NATURAL RESOURCE M ODELLINGVolum e 21, Number 1, Spring 2008

A STATE SPACE BIOECONOMIC MODEL OF PACIFICHALIBUT

KEITH R. CRIDDLESchool of Fisheries and Ocean Sciences

University of Alaska FairbanksJuneau, AK 99801

E-mail: Keith.Criddle@uaf.edu

MARK HERRMANNDepartment of Economics

University of Alaska FairbanksFairbanks, AK 99775-6080

E-mail:ffmlh@uaf.edu

Abstract. Evaluation of potential economic conse-quences of alternative management actions requires an un-derstanding of how the biological stock will be affected by themanagement action and an understanding of the response ofeconomic systems to changes in the timing, magnitude, andsize distribution of harvests and changes in the location andcatchability of the biological stock. We use a hybrid structuraltime series model to represent Pacific halibut (Hippoglossusstenolepis) stock and recruitment dynamics and a system ofstructural equations to represent supply and demand relation-ships for Pacific halibut from Alaska and British Columbia.Model simulations explore the economic effects of changes inrecruitment success, growth rate, and carrying capacity, andchanges in international supplies of halibut.

Key Words: Pacific halibut, population dynamics,bioeconomics, state space model.

1. Introduction. US and Canadian stocks of Pacific halibut (Hip-poglossus stenolepis) are jointly managed under a bilateral treaty—theHalibut Convention of 1923. Under the treaty, responsibility for con-ducting stock assessments and recommending conservation measuresis delegated to the International Pacific Halibut Commission (IPHC).Each nation is responsible for allocating catches among its varioususer groups while ensuring that the sum of commercial, sport, andother removals within its jurisdictional waters does not exceed theregion-specific IPHC quotas. The organization and management of the

Received by the editors on September 23, 2005, and in revised form on April 3,2006

Copyright c©2008 B lackwell Publish ing, Inc.

117

118 K.R. CRIDDLE AND M. HERRMANN

commercial fishery for halibut has recently metamorphosed from regu-lated open access to dedicated access privileges. This transition beganwith the introduction of individual vessel quotas (IVQs) for commer-cial fishing vessels in British Columbia in 1991, followed by individualfishing quotas (IFQs) for commercial fishing vessel owners in Alaska in1995; a proposal to extend IFQs to the sport-charter fleet in Alaska iscurrently under consideration.

Although theoretical arguments for the economic merits of dedicatedaccess privileges have been established by, among others, Moloney andPearse [1979], Pearse [1980], Morey [1980], Wilen [1985], Stollery [1986],and Scott [1988], other authors have expressed concerns about the fea-sibility of developing a system of dedicated access privileges capable ofyielding maximal economic benefits over time. For example, Johnsonand Libecap [1982], Keen [1983], Boyce [1992], and NRC [1999] ar-gue that although dedicated access privileges may abate within-seasonincentives to race for catches, the absence of right to capital (Honore[1961]) may lead to stock externalities as individuals are unable to fullycapture the benefits of actions taken to conserve stocks. Matulich etal. [1996] and Matulich and Sever [1999] suggest that dedicated accessprivileges allocated to harvesters disadvantage processors who have in-vested in nonmalleable capital. NRC [1999] notes that crew, skippers,expediters, shippers, input suppliers, and communities may have simi-lar sunk investments in physical, human, and social capital that couldbe similarly disadvantaged. Criddle and Macinko [2000] suggest thatrent seeking may dissipate the net benefits of dedicated access privilegesif the initial allocation is realized through political processes. Macinko[2005] faults the permanent allocation of dedicated access privilegesas unnecessary to ending the race for fish and a stimulus for waste-ful rent seeking. However, although this argument has merit in so faras it applies to the within-year race for fish, NRC [1999] notes thatshort-term durability of capital can exacerbate stock externalities andlead to suboptimal investment in fishing, processing, and related infras-tructure. Ultimately, determination of the magnitude of net economicbenefits of dedicated access privileges is an empirical question, one thathas been difficult to tease out of background variation in stock abun-dance and markets, and general lack of before and after data on thesocial and economic wellbeing of participants in the fishery and theircommunities.

A STATE SPACE BIOECONOMIC MODEL 119

Although earlier studies by Crutchfield and Zellner [1962], Cook andCopes [1987], Lin et al. [1988], NPFMC [1991], Homans [1993], Schell-berg [1993], Criddle [1994], and Love et al. [1995] offer some helpful in-sights into the operation of the regulated open access fishery for Pacifichalibut and the changes anticipated under dedicated access privileges,they are not useful descriptions of the fishery following IFQ implemen-tation and offer, at best, rudimentary models of the exvessel marketstructure. Postimplementation analyses of the Canadian (Conklin andKolberg [1994], Casey et al. [1995], Homans and Wilen [1997], Wilenand Homans [1998]) and Alaskan (Dinneford et al. [1999], NRC [1999])fisheries ignore or offer similarly naıve models of the economic behaviorof markets for Pacific halibut. Herrmann [1996], Herrmann [2000], Mat-ulich and Clark [2003], and Herrmann and Criddle [2006] provide morecomprehensive representations of market relationships, but they do notdirectly characterize halibut population dynamics. To model optimalcapital investment strategies, Singh et al. [2006] combine informationon the exvessel demand for halibut from Herrmann and Criddle [2006]with information on harvesting costs and capital adjustment costs, andquadratic stock and recruitment functions with an adjustment for se-rial correlation, and conclude that constant catch policies are preferredto constant exploitation rate strategies when capital adjustment iscostly.

The wealth of available biological data on Pacific halibut and thescientific foundation of management espoused in the structure andoperation of IPHC have served as stimuli for innovative develop-ments in population modeling. Clark [2003] recounts key concep-tual and practical innovations in population modeling at the IPHCfrom its inception to the present. Quinn [2003] provides a retro-spective review of the naissance of population modeling and theevolution of models of fish population dynamics. With apologies toClark, Quinn, and the many other unnamed titans upon whose shoul-ders we stand, we will brush aside the critical questions of esti-mating biomass and catches from fisheries-dependent and fisheries-independent data sources and focus instead on the simpler task ofusing those uncertain estimates of biomass and catches to model thedynamics of the halibut population and the bioeconomics of the hal-ibut fishery. Following Berck and Johns [1985], Noakes et al. [1987],and Criddle and Havenner [1989, 1991], we characterize population

120 K.R. CRIDDLE AND M. HERRMANN

dynamics as the realization of observable and latent time-dependentprocesses.

2. The Population Model. Pacific halibut is managed as a sin-gle stock throughout its range in United States and Canadian waters.Although Pacific halibut are also found in the Western Bering Sea,Western North Pacific Ocean, and the Sea of Okhotsk, there is no directcooperation in the management of halibut stocks among the nationsbordering those waters and the United States and Canada. Moreover,estimates of halibut populations suggest that the center of abundanceis in the Eastern Pacific, Gulf of Alaska, and Eastern Bering Sea, wa-ters covered by the Halibut Convention. Consequently, the biomass ofhalibut can be represented as:

Xt =n∑

i=1

xit ,

where Xt is the coastwide biomass, xit is the biomass of halibut in areai in time t, and n represents the number of IPHC management areas2A, 2B, 2C, 3A, 3B, and 4A-E (with 2A = 1, 2B = 2, and so on)(Figure 1).

Because of data limitations, IPHC stock assessments have been basedon models of the dynamics of the stock in the center of its distribution(areas 2B, 2C, and 3A) expanded to the full range based on surveycatch rates (Sullivan et al. [1999]). Thus biomass can be represented as

Xt = Xt + ut,

where

Xt =4∑

i=2

xit ,

and

ut = Xt − Xt = x1t + x5t + x6t + x7t + x8t + x9t + x10t .

A STATE SPACE BIOECONOMIC MODEL 121

FIGURE 1. IPHC regulatory areas.(Source:http://www.iphc.washington.edu/halcom/images/RegAreasbig.gif.)

The biomass of fish populations changes over time as a result ofgrowth, recruitment, predation and other natural mortality, and inten-tional and incidental harvesting. Unfortunately, although harvesting is,in principle, observable and current biomass can be sampled, naturalmortality, predation, and recruitment are latent variables; we relatethese latent variables to present and past values of the observables.Our model comprises four equations: a stock dynamics equation, a re-cruitment equation, and two matrix equations that use state variablesto represent dynamic processes that result from specification error inthe stock dynamics and recruitment equations.

Stock dynamics are modeled as a delay-difference process describedby:

Xt = β1Xt−1 + β2X2t−1 + β3rt − ht−1 + υt,(1)

where rt is the number of age-8 recruits in millions, ht−1 is the sumof direct and incidental commercial, sport, and personal use removals,and υt is a normal random variable. Equation (1) can be expressedin terms of the explicitly and implicitly modeled components of the

122 K.R. CRIDDLE AND M. HERRMANN

population:

Xt + ut = β1(Xt−1 + ut−1) + β2(Xt−1 + ut−1)2 + β3rt − ht−1 + υt,

or

Xt = β1Xt−1 + β2X2t−1 + β3rt − ht−1 + υt − ut

+β1ut−1 + β2(2Xt−1ut−1 + u2

t−1)

= β1Xt−1 + β2X2t−1 + β3rt − ht−1 + εt ,

(2)

where εt includes contemporaneously and serially correlated sample,observation, and specification error processes.

Clark and Hare [2002] model halibut recruitment as a function oflagged values of biomass and latent processes that are correlated withlagged values of the Pacific decadal oscillation (PDO), a proxy forchanges in oceanographic conditions that affect halibut recruitment,growth rates, or carrying capacity. Rather than impose a particularfunctional form and dynamic structure on the linkage between recruit-ment and the PDO, we have chosen to model recruitment using a Rickerfunction

ln (rt) = ln(Xt−8) + γ0 + γ1Xt−8 + ηt ,(3)

and to include the PDO as a covariate in a multivariate time serieserror-correction model. The advantage of our approach is that we donot need to assume a prior specification of the functional form or laglength of the association between recruitment and the PDO. In addi-tion, our approach also allows for possible direct associations betweenthe PDO and the dynamics of the halibut stock. Moreover, our ap-proach does not imply that there is a direct causal relationship betweenthe PDO and stock abundance or recruitment, but is instead consistentwith the use of the PDO as a proxy for latent processes that influencethe dynamics of recruitment and stock abundance.

Although the residuals to equations (2) and (3) are characterized bycomplex contemporaneous and serial correlations, they can be regardedas draws from stationary stochastic processes. Because even nonlinear

A STATE SPACE BIOECONOMIC MODEL 123

stationary time series have linear state space representations (Aoki[1990]), the dynamic factors that are important in the residuals ofequations (2) and (3) can be modeled using innovation form equations(4) and (5):

ωt = ( ηt εt PDOt )′ = Czt + et ,(4)

zt+1 = Azt + Bet .(5)

Equations (4) and (5) form a system of matrix equations that canbe solved for latent state variables, zt , that represent dynamic factorspresent in the covariates, ωt . The state variables are unobservable butcan be determined after the model parameters have been estimatedand are constructed to be minimum sufficient statistics for the pastrealizations of the ωt . The number of states can be less than, equalto, or greater than the number of series modeled depending on thedegree to which the series are correlated and on the complexity of theunderlying dynamics. Equation (4) projects the residuals of equations(2) and (3) and the PDO onto the latent state variables. Equation(5) describes the dynamics of the state variables and is written as afirst-order difference equation because equations (4) and (5) can beaugmented to first order by defining states equal to the lag of otherstates. The matrices A and B represent the intertemporal dynamiclinkages among state variables. The deterministic component of systemdynamics are embodied in A, whereas B is analogous to the Kalmanfilter update matrix. Because the states extract all of the informationin the past, only serially uncorrelated innovations (εt) orthogonal tothe states remain. The solution to equations (4) and (5) is developedin Aoki [1990] and Aoki and Havenner [1991].

Because the coefficients in equations (2) and (3) are correlated withtheir residuals, efficient estimation requires joint estimation of thecoefficients and covariances. We resolved this problem by iteratingequations (2) to (5) to convergence in a manner analogous to thecoefficient-covariance iteration typically used to solve systems of seem-ingly unrelated regressions. With estimates of the elements in βi , γi ,

124 K.R. CRIDDLE AND M. HERRMANN

A, B, and C, and a set of initial conditions z0 , equations (2)–(5), canbe solved for estimates of the state variables, biomass, recruitment, andthe PDO.

The data used for estimating the parameters of the population modelare included in Table 1 and Tables 2–4, in the Appendix. The resultswe report are based on 75 iterations with convergence to eight decimalsin estimates of the elements of the coefficient and covariance matrices.The coefficient estimates for equation (2) are

Xt = 1.3356(0.0402)

(Xt−1

)− 6.488 × 10−4

(8.318×10−5 )(X2

t−1)

+ 8.9643(1.8950)

(rt) − ht−1 + εt R2 = 0.995.(6)

All of the estimated coefficients of the stock dynamics equation havep-values less than 0.01. The value of R2 represents the fit of equation(2) conditioned on the fit to equations (3), (4) and (5). Without theerror correction embodied in equations (4) and (5), the fit to equation(2) is reduced to R2 = 0.994. The coefficient estimates for equation (3)are

ln(rt) = ln(Xt−8)−3.1170(0.0960)

−3.009 × 10−3

(2.868×10−4 )Xt−8 + ηt ,(7)

or

rt = Xt−8exp(−3.1170 − 3.009 × 10−3Xt−8 + ηt

)× exp (υt) R2 = 0.519,

(7′)

where

υt =( 1

2

)σ2 −

( 12

)σ2 (

1 Xt−8) (

n XX′ X′X

)(1

Xt−8

),

σ = 0.141, and X is the matrix of observed values of biomass. The valueυt is a transformation necessary to eliminate a bias that results fromusing ordinary least squares to obtain estimates of the coefficients of anexponential model (Kennedy [1983]). All of the estimated coefficientsof the recruitment equation have p-values less than 0.01.The value ofR2 represents the fit to equation (3) conditioned on the fit to equations

A STATE SPACE BIOECONOMIC MODEL 125

(2), (4), and (5). Without the error correction embodied in the statespace equations, the fit of the recruitment equation is R2 = 0.087. (Theuncorrected fit of equation (3), expressed in natural logarithms, is R2

= 0.682.)

The state space time series equation system was specified with a sys-tem lag length of 4 years to allow lags of up to 15 years on the individualtime series. Specification of the state space equations (equations [4] and[5]) consists of determining the minimum number of states needed toreflect the latent dynamic processes. Aoki [1990] demonstrates that thenumber of states needed to represent the latent dynamics is equal tothe rank of a matrix of autocorrelations among the modeled time se-ries. If the autocorrelations were observed exactly, any singular valuesof the autocorrelation matrix beyond the model order would be exactlyzero. Because the autocorrelations are sample estimates, they containsampling error, as do the singular values of the sample autocorrelationmatrix they make up. Following Aoki and Havenner [1991], we selectedthe number of singular values to be included in the model such thatthe ratio of the first excluded singular value to largest singular value ison the order of the square root of the number of observations, a crite-rion analogous to the condition number. For our model, we determinedthat there were three nonzero singular values and thus three uniquelatent dynamic processes. Based on this determination, the coefficientestimates for equations (4) and (5) are:

ωt =

(ηt

εt

PDOt

)=

(−0.3969 0.0423 0.8187−3.4244 −5.2755 −2.1925−0.5778 0.0342 −0.0984

)zt + et

zt+1 =

( 0.3387 −0.6356 −0.15510.7526 0.1342 0.0334

−0.1170 −0.6265 0.2623

)zt

+

(−0.4406 −0.0356 −0.47340.2788 0.0089 0.61370.5775 0.0227 −0.3676

)et .

Consideration of the estimated C matrix suggests that the resid-uals of the structural equations and the PDO are all related to all

126 K.R. CRIDDLE AND M. HERRMANN

three of the state variables. This suggests that the PDO reflects latentprocesses present in the residuals of stock dynamics and recruitmentequations and that models that explicitly represent the PDO in the re-cruitment equation should also include the PDO in the stock dynamicsequations. Examination of the estimated elements of the B matrix in-dicates that errors in the estimates of the residuals of the structuralequations and the PDO include information that is useful for adjust-ing future values of the state variables and thus predictions of stockabundance, recruitment, and the PDO. Together, the estimated ele-ments of the B and C matrices suggest that the two structural equa-tions and the PDO time series form an integrated system and thatwould not be as effectively modeled as a system of three independentequations.

The A matrix describes the dynamic linkages among the state vari-ables. The three eigenvalues of A are:

Eigenvalues of A Moduli

0.1576 ± 0.7024i 0.71990.4200 0.4200

Because the moduli of the roots of A are less than one, the dynam-ics of the residuals to equations (2) and (3) and the PDO reflect adampened oscillation. The persistence of the latent dynamics can berepresented by allowing the final values of the estimated states to decayfollowing a one-time perturbation. The decay paths of the state vari-ables suggest that dynamic adjustments of the estimates of recruitmentand stock biomass to account for the influence of biophysical processescorrelated with the PDO and to account for estimation errors havelittle effect beyond 5 years.

The expected value of steady-state biologically sustainable yields andthe expected value of recruitment levels are represented in Figure 2.The coastwide expected steady-state sustainable yield reaches a maxi-mum of 91 million pounds at population biomass of about 280 millionpounds.

A STATE SPACE BIOECONOMIC MODEL 127

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Sus

tain

able

Yie

ld (

mil

lion

po

unds

)

0

2

4

6

8

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Age

8 R

ecru

its

(mil

lion

)

FIGURE 2. Coastwide sustainable yields (million pounds) and recruitment(millions) of Pacific halibut.

The ocean environment is dynamic, and the productivity of natu-ral species is subject to interannual and decadal scale variation. Themagnitude of recruitments is dependent on the population of halibut.Consequently, the expected recruitments associated with steady-statebiomass levels will also change as ocean productivity changes. Changesin the productivity of the ocean environment affect the magnitude ofsustainable harvests and associated revenues. Figure 3 represents a 10%increase and a 10% decrease in the carrying capacity for halibut in theNorth Pacific.

Simulated increases (decreases) in ocean carrying capacity leadto increases (decreases) in the expected value of sustainable yield

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Sus

tain

able

Yie

ld (

mil

lion

lbs)

0

2

4

6

8

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Age

8 R

ecru

its

(mil

lion

)

FIGURE 3. Coastwide sustainable yields (million pounds) and recruitment(millions) of Pacific halibut under baseline (thick line) conditions and con-ditions that result in ±10% changes in ocean carrying capacity for Pacifichalibut.

128 K.R. CRIDDLE AND M. HERRMANN

0

2

4

6

8

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Age

8 R

ecru

its

(mil

lion

)

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Age 8+ Biomasss (million pounds)

Sus

tain

able

Yie

ld (

mil

lion

lbs)

FIGURE 4. Coastwide recruitment (millions) and sustainable yields (millionpounds) of Pacific halibut under baseline conditions (thick line) and conditionsthat result in a ± 10% change in recruitment success.

and increases (decreases) in the expected value of adult biomass re-quired to produce the maximum sustainable yield. Simulated increases(decreases) in ocean carrying capacity lead to increases (decreases)in the expected number of recruits, but do not result in perceptiblechanges in the expected value of adult biomass required to producethe maximum number of recruits.

Instead of altering the overall carrying capacity for halibut, changesin the ocean environment could affect recruitment success and the mag-nitude of sustainable yields. The expected recruitments and sustainableyields associated with baseline and 10% increases or decreases in re-cruitment success are represented in Figure 4.

Simulated increases (decreases) in recruitment success lead to in-creases (decreases) in the expected number of recruits but do notresult in perceptible changes in the expected value of adult biomassrequired to produce the maximum number of recruits. Simulated in-creases (decreases) in recruitment success lead to increases (decreases)in the expected value of sustainable yield and increases (decreases) inthe expected value of adult biomass required to produce the maximumsustainable yield.

Yet another way that halibut could be affected by changes in theocean environment is for the environmental changes to lead to changesin the growth rate of halibut. Figure 5 represents the steady-state base-line sustainable yields and associated expected recruitment levels for10% increases and decreases in the growth rate of halibut.

A STATE SPACE BIOECONOMIC MODEL 129

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Sus

tain

able

Yie

ld (

mil

lion

lbs)

0

2

4

6

8

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Age

8 R

ecru

its

(mil

lion

)

FIGURE 5. Coastwide sustainable yields (million pounds) and recruitment(millions) of Pacific halibut under baseline conditions (thick line) and condi-tions that result in 10% changes in the growth rate of Pacific halibut.

Simulated increases (decreases) in the growth rate of Pacific halibutlead to increases (decreases) in the expected value of sustainable yieldbut do not result in perceptible changes in the expected value of adultbiomass required to produce the maximum sustainable yield. Simu-lated increases (decreases) in the growth rate of Pacific halibut leadto increases (decreases) in the expected number of recruits but do notresult in perceptible changes in the expected value of adult biomassrequired to produce the maximum number of recruits.

Changes in the PDO are also associated with changes in recruit-ment and biomass. Figure 6 represents the steady-state baseline sus-tainable yields and associated expected recruitment levels for +1◦ and−1◦ changes in the PDO.

0

2

4

6

8

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Age

8 R

ecru

its

(mil

lion

)

0

20

40

60

80

100

120

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

Sus

tain

able

Yie

ld (

mil

lion

lbs)

FIGURE 6. Coastwide sustainable yields (million pounds) and recruitment(millions) of Pacific halibut under baseline conditions (thick line) and condi-tions that result from +1◦ or −1◦ anomalies in the PDO.

130 K.R. CRIDDLE AND M. HERRMANN

British Columbia halibut capture fishery

U.S. (Alaska and Washington) Pacific halibut capture fishery

U.S. market

U.S. inventories U.S. processing Canadian processing

Other minor markets

Other minor supplies

FIGURE 7. Product and financial flows in the Alaska and British Columbiahalibut fisheries. Solid arrows represent product flows and dashed arrows rep-resent financial flows. Light gray flows not modeled. Source: Herrmann andCriddle [2006].

Simulated positive (negative) anomalies in the PDO result in de-creases (increases) in expected recruitment and increases (decreases)in the expected value of sustainable yield but do not result in percep-tible changes in the expected value of adult biomass required to pro-duce the maximum sustainable yield. These changes in the PDO areconsistent with the PDO serving as a proxy for latent environmentalprocesses that influence the growth rate of halibut and are consistentwith observed variations in halibut size at age.

3. The Market Model. Alaska and British Columbia are theprincipal sources of Pacific halibut and the U.S is the principal marketfor halibut (Herrmann [1996, 2000]) (Figure 7). Consequently, it will beassumed that the world price of Pacific halibut is determined in the USmarket. Although harvest limits are set by the IPHC, actual harvestscan be less than the upper limit, and the allocation of these harvests(to immediate consumption or inventory) is also an endogenous choice.The structure and estimation of the parameters of the market modelare described in Herrmann and Criddle [2006]. The economic modelcan be characterized as a flow of products and payments.

The principle product and financial flows can be represented by ajointly determined system of six behavioral equations and six marketclearing identities. Based on observations from 1976 to 2002, the coef-ficients were estimated using three-stage least squares (3SLS), a full-information systems method in that uses a variance–covariance matrix

A STATE SPACE BIOECONOMIC MODEL 131

to adjust coefficient estimates across all of the structural equations.The equations presented below include the estimated standard errorsin parentheses below the corresponding parameters.

3.1 Demand and supply equation system. The US-derivedwholesale demand for Pacific halibut is modeled with the real Pa-cific halibut wholesale price being a linear function of US per-capitaconsumption of Pacific halibut from domestic and imported sources(British Columbia), Alaskan Pacific halibut season length (area 3A),the real producer price index for meat, the real producer price index forfuel, and real per-capita income (standard errors are shown in paren-theses):

PUSR Wt = − 0.0094

(0.0060)− 0.0356

(0.0084)

(QCUS

t

)− 0.1213

(0.0581)

(QCBC

t

)+ 8.107 × 10−6

(5.339×10−6 )

(SEASAK

t

)+ 0.0357

(0.0070)

(PPIUS MeatR

t

)− 0.0141

(0.0022)

(PPIUSFuelR

t

)+ 87, 716

(22,662)

(INCUSR

t

)R2 = 0.871.

(8)

US Pacific halibut inventory demand is modeled as a linear functionof US landings, current and lagged US wholesale prices, and Alaskaseason length:

INV USt = 5.515 × 106

(4.078×106 )+ 0.2334

(0.0342)

(LAN US

t

)− 7.781 × 108

(2.378×108 )

(PUSW

t

)+ 6.559 × 108

(2.167×108 )

(PUSW

t−1)− 22, 032

(7,059)

(SEASAK

t

)R2 = 0.446.

(9)

British Columbian exports of Pacific halibut to the United States aremodeled as a linear function of British Columbia landings and the realUS import price of British Columbian halibut:

QSBCt = − 4.614 × 106

(1.344×106 )+ 0.5871

(0.1650)

(LANBC

t

)+ 2.211 × 108

(3.431×107 )

(P ImportR

t

)R2 = 0.795.

(10)

132 K.R. CRIDDLE AND M. HERRMANN

Real import prices of Pacific halibut from British Columbian aremodeled as a linear function of US real wholesale prices and the seasonlengths in British Columbia and Alaska:

P ImportRt = − 0.0044

(0.0015)+ 1.1745

(0.0817)

(PUSRW

t

)+ 2.390 × 10−5

(2.818×10−6 )

(SEASBC

t

)− 1.330 × 10−5

(3.259×10−6 )

(SEASAK

t

)R2 = 0.946.

(11)

The exvessel derived demand for Pacific halibut from Alaska is mod-eled with Alaska Pacific halibut real exvessel prices as a linear functionof US real wholesale prices, Alaskan landings, and the real producerprice index for fuel:

PAKExvRt = 8.000 × 10−5

(0.0022)+ 0.9008

(0.0733)

(PUSRW

t

)− 3.090 × 10−11

(1.530×10−11 )

(LANAK

t

)− 0.0027

(0.0019)

(PPIUSFuelR

t

)R2 = 0.899.

(12)

The exvessel derived demand for Pacific halibut from BritishColumbia is modeled with Canadian Pacific halibut real exvessel pricesas a linear function of the natural logarithm of the import price forBritish Columbian halibut, the natural logarithm of Canadian real en-ergy prices, and Alaska season length.

PBCExvRt = 0.1328

(0.0090)+ 0.0305

(0.0025)

(ln

(P ImportR

t

))− 0.0165

(0.0020)

(ln

(CPICanFuelR

t

))− 1.521 × 10−5

(6.947×10−6 )

(SEASAK

t

)R2 = 0.882.

(13)

3.2 Market clearing identities. US per capita consumption ofdomestic halibut is defined as the quotient of US landings and changesin domestic inventories and the US population:

A STATE SPACE BIOECONOMIC MODEL 133

QCUSt =

(LANAK

t + LANWAt + INV US

t−1 − INV USt

)/POPUS

t .(14)

US per capita consumption of imported halibut is defined as the quo-tient of British Columbian exports to the US and the US population:

QCBCt = QSBC

t

/POPUS

t .(15)

The real wholesale price of halibut is defined as the quotient of thenominal wholesale price and the US producer price index for interme-diate food and feeds:

PUS R Wt = PUS W

t

/PPIUS IFF

t .(16)

The real exvessel price of Alaskan halibut is defined as the quotientof the nominal exvessel price and the US producer price index for in-termediate food and feeds:

PAK Exv Rt = PAK Exv

t

/PPIUS IFF

t .(17)

The real import price for British Columbian halibut is defined as thequotient of the nominal import price for halibut and the US producerprice index for intermediate food and feeds:

P Import Rt = P Import

t

/PPIUS IFF

t .(18)

The real export price for British Columbian halibut is defined asthe product of the exchange rate ($/CDN$) and the US import pricedivided by the Canadian consumer price index for food:

PBC Exp Rt = (EXCHt) P Import

t

/CPICan Food

t .(19)

The market model performed well on individual equations and as asystem, with the correlation between actual and predicted variable lev-els ranged between 0.76 and 0.96 for simulations of the full equation

134 K.R. CRIDDLE AND M. HERRMANN

system. (See Herrmann and Criddle [2006] for a detailed descriptionof the variables, coefficient estimates, and model performance statis-tics.)

3.3 Total revenues and optimal static harvest levels. Theexvessel revenues that fishermen receive depend on the overall catchquotas set by the IPHC and the fractions of those quotas harvested inthe commercial fishery, the conditions under which shares of the over-all quota are allocated to fishermen, the relative bargaining strengthof fishermen with buyers (normally processors), the cost of processingand delivering the processed product to end-markets, and the strengthof the market for the final product. Estimates of the static total rev-enue curve can be obtained by simulating changes in Pacific halibutprices given changes in Pacific halibut landings from 2002 baseline lev-els. As landings vary, revenues vary as a direct consequence of changesin harvest levels and as a consequence of changes in exvessel pricesas simulated from all the interactions reflected in the system of equa-tions. Total exvessel revenues for Alaska fishermen were estimated byvarying coastwide (Alaska, British Columbia, and Washington) halibutlandings away from their 2001 and 2002 levels, while holding all othervariables at their actual 2002 levels. These changes in exvessel revenuesfor coastwide changes in halibut landings can be described by:

TRAllt = −2.52126 + 3.23704

(LANAll

t

)− 0.01510

(LANAll

t

)2.(20)

As landings increase from 2002 base levels, revenues also increase,but as landings continue to increase, price becomes increasingly sen-sitive to equal percentage increases in landings. If increased landingsare distributed proportionally throughout Alaska, British Columbia,and Washington, Alaskan exvessel revenues would peak at $142.2 mil-lion when Alaskan landings total 86.7 million pounds and coastwidelandings total 104.6 million pounds.

4. The Bioeconomic Model. A bioeconomic model of the hal-ibut fishery can be formed from the population and market models.Figure 8 represents coastwide sustainable revenues, the exvessel rev-enue (equation [20]) evaluated across the range of sustainable yields(the expectation of equations [2]–[5]).

A STATE SPACE BIOECONOMIC MODEL 135

$0

$50

$100

$150

$200

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

$US

mil

lion

FIGURE 8. Coastwide sustainable exvessel revenues of Pacific halibut.

Comparison of the sustainable revenues (Figure 8) with sustainableyields (Figure 2) suggests that sustainable revenues are less variablethan sustainable yields. This occurs because the inverse relationshipbetween price and commercial landings helps offset the reduction insustainable yield over a broad range of biomass levels. For example,over the biomass range of 166 to 402 million pounds, sustainable rev-enues only vary by 5% whereas sustainable yields vary by 15%. Becausecatch-per-unit effort is inversely related to biomass, the marginal cost ofharvesting sustainable yields is lower at population levels above BMSY ,the biomass level associated with the maximum sustainable yield, thusthe population level associated with the sustainable net revenue max-imizing yield, BMSNR, will be above BMSY .

The steady-state bioeconomic model can also be used to examinethe economic implications of changes in the ocean environment thatlead to changes in the carrying capacity for Pacific halibut. Figure 9includes baseline steady-state sustainable revenues and sustainable rev-enues under conditions that result in a 10% increase or 10% decreasein the carrying capacity of halibut stocks in the North Pacific.

Although increases (decreases) in carrying capacity lead to increased(decreased) levels of sustainable harvests, the inverse relationship

136 K.R. CRIDDLE AND M. HERRMANN

$0

$50

$100

$150

$200

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

$US

mil

lion

$0

$50

$100

$150

$200

0 25 50 75 100 125 150

Coastwide Landings (million pounds)

$US

mil

lion

+10

%

+10

%

FIGURE 9. Coastwide sustainable exvessel revenues under baseline conditionsand conditions that result in a 10% increase or 10% decrease in the carryingcapacity of halibut in the North Pacific.

between exvessel price and the poundage of halibut landings moder-ates the effect of changes in carrying capacity on the magnitude ofsustainable revenues. Although the difference in revenues is relativelymodest (+2.1% to −4.6%) at BMSY , the differences become substan-tially more pronounced at higher biomass levels. Under the baselinemodel, BMSY is 280 million pounds. With a 10% increase in carryingcapacity, BMSY is 305 million pounds. With a 10% decrease in carryingcapacity, BMSY declines to 254 million pounds. As carrying capacityincreases (decreases), the sustainable revenue curve forms more (less)of a plateau. Under the baseline, sustainable revenues are within 2%of their maximum for biomasses between 196 and 367 million pounds.Under conditions that result in a 10% increase in carrying capacity, therevenue maximizing sustainable yield increases from 91 to 102.4 millionpounds, and the revenue plateau extends over biomasses between 189and 430 million pounds. Under conditions that result in a 10% decreasein carrying capacity, the revenue maximizing sustainable yield declinesto 80.2 million pounds and the range of the revenue plateau shrinks tobetween 189 and 319 million pounds. Because BMSNR is above BMSYand because sustainable revenues are more sensitive to changes in car-rying capacity at elevated levels of biomass, BMSNR will be highly sen-sitive to environmental regime shifts that result in long-term changesin carrying capacity.

Figure 10 includes baseline steady-state sustainable revenues and sus-tainable revenues under conditions that result in a 10% increase or 10%decrease in recruitment success.

A STATE SPACE BIOECONOMIC MODEL 137

$0

$25

$50

$75

$100

$125

$150

$175

$200

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

$US

mil

lion

$0

$50

$100

$150

$200

0 25 50 75 100 125 150

Coastwide Landings (million pounds)

$US

mil

lion

-10%

+10

%

FIGURE 10. Sustainable coastwide exvessel revenues under baseline condi-tions and conditions that result in a 10% increase or 10% decrease in recruit-ment success.

Increases (decreases) in recruitment success lead to increased (de-creased) levels of sustainable harvests. The change in recruitment suc-cess is tempered by the inverse relationship between exvessel priceand the poundage of halibut landings. With a 10% increase in re-cruitment success, BMSY rises from 280 to 309 million pounds; witha 10% decrease in recruitment success, BMSY declines to 247 mil-lion pounds. At BMSY , the percentage differences in expected exves-sel revenues are +2.2% for a 10% increase in recruitment successand −6.5% for a 10% decrease in recruitment success. Under condi-tions that result in a 10% increase in recruitment success, the revenuemaximizing sustainable yield increases to 103.8 million pounds and therevenue plateau extends over biomasses between 187 and 441 millionpounds. Under conditions that result in a 10% decrease in recruitmentsuccess, the revenue maximizing sustainable yield declines to 77.5 mil-lion pounds, and the range of the revenue plateau shrinks to between187 and 309 million pounds. Because BMSNR is above BMSY and be-cause sustainable revenues are more sensitive to variations in recruit-ment success at elevated levels of biomass, BMSNR will be highly sen-sitive to changes in environmental conditions that lead to long-termchanges in recruitment success.

Figure 11 includes baseline steady-state sustainable revenues and sus-tainable revenues under conditions that result in a 10% increase or 10%decrease in the growth rate of halibut.

138 K.R. CRIDDLE AND M. HERRMANN

$0

$25

$50

$75

$100

$125

$150

$175

$200

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

$US

mil

lion

$0

$50

$100

$150

$200

0 25 50 75 100 125 150

Coastwide Landings (million pounds)

$US

mil

lion

-10%

+10

%

FIGURE 11. Sustainable coastwide exvessel revenues under baseline condi-tions and conditions that result in a 10% increase or 10% decrease in halibutgrowth rates.

Increased (decreased) growth rates lead to larger (smaller) biomassand increased (decreased) levels of sustainable harvests. The range ofvariation BMSY is small for changes in the growth rate. With a 10% de-crease (increase) in the growth rate, BMSY is 302 (278) million pounds.At BMSY , the percentage differences in expected exvessel revenues are+2.4% for a 10% increase in the growth rate and −6.3% for a 10%decrease in the growth rate. Under conditions that result in a 10%increase in the growth rate, the revenue maximizing sustainable yieldincreases to 107.2 million pounds and the revenue plateau extends overbiomasses between 161 and 412 million pounds. Under conditions thatresult in a 10% decrease in the growth rate, the revenue maximizingsustainable yield declines to 76.3 million pounds and the range of therevenue plateau shrinks to between 210 and 348 million pounds. Un-like the simulations of varied carrying capacity and varied recruitmentsuccess, the percentage differences in exvessel revenues between highgrowth rate and low growth rate scenarios do not increase at elevatedlevels of biomass. Because sustainable net revenues are not markedlymore sensitive to changes in the growth rate of halibut at high popu-lation levels than at low or intermediate population levels, BMSNR willnot be particularly sensitive to changes in environmental conditionsthat lead to long-term changes in growth rates.

Changes in the PDO are also associated with changes in recruitmentand biomass. Figure 12 represents the steady-state baseline sustainableyields and associated expected recruitment levels for 1◦ positive andnegative anomalies in the PDO index.

A STATE SPACE BIOECONOMIC MODEL 139

0

50

100

150

200

0 100 200 300 400 500 600 700

Age 8+ Biomass (million pounds)

$US

(m

illi

on)

$0

$50

$100

$150

$200

0 25 50 75 100 125 150

Coastwide Landings (million pounds)

$US

mil

lion

-1 +1

FIGURE 12. Coastwide sustainable exvessel revenues under baseline condi-tions and conditions that result from +1◦ or −1◦ anomalies in the Pacificdecadal oscillation.

Simulated 1◦ positive (negative) anomalies in the PDO index lead todecreases (increases) in expected recruitment and increases (decreases)in the expected value of sustainable yields. The simulated levels ofBMSY are unaffected by assumed 1◦ positive or negative anomalies inthe PDO. The differences in sustainable revenues are also small (+1.9%for a 1◦ positive anomaly and −3.4% for a 1◦ negative anomaly). Underconditions that result in a 1◦ positive anomaly, the revenue maximizingsustainable yield is 100 million pounds and the revenue plateau extendsover biomasses between 176 and 390 million pounds. Under conditionsthat result in a 1◦ negative anomaly, the revenue maximizing sustain-able yield declines to 82 million pounds and the range of the revenueplateau shrinks to between 210 and 352 million pounds. Because sus-tainable net revenues are not markedly more sensitive to anomalies inthe PDO at high population levels than at low or intermediate pop-ulation levels, BMSNR will not be particularly sensitive to changes inthe PDO index. The responses of sustainable yields and sustainablerevenues to changes in the PDO index are more similar to the responseof sustainable yields and sustainable revenues to changes in the growthrate of halibut than to changes in recruitment success or changes inthe carrying capacity. This suggests that the mechanism underlyingthe relationship of the PDO index and halibut population dynamicsis related to changes in halibut growth rates rather than changes inrecruitment success or changes in carrying capacity.

5. Discussion. The parable of the Pacific halibut fishery is a taleoft told, the moral of which usually professes a demonstration of thetriumph of science-based management strategies against a backdrop

140 K.R. CRIDDLE AND M. HERRMANN

of decadal-scale environmental variability. Although the halibut stockis undeniably healthy and the current fishery is profitable, there areunique characteristics of the halibut fishery that may predispose suc-cess, conditions that may not characterize other fisheries and may limitthe relevance of the halibut fishery as a type for the response of otherfisheries to the application of similar management measures. For ex-ample, the average sustainable yield of halibut is nearly constant overa wide range of fishing mortality rates (Sullivan et al. [1999]). In ad-dition, although there is clear evidence of decadal-scale environmentalinfluence on recruitment success, the effect seems to be substantiallyoffset by density dependence in growth (Clark and Hare [2002]). More-over, revenues are constrained by the sustainable yield of halibut not byconsumer demand (Herrmann and Criddle [2006]). That is, althoughincreases in the quantity of halibut released into the exvessel and whole-sale markets will result in decreased prices, the increased quantity morethan offsets the decreased price: thus, revenues will increase as land-ings increase up to the maximum sustainable level of landings undercurrent environmental conditions. In addition, as shown above, sus-tainable revenues are also relatively insensitive to changes in halibutgrowth rates and variations in halibut biomass associated with varia-tions in the PDO index. These characteristics of the halibut resourceand the market for halibut serve as biological and economic buffersfor the halibut stock and fishery. The usual moral of the halibut para-ble is that good management and good governance can yield success.Perhaps the moral should be that successful management is easier toobtain when the Fates bless us with a fish stock that is robust in re-sponse to environmental variation and variations in fishing pressure,has markets that are robust in their ability to absorb as much as na-ture will give, and has a governance system that does not foster a racefor catch shares.

Acknowledgments. This paper is a result of work funded fromthe National Oceanic and Atmospheric Administration Office of SeaGrant, Department of Commerce, under grant no. NA 16RG2321project no. RR/32-02, and from the University of Alaska with fundsappropriated by the state, and from the Utah Agricultural ExperimentStation. All opinions are the authors and do not necessarily representthe Alaska Sea Grant College Program.

A STATE SPACE BIOECONOMIC MODEL 141

Appendix

TABLE 1. Landings of Pacific halibut (million pounds).

WA CA AK Total

1974 0.515 4.624 16.167 21.3061975 0.460 7.127 15.278 22.8651976 0.238 7.283 14.797 22.3181977 0.207 5.427 12.430 18.0641978 0.097 4.607 13.034 17.7381979 0.046 4.857 15.069 19.9721980 0.022 5.650 11.432 17.1041981 0.202 5.654 16.789 22.6451982 0.211 5.236 20.423 25.8701983 0.265 5.436 31.911 37.6121984 0.431 9.054 35.420 44.9051985 0.493 10.389 45.211 56.0931986 0.581 11.225 57.819 69.6251987 0.592 12.246 56.511 69.3491988 0.486 12.858 61.044 74.3881989 0.472 10.431 56.017 66.9201990 0.203 8.574 52.590 61.3671991 0.233 7.191 49.520 56.9441992 0.282 7.626 51.813 59.7211993 0.366 10.573 48.104 59.0441994 0.380 9.916 44.404 54.7001995 0.270 9.524 33.978 43.7721996 0.290 9.557 37.272 47.1191997 0.760 12.420 52.293 65.4731998 0.460 13.150 55.126 68.7361999 0.454 12.704 60.099 73.2572000 0.483 10.811 55.629 66.9232001 0.680 10.288 58.897 69.865

142 K.R. CRIDDLE AND M. HERRMANN

TABLE 2 Coastwide (Alaska, British Columbia, Washington) removals of Pacifichalibut (million pounds).

Personal TotalCommercial Sport use Wastage Bycatch removals

1974 21.310 0.000 0.000 0.000 10.708 32.0181975 27.620 0.000 0.000 0.000 6.973 34.5931976 27.540 0.000 0.000 0.000 7.562 35.1021977 21.880 0.289 0.000 0.000 8.272 30.4411978 22.000 0.378 0.000 0.000 8.166 30.5441979 22.540 0.563 0.000 0.000 10.207 33.3101980 21.870 0.845 0.000 0.000 12.695 35.4101981 25.720 1.112 0.000 0.000 10.642 37.4741982 29.010 1.299 0.000 0.000 8.574 38.8831983 38.380 1.616 0.000 0.000 6.517 46.5131984 44.960 1.840 0.000 0.000 5.697 52.4971985 56.110 2.355 0.000 1.600 4.385 64.4501986 69.620 3.177 0.000 3.200 4.501 80.4981987 69.480 3.509 0.000 2.722 5.906 81.6171988 74.350 4.877 0.000 1.953 6.669 87.8491989 66.930 5.233 0.000 2.025 5.361 79.5491990 61.590 5.586 0.000 1.655 9.149 77.9801991 57.081 6.509 2.000 2.227 9.385 77.2021992 59.891 6.179 1.100 1.255 8.943 77.3681993 59.268 7.725 0.920 0.814 5.671 74.3981994 54.731 7.065 0.920 1.289 7.996 72.0011995 43.882 7.447 0.528 0.257 6.942 59.0561996 47.343 8.083 0.528 0.347 7.279 63.5801997 65.197 9.025 0.528 0.290 7.106 82.1461998 69.757 8.586 0.528 0.359 7.636 86.8661999 74.305 7.379 0.730 0.395 7.011 89.8202000 68.305 9.016 0.730 0.222 6.615 84.8882001 70.715 8.106 0.730 0.238 7.088 86.877

A STATE SPACE BIOECONOMIC MODEL 143

TABLE 3 Data used for biological modeling.

Area 2B, 2C, 3A Area 2B, 2C, 3ACoastwide total age 8 recruits age 8+ biomass Pacific decadal

removals (million) (million lb net) oscillation

1974 32.018 1.290 140.400 −0.341975 34.593 1.520 145.590 −1.101976 35.102 1.690 152.180 0.011977 30.441 1.870 166.520 0.231978 30.544 2.310 190.800 0.241979 33.310 2.270 218.100 0.341980 35.410 2.740 242.570 0.601981 37.474 3.420 273.560 0.921982 38.883 3.350 308.480 0.111983 46.513 3.620 345.130 1.651984 52.497 4.380 364.530 0.841985 64.450 5.810 393.120 0.451986 80.498 4.930 425.470 1.241987 81.617 5.930 432.510 1.821988 87.849 6.700 438.030 0.531989 79.549 5.310 429.460 −0.181990 77.980 4.830 426.060 −0.361991 77.202 6.390 420.790 −0.421992 77.368 5.690 411.350 0.931993 74.398 4.720 400.560 1.421994 72.001 4.560 425.150 −0.151995 59.056 8.590 443.530 0.641996 63.580 7.310 458.810 0.641997 82.146 4.830 450.710 1.461998 86.866 4.380 426.670 0.251999 89.820 3.260 392.710 −1.062000 84.888 3.480 358.570 −0.592001 86.877 3.070 334.560 −0.56

144 K.R. CRIDDLE AND M. HERRMANNT

AB

LE

4IP

HC

com

mer

cial

catc

hlim

its

(mill

ion

lbs)

byre

gula

tory

area

.

Reg

ulat

ory

area

s

2A2B

2C3A

3B4A

4B4C

DE

Are

a4

Tot

al

1982

0.20

05.

400

3.40

014

.000

3.00

01.

500

27.5

0019

830.

200

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03.

400

14.0

005.

000

1.20

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0.60

02.

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30.6

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840.

300

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18.0

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1.20

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0.85

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43.1

5019

850.

500

10.0

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23.0

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1.70

01.

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1.25

04.

250

55.7

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860.

550

11.2

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28.1

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.300

2.00

01.

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1.35

05.

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66.4

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870.

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11.5

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31.0

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1.75

01.

750

1.27

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880.

750

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36.0

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02.

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1.50

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74.1

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890.

650

10.0

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31.0

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1.80

01.

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1.30

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64.6

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900.

520

7.80

08.

000

31.0

007.

200

1.50

01.

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1.10

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58.6

2019

910.

450

7.40

07.

400

26.6

008.

800

1.70

01.

700

1.30

04.

700

55.3

5019

920.

650

8.00

010

.000

26.6

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800

2.30

02.

300

1.73

06.

330

60.3

8019

930.

600

10.5

0010

.000

20.7

006.

500

2.02

02.

300

1.72

06.

040

54.3

4019

940.

550

10.0

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.000

26.0

004.

000

1.80

02.

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1.50

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56.9

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950.

520

9.52

09.

000

20.0

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1.95

02.

310

1.66

05.

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48.6

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9.52

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A STATE SPACE BIOECONOMIC MODEL 145

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