incentives under rights-based management aﬀect ﬁshers’ behavior. The result, as

we demonstrate, is that the predictions of these models could be highly misleading.

To address this deﬁciency, we extend random utility maximization (RUM) models of

spatiotemporal ﬁshing behavior (e.g., Eales and Wilen,1986;Holland and Sutinen,2000;

Smith,2005;Haynie et al.,2009;Abbott and Wilen,2011), which are the dominant form

of management models used to predict the consequences of proposed ﬁshery policies, such

as spatial regulations (Smith and Wilen,2003;Berman,2006;Haynie and Layton,2010;

Hicks and Schnier,2010). Conventional RUM models of ﬁshing behavior do not consider the

implications of individualized (and often transferable) quotas of catch entitlements within a

season, which create a shadow value reﬂecting the opportunity cost of quota. We incorporate

2

the dynamic and general equilibrium elements of ﬁsheries with tradable short-term rights

of annual catch entitlements by introducing a lease-market for quota, which we model as a

pure exchange economy. Fishers are assumed to be forward-looking within the ﬁshing season

and form expectations over future quota usage when considering contemporaneous quota

supply and demand decisions. Under the assumption of rational expectations, each ﬁsher’s

stochastic dynamic programming problem reduces to a period-by-period static maximization

problem given a set of equilibrium quota prices. Critically, expectations are updated in each

period, leading to a new set of equilibrium quota prices to reﬂect the changing relative

scarcity of quota in a stochastic production environment.

We demonstrate the utility of our estimation strategy—dubbed the rational expectations

RUM (RERUM)—for both parameter estimation and out-of-sample prediction through nu-

merical simulations and Monte Carlo analyses. We ﬁrst show that the omitted nature of

quota lease prices in the conventional RUM approach leads to a form of omitted variable

bias (or, alternatively, non-classical measurement error). These biases could jeopardize the

estimation of shadow values or welfare estimates (e.g., Abbott and Wilen,2011;Haynie

et al.,2009;Hicks and Schnier,2006). We demonstrate that substitution of high-resolution

lease prices as data into the conventional RUM is able to eliminate estimation bias of behav-

ioral parameters. Unfortunately, thin markets combined with conﬁdentiality concerns rarely

allow for such an approach. Imputing annual average prices—which are more commonly

available—oﬀers only a partial mitigation of the bias, since it fails to capture dynamic ad-

justments of behavior within the ﬁshing season. Furthermore, even if high-resolution lease

prices are available, prediction for out-of-sample policy scenarios requires the imputation of

counterfactual lease prices that are consistent with the stochastic production environment

and the changes in market, ecological, or policy conditions embodied in the scenario.

Our estimation approach imputes quota-lease prices via a market simulator at the core

of the estimation procedure, whereby a ﬁxed-point problem is solved to determine state-

contingent equilibrium lease prices in every period. Thus, the RERUM estimator does not

rely on the availability of high-resolution lease-price data and can produce counterfactual

lease prices that are consistent with the structure of ﬁshers’ dynamic decision problem and

observed ﬁsher behavior. Moreover, our approach does not suﬀer from the curse of dimen-

3

sionality because the dimensions of the ﬁxed-point problem increase linearly, as opposed to

exponentially, with the number of quota-constrained species. Thus, we are able to solve the

behavioral model exactly and recover the structural parameters through a nested ﬁxed-point

(NFXP) maximum likelihood procedure (Rust,1987).

We conduct numerical simulations to demonstrate how our model can be used for ex

ante evaluation of ﬁshery policies, such as spatial closures or reductions in the total allow-

able catch. We illustrate this point through a Monte Carlo analysis and investigate data-

generating environments for which our approach matters most for out-of-sample predictions.

We ﬁnd that our approach matters more as counterfactual policy changes lie increasingly

out-of-sample, as measured by the degree to which lease prices are responsive to the counter-

factual policy. For counterfactuals that have only a marginal inﬂuence on quota-lease prices,

reduced-form approaches that approximate the state-contingent equilibrium lease prices can

be suﬃcient for out-of-sample predictions.

Finally, we note that while our estimator is tailored speciﬁcally to the production process

and institutions of modern-day ﬁsheries, our work has broader relevance for other industrial

and institutional settings—particularly for industries characterized by stochastic produc-

tion processes and managed under quotas (or quantity controls) with transferable property

rights. For example, cap-and-trade systems for controlling greenhouse gas emissions are

typically comprised of ﬁrms that make dynamic production decisions under uncertainty of

future abatement costs while balancing emissions and permits over a ﬁxed regulatory horizon

(Rubin,1996;Kling and Rubin,1997;Fell,2016;Kollenberg and Taschini,2016). As in our

setting, binding quota allocations create shadow values that reﬂect the opportunity cost of

such constraints, and these shadow values are harmonized through the coordinating mecha-

nism of the quota market. Any proposed policy that inﬂuences these shadow values will thus

be reﬂected in the equilibrium quota prices. Thus, quota prices are not policy invariant, and

therefore, models of endogenous quota prices are likely required for counterfactual policy

evaluations.

The course of the paper is as follows. Section 2discusses the relevant literature and

the institutional background of rights-based management of commercial ﬁsheries. Sections

3and 4present the structural behavioral model and the estimation strategy of the RERUM

4

estimator. Section 5demonstrates the utility of the RERUM model for predicting realistic

policy changes, such as quota reductions and spatial closures, and provides Monte Carlo

simulation evidence of the estimation performance and predictive utility of the RERUM

model in comparison to alternative RUM model speciﬁcations. Section 6concludes the

paper.

2. Background and Related Literature

The governance of many nation states’ ﬁsheries has been transformed in recent decades—

from the “tragedies” of open access and input regulation to a range of governance structures

based upon individual or collective extractive rights. By one estimate, approximately 20%

of global catch comes from ﬁsheries managed under individual transferable quotas (Costello

and Ovando,2019)—a number that only partially accounts for the full spectrum of rights-

based management approaches, including ﬁshing cooperatives (Deacon,2012) or TURFs

(Wilen et al.,2012). Rights-based management (RBM) is particularly common in the Global

North where it is facilitated by strong scientiﬁc input and adequate governance. RBM, in

combination with scientiﬁcally-based quotas and sound enforcement, has played a prominent

role in reversing overﬁshing and improving economic eﬃciency in many ﬁsheries (Worm et al.,

2009;Grafton et al.,2006;Hilborn et al.,2005).

Despite these successes, RBM has not reduced the role of ﬁsheries managers to merely

conducting stock assessments and setting seasonal quotas. Catch shares, especially individual

quotas, may leave signiﬁcant in-season externalities unaddressed (Boyce,1992;Costello and

Deacon,2007), forcing managers to deploy additional management measures to address

concerns such as growth overﬁshing or in-season rent dissipation. Furthermore, many of the

concerns of ecosystem-based management—e.g., protection of spawning stocks or vulnerable

life stages, reducing external impacts on unﬁshed stocks or species of conservation concern,

and habitat protection—are outside the scope of most RBM systems (Holland,2018).

As a result of these concerns, managers use a wide range of tools, including input restric-

tions, protected areas, time-area closures, and dynamic ocean management (Maxwell et al.,

2015), in addition to RBM systems. Economists have informed managers of the potential

consequences of these actions by developing positive bioeconomic models (e.g., Smith and

5

Wilen,2003;Hutniczak,2015;Lee et al.,2017;Holland,2011;Huang and Smith,2014) that

predict how changes to policy design may change catch, eﬀort, proﬁts, employment, or eco-

logical impacts. However, the continued adoption of RBM presents a signiﬁcant challenge

to ﬁsheries policy modeling in that the overwhelming majority of empirical models used to

inform in-season management measures fail to consider the implications of individualized

(and often transferable) catch rights within a season. Catch share ﬁsheries deﬁne individu-

alized (or sometimes cooperative-based) quota constraints, and the shadow values that arise

from such constraints are coordinated through within-season quota trading in a shared lease

market. Experience has demonstrated that in-season behavior is often drastically altered by

catch shares. This is particularly likely in terms of the allocation of ﬁshing “eﬀort” in both

space and time (Reimer et al.,2014;Abbott et al.,2015;Birkenbach et al.,2017;Miller and

Deacon,2017). Fishers may spread their eﬀort temporally and reallocate where they ﬁsh

to enhance revenues or reduce costs. More complex patterns may emerge in multispecies

catch-share ﬁsheries as vessels utilize space and time to maximize the proﬁt associated with

their quota portfolios (Birkenbach et al.,2020). However, models of commercial ﬁsheries

often do not capture the behavioral mechanisms that arise under RBM institutions, with

the result that their predictions could be highly misleading (Reimer et al.,2017b).

Our econometric estimation approach is not the ﬁrst to include dynamic or stochastic

elements of within-season ﬁshing behavior. Models of within-trip behavior have been ex-

tended to consider the logistical problem of the optimal trajectory of ﬁshing decisions within

a trip. Optimal within-trip behavior is therefore cast as a dynamic programming problem,

with estimation of model parameters coinciding with the solution (Hicks and Schnier,2006,

2008) or approximation (Curtis and Hicks,2000;Curtis and McConnell,2004;Abe and An-

derson,2020) of the dynamic programming problem. Such models, however, do not capture

the overriding dynamic concern that we would expect to emerge under catch shares—the

management of a portfolio of quotas over the course of an entire season, where the state vari-

ables that provide the information set for ﬁshermens decisions (i.e., expected catch, quota

balances) evolve in a partially stochastic manner.

A handful of papers have tackled seasonal ﬁshing supply decisions dynamically (Provencher

and Bishop,1997;Smith and Provencher,2003;Huang and Smith,2014). However, the

6

stochastic evolution of the state variables coupled with the need to solve a ﬁsher’s seasonal

optimization repeatedly in the estimation process through stochastic dynamic programming

has resulted in the imposition of very strong assumptions on the models to maintain com-

putational tractability. This has usually taken the form of severely limiting the number of

spatial locations available to ﬁshermen and curtailing the horizon of decision making in order

to reduce the “curse of dimensionality.” Indeed, while notable advances have been made in

reducing these computational burdens, the dimensionality of most applied dynamic discrete

choice models remains quite small (Aguirregabiria and Mira,2010). As we explain below, the

coordinating mechanism of the quota lease market allows us to specify production decisions

over a realistic spatial and temporal scale and number of state variables (species).

3. A Model of a Catch-Share Fishery

Our objective is to build a model of within-season ﬁshing behavior that generates exter-

nally valid ex ante predictions of ﬁshery policies in a catch-share ﬁshery. This prospective

model must be structural or mechanistic, in the sense that it identiﬁes policy-invariant pa-

rameters that can be safely transported into “out-of-sample” environments, facilitating the

job of ex ante prediction (Heckman and Vytlacil,2007;Heckman,2010). Structural mod-

els achieve this ﬂexibility through explicitly modeling the hypothesized decision process of

agents in response to their decision context, usually through a constrained optimization ap-

proach. This diﬀers from estimating a reduced-form decision rule in that the latter runs

the risk of fragility since underlying ecological, economic, or policy state variables may be

subsumed into the estimated reduced form parameters (Fenichel et al.,2013).

Our model must satisfy several criteria. First, it must capture the primary within-

season mechanisms ﬁshermen use to shape economic returns and catch compositions. While

some aspects of input usage (e.g., bait or crew staﬃng) may be somewhat variable within a

season, the primary short-run mechanisms inﬂuencing vessel output are where and when to

ﬁsh (Abbott et al.,2015;Reimer et al.,2017a;Scheld and Walden,2018). Second, the model

must be both dynamic and stochastic. Dynamic models consider that ﬁshermen allocate

their portfolio to maximize seasonal returns so that current ﬁshing decisions depend on

expectations of ﬁshery conditions later in the season. Stochasticity implies that planning

7

will not be perfect—catch, and hence quota balances, will not exactly match expectations.

Third, the model must easily accommodate realistic changes to management policies—such

as catch limits and time/area closures. Finally, estimation and simulation of the model must

be achievable from available data with reasonable technology and computing time.

Structural models face a trade-oﬀ between realism and computational tractability, re-

quiring that modeling decisions preserve realism where it is fundamental to the nature of

agents’ decision problem and predicted outcomes while sacriﬁcing it elsewhere. In our case,

the most fundamental decision concerns the modeling of the quota lease-market, for which

we make two simplifying assumptions. First, we assume that ﬁshers must have enough quota

at the end of the ﬁshing season to cover their cumulative catch. Accordingly, the market

for leasing quota clears at the end of the season, and ﬁshers’ expectations regarding end-

of-season quota demand and supply form the basis for within-season quota prices. Second,

we assume the market for quota is competitive. That is, ﬁshers’ treat their expectations of

quota-lease prices as given, even though prices are endogenously determined by the aggregate

behavior of all ﬁshers. Given the incentives embodied in these expected prices, ﬁshers carry

out individually optimal “on-the-water” plans by allocating their eﬀort over a discrete num-

ber of ﬁshing sites and time periods. We close the model under the assumption of rational

expectations so that equilibrium quota prices are consistent with ﬁshers’ beliefs.

3.1. A ﬁsher’s dynamic programming problem

Consider agent (i.e., the ﬁsher) i, who has preferences deﬁned over a sequence of states

of the world zi,t from period t= 1 until period t=T+ 1. In periods t≤T, agents choose

a ﬁshing location a∈A={0,1, ..., J }, where a= 0 represents the option of not ﬁshing. In

the ﬁnal period t=T+ 1, the agent incurs costs or receives revenues from buying or selling

quota in the leasing market according to their cumulative quota usage. In any given time

period, ﬁshers must account for the opportunity cost of using quota—whether it is best to

use quota today for the proﬁts it generates or preserve it for sale in the competitive quota

market. The problem is stochastic because ﬁshers do not know exactly what they (or others)

will catch at each location and time period, and thus, they form expectations over ﬂeet-wide

catch realizations and the resulting end-of-season excess demand for quota. We assume that

8

the number of ﬁshers is large enough that any single ﬁsher perceives their eﬀect on aggregate

harvest and the quota lease price as negligible. Therefore, ﬁshers’ expectations of quota

prices are formed exogenously to their own decisions.

We make a number of simplifying assumptions for the sake of tractability. First, the state

of the world at period tfor agent iis assumed to consist of two components: zi,t = (xi,t , εi,t).

The subvector εi,t is private information known only by agent iat the time of decision, and is

assumed to be exogenous. The subvector xi,t is an endogenous and stochastic state variable

representing an agent’s S-dimensional vector of cumulative catch prior to making a decision

in period t:xi,t =fx(xi,t−1) = Pt−1

k=1 yi,k =xi,t−1+yi,t−1, where yi,t =Y(ai,t, ξi,t ) represents

ﬁsher i’s S-dimensional vector of catch in period t.1The term ξi,t represents the stochastic

component of catch, which we assume to be serially uncorrelated and unknown to any ﬁsher

at the time a decision is made in period t. We denote xt=P∀ixi,t as the vector of ﬂeet-wide

cumulative catch at the beginning of period tfor all species, which we assume to be common

knowledge to all ﬁshers.

Second, we assume that an agent’s contemporaneous utility function for location ai,t is

additively separable in the observable and unobservable components:

U(ai,t, zi,t ) =

u(ai,t, p0yi,t ) + εi,t (ai,t) if t∈ {1, ..., T }

u(0, w0(ωi−xi,T +1)) if t=T+ 1,

(1)

where ωidenotes a vector of quota endowments possessed by ﬁsher iat the beginning of the

season, wdenotes a vector of quota-lease prices, and pdenotes a vector of ex-vessel prices.

An agent’s utility in the ﬁnal period T+ 1 is evaluated at port (a= 0) with revenue equal

to the value of their remaining endowment of quota.2

Third, we assume that the unobserved state variables εi,t are independently and identi-

cally distributed (iid) across agents, time, and locations, and have an extreme-value type 1

1Note that the time index tshould also be a component of the state vector, but we omit it here for the

sake of keeping notation as simple as possible.

2It can be shown that the indirect utility function in period T+ 1 follows from an agent choosing con-

sumption and an amount of quota to maximize utility, subject to a budget constraint (see section Appendix

Bfor details).

9

distribution that is common knowledge across ﬁshers.

Fourth, we assume that catch yis independent of the unobserved state variables εand the

observed endogenous state variables x, conditional on the location choice a. This assumption

implies that the stochastic component of catch ξis conditionally independent of past, present,

and future values of εand x, so that: E(yi,t |ai,t , xi,t, εi,t) = E(yi,t |ai,t ).Practically speaking,

this assumption has several implications. First, a ﬁsher’s private information about a location

choice does not aﬀect catch (or expectations of catch) once the ﬁsher’s choice has been

made—i.e., private information only inﬂuences catch by inﬂuencing a ﬁsher’s choice. Second,

cumulative catch, as reﬂected in xi,t, does not inﬂuence the distribution of contemporaneous

catch—i.e., within-season spatiotemporal stock dynamics are exogenous to ﬁshing behavior.

Finally, this assumption also implies that the next-period cumulative catch xj,t+1 of any

ﬁsher jis independent of ﬁsher i’s current period unobserved state variable εi,t, conditional

on the values of the decision ai,t and state variable xi,t. Together, these assumptions deﬁne

what is often referred to as the dynamic programming conditional logit model (Rust,1987).

In periods t≤T, an agent observes the vector of state variables zi,t and chooses an action

ai,t ∈Ato maximize expected utility

E T+1−t

X

j=0

U(ai,t+j, zi,t+j)ai,t, zi,t !.(2)

The decision at period taﬀects the evolution of future values of the state variables xi,t, but

the agent faces uncertainty about these future values due to the unknown nature of future

catch. The agent forms beliefs about future states, which are objective beliefs in the sense

that they are the true transition probabilities of the state variables. By Bellman’s principle

of optimality, the value function during the ﬁshing periods t≤Tcan be obtained using the

recursive expression:

V(zi,t) = max

a∈AU(a, zi,t) + EzV(zi,t+1)a, zi,t,(3)

where Ezdenotes the expectations operator with respect to the state vector z.3

3Note that we do not include a discount factor.

10

Unfortunately, there is typically no analytical form for the expected value function, and

computationally expensive numerical and recursive methods are often needed to solve the

Bellman equation instead. The restrictions these methods place on the dimensionality of

the state space have often limited the empirical relevance of dynamic programming models

of ﬁsher behavior. Thankfully, the assumptions underlying the dynamic programming con-

ditional logit model, combined with the additional assumption that ﬁshers are risk-neutral,

imply that ﬁsher i’s optimal decision rule in each period is dramatically simpliﬁed. The ex-

pected quota-lease price win period tacts as a shadow price of quota, which is harmonized

across ﬁshers given the transferability of quota.4Conditional on expected lease prices w, the

solution of Eq. (3) takes on a simple, static form:5

α(zi,t |w) = argmax

a∈Aua, (p−w)0Eyi,t a+εi,t (a).(4)

Notably, the policy function has a simple analytical form that does not depend on the endoge-

nous state variable xi,t. Rather, it depends only on the ﬁsher’s current private information

εi,t and the expected quota-lease price w, both of which are exogenous. Intuitively, the

quota-lease price embeds all relevant information regarding expected future quota scarcity

needed to inform the present-day decision.6Functionally, this means that, given a perceived

quota-lease price, the location-choice problem in equation (2) reduces to a tractable period-

by-period static maximization problem that does not require recursively solving the Bellman

equation.

3.2. Rational Expectations Equilibrium Quota Prices

Eq. (4) presents a ﬁsher’s optimal decision rule for a given expected quota-lease price w.

Fishers determine their current and future optimal location choices given perceived quota

prices was speciﬁed by the policy function α(zi,t |w) in equation (4). In this sense, quota

4The assumption of risk neutrality has the practical implication that revenue enters utility linearly and

is additively separable from the rest of utility.

5See Appendix C for a formal derivation.

6The policy function in equation (4) takes on a similar form to the utility function used by Miller and

Deacon (2017).

11

prices determine ﬁsher behavior. At the same time, given ﬁshers’ decision rules α(zi,t |w),

the quota market determines expected quota prices in each period so that aggregate ﬁsher

behavior determines the equilibrium quota prices. Rational expectations states that the

market-clearing quota prices implied by ﬁsher behavior are the same as the quota prices on

which ﬁshers’ decisions are based. That is, the market-clearing equilibrium quota prices are

consistent with ﬁshers’ quota-price expectations.

The expected quota-price vector wis the equilibrium price that clears a seasonal compet-

itive market for quota leasing, which is assumed to be frictionless and without transaction

costs. Let Ω = P∀iωidenote the vector of ﬂeet-wide quota endowments for all species.

Then the seasonal excess demand for quota for species scan be written as es=xs,T +1 −Ωs.

In any given period t≤T, a ﬁsher does not know with certainty what the demand for quota

will be at the end of the season; thus, forward-looking ﬁshers form expectations over excess

demand given a perceived wand the state of the world in period t:

E(es|w, xt) = E(xs,T +1 |w, xt)−Ωs

=T

P

k=tP

∀iP

∀a∈A

f(a|w)E(yi,s,k |a)+xs,t −Ωs,(5)

where f(·) denotes the probability mass function for the discrete location-choice variable a

and the bracketed term represents the expected catch for all ﬁshers in the remaining periods.7

Given the assumption that ﬁshers know the distribution of private information for all agents,

f(·) can be derived by integrating the policy function (4) over the unobserved state variable:

f(a|w) = ZI[α(z|w) = a]g(ε)dε,

where I[·] is an indicator function and g(·) is the probability density function of ε. The

expected equilibrium quota-lease prices in period tcan then be deﬁned as those that satisfy

7For simplicity, we have implicitly assumed that a ﬁsher forms their expectation of excess demand before

they observe their private information ε. For a large number of ﬁshers, as we’ve assumed here, this has

a negligible inﬂuence on our results; it is, however, trivial to relax this assumption at the cost of model

presentation.

12

the following market-clearing conditions:

E(es|w, xt) = 0 for ws>0

E(es|w, xt)≤0 for ws= 0.

(6)

That is, in equilibrium, prices will adjust so that positive prices achieve zero expected excess

quota demand for scarce species, while prices fall to zero for species in excess supply (i.e.,

“free goods”). The equilibrium quota prices that solve the market-clearing conditions in the

system of equations (6) are state-contingent—i.e., they are a function of the observed (and

common knowledge) state of the world in period t. We denote the equilibrium quota-lease

price vector as ˜w(xt).

Under the assumption of rational expectations, ﬁshers’ beliefs are consistent with the

market-clearing conditions in (6). Thus, to close the rational expectations model, we sub-

stitute the equilibrium quota prices ˜w(xt) into a ﬁsher’s optimal decision rule:

α(zi,t) = argmax

a∈Aua, (p−˜w(xt))0Eyi,t a+εi,t (a),(7)

Eq. (7) serves as the basis for our rational-expectations RUM (or RERUM) model.

We emphasize here that the state-contingent equilibrium prices ˜w(xt) reﬂect the scarcity

of quota that exists in time tgiven expectations regarding optimal future behavior and

harvesting conditions. Thus, while the equilibrium quota prices are determined by a market-

clearing condition at the end of the season, ˜w(xt) are the equilibrium prices that emerge in

period tas quota is exchanged. We further note that since equilibrium quota prices are

determined by common knowledge of aggregate cumulative catch xt, and not knowledge of

individual catch xi,t, it is not necessary to track quota within-season quota exchanges.

4. Estimation

We wish to estimate a vector of structural parameters in the utility function θutilizing

panel data for Nindividuals who behave according to the decision model described in Section

3. For every observation (i, t) in this panel dataset, we observe the individual’s action ai,t,

the payoﬀ variable yi,t, and the subvector xi,t of the state vector zi,t = (xi,t, εi,t). Because the

subvector εi,t is observed by the agent but not by the researcher, εi,t is a source of variation

13

in the decisions of agents conditional on the variables observed by the researcher. It is the

model’s econometric error, which is given a structural interpretation as an unobserved state

variable.

Assuming that the data are a random sample over individuals, the log-likelihood function

is PN

ili(θ), where li(θ) is the contribution to the log-likelihood function of i’s individual

history:8

li(θ) = log Pr ai,t :t= 1, ..., T yi,t, xt, θ

= log Pr ai,t =α(xi,t, εi,t, θ) : t= 1, ..., T yi,t, xt, θ(8)

=

T

X

t=1

log f(ai,t |xt, θ).

Closed-form expressions for f(·) follow from the iid extreme value type 1 distribution we’ve

assumed for εi,t, which produces the conventional logit probabilities:

f(a|xt, θ) = eu(a,(p−w(xt))0E(y|a))

P∀keu(k,(p−w(xt))0E(y|k)).(9)

This expression is predicated on knowledge of the quota price rules w(xt). Therefore, we

need to either observe the state-contingent quota prices or come up with a strategy for

determining the implied quota prices within the estimation process. In the former case,

observed quota prices can simply be inserted into the choice probabilities in equation (9)

and maximum likelihood estimation can proceed as usual. However, in many cases, these

lease prices are not observed due to limitations on data disclosure or because only average

prices are reported, as opposed to state-contingent prices. Given this missing data problem,

we propose solving for the rational expectations equilibrium prices for each trial value of θ.

The nested ﬁxed-point algorithm (NFXP) pioneered by Rust (1987) is a search method

for obtaining maximum likelihood estimates of the structural parameters, which combines

8Note that we are estimating the structural parameters θtaking the harvest variable yi,t and state variable

xtas given. Thus, we are taking a partial MLE approach here. In theory, it is possible to jointly estimate

the structural parameters of both the harvesting and utility functions in a full MLE approach; however, for

the sake of simplicity, we leave that for future research.

14

an “outer” algorithm that searches for the root of the likelihood equations with an “inner”

algorithm that solves for the ﬁxed-point of the rational expectations equilibrium for each

trial value of the structural parameters. Speciﬁcally, consider an arbitrary value of θ, say ˆ

θ0.

Conditional on ˆ

θ0, the inner algorithm solves for the wtthat solves the ﬁxed-point problem

in equation (6) given optimal ﬁsher behavior described in equation (5). This produces an

equilibrium vector of quota prices ˜w(xt) for each observation in our data, which can be

substituted into equation (9) to form the choice probabilities fai,t |xt,ˆ

θ0. Next, the

outer algorithm uses the gradient of the log-likelihood function with the choice probabilities

in equation (9) to start a new iteration with a new structural parameter ˆ

θ1.This process

continues until either ˆ

θor the log-likelihood converges based on a pre-speciﬁed convergence

tolerance.9

5. The RERUM Estimator: Demonstration and Evaluation

In this section, we demonstrate how the RERUM can be used for predicting counterfac-

tual ﬁshery policies, and evaluate the estimation and predictive performance of the RERUM

estimator. We ﬁrst simulate the structural model with known parameter values to demon-

strate the utility of the RERUM model for predicting realistic policy changes, such as quota

reductions and spatial closures. We then evaluate the estimation and predictive perfor-

mance of the RERUM through a Monte Carlo analysis, where we estimate the RERUM

model over several draws from a data generating process with a random parameter space.

Finally, we evaluate the in- and out-of-sample predictive performance of the RERUM model

and alternative RUM models to investigate the biological and regulatory conditions under

which alternative RUM models may provide adequate predictions of ﬁshing behavior under

rights-based management.

5.1. Fishery setting and the data-generating process

We consider a ﬁshery in which ﬁshers receive individual quotas for two species that are

jointly harvested, but only one of these species (Species 1) has an ex-vessel value to a ﬁsher—

i.e., Species 2 can be considered a bycatch species. We simulate two forms of hypothetical

9For more details on the the NFXP algorithm, see Appendix D.

15

policies designed to reduce bycatch: (1) reductions to the quota for the bycatch species, and

(2) bycatch hot-spot area closures.

The data generating process (dgp) is purposefully simple to facilitate our understanding

of the model predictions. We assume ﬁshers begin each period in port and choose from a

n×ngrid of ﬁshing locations. The observable component of a ﬁsher’s contemporaneous

expected utility function in equation (1) for location ais speciﬁed as:

E(ui,t) = θRev p0E(yi,t |a) + θDistDist(a),

where Dist(a) represents the distance from port to location a. A ﬁsher’s optimal location

choice is determined by equation (7), which takes on the speciﬁcation

α(zi,t) = argmax

a∈A

{θRev(p−˜w(xt))0E(yi,t |a) + θDistDist(a) + εi,t (a)},

where the rational-expectations quota prices ˜w(xt) are determined by equation (6).10

We model ﬁsher i’s catch of species s∈ {1,2}in period tand location aas ys,i,t =

Y(a, ξs,i,t) = qs,i exp {ξs,i,t (a)}, where qs,i ∈(0,1) denotes ﬁsher i’s catchability coeﬃcient

and ξs,i,t(a) is a normally distributed random variable with location-speciﬁc mean param-

eters µs(a) and a common variance σ2. Catch is thus a log-normal distributed random

variable with mean E(ys,i,t |a) = qs,i exp{µs(a) + σ2/2}.11 For simplicity, µs(a) and σ2(and

thus expected catch) are assumed to remain constant over all individuals and time periods;

however, realized catch varies across all individuals and time periods due to the individual-

and time-speciﬁc nature of the idiosyncratic shock ξs,i,t(a).12

10In general, quota prices are sensitive to the data-generating parameters, as depicted in Figure A.1, and

have comparative statics that are consistent with theory: quota prices increase with ex-vessel prices, quota

scarcity, and the marginal utility of revenue. Note that the latter is only true for the target species. Quota

prices decrease with the marginal utility of revenue if a species’ net price (ex-vessel price minus quota lease

price) is negative. In this case, ﬁshers will try to avoid catching this species, decreasing demand for it’s

quota.

11The mean parameters µs(a) vary over the grid according to distinct two-dimensional normal density

functions for both species.

12This example does not incorporate stock depletion or other spatial/temporal variability in expected catch

over the course of the season. We do so to focus attention on the dynamics generated by the opportunity

16

We consider two diﬀerent biological scenarios with diﬀerent spatial distributions for each

species, producing the global production sets depicted in Figure 1. In the ﬁrst scenario, the

two species have minimal spatial overlap, and thus, ﬁshers are able to substitute between

species relatively easily. In contrast, ﬁshers are more constrained by the bycatch species in

the second scenario as there is greater spatial overlap between species and ﬁshers must travel

further away from port to avoid bycatch.

5.2. Simulating bycatch quota reductions and hot-spot closures

We reduce the bycatch quota and the area open to ﬁshing, respectively, by increments

of 5% to a minimum of 25% of their baseline levels. For the area closures, we emulate a

hot-spot closure policy by closing areas to ﬁshing that experience the highest amount of

bycatch in the baseline simulations.13 Harvest and utility shocks (ξand ε) are drawn from

their respective probability distributions, and state variables are endogenously updated in

each time period. The remaining data-generating parameter values are known and remain

ﬁxed across all policy simulations (presented in column 1 of Table 1).

Results from the policy simulations are presented in Figure 2, where we’ve simulated

200 counterfactual seasons under each policy. Under the baseline policies, the quota for

the bycatch species (s= 2) is binding in both biological scenarios, resulting in a positive

quota-lease price in all simulated seasons. In scenario 1, the lease price for the target species

(s= 1) is consistently positive as well, pointing toward the dominance of interior solutions in

the quota market. In contrast, the target species almost always has a non-positive lease price

in scenario 2, where the bycatch species consistently acts as a choke species, preventing the

full harvest of the target species quota. This diﬀerence largely stems from the higher spatial

overlap between the target and bycatch species in scenario 2, making bycatch avoidance so

costly that it is not possible to fully utilize the target species quota.

cost of quota. It is a relatively straightforward extension of our approach to include these extensions, so long

as ﬁshers consider stock depletion and other non-stationarities to be an exogenous process in their planning

behavior.

13For example, if 75% of a 100-location grid is closed to ﬁshing, we close the 75 cells that have the highest

amount of bycatch from a baseline simulation with no spatial closures.

17

The eﬀect of the bycatch reduction policies diﬀers across both biological scenarios and

policy types. Not surprisingly, the quota reductions are eﬀective at achieving desired bycatch

reductions: bycatch falls at a 1:1 ratio with the bycatch quota as the quota remains binding

over all reductions. The lost utility from achieving a given level of bycatch reduction is

considerably higher in scenario 2 because of the higher cost of bycatch avoidance. In scenario

2, the primary cost of bycatch reduction is foregone catch of the target species, as the bycatch

quota continues to bind before the target-species quota is harvested. By contrast, the primary

cost in scenario 1 is traveling greater distances to avoid bycatch: there is minimal foregone

target species catch in scenario 1 and the target species quota price declines very slowly on

average while the price of bycatch quota rises steadily with increased scarcity.

Hot-spot closures, on the other hand, have virtually no impact on bycatch in either

scenario over the examined range of closures. In fact, hot-spot closures have the eﬀect of

pushing ﬁshers into areas with higher bycatch-to-target species ratios. Since ﬁshers are

already avoiding bycatch under the baseline policy, bycatch is being generated in areas with

relatively low bycatch-to-target species ratios; hot-spot closures therefore push ﬁshers out of

relatively cleaner areas, thereby increasing bycatch per unit of target species catch.

The key diﬀerence between the two bycatch-reduction policies is reﬂected in the quota-

lease prices: quota reductions signal scarcity to ﬁshers through increased quota-lease prices,

and ﬁshers have the incentive to reduce bycatch in the most cost-eﬀective manner given their

information about catch rates. Hot-spot closures, on the other hand, do not signal bycatch

scarcity over a wide spectrum of policy severity when bycatch quota is already suﬃciently

scarce under the baseline scenario to command a positive price. Instead, for ﬁsheries where

bycatch species does not consistently act as a choke species (scenario 1), the closures decrease

the value of the target species quota price by pushing ﬁshers into increasingly sub-optimal

ﬁshing locations. In fact, quota prices for the bycatch species are only responsive to the

closures in scenario 1 once the target-species quota can no longer be harvested before the

bycatch quota binds.

Altogether, these policy simulations demonstrate the utility of modeling the spatiotem-

poral production decisions of harvesters under the dynamically evolving constraints imposed

by the seasonal quota market. The structural model can yield counterfactual policy predic-

18

tions of ﬁsher welfare, catch rates, and lease price behavior for changes in both rights-based

management parameters (i.e., quota allocations) and “ecosystem based” policies targeting

the spatiotemporal footprint of ﬁshing eﬀort. The simulation results also highlight the role

that lease prices play in relaying signals of quota scarcity, and how policies that fail to inﬂu-

ence the relative scarcity of quota in the desired direction as reﬂected in these relative prices

are likely to fall short of their intended objectives.

5.3. Evaluating the RERUM and alternative RUM models: A Monte Carlo analysis

We now evaluate the estimation and predictive performance of the RERUM model and

other alternative RUM model speciﬁcations through a Monte Carlo analysis. Speciﬁcally, we

evaluate the in- and out-of-sample predictive performance of the RERUM and common RUM

model speciﬁcations to investigate the biological and regulatory conditions under which these

models provide adequate in- and out-of-sample predictions of ﬁshing behavior within a catch-

share program. We estimate the RERUM and alternative RUM models on each draw from

the data-generating process described in Section 5.1. To judge each estimator’s in-sample

predictive performance across diﬀerent data-generating and sampling environments, we also

draw randomly from the data-generating parameter space (e.g., θ, µ, σ) and the sampling

parameter space (e.g., T, N, S). To evaluate out-of-sample prediction performance, we sim-

ulate the same counterfactual bycatch-reduction policies as in Section 5.2 using estimated

parameters from the RERUM and alternative RUM estimators.14

We note that the RERUM estimator is an unbiased estimator of the true parameters by

construction, so long as the NFXP maximum likelihood algorithm converges to it’s global

maximum. Thus, the Monte Carlo results for the RERUM estimator are useful for ensuring

that the NFXP algorithm works appropriately and for investigating the properties of the

estimator (e.g., precision and identiﬁcation) under realistic data settings.

14Monte Carlo simulations were conducted using Matlab (Version 2019a) with parallel computing (18

workers) running on an Amazon EC2 instance (c4.8xlarge) with an Intel Xeon E5-2666 v3 proces-

sor (2.9 GHz) and 60 GiB of memory. Code for reproducing Monte Carlo results can be found at

https://github.com/mnreimer/RERUM.git.

19

5.3.1. Alternative RUM model speciﬁcations

We consider the following alternative RUM model speciﬁcations, which diﬀer in their

treatment of the shadow cost of quota in the speciﬁcation of a ﬁsher’s optimal location

choice:

Static RUM (SRUM):

αi,t = argmax

a∈A

{θRevp0E(yi,t |a) + θDistDist(a) + εi,t(a)};

Quota-Price RUM (QPRUM):

αi,t = argmax

a∈AθRev (p−wt)0E(yi,t |ai,t) + θDistDist(ai,t) + εi,t (a),

where wt= observed quota-lease prices;

Approximate Rational Expectations RUM (ARUM):

αi,t = argmax

a∈AθRev (p−ˆwt)0E(yi,t |ai,t) + θDist Dist(ai,t ) + εi,t(a),

where ˆws,t =γ0,s +γ0

1,szt+z0

tγ2,szt, z0

t= [x1,t, x2,t , t], s = 1,2,

and xs,t denotes the proportion of remaining ﬂeet-wide quota for species sin period t. The

parameters θ= [θRev, θDist] are the structural preference parameters of interest and are

estimated alongside the vector [γ0,s, γ1,s] and symmetric matrix γ2,s .

The ﬁrst speciﬁcation (SRUM) is a static RUM approach that does not account for

the forward-looking thinking of ﬁshers, and thus, estimates a policy function that does not

deduct the shadow cost of quota from expected revenues. So long as the TAC has a non-

zero probability of binding for at least one species, the SRUM model will underestimate the

expected revenue coeﬃcient θRev. Moreover, to the extent that a location’s distance from

port is correlated with the expected catch of a species with binding quota, the estimate of

the distance coeﬃcient θDist will also be biased (upwards or downwards, depending on the

direction of the correlation).

The second speciﬁcation (QPRUM) represents the approach one would take to address

the bias of the SRUM model if quota-lease prices were observed—that is, include the observed

prices wtdirectly into the policy function. We consider two versions of this approach, one

which uses the period-speciﬁc quota-lease prices wt(QPRUM1, the best-case scenario) and

another which uses the seasonal average quota price ¯w(QPRUM2, a more likely scenario).

20

The third speciﬁcation (ARUM) attempts to address the bias of the SRUM model with-

out the luxury of having quota-lease prices. Speciﬁcally, the ARUM model introduces a

reduced-form quadratic approximation of quota-lease prices by interacting expected catch

with observed state variables meant to reﬂect the scarcity of quota, including the proportion

of remaining quota xs,t and time period t.15 Similar approaches have been followed previ-

ously, for example, to estimate the implicit cost of ﬂeet-wide bycatch quotas (Abbott and

Wilen,2011) and to estimate the extent of cooperation in a common-pool ﬁshery (Haynie

et al.,2009). The ARUM model approximates the shadow value of quota using both species’

cumulative catch information. Note that without temporal variation in the ex-vessel price p,

it is not possible to identify the constant γ0,s in the ARUM model. In practice, it is rare to

observe within-season variation in prices; thus, we omit γ0,s from the ARUM speciﬁcation,

and note that only the diﬀerences in quota prices wacross the state space are identiﬁed, as

opposed to the absolute level of quota prices. As we discuss below, this has implications for

identifying the structural parameter θRev, but has no implications for prediction.

5.3.2. Estimation and in-sample performance

For each of 200 independent Monte Carlo draws, we estimate the parameters of the

RERUM and the alternative RUM models, and calculate parameter bias and the root-mean-

squared-error (RMSE) of predicted location-choice probabilities, relative to the true model.

Column 2 of Table 1provides the range of parameter values for the random parameter space

that we sample from (uniformly).

As expected, both the RERUM and QPRUM1 estimators are able to recover the struc-

tural parameters θdue to explicitly accounting for the evolving shadow-cost of quota (either

imputed or observed, respectively) in the estimation process (Figure 3). The QPRUM2

estimator, which accounts for only the seasonal average quota price, also provides a rela-

tively unbiased estimator θRev. In contrast, the SRUM speciﬁcation underestimates θRev , as

predicted for situations in which the shadow cost of quota is strictly positive. The ARUM

speciﬁcation does not improve the estimation performance of θRev over the SRUM because

15We also considered ﬂeet-wide cumulative catch as a state variable, but the proportion of remaining quota

was selected for the ARUM model due to it’s superior predictive performance.

21

it is unable to identify the absolute level of the quota prices (γ0) due to the time-invariant

nature of prices p. Instead, γ0is subsumed into the estimate of θRev, resulting in a underes-

timation of θRev. Moreover, including an approximation of the shadow cost of quota creates

challenges for precision, as reﬂected in the wide distributions of ˆ

θRev for the ARUM speciﬁca-

tion. All ﬁve models have relatively good estimation performance for θDist , which is expected

when the distance from port to areas with high expected catch is symmetric across species.16

Altogether, despite having trouble using variation in observed state variables to identify θRev,

the ARUM model oﬀers an improvement over the SRUM model for in-sample predictions

according to the RMSE of choice probabilities. By contrast, the QPRUM2 estimator does

not provide much improvement over the SRUM estimator for in-sample predictions because,

despite incorporating quota price information into the estimation process, it does not account

for the within-season evolution of the quota shadow costs.

In Figure 4, we investigate whether there are any particular areas of the data-generating

and sampling parameter space in which the RERUM estimator performance is worse at

recovering estimates of θRev. The median bias of θRev for the RERUM estimator is unsur-

prisingly zero across the parameters space; however, heterogeneity in the spread between

the 10th and 90th percentiles indicates that there are some areas of the parameter space

in which the sampling distribution of the RERUM estimator is more diﬀuse. Most notably,

the RERUM estimator tends to perform better when there are a larger number of species S

and a larger level of harvest variance σ2. With more species, there is potential for greater

spatiotemporal variation in “net revenue”—i.e., (p−˜wt)0E(yi,t)—that can be used to iden-

tify θRev, especially if quota prices vary asynchronously over time across species.17 A similar

argument can be made regarding σ2: with low σ2, quota prices tend to be relatively stable

over time, providing less spatiotemporal variation for identifying θRev. In general, Monte

16This symmetry is exhibited, on average, in our Monte Carlo sample since we allow for the spatial overlap

of species to be randomly determined when drawing from the data-generating parameter space.

17As an example, in the extreme case with S= 1, the relative ﬁshing payoﬀs over space do not change over

time because the quota price aﬀects all locations the same, regardless of how much the quota price changes

over time. With more species, the relative payoﬀs do change over time, so long as the quota prices for each

species do not vary synchronously over time.

22

Carlo draws that have small Sand/or small σ2tend to have a ﬂatter log-likelihood function,

resulting in less precise estimates.

We also consider practical issues regarding estimation of the RERUM model. To investi-

gate the potential for convergence issues of the NFXP algorithm, we estimate the RERUM

parameter vector multiple times for each Monte Carlo draw starting from diﬀerent initial

values.18 While the algorithm displays occasional convergence issues, the RERUM estimator

behaves reasonably well, with approximately 90% of the Monte Carlo draws appearing to

converge to a global maximum.19 Convergence issues generally occur under the same con-

ditions that produce a ﬂat log-likelihood function—i.e., when the number of species (S) or

the variance of the stochastic harvesting component (σ2) are small. Measures of estimation

time demonstrate that while the computational burden of the RERUM estimator increases

with the number of observations per year (N×T) and the number of species (S), it does so

at a rate that is more-or-less linear in Sand slightly convex in N×T(Figure A.3).20 Alto-

gether, the computational costs of the RERUM estimator do not appear to be prohibitively

burdensome within the range of sample sizes and numbers of quotas/species encountered by

practitioners on a regular basis.

5.3.3. Out-of-sample Performance

For both forms of policy counterfactuals considered in Section 5.2, we simulate an entire

ﬁshing season with stochastic harvest and state variables that are endogenously updated in

each time period. Fishers make location choices according to their policy-function speciﬁca-

tion (i.e., SRUM, QPRUM, ARUM, or RERUM). For both the ARUM and RERUM models,

18Speciﬁcally, for each Monte Carlo draw, we estimate the RERUM model starting from nine diﬀerent

initial guesses arranged in a grid centered on the true data-generating parameter values. The parameter

vector(s) associated with the largest log-likelihood value is the RERUM estimate.

19The proportion of estimates that converged to the same maximum log-likelihood value is presented in

Figure A.2

20In theory, the computational burden of the RERUM estimator (above that for a static RUM) is a function

of the number of rational-expectations equilibrium quota prices that need to be computed. Let time(T, N , J)

represent the time it takes to solve for a single quota price, which is increasing linearly in the number of

individuals (N), time periods (T), and locations (J) (see equation 5). Then the computation time devoted

to solving for quota prices is equal to time(T, N, J )×T×S×Y rs.

23

the quota-lease price is updated in each period using each model’s respective quota-price

rule. For example, the ARUM model inserts the observed state variables into the quadratic

quota-price approximation function, while the RERUM model updates the quota-lease price

using the observed state variables and solving for the rational-expectations equilibrium quota

prices in equation (6). In contrast, the SRUM and QPRUM models are static, and do not

update each period to reﬂect the evolving shadow cost of quota. The SRUM model uses no

quota prices while the QPRUM models use the observed quota prices from the estimation

sample, essentially considering them exogenous to the counterfactual policies under consid-

eration. For each counterfactual policy, we generate 200 independent draws from the dgp.

Process error is introduced through harvest and utility shocks (ξand ε), which are drawn

from their respective probability distributions. Sampling error is introduced by drawing

utility parameters from simulated sampling distributions, which are generated by estimating

the parameters of the RERUM and the alternative models using 500 independent draws

from the dgp under the baseline policy. More details concerning the process for generating

out-of-sample simulations is contained in Appendix E.

In general, the alternative RUM models perform well in predicting changes in expected

utility for small changes from the baseline policy, but get progressively worse as counter-

factual policies move farther away from the baseline (Figure 5).21 In both scenarios, the

alternative RUM models tend to overestimate the cost of reducing the bycatch TAC. The

SRUM and QPRUM models have no method of accounting for increased shadow prices from

TAC reductions; thus, ﬁshers are predicted to ﬁsh business-as-usual until the season ends

from a binding TAC. As a result, predicted changes in expected utility under the SRUM

and QPRUM models are proportional to bycatch TAC reductions. The ARUM model does

account for changes in bycatch quota scarcity through the approximated quota-lease prices,

and in turn, ﬁshers are predicted to ﬁsh in diﬀerent locations with less expected bycatch.

As a result, early-season endings from hitting the bycatch TAC are avoided and predicted

changes in expected utility are relatively close to the truth, at least for small reductions in

21Given the similarity in the out-of-sample predictions for the QPRUM1 and QPRUM2 models, we only

present the results for QPRUM1.

24

the TAC.

The alternative RUM models tend to do better predicting changes in expected utility

from the hot-spot closures. The performance of the SRUM and QPRUM models tend to

be inferior to the ARUM model, although they are still capable of producing reasonable

predictions for a small number of closures. Predictions from the ARUM model are quite

good for the hot-spot closures, particularly for scenario 2; ARUM predictions are close to

the true model, on average, even for large changes from the baseline. However, sampling

error in the lease-price parameters leads to considerably more variation in the ARUM model’s

prediction error, demonstrating a potential drawback of using the reduced-form approach to

approximate the quota-lease prices.

The out-of-sample predictions we consider here produce two important insights. First,

despite being able to recover structural parameters reasonably well, static RUM models that

incorporate observed quota-lease prices in the estimation process do not produce good out-

of-sample predictions if quota-prices are not allowed to adjust to the market, ecological, or

regulatory conditions of the counterfactual policy. This is true even for policies such as the

bycatch hot-spot-closure policy for scenario 2, which does not induce large changes in quota

prices, on average (Figure 2). The reason lies in the stochastic realizations of production,

which are embodied in the observed quota prices but are not expected to be the same as

those observed in the estimation sample. Thus, quota prices that do not update to reﬂect

the prevailing state-of-the-world under counterfactual policies will not accurately predict

behavior.

Second, RUM models that incorporate a state-contingent, reduced-form approximation

of the quota-price, such as the ARUM, are capable of improving out-of-sample predictions

over static RUM models. However, this improvement is limited to only certain situations.

The reason largely lies in the quota-price responses to the policy change (Figure 2): as quota

prices move further away from those observed in the estimation sample, predictions from

the reduced-form models tend to move further away from the truth. For example, hot-spot

closures in scenario 2 have almost no eﬀect on quota prices. Accordingly, the ARUM model

does very well at predicting out-of-sample in this case since the lease-price parameters of

the ARUM are calibrated to replicate the in-sample behavior under economically equivalent

25

scenarios. In contrast, TAC reductions in scenario 1 have the largest inﬂuence on quota

prices, and in turn, predictions from the ARUM model are only acceptable for small changes

in the TAC.

6. Conclusion

We develop a model of spatiotemporal ﬁshing behavior that incorporates the dynamic and

general equilibrium elements of catch-share ﬁsheries. Our approach extends the traditional

RUM framework for estimating ﬁshing location choices by incorporating a within-season

market for quota exchanges, which determines equilibrium quota-lease prices (or, equiv-

alently, quota shadow costs) endogenously. Our proposed estimation strategy is able to

recover structural behavioral parameters under reasonable sample sizes and speciﬁcations of

the data generating process, even when quota-lease prices are unobserved. We demonstrate

the use of our model for predicting behavioral responses to ﬁshery policies, such as spatial

closures and TAC reductions, within a catch-share ﬁshery and illustrate the importance of

allowing quota-prices to be endogenous for conducting out-of-sample policy evaluations.

Our study provides several insights. First, the inclusion of quota-prices, either observed

or imputed, in the speciﬁcation of RUM models is necessary to identify structural parameters.

However, identifying the structural parameters of the RUM model is not suﬃcient for making

accurate out-of-sample predictions of counterfactual policy changes. Rather, suﬃciency lies

in determining what quota prices would be under the counterfactual policy change. Thus,

even if practitioners observe quota prices and use them to recover the structural behav-

ioral parameters, a model of endogenous quota prices is necessary for counterfactual policy

evaluations. In other words, quota prices themselves are not policy invariant.

Second, in the absence of a structural model for quota-lease prices, a reduced-form ap-

proximation of state-contingent quota-lease prices can perform well in evaluating out-of-

sample policy changes, provided there is adequate quota-price variation in the sample, rela-

tive to the range of price variation induced by the counterfactual policy. Changes in quota

prices reﬂect the realized magnitude of the eﬀect of the policy on economic incentives, and

therefore function as suﬃcient statistics for whether a particular policy/economic/biological

regime is suﬃciently “in sample” to be evaluated using a reduced-form model. The challenge

26

is knowing ahead of time whether a policy change of interest will result in quota-prices that

lie out-of-sample. As we demonstrate in Section 5, even seemingly “marginal” policy changes

can result in large quota-price changes. Without knowing how quota prices will respond to a

policy change, it is hard to determine ex ante whether a reduced-form approach will produce

adequate policy evaluations.

In short, the layering of spatial closures and a host other policies on top of RBM systems

creates unavoidable feedbacks to seasonal quota markets. These prices, or internal shadow

prices for systems that disallow leasing, are the endogenous mechanisms by which RBM alters

the responses of ﬁshers to these scenarios. Our model has shown the crucial importance of

drawing upon structural models of the quota-price determination process for prediction—

whether or not these models are used to estimate ﬁshers’ underlying behavioral parameters.

Failure to do so will fundamentally limit the ability of economists to answer crucial “what

if” questions posed by ﬁshery managers.

27

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33

Tables

Parameter Values

Parameter KnownaRandombDescription

θRev 1 [0.5,1.5] True preference parameter for expected revenue

θDist -0.4 [-0.5,-0.1] True preference parameter for distance

J100 [36,144] Number of locations

N20 [10,40] Number of individual ﬁshers

T50 [25,60] Number of time periods in a year

S2 [1,4] Number of species

Y rs 1 [1,5] Number of years

p(1000,0) [500,1500] Ex-vessel price vector

q10−3[0.15,5.8]×10−3Catchability coeﬃcient

σ23 [0.1,5] Variance of random harvest component (ξ)

T AC (13,7)×10−3[0.8,1.5]×10−3Total allowable catch (proportion of abundance)

aDenotes the parameter values (species-speciﬁc, where applicable) for the data generating process with

known (and ﬁxed) parameter values.

bDenotes the range of parameter values for the data generating process with a random parameter space.

Parameter values are drawn randomly from a uniform distribution.

Table 1: Parameter values and descriptions for the data generating process.

35

Figures

Figure 1: Spatial distribution of expected catch for species 1 (left) and 2 (center) with port located in the

upper left-hand corner in cell [1,1]; expected global production set (right) with the total allowable catch

(black dot and dashed lines).

36

Figure 2: Numerical simulation outcomes—bycatch hot-spot closures (left column) and bycatch TAC re-

ductions (right column) for two biological scenarios (blue and red). The median (solid line) and 25th-75th

percentile range (shaded area) are presented using 200 draws from the data-generating process.

37

SRUM

QPRUM1

QPRUM2

ARUM

RERUM

-160

-140

-120

-100

-80

-60

-40

-20

0

SRUM

QPRUM1

QPRUM2

ARUM

RERUM

-80

-70

-60

-50

-40

-30

-20

-10

0

SRUM

QPRUM1

QPRUM2

ARUM

RERUM

1

2

3

4

5

6

7

8

9

Figure 3: Parameter estimation and in-sample predictive performance—percent bias in utility parameter

estimates (left and center columns); root-mean-square error (RMSE) between estimated and population

choice probabilities (right column). Markers denote median values and error bars denote the 25th and 75th

percentiles. Distributions generated from 200 draws from the data-generating process with random draws

from the data-generating and sampling parameter space.

Figure 4: RERUM parameter bias for θRev across four parameter spaces: number of observations per year

(far left), number of years (mid left), number of species (mid right), and the variance of the stochastic

harvest component (far right). The lines denote quantile regression predictions for the 10th, 50th, and 90th

quantiles. Distributions generated from 200 draws from the data-generating process with random draws

from the data-generating and sampling parameter space.

38

0 15 30 45 60 75

Closed areas (%)

-60

-40

-20

0

20

Prediction Error

0 15 30 45 60 75

Closed areas (%)

-20

-10

0

10

20

30

0 15 30 45 60 75

Reduction in TAC2 (%)

-70

-60

-50

-40

-30

-20

-10

0

Prediction Error

0 15 30 45 60 75

Reduction in TAC2 (%)

-40

-30

-20

-10

0

SRUM ARUM QPRUM RERUM

Figure 5: Out-of-sample prediction errors: percentage change in expected utility. Top: bycatch hot-spot clo-

sures. Bottom: bycatch TAC reductions. Markers denote median values and error bars denote the 25th and

75th percentiles. QP-SRUM model uses period-speciﬁc quota-prices from estimation sample. Distributions

generated from 200 draws from the data generating process and sampling distributions of utility parameter

estimates.

39

Appendix A. Supplementary Figures

Figure A.1: Quota prices in period t= 1 as a function of ex-vessel prices (p1and p2, row 1), total allowable

catches (T AC1and T AC2, row 2), and preference parameters (θRev and θDist, row 3). Dashed lines indicate

the data-generating parameter values.

40

Figure A.2: Global convergence of the RERUM estimator—the proportion of maximum-likelihood searches,

for each draw from the data generating process, that converged to the same maximum. Distribution generated

by 200 independent draws from the data-generating process and 9 initial values for each draw.

41

Figure A.3: RERUM estimation time across four parameter spaces: number of observations per year (far

left), number of years (mid left), number of species (mid right), and the variance of the stochastic harvest

component (far right). The lines denote quantile regression predictions for the 10th, 50th, and 90th quantiles.

Distributions generated from 200 draws from the data-generating process with random draws from the data-

generating and sampling parameter space.

42

Appendix B. Deriving the Last-Period Utility Function

The indirect utility function in period T+ 1 in equation (1) can be derived as follows.

Each agent is endowed with an S×1 vector of quota ωi, which can be used to fund harvests

over the season or be leased in the competitive quota market. The agent buys a vector of

quota qiafter observing their cumulative harvest xi,T +1. The agent’s objective in period

T+ 1 is to maximize utility with respect to consumption c, subject to a budget constraint:

max

c,q u(0, c) subject to c≤w0(ωi−q) + mi;q≥xi,T +1 ,

where the consumption good is the numeraire good whose price is normalized to one, w

denotes a vector of quota lease prices, u(·) is equivalent to the utility function in equation

(1) evaluated at a= 0 (i.e., port), and midenotes agent i’s exogenous component of income.

The constraints act to restrict the agent from consuming more than their net income, while

also ensuring that the owner has enough quota to cover their annual harvests. Assuming that

u0(c)>0 for c > 0 and that miis large enough to allow for positive consumption, then the

budget constraint will be binding and the agent will choose quota such that q∗

i(w) = xi,T +1.

Thus, the agent’s indirect utility function can be expressed as

V(zi,T +1) = u(0, w0(ωi−xi,T +1)) ,

which gives us the indirect utility function for period T+ 1 in equation (1). For supplemental

derivations, it is useful to simplify this expression further as

V(zi,T +1) = u(0) + v(w0(ωi−xi,T +1 ))

=v(w0(ωi−xi,T +1)) ,(B.1)

where the ﬁrst equality follows from the assumption that revenue is additively separable from

the rest of utility and the second equality follows from using location a= 0 as the baseline

choice alternative.

Appendix C. Derivation of the Policy Function

Consider the Bellman equation in (3) given the state of the world zi,t = (xi,t, εi,t), where

we substitute in the assumed utility function (1) and, since catch is not known ex ante, we

43

replace it with the expected catch:22

V(zi,t) = max

a∈Aua, p0Eyi,t a+εi,t(a) + EzV(zi,t+1 )a, zi,t.

To see that the policy function takes the form presented in equation (4), note that the next-

period expected value function in the last ﬁshing period Tcan be written in the following

way:

EzV(zi,T +1)ai,T , zi,T =vw0ωi−Exxi,T +1 ai,T , xi,T

=v(w0(ωi−xi,T )) −vw0Eyyi,T ai,T .

The ﬁrst equality follows from substituting the indirect utility function in period T+ 1 (equa-

tion B.1) into the expectation of the last-period value function, while the second equality

follows from the transition equation, xi,T +1 =xi,T +yi,T , and the linear nature of v(·). Notice

that vw0Eyyi,T ai,T —i.e., the marginal eﬀect of location choice on the value of remain-

ing quota used in the last period—is the only term that aﬀects the optimal location choice in

period T. In contrast, the term v(w0(ωi−xi,T ))—i.e., the value of remaining quota—is sunk

and does not inﬂuence the contemporaneous location choice. Substituting the derivation of

the next-period expected value function into the Bellman equation for the last ﬁshing period

T, we have:

V(zi,T ) = max

ai,T ∈Auai,T , p0Eyyi,T ai,T +εi,T (ai,T )

−vw0Eyyi,T ai,T +v(w0(ωi−xi,T ))

= max

ai,T ∈Au(ai,T ) + vp0Eyyi,T ai,T +εi,T (ai,T )

−vw0Eyyi,T ai,T +v(w0(ωi−xi,T ))

= max

ai,T ∈Au(ai,T ) + v(p−w)0Eyyi,T ai,T

+εi,T (ai,T )}+v(w0(ωi−xi,T ))

= max

ai,T ∈Auai,T ,(p−w)0Eyyi,T ai,T +εi,T (ai,T )

+v(w0(ωi−xi,T )) ,

(C.1)

22This substitution is justiﬁed on the basis that revenues are assumed to be enter linearly into utility (i.e.,

risk neutrality).

44

where we’ve used the fact that utility is linear in revenue and revenues are additively separa-

ble from non-revenue aspects in utility. The optimal location choice in period Tis therefore

deﬁned as:

α(εi,T |w) = argmax

ai,T ∈Auai,T ,(p−w)0Eyyi,T ai,T +εi,T (ai,T ).

Moving to the penultimate ﬁshing period T−1, we can write the next-period expected

value function in the Bellman equation as:

Ez(V(zi,T ai,T −1, zi,T −1=Ex,ε max

ai,T ∈Auai,T ,(p−w)0Eyyi,T ai,T

+εi,T (ai,T )}+v(w0(ωi−xi,T )) ai,T −1, xi,T −1, εi,T −1.

Let Λi,T = max

ai,T ∈Auai,T ,(p−w)0Eyyi,T ai,T +εi,T (ai,T )for notational simplicity. Be-

cause wis considered exogenous by ﬁshers and yis conditionally independent of x, Λi,T is not

inﬂuenced by the location choice ai,T −1. Thus, we can write Ex,ε (Λi,T |ai,T −1, xi,T −1, εi,T −1) =

Eε(Λi,T ) and simplify the next-period expected value function in the Bellman equation as:

Ez(Vzi,T ai,T −1, zi,T −1

=Ex,ε Λi,T +v(w0(ωi−xi,T )) ai,T −1, xi,T −1, εi,T −1

=Ex,ε Λi,T +v(w0(ωi−xi,T −1−yi,T −1)) ai,T −1, xi,T −1, εi,T −1

=−vw0Eyyi,T −1ai,T −1+v(w0(ωi−xi,T −1)) + Eε(Λi,T ).

As in period T, the only component of next-period’s value function that varies with ais its

eﬀect on the value of remaining quota in the ﬁnal period: vw0Eyyi,T −1ai,T −1. Thus,

the optimal decision rule in period T−1 is fully characterized by

α(εi,T −1|w)

= argmax

ai,T −1∈Auai,T −1,(p−w)0Eyyi,T −1ai,T −1+εi,T −1(ai,T −1).

45

Repeated substitution into earlier periods generalizes this result to any decision period t,

giving us the optimal decision rule in equation (4). Ultimately, it is the conditional indepen-

dence assumption for y, the assumption that utility is linear in revenue (and therefore also

additively separable in non-revenue components in utility), and the assumption that ﬁshers

consider their eﬀect on the quota price wto be negligible that allow us to reduce a ﬁshers

optimal decision rule to something tractable and easily solvable (conditional on w).

Appendix D. The Nested Fixed-Point (NFXP) algorithm

Appendix D.1. Inner algorithm: the ﬁxed-point problem

A rational expectations equilibrium for the inner algorithm is a vector-valued function

of quota prices w(xt|θ) that solves the market clearing conditions in (6) subject to ﬁshers

making their optimal ﬁshery choices according to equation (4) for a given vector of structural

parameters θ. Our goal is to ﬁnd ˜w(xt|θ) such that:23

F( ˜w(xt|θ)) = max {E(es|˜w(xt|θ), xt),−˜w(xt|θ)}= 0 ∀s∈ {1, ..., S },(D.1)

where esis the end-of-season excess demand function for species squota. Since we are

solving for Squota lease prices that satisfy Sequilibrium equations, the system of equations

in (D.1) is just identiﬁed.

Appendix D.1.1. Algorithm

Consider an arbitrary initial vector of quota prices w0. Then the rational equilibrium

quota prices ˜w(xt|θ), conditional on a given vector of structural parameters θ, can be

determined by the following algorithm:

1. For each time period tin the data, use the observed state variable xtto calculate the

cumulative ﬂeet-wide catch for each species, Xs,t.

2. Calculate the choice probabilities f(ai,t |xt, w0).

23This is actually a complementarity problem, as opposed to a ﬁxed-point problem. See page 44 in Miranda

and Fackler (2002) for more details.

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3. Calculate the expected end-of-season excess demand E(es|w0, xt) for each species

s∈ {1, ..., S}using Xs,t from step 1 and f(ai,t |xt, w0) from step 2.

4. Given the expected excess-demand functions from step 3, compute the system of equa-

tions F(w0) in (D.1).

5. In general, F(w0) will not equal 0, as required by the equilibrium conditions in (D.1).

Generate a new value of w, say w1, using a Newton step (or some other method).

6. Repeat steps 2 to 5 until F(wk) = 0.

7. Repeat steps 2 to 6 for all time periods tin the data.

8. Use the resulting equilibrium quota-price vector ˜w(xt|θ) to calculate the rational

expectations choice probabilities (equation 9) and pass them to the outer algorithm.

Appendix D.2. Outer algorithm: maximum likelihood estimation

The goal of the outer algorithm is to ﬁnd a value for the vector of parameters ˆ

θthat max-

imizes the log-likelihood function P∀ili(θ) while allowing the rational-expectations quota

price ˜w(xt|θ) to be endogenous to the structural parameter vector θ. Consider an arbitrary

value of θ, say ˆ

θ0. Then NFXP maximum likelihood parameter ˆ

θis determined as follows:

1. Pass ˆ

θ0to the inner algorithm, which will return the choice probabilities nfai,t |xt,ˆ

θ0o∀i,t .

2. Use the choice probabilites in step 1 to evaluate the log-likelihood l(ˆ

θ0) = P∀ili(ˆ

θ0)

and it’s gradient, where li(·) is given in equation (8).24

3. Use the gradient from step 2 to obtain a new structural parameter vector, say ˆ

θ1.

4. Repeat steps 1 through 3 until either ˆ

θkor l(ˆ

θk) converges based on a pre-speciﬁed

convergence tolerance.

Appendix E. Out-of-Sample Policy Simulations

The out-of-sample policy simulations presented in Section 5.3 are generated in the fol-

lowing way. We ﬁrst generate sampling distributions for the structural parameter estimates

24While the gradient of the log-likelihood function, conditional on w, has a closed-form expression under

the DP conditional logit assumptions, the gradient of w(xt|θ) does not; thus, the gradient of the log-likelihood

function must be computed using numerical methods. This means that each time θis ‘perturbed’ to obtain

the numerical gradient, a new solution for the rational-expectations quota prices is required.

47

ˆ

θand ˆγ(where applicable) under the baseline policy scenarios using the data-generating

parameter values reported in Column 1 of Table 1. The sampling distributions are created

using 500 independent samples from the dgp, where draws diﬀer due to harvest and util-

ity shocks (ξand ε). To simulate outcomes under the counterfactual policies, we use the

following procedure:

1. For each counterfactual policy (including the baseline) and RUM model (including

the RERUM model): draw parameter values from their respective simulated sampling

distribution, draw harvest and utility shocks from the dgp, and simulate an entire

ﬁshing season.

2. Compare a model’s simulated outcome against its baseline counterpart to generate a

“relative impact”. For example, for the SRUM model under the 5% TAC reduction

policy, compare a ﬁsher’s expected utility u(α(zi,t), p0E(yi,t |α(zi,t)) from draw 1 to the

expected utility predicted by the SRUM model for draw 1 under the baseline policy.

3. Compare a model’s relative impact against the relative impact from the true model to

come up with the “impact prediction error.”

4. Repeat Steps 1-3 200 times.

Note that for a given draw, the set of harvest and utility shocks are the same for each of the

counterfactual policies and RUM models, so the only diﬀerences across policies and models

are the policy and model parameters.

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