Download Linking Dynamic Economic and Ecological General Equilibrium

Technical Appendix:
Linking Dynamic Economic and Ecological General Equilibrium Models
1.
The Ecology Model
GEEM is applied here to a frequently-studied marine ecosystem comprising Alaska's Aleutian Islands
(AI) and the Eastern Bering Sea (EBS). All energy in the system originates from the sun it is turned into
biomass through plant photosynthesis. Photosynthesis is carried out in the AI by individuals of various species
of algae, or kelp, and in the EBS by individuals of various species of phytoplankton. All individual animals in
the system depend either directly or indirectly on the kelp and phytoplankton plant species. In the EBS,
zooplankton prey on phytoplankton and are prey for pollock. The pollock are a groundfish that support a very
large fishery. Steller sea lions, an endangered species, prey on pollock, while killer whales prey on the sea lions.
In the AI, killer whales also prey on sea otter that in turn prey on sea urchin that in turn prey on kelp.
GEEM combines two disparate ecological modeling approaches: optimum foraging models and
dynamic population models. The former approach has been likened to consumer theory (Stephens and Krebs,
1986) and describes how individual predators search for, attack and handle prey to maximize net energy intake
per unit time. The approach does not account for multiple species in complex food webs and it does not track
species population changes. The latter approach tracks population changes by using a difference or differential
equation for each species. The familiar logistic-growth model used extensively in the economics literature is the
simplest example, although extensions include resource competition models (e.g., Gurney and Nisbet 1998) and
the Lotka-Volterra predator/prey model and its variations. However, the parameters in the dynamic equations
represent species-level aggregate behavior: optimization by individual plants or animals or by the species is
absent. Alternatively, GEEM employs optimization at the individual level as in foraging models, and uses the
results of the optimization to develop difference equations that track population changes.
In GEEM, demand and supplies are developed somewhat similarly to CGE. Species are analogous to
industries, and individual plants and animals are analogous to firms. Where perfectly competitive firms sell
2
outputs and buy inputs taking market-determined prices as signals; plants and animals transfer biomass from
prey to predators taking ‘energy prices’ as signals. (Plants can be thought of as preying on the sun.) An energy
price is the energy a predator loses to the atmosphere when searching for and capturing prey. A key difference
between economic markets and ecological transfers, however, is that in the latter the prey does not receive this
energy price. Therefore, the biomass transfer is not a market because there is no exchange. Nevertheless,
predators’ demands and preys’ supplies are functions of the energy prices as explained in Tschirhart (2003).
Plants and animals are assumed to behave as if they maximize their net energy flows and a general expression
for the net energy flow through a representative animal from species i is given by:
i 1
Ri   [e j  eij ]xij 
j 1
m
 e [1  t e
k  i 1
i
i ki
i 1
] yik  f ( xij )   i ,
i
(1.1)
j 1
where Ri is in power units (e.g., Watts or kilocalories/time).1,2 The species in (1.1) are arranged so that members
of species i prey on organisms in lower numbered species and are preyed on by members of higher numbered
species. The first term on the right side is the inflow of energy from members of prey species (including plants)
to the representative individual of species i. The choice variables, xij, are the biomasses (in kilograms/time)
transferred from the member of species j to the member of species i, ej are the energies embodied in a unit of
biomass (e.g., in kilocalories/ kilogram) from a member of species j, and eij are the energies the member of
species i must spend to locate, capture and handle units of biomass of species j. These latter energies are the
energy prices. There is one price for each biomass transfer between a predator and prey species. As in economic
CGE models, the prices play a central role in each individual’s maximization problem, because an individual’s
choice of prey will depend on the relative energy prices it pays. Individuals are assumed to be price takers: they
have no control over the energy price paid to capture prey, because each is only one among many individuals in
a predator species capturing one of many individuals in a prey species. However, within the ecosystem the
prices are endogenous, being determined by demand and supply.
1
According to Herendeen (1991) energy is the most frequently chosen maximand in ecological maximization models, and
energy per time maximization as adopted here originates with Hannon (1973, 1976) and expanded to multiple species in
Crocker and Tschirhart (1992) and to the individual level in Tschirhart (2000). Energy per time is also the individual’s
objective in the extensive optimum foraging literature (e.g., Stephens and Krebs 1986).
2
Plants are somewhat different in that they “prey” on the sun, they occupy a fixed area that dictates the supply of available
solar energy, they are in resource competition for the solar energy, and their choice variable, xij, is biomass of the plant
instead of a biomass flow as it is for animals (Tschirhart 2002).
2
3
The second term is the outflow of energy to animals of species k that prey on i. The ei is the embodied
energy in a unit of biomass from the representative individual of species i, and yik is the biomass supplied by i to
k. The term in brackets is the energy the individual uses in attempts to avoid being preyed upon. It is assumed to
be a linear function of the energy its predators use in capture attempts: the more energy predators expend, the
more energy the individual expends escaping. In a sense, ti is a tax on the individual because it loses energy
above what it loses owing to being captured. The third and fourth terms in (1.1) represent respiration energy lost
to the atmosphere. Following Gurney and Nisbet (1998), respiration is divided into two parts. First, the f i() that
depends on energy intake and includes feces, reproduction, defending territory, etc., and the second, the i is
basal metabolism and it is independent of energy intake.
As prey, the individual would prefer to supply zero biomass (yik = 0) to predators since outflows reduce
net energy. However, the only way the individual can supply zero biomass is if it demands zero biomass,
because to capture prey biomass the individual risks being captured by predators. The more biomass an
individual captures the more it is exposed and the more biomass it supplies to predators.3 Because individuals
are subject to predation risk, the biomass they supply can be written as a function of their demand4,
i 1
yik  yik ( xij ), for i  p,..., m ,
(1.2)
j 1
(This representation is similar to, but in reverse from, a firm whose supply of output determines its demand for
inputs.)
Assuming second-order sufficient conditions for a maximum are satisfied (Tschirhart, 2000), the firstorder conditions from maximizing (1.1) can be solved for the xij as functions of the energy prices to yield i’s
demands,
xij(ei)  xij(ei1,…, ei,i-1), for j = 1,…, i-1,
(1.3)
where the boldface ei = (ei1,…, ei,i-1) is a vector of energy prices. Substituting these demands into the yik yields
3
This tradeoff between foraging gains and losses is called predation risk (See, e.g., Lima and Dik (1990)).
A problem of interpretation arises in the maximization problem, because once an animal is successfully preyed upon, it is
gone. To avoid this discrete, zero/one issue, the maximization problem is assumed to represent the ‘average’ member of the
species. Thus, when the animal is captured it does not lose its entire biomass, but rather it loses biomass equal to the mean
loss over all members of its species.
4
3
4
the representative animal i’s supplies,
yik(xi(ei))  yik(xi1(ei1,…, ei,i-1),…, xi,i-1(ei1,…, ei,i-1)), for k = i+1,…m,
(1.4)
where the boldface xi = (xi1,…, xi,i-1) is a vector of demands.
1.1 Equilibrium and Dynamics Time in the Alaskan ecosystem is divided into yearly reproductive
periods, although some species do reproduce more than once annually. Each year a GEEM is determined
wherein the populations of all species are constant, each plant and animal is maximizing its net energy, and
aggregate demand equals aggregate supply between each predator and prey species. To find a set of prices that
equate demands and supplies, an equilibrium equation is needed for each price. Because each animal in (2.1) is
assumed to be a representative individual from that species, all animals in each species are identical; therefore,
the demand and supply sums are obtained by multiplying the representative individual’s demands and supplies
by the species populations. Letting Ni be the population of species i, the equilibrium conditions are:
Nixij(ei) = Njyji(xj(ej))
(1.5)
for i = p,…, m and j = 1,…, i-1. There is one condition for each biomass transfer. The left side of (1.5) is the
total demand of species i for species j, and the right side is the total supply of species j to species i. Because
plants ‘prey’ on the sun, their equilibrium conditions differ somewhat, details can be found in Tschirhart (2002).
A representative plant or animal and its species may have positive, zero or negative net energy in
equilibrium. Positive (zero, negative) net energy is associated with greater (constant, lesser) fitness and an
increasing (constant, decreasing) population between periods. (The analogy in a competitive economy is the
number of firms in an industry changes according to the sign of profits.) Net energies, therefore, are the source
of dynamic adjustments. If the period-by-period adjustments drive the net energies to zero, the system is moving
to stable populations and a steady state. The predator/prey responses to changing energy prices tend to move the
system to steady state.
The update equations are derived as follows. Consider population changes for species i that, for
simplicity, is assumed to be a top predator (a top predator is simpler to work with because there are no predation
terms). In steady-state it must be the case that births equals deaths. If si is the lifespan of the representative
individual, then the total number of births and deaths must be Ni/si, with per capita steady-state birth and death
4
5
rates of 1/si. The top predator’s maximized net energy is given by Ri(xij; Nt) = Ri () which is obtained by
substituting the optimum demands as functions of energy prices into objective functions (1.1). Ni t is a vector of
all species' populations and it appears in Ri () to indicate that net energies in time period t depend on
populations in time period t. In the steady state, Ri ()  0 . Reproduction requires energy and, by the definitions
of the terms in (1.1), that energy is contained in the variable respiration. Let vi ss be the steady-state variable


respiration, and let i N it , N iss viss be the proportion of this variable respiration devoted to reproduction, where
i is a function of the population in time period t and the steady state population. Thus, in steady state the


energy given by i N it , N iss viss over all members of the species yields the number of births that exactly offset


deaths, Births = Deaths ==> N it  i N it , N iss viss   N it 1 si  .




The term i N it , N iss  1 viss si found from this relationship converts reproductive energy into
individuals. The conversion of energy into individuals is assumed to depend on the relationship between actual
population and the steady state population. In particular, less reproductive energy is needed per offspring when
the population is at their steady state. This accounts for increased difficulty in finding a mate if the population is
below the steady state, and the competition costs of finding a mate if the population is above the steady state. To
implement this assumption let the functional form of i be quadratic with a maximum when Nit = Niss (the
familiar logistic). Using the above development and the functional form it is easy to show,
i 
2 N it
v ss si N iss

N it 

1

ss 
 2 Ni 
(1.6)
which collapses to i  1 viss si when Nit = Niss.
Next, suppose the top predator is not in steady state and that Ri ()  0 and variable respiration is vi.
Assuming that the proportion of Ri () that is available for reproduction is the same as the proportion of variable
respiration available for reproduction, the energy now available for reproduction is i  Ri ()  vi  . Finally,
assuming that reproduction is linear in available energy, then it follows that if i viss yields a per capita birth rate
5
6
 2 N it  
N it 

R (.)  vi  .
1
ss
ss  
ss   i
s
v
N
N
2
i 
i 
 i
1/si, then i  Ri ()  vi  yields a per capita birth rate of 
The change in population is obtained by multiplying the population by the difference between the birth
and death rates, where the latter rate is assumed independent of energy available for reproduction. Therefore the
population adjustment equation is
 2N t 
Nt 
1
N it 1  N it  N it  ss i ss 1  i ss   Ri ()  vi   
si 
 si vi N i  2 N i 
(1.7)
Expression (1.7) reduces to the steady state if Ri ()  0 (in which case vi = viss and Nit = Niss), and the second
and third terms sum to zero. Because the biomass demands depend on the period t populations of all species, the
population adjustment for species i depends on the populations of all other species. In addition, out of steady
state Ri () and vi change across periods. These changes distinguish the GEEM approach from most all
ecological dynamic population models, because the latter rely on fixed parameters in the adjustment equations
that do not respond to changing ecosystem conditions.
If the species is not a top predator and is prey for another species, then in steady state the births must
equal the deaths plus any individuals lost to predation. From the net energy expression, we know each individual
loses yik(xi(ei)) kg to predation. The summation of all individual losses to predation gives the total biomass lost
to predation, and total biomass divided by weight of a representative individual provides the number of
individuals lost to predation,
Nit(yik(xi(ei))/wi).
(1.8)
Thus, in the steady state, the number of births, from respiration energy, equals the sum of deaths from predation
and deaths from natural mortality net of losses to predation,
N it  i viss   N it
 y
ik
( xijss (e) / wi  1  (1/ si )   1/ si

(1.9)
The parameter i can again be found from this relationship and the assumption of quadratic reproductive
energy efficiency, although now it includes predation and natural mortality,
 ( yik ( xijss (e) / wi )(1  (1/ si ))  (1/ si )  2 N it 
N it  
1
i  
.



ss  

  v ss s N ss
viss

  i i i  2 Ni  
6
(1.10)
7
Again, i converts watts into individuals. Note the use of steady state predation supply. Again, assuming that
the proportion of Ri   available for reproduction is the same as the proportion of variable respiration available
for reproduction, when the species is not in the steady state the population update function becomes,


 1 1 
ss
 yik  xij (e) wi 1    

t
t
 yik ( xij (e))  1  1 
si  si   2Ni 
Ni  

t 1
t
t 
Ni  Ni  Ni
R (.)  vi   
1
1     .
 w

  siviss Niss  2Niss   i
viss
i
 si  si 



 





(1.11)
Note that in steady state Ri   = 0, xij = xijss , vi = vi ss and Nit = Niss so that (1.11) reduces to N it 1  N it . Finally,
humans can enter the ecological model in a variety of fashions, but in what follows the only consumptive human
impacts are through fishery harvests, which serve to lower the net growth of pollock.
2
The Economy Model
2.1
Recreation Sector The recreation sector is reliant on the state of the Alaska's natural resources,
a component of which are the stocks of sea lions and killer whales in the Bering Sea, and sea otters of the
Aleutian Islands. For analytical simplicity the primal formulation of profit maximization is also maintained in
this sector.
In providing their recreational goods and services, firms in the industry combine labor and capital with
the aggregate stock of natural resources to deliver a "recreational experience." Firms are assumed to use these
stocks in a nonconsumptive, costless fashion. Recreational firms maximize profits subject to a Cobb-Douglas
production function,
max  h  PR X R  WLR  RK R
LR , K R
R
R
R
c
s.t. X R  d R LaR K Rb N AT
(2.1)
where aR, bR, cR, and dR are parameters, R subscripts for the recreation sector, and NAT the aggregate natural
resource measure. The values of these inputs are realized through productivity changes in the primary factors.
We deliberately separate out this effect in this analysis to focus on this point of contact between the economy
and the natural system.
7
8
Recreation firms choose levels of labor and capital inputs. From the maximization the production
function and ratio of first order conditions determine efficient levels of factor employment.
As the recreational use of natural resources is non-consumptive, NAT enters recreation production as a shift
parameter, exogenous to the recreation firms but endogenous to the joint system.
2.2
Composite Sector The composite sector has no exogenous inputs, which require the explicit
incorporation of physical balances. This allows the dual formulation, as described in processing, to be employed
(Ginsburgh and Keyzer, 1997). Cost minimization takes place by firms smoothly substituting between factors of
production by means of a CES production function, VAC,
(
1
C
VA C  C [ C
 1
( C
)
LC
C
(
)
 (1   C )
1
C
 1
)
( C
KC
C
)
(
]
C
 C 1
)
.
(2.2)
Composite factor employment levels are defined equivalently as in other sectors, while φC, δC and σC are sectoral
efficiency, distribution, and elasticity of substitution parameters.
Under the assumptions of constant returns to scale and perfect competition, the industry minimizes costs
subject to (2.8). Through this optimization the industry cost function CVC can be derived as,
CVC 
QCD
C
1
[ CW
(1 C )
 (1   C ) R
(1 C ) 1 C
]
(2.3)
Through the application of Shepard's Lemma to the cost function, and given factor prices, sectoral factor
demand functions can be readily derived.
2.3
Household Behavior Households consume fish (pollock), recreation, composite goods and
save for future consumption. To demonstrate the fundamental dependence of the economy on the ecosystem,
households are assumed to require at least subsistence levels of fish to survive. Within each period, household
behavior is modeled through a tri-level nesting structure that allows smooth substitution between consumption
goods and recreation, but deliberately differentiates between the two.5 Individuals derive their gross income IAK
from the sales/rental of their current labor LAK and capital KAK endowments. Given their gross income, in the
top nest individuals' first choose the proportions allocated between consumption today CT and future
consumption CF, given prices PT and PF. In the second nest, income for current consumption is divided between
8
9
expenditures on recreation CTR and an aggregate consumption commodity CTA, at prices PQR and PA. In the
bottom nest, income allocated for expenditures on CTA is divided between purchases of fish, CTAF and the
composite good CTAC, at prices PQF and PQC.
Future consumption is funded through household savings S, costing PS. Savings decisions are made over
expected increments to a stream of consumption in future periods (CF1, CF2, ...) where consumer expectations are
assumed myopic in that consumers expect current prices to remain constant in all future periods.6 Savings are
used to buy investment goods ,7 which add to the stock of household capital for future periods.8 Capital service
income finances future consumption at the level that equates the value of savings to the present value of
expected future consumption. 9
In the bottom nest, households decide how to allocate disposable income over consumption of fish and
composite goods through the optimization of a Stone Geary sub-utility function. The Stone Geary functional
form incorporates subsistence levels of consumption and has been used in a similar fashion to capture the nonseparablity of environmental quality and preferences (Espinosa and Smith, 2002). Our use of this form
underscores the reliance of the economy on the ecosystem.
Given this development, the top nest of household behavior becomes an optimization of overall utility U
subject to the gross budget constraint, where U is given by a CES utility function,
1
 2R 1
1
 2R 1
 2R
 2R
 2R
 2R
max U  [ R C T
 (1   R ) C F
 2R
R
] 2 1 s.t.
I AK  PT C T 
Ps PT F
C .
R
(2.4)
The R is the distributional share of current consumption, and R2 the relevant elasticity of substitution. From
the maximization, demand functions Cˆ T , Cˆ F can be found,
5
CES utility functions are employed in the top two nests, a Stone-Geary form in the bottom nest.
Given a composite price for CTA of PA, under myopic expectations this is also the expected price for CF.
7
Investment demands are modeled as fixed proportions of output, Ii = aiIXi, and    I i , i  {F, R, C}
6
i
Savings are translated into capital services in future periods by the proportion  so that  t   St . Ballard et al define  as
the initial real rate of return as rental rate of capital R is normalized in the benchmark.
8
9
So that Ps S 
Ps PT F
C .
R
9
10
Cˆ T 
Cˆ F 
 R I AK
 2R
(1 2R )
R T
PT [ P
R
PP
 (1   R )( s T )(1 2 ) ]
R
,
(1   R ) I AK
R
R
PP R
PP
( s T ) 2 [ R PT(1 2 )  (1   R )( s T )(1 2 ) ]
R
R
(2.5)
.
(2.6)
The demand for savings is easily found from (2.6) and the value of savings as given by footnote (9) as,
(1   R ) I AK
Sˆ 
Ps
 2R
R
R
R
P
PP
( T )( 2 1) [ R PT(1 2 )  (1   R )( s T )(1 2 ) ]
R
R
.
(2.7)
Following the savings decision, in the second nest households choose between expenditures over the
composite commodity and recreation by maximizing a CES sub-utility function CT given the proportion of gross
income available for current consumption,
1
1R
max C  [(1   R ) C
T
1R 1
T 1R
A
1R 1
1
 2R
 R C
T 1R
R
1R
1R 1
]
s.t.
I AK  Ps S  PAC AT  PRQCRT ,
(2.8)
where  R is the distributional share of recreation and R1 the elasticity of substitution between the composite
consumption commodity and recreation. Optimization of this system derives the demand functions Cˆ AT and Cˆ RT ,
Cˆ AT 
Cˆ RT 
(1   R )( I AK  PS S )
R
R
R
PA1 [(1   R ) PA(11 )   R PRQ (11 ) ]
R
PRQ1
,
(2.9)
 R ( I AK  PS S )
.
[(1   R ) PA(1 )   R PRQ (1 ) ]
R
1
(2.10)
R
1
In the third and bottom nest, following the savings and recreation decisions, households decide how to
allocate disposable income over consumption of fish and composite goods through the optimization of a Stone
Geary utility function C AT ,
T
max  C AF
 FH 
CH
 CACT  CH 
CH
s.t.
T
T
I AK  Ps S  PRQCRT  PFQC AF
 PCQC AC
.
10
(2.11)
11
The parameters FH and CH are parameters governing the minimum subsistence levels of each commodity.
Parameters  FH and  CH are specific to the Stone Geary form and discussed in the calibration section below.
T
T
From the maximization the resulting demand functions Cˆ AF
and Cˆ AC
are,
 H 
T
 FH   FQ   I AK  Ps S  PRCRT   FH PFQ  CH PCQ ,
Cˆ AF
 PF 
(2.12)
 H 
T
 CH   CQ   I AK  Ps S  PRCRT   FH PFQ  CH PCQ .
Cˆ AC
 PC 
(2.13)




Give the choice of functional form, all of the demand functions fulfill the necessary restrictions and are therefore
"continuous, nonnegative and homogeneous of degree zero (in absolute prices)" (de Melo and Tarr, pg 43).
The welfare impacts of alternative policies are evaluated in terms of modified Hicksian equivalent
variational measures similar to those developed in Ballard et al. Each policy change leads to changes across
prices and income in relation to a reference/benchmark sequence of business as usual. Let the Hicksian
expenditure function associated with consumption in period t be given by Mt(.). Indirect utility in any period t is
given by Vt where,
1
Vt  ( I
AK
(1 1R )
A
 PS S )[(1   R ) P
Q (1 1R )  1R 1
R R
 P
]
(2.14)
Solving Vt for disposable income provides the expenditure function Mt.
Vectors of prices in any period t of the reference scenario b or policy alternate a are given by Pbt and Pat,
with corresponding indirect utility functions Vbt and Vat. In the results we employ annual equivalent variations
EVt  M t ( Pt b ,Vt a )  M t ( Pt b ,Vt b ) 10 to calculate welfare changes for any single period across policy scenarios.
Cumulative aggregate (or multi-market) welfare measures are found using discounted summations of EVt ( PEV )
which is possible as the measure is based upon a common baseline price vector. Future welfare changes are
discounted both by consumers' rate of time preference and by the human population growth rate. Also, given the
10
Defined as the difference between initial expenditure and that expenditure necessary to achieve the post-policy level of
satisfaction at initial prices.
11
12
exogenous time horizon a termination term is added to account for welfare impacts after the final period T. In
this we assume that by T the economy is close to a steady state.
2.4
Trade The model involves trade in both commodities and factors between the region and the
"rest of the world" or foreign sector. Product differentiation for all production sectors i is achieved in demand
through the use of the "Armington assumption" (Armington 1969). For commodities, gross regional demand Qi
is a composite of regionally produced goods QDi and imports QMi, where the differentiation is assumed to occur
in perfectly competitive international markets. The blend of regional and imported goods is found through
households minimizing the costs to them of meeting their composite commodity demands. Given relative prices
and substitution possibilities (from a CES function), the cost minimization is,
min
Pi QQi  Pi DQiD  Pi M QiM
subject to
Qi   [ Q
C
i
i 
C
i
M
i
(
 iC 1
 iC
)
 (1   )Q
C
i
F, R, C
D
i
(
 iC 1
 iC
(
)
]
 iC
 iC 1
(2.15)
)
where iC and  iC are sector specific parameters, and iC is elasticity of transformation. PQi, PDi, and PMi are
composite, regional, and import prices of good i. Manipulating the first order conditions finds,
QiD
Pi M 1   iC  iC

[(
)( C )] ,
QiM
Pi D
i
(2.16)
which determines the mix of imports to regional production.
Product differentiation is introduced to the commodity markets supply side through the use of a CET
function following De Melo and Tarr (1992). Industry output for all sectors Xi is allocated to regional
consumption XDi or export XEi through constrained maximization of industry revenues given regional and export
prices, subject to a CET function,
max
Pi X X i  Pi D X iD  Pi E X iE
subject to
X i   [ X
T
i
i 
T
i
E
i
F,R,C
(
1 iT
 iT
)
 (1   ) X
T
i
D
i
(
1 iT
 iT
(
)
]
 iT
)
1 iT
,
12
(2.17)
13
where iT and  iT are sector specific parameters, and hT is elasticity of transformation. PXi and PEi are composite
and export prices of good i. The resultant first order conditions determine the level of exports,
X iD
Pi E 1   iT  iT

[(
)( T )] ,
X iE
Pi D
i
(2.18)
The optimization allows for substitution between production for regional and export markets, driven by
the relative prices of regional goods and exports, and the magnitude of substitution possibilities given by the
elasticity of transformation, iT.
Factor inputs in the fishery are differentiated in the same fashion as commodity demands. Factor input
are a composite of regional and foreign factors. Total fishery demand for capital KF (labor LF) is a composite of
regional provisions KDF (LDF) and foreign KDF (LDF), where the differentiation is assumed to occur in perfectly
competitive international markets. The blend of regional and foreign factors is found through the industry
minimizing costs of meeting factor demands, given relative prices and substitution possibilities (given by a CES
function). For capital, the optimization is given by,
min
RK F  R D K FD  R M K FM
subject to
(2.19)
K F   [ K
K
F
K
F
M
F
(
 FK 1
 FK
)
 (1   ) K
K
F
D
F
(
 FK 1
 FK
)
]
K
( F )
 FK 1
where FK and FiK are parameters, and FK is elasticity of transformation. R, RD, and RM are the composite,
regional, and foreign capital service price. The mix of foreign to regional capital can be found from the first
order conditions,
K iD
R M 1   FK  FK

[(
)( K )] .
K iM
RD
F
(2.20)
An identical process is followed for fishery labor.
2.5
Price Relationships In modeling a "small" region (in relation to the rest of the world), the
price consumers' face are indices of import and regional prices, with import prices taken exogenously,
Pi Q 
( Pi DQiD  Pi M QiM )
,
Qi
(2.21)
13
14
Similarly, the prices producers receive are composites of domestic and parametric export prices,
( Pi D X iD  Pi E X iE )
.
Xi
Pi X 
(2.22)
Factor prices indices in the fishery are found in an identical fashion.
2.6
Equilibrium Conditions The system is in a temporary equilibrium when households and firms
optimize, there exists a set of prices and level of output at which all firms just break-even, Walras Law holds,
and all markets clear. Following from the restrictions of perfect competition and constant returns to scale brings
a competitive equilibrium condition of zero profits. The condition forces prices equal to average costs and so
equal to marginal costs due to constant returns to scale. Walras Law is fulfilled by consumer income (net of
savings) being equated to expenditures.
Incomes are derived through a two-stage process. Regional households are endowed with labor LAK and
capital KAK . These factors are exchanged in factor markets, and through the production process generate value
added. An important facet of our model is that we allow for the employment of foreign capital and labor in the
region. While foreign value added expenditures accumulate out of the region, regional value added expenditures
flow first to the factor "institutions", and are then redistributed to households. The labor and capital payments
that sum together to give household income IAK. Walras Law is maintained when there is equilibrium over all
markets (commodities and factors) and a set of prices exists at which all commodity and factor markets clear.
Model closure follows Waters, Holland and Weber (1997) and Coupal and Holland (2002) where net
financial inflows EXS (from foreign and domestic sources) adjust to balance the regional investment savings
balance.
Q
M
i
i
Pi M  W M LMF  R M K FM   X iE Pi E  ESX
i  {F, R, C} .
(2.23)
i
Gross regional investment is equated to the sum of regional savings, net of foreign borrowing or lending,
P
i
X
I i  ESX  PS S i   F , R, C .
(2.24)
i
3.
Calibration
14
15
3.1 Ecosystem Calibration In applying GEEM to the Alaskan ecosystem, ecological studies of the
Alaskan and other ecosystems were used to construct the benchmark dataset shown in Table 1. A reasonable
time series of pollock biomass estimates exists for the period 1966 through 1997, and the rest of the data are
from 1966 or interpolated to that date. Data were obtained for plant and animal populations, benchmark plant
biomasses and animal biomass demands, and parameters that include embodied energies, basal metabolisms,
and plant and animal weights and lifespans. Sources include numerous National Marine Fisheries Service
publications and ecological journal articles.
In the calibration procedure we assume that benchmark values of populations and biomass demands
may or may not be at their steady state levels. If the benchmark year were to be a steady state (with all net
energies equal to zero by definition), the method would consist of simultaneously solving the net energy
expressions, first order conditions and equilibrium conditions for each species to find benchmark values of
energy prices eij's, and parameters dik's and ri's. But, in cases where the benchmark is known to not have each
species at zero net energies, a certain amount of adaptation is required. For the reported results the calibration
procedure employs a 1966 calibration year with significant pollock harvests. Under this specification several
assumptions are necessary.11
For harvesting to not drive the pollock population to extinction there must be a "surplus" production of
pollock, implying positive net energy. Harvesting therefore lowers the pollock population below its natural
steady state levels. A crucial assumption is made at this stage. In this, xb42 (where the b superscript indicates a
benchmark value, as given in Table 1.) is assumed as the biomass demand of pollock in an intervened
ecosystem. The need of this distinction will become obvious below.
Under harvesting and the necessary excess production with its associated positive net energy, the net
energy expression and first order condition for pollock are,
11
b
b
b 2
b
b .5
R4b  [e3  e43
]x43
 r4 x43
  4  e4 [1  t4 e54
]d 45 x43
(3.1.1)
b
b
b
b .5
[e3  e43
]  2r4 x43
 .5e4 [1  t4 e54
]d 45 x43
0,
(3.1.2)
All other species (including sea lions) are assumed to be in a steady state in 1966.
15
16
where b superscripts denote benchmark values. But, in these equations there are three unknowns, Rb4, eb43, and
r4. To solve for these three variables, the effects of harvesting are introduced by our assumption of a steady
pollock population, implying growth (from the pollock population update equation) balancing harvests,


1 1 
ss
 y45  x43  w4 1    
b
b
 s4  s4   2 N 4 1  N 4
N 4b 

  s4v4ss N 4ss  2 N 4ss
v4ss

 






   R4 (.)  v4   


,


1  1 
y45 ( x43 (e)) 
b
1       TAC
w4
 s4  s4  

(3.1.3)
where,
b 2
ss 2
y45  d 45 x42.5 and v4  r4 x42
, v4ss  r4 x42
.
Notice the differentiation between xb42 and xss42. While xb42 is the biomass demand of an individual pollock in an
intervened ecosystem, xss42 is the magnitude of the demand in a natural ecosystem (under no intervention). As
b
ss
 x43
.
there is a positive net energy, V4  Vˆ4  x43
b
ss
b
ss
 x43
 e43
 e43
,
Given the differentiation in demands there must also be a differentiation in price, x43
leaving the three equations possessing six unknowns: Rb4, eb43, d45, r4, xss43, and ess43. While d45 would be
determined from the associated equilibrium conditions, we have to find the steady state values xss43, and ess43,
which imply steady state populations which may not coincide with the benchmark. We therefore have to
simultaneously solve for steady state equilibrium biomass demands, energy prices and species populations. In
this sub-component of the system, all individuals of all species have zero net energies, each individual optimizes
according to the first order conditions, and the equilibrium conditions hold (where we assume that the
parameters from the benchmark are the same as those in the steady state).
The complete method consists of the simultaneous solution of the benchmark and natural steady state
components as shown below.
16
17
Benchmark Sub-Component:
Net energy expressions, first order conditions and equilibrium conditions, with given 1966 populations,
biomass demands, embodied energies and basal metabolism. Note that all species apart from pollock
(due to harvests) have zero net energies.
Phytoplankton
b
R1b  0  [e01  e10b ]x10b  e1[1  t1e31
]d13 x10b .5  r1 x10b 2  1
(3.1.4)
b
[e01  e10b ]  .5e1[1  t1e31
]d13 x10b .5  2r1 x10b  0
(3.1.5)
Kelp
b
b
b
b .5
b
R2b  0  [e02  e20
]x20
 .5e2 [1  t2e62
]d 26 x20
 2r2 x20
 2
(3.1.6)
b
b
b .5
b
[e02  e20
]  .5e2 [1  t2e62
]d 26 x20
 2r2 x20
0
(3.1.7)
Zooplankton
b
b
b
b .5
b2
R3b  0  [e1  e31
]x31
 e3[1  t3e43
]d34 x31
 r3 x31
 3
(3.1.8)
b
b
b .5
b
[e1  e31
]  .5e3[1  t3e43
]d34 x31
 2r3 x31
0
(3.1.9)
Sea urchin
b
b
b
b .5
b 2
R6b  0  [e2  e62
]x62
 e6 [1  t6e76
]d 67 x62
 r6 x62
 6
(3.1.10)
b
b
b .5
b
[e2  e62
]  .5e6 [1  t6e76
]d 67 x62
 2r6 x62
0
(3.1.11)
Pollock
b
b
b 2
b
b .5
R4b  [e3  e43
]x43
 r4 x43
  4  e4 [1  t4 e54
]d 45 x43
(3.1.12)
b
b
b
b .5
[e3  e43
]  2r4 x43
 .5e4 [1  t4 e54
]d 45 x43
0,
(3.1.13)
Stellar Sea Lions
b
b
b
b .5
b 2
R5b  [e4  e54
]x54
 e5 [1  t5e85
]d58 x54
 r5 x54
 5
(3.1.14)
b
b
b .5
b
[e4  e54
]  .5e5 [1  t5e85
]d58 x54
 2r5 x54
0
(3.1.15)
Sea otter
17
18
b
b
b
b .5
b 2
R7b  0  [e6  e76
]x76
 e6 [1  t7 e87
]d 78 x76
 r7 x76
 7
(3.1.16)
b
b
b .5
b
[e6  e76
]  .5e6 [1  t7 e87
]d 78 x76
 2r7 x76
0
(3.1.17)
Killer whale
b
b
b
b
R8b  [e5  e85
]x85
 [e7  e87
]x87
(3.1.18)
b
b
b b
b 2
b 2
 r8 ( x87
 x85
)  .5r8 ( x87
x85  x85
 x87
)  8
b
b
b
[e5  e85
]  r8  .5r8 ( x87
 2 x85
)  0,
(3.1.19)
b
b
b
[e7  e87
]  r8  .5r8 ( x85
 2 x87
)0,
(3.1.20)
Equilibrium Conditions
N1b x10b   11300000 ,
(3.1.21)
b
N 2b x20
  2 26000 ,
(3.1.22)
b
N 3b x31
 N1b d13 x10b .5 ,
(3.1.23)
b
b .5
N 4b x43
 N 3b d34 x31
,
(3.1.24)
b
b .5
,
N 5b x54
 N 4b d 45 x43
(3.1.25)
b
b .5
N 6b x62
 N 2b d 26 x20
,
(3.1.26)
b
b .5
N 7b x76
 N 6b d 67 x62
(3.1.27)
b
b .5
N8b x85
 N 5b d58 x54
(3.1.28)
b
b .5
,
N8b x87
 N 7b d 78 x76
(3.1.29)
Pollock Harvesting


1 1 
ss
 y45  x43  w4 1    
s4  s4   2 N 4b 
N 4b

b 
1
N4

  s4v4ss N 4ss  2 N 4ss
v4ss









   R4 (.)  v4   




1  1 
y45 ( x43 (e)) 
b
1       TAC
w4
 s4  s4  

(3.1.30)
Natural Steady State Sub-Component:
18
19
In the natural steady state all species will have zero net energy, all first order conditions will hold and
all equilibrium conditions will be satisfied. The equations for the steady state in this stage are therefore:
Phytoplankton
R1ss  0  [e01  e10ss ]x10ss  e1[1  t1e31ss ]d13 x10ss.5  r1 x10ss 2  1
(3.1.31)
ss
[e01  e10ss ]  .5e1[1  t1e31
]d13 x10ss .5  2r1 x10ss  0
(3.1.32)
Kelp
ss
ss
ss
ss .5
ss 2
R2ss  0  [e02  e20
]x20
 e2 [1  t2e62
]d 26 x20
 r2 x20
 2
(3.1.33)
ss
ss
ss .5
ss
[e02  e20
]  .5e2 [1  t2e62
]d 26 x20
 2r2 x20
0
(3.1.34)
Zooplankton
ss
ss .5
ss 2
R3ss  0  [e1  e31ss ]x31ss  e3[1  t3e43
]d34 x31
 r3 x31
 3
(3.1.35)
ss
[e1  e31ss ]  .5e3[1  t3e43
]d34 x31ss .5  2r3 x31ss  0
(3.1.36)
Sea urchin
ss
ss
ss
ss .5
ss .5
R6b  0  [e2  e62
]x62
 e6 [1  t6e76
]d 67 x62
 r6 x62
 6
(3.1.37)
ss
ss
ss .5
ss
[e2  e62
]  .5e6 [1  t6e76
]d 67 x62
 2r6 x62
0
(3.1.38)
Pollock
ss
ss
ss 2
ss .5
R4ss  0  [e3  e43
]x43
 r4 x43
  4  e4 [1  t4 e54ss ]d 45 x43
(3.1.39)
ss
ss
ss .5
[e3  e43
]  2r4 x43
 .5e4 [1  t4 e54ss ]d 45 x43
0,
(3.1.40)
Stellar Sea Lions
R5ss  0  [e4  e54ss ]x54ss  e5 [1  t5e85ss ]d58 x54ss .5  r5 x54ss 2  5
(3.1.41)
[e4  e54ss ]  .5e5 [1  t5e85ss ]d58 x54ss .5  2r5 x54ss  0
(3.1.42)
Sea otter
ss
ss
ss .5
ss 2
R7ss  0  [e6  e76
]x76
 e6 [1  t7 e87ss ]d 78 x76
 r7 x76
 7
(3.1.43)
ss
ss .5
ss
[e6  e76
]  .5e6 [1  t7 e87ss ]d 78 x76
 2r7 x76
0
(3.1.44)
19
20
Killer whale
R8ss  0  [e5  e85ss ]x85ss  [e7  e87ss ]x87ss
(3.1.45)
 r8 ( x87ss  x85ss )  .5r8 ( x87ss x85ss  x85ss 2  x87ss 2 )  8
[e5  e85ss ]  r8  .5r8 ( x87ss  2 x85ss )  0 ,
(3.1.46)
[e7  e87ss ]  r8  .5r8 ( x85ss  2 x87ss )  0 ,
(3.1.47)
Equilibrium Conditions
N1ss x10ss   11300000 ,
(3.1.48)
ss
N 2ss x20
  2 26000 ,
(3.1.49)
N 3ss x31ss  N1ss d13 x10ss.5 ,
(3.1.50)
ss
N 4ss x43
 N 3ss d34 x31ss.5 ,
(3.1.51)
ss .5
,
N 5ss x54ss  N 4ss d 45 x43
(3.1.52)
ss
ss .5
N 6ss x62
 N 2ss d 26 x20
,
(3.1.53)
ss
ss .5
,
N 7ss x76
 N 6ss d 67 x62
(3.1.54)
N8ss x85ss  N 5ss d58 x54ss .5
(3.1.55)
ss .5
,
N8ss x87ss  N 7ss d 78 x76
(3.1.55)
The calibration consists of simultaneously solving the above fifty three equations for fifty three endogenous
variables, i's, dik's, ri's, ebij's, essij's, xssij's, and Niss's, where the i's, dik's, and ri's are held constant across
benchmark and natural steady state sub-components.
3.2 Economic Calibration The economic specification is based on a benchmark of 1997. The
benchmark dataset is shown in Table 1. Data sources are given in the Table and include reports from the U.S.
Department of Commerce, Bureau of Census, Bureau of Labor Statistics, the Alaskan Bureau of Economic
Analysis, and others.
The calibration routine sets benchmark input and output prices equal to unity (by constant returns to
scale and the units of the initial data being in value terms). Using all first-order conditions from profit
20
21
maximization, cost minimization, and utility maximization; and the benchmark data and prices, most parameters
apart from the elasticities of substitution are found.
Calibration equations for the first and second nests of household behavior and those for composite
goods are exactly the same as Ballard et al. (1985) and De Melo and Tarr (1992) and omitted. Calibrated
parameters are displayed in Table 3.
In parameterizing households, gross annual income is the sum of all payments to capital and wage and
salary disbursements by place of work for the region. Savings are determined using Consumer Expenditure
Survey data for the region, following the methods of Ballard et al., and recreation expenditures are from Bureau
of Census data for wildlife related recreation. Using the Survey data, all fish consumption in the region is
assumed to be comprised of pollock, and composite goods consumption determined as a residual.
The parameterization of the bottom nest of household behavior follows directly from De Melo and Tarr
(1992). The first step to finding FH, CH,  FH and  CH is to find the  iH knowing they must sum to unity. Given
the definition for the income elasticity of demand in the Stone Geary it is possible to solve for,
iH 
iH PQib HX ib
,
b
I AK
 Psb S b  PRb H Rb
(3.2.1)
where Hi is the income elasticity of demand and all b superscripts indicate benchmark values of variables. We
use the income elasticity estimate from De Melo and Tarr for Food of 0.3 (page 235) as our value for HF. The
second step is then to find the Stone Geary subsistence parameters iH. These are calibrated according to the
formula,
  H , b  I b  Psb S b  PRb H Rb 
 iH  HX ib   i b  AK
 i  F, C
FRISCH

 PQi  
(3.2.2)
where De Melo and Tarr define the FRISCH parameter as the marginal utility of income with respect to income.
We follow those authors and employ their central elasticity case of -1.09, allowing the full specification of
household behavior.
We parameterize the quota function in a similar fashion to Finnoff and Tschirhart (2003). Using
biomass estimates, total allowable catches and actual catches from Witherell (2000) we estimated the quota
function (with additive errors assumed iid) by ordinary least squares using LIMDEP software. Estimating the
21
22
quota function presented difficulty due to fairly constant TAC's since the fisheries inception (with relatively
stable, large populations). In this case the safe minimum population is very low (unrealistically) just because of
the historical records having fairly uniform harvests for high population levels. Normalizing the data around its
"lower left hand value" (ie the lowest biomass harvest combination) shifts both axes to this point in the data. The
quota function was then estimated for this "normalized range". We corrected for first order autocorrelation
found in the quota function, and obtained significant parameter estimates employed in the reported results. This
provides a quota function for populations greater than the normalization point. For populations less than the
normalization point, 1000000 tons was arbitrarily chosen as reasonable. Given this, a linear function was
employed between this point and the normalization point. We corrected for first order autocorrelation found in
the quota function, and obtained significant parameter estimates displayed in Table 2.
The harvest function is estimated using a linear transformation of (2.1) and a time series of harvests,
aggregate effort, and biomass estimates gathered from Northern Economics Inc. and EDAW (2001), Northern
Economics (2001) and Witherell (2000). OLS regressions directly yield aF, and allowed dF to be recovered from
the estimate of the intercept12.
Following standard practice in CGE, factor prices are set to one in the non-fishery sectors, but the prices
in the fishery are set according to (2.4). An estimate of  is obtained by noting that in 1997 the season was 111
days, and if a fully employed year is taken as 250 days,  = 111/250  0.444. Labor is used to obtain ,
although as stated above capital is assumed to exhibit the same divergence. The fishery employed 6035 full-time
equivalent workers (FTE) in 1997 (Northern Economics Inc. and EDAW, 2001) and the total labor bill was
$293,570,000, yielding $48,645 per FTE. The annual wage per FTE in all Alaskan sectors for 1997 was
$43,368.13 The greater fishery wage reflects the rents as explained above, and the divergence measure is  =
$48,645/$43,368  1.12. Finally, the factor payments for labor and capital are   W =   R = 0.497 indicating
that fishery factor payments relative to the payments in other sectors are offset downward owing to the shorter
season, but offset upward owing to the rents.
Only catcher processor CP results were significant, providing an estimate of aF of 0.8997 with standard error 0.2324 (pvalue of 0.0061) and an adjusted R2 of 0.636. We assume both harvesting sectors have a homogeneous technology.
13
Institute of Social and Economic Research, University of Alaska Anchorage,2003.
http://citizensguide.uaa.alaska.edu/4.COMPARISON_TO_OTHER_PLACES/4.5.PublicEmployeesandWages.htm
12
22
23
For the recreation sector, given Cobb-Douglas production functions, the first order conditions are
usually used to find the distributional shares over labor, capital and ecosystem inputs NAT0. But, the presence of
the ecosystem inputs creates problems. Under the zero profit assumptions employed in the construction of the
dataset any costless inputs to production are ignored. In effect this will assign a zero value to cR! To circumvent
this issue, using the CRS identity,
a R  bR  cR  1
(3.2.3)
we alternately substitute for aR and bR in the manipulation of the first order conditions to derive,
a 
R
WG0 LR ,0 (1  c R )
WG0 LR ,0  RN 0 K R ,0
b 
R
RN 0 K R ,0 (1  c R )
WG0 LR ,0  RN 0 K R ,0
.
(3.2.4)
This allows us to exogenously assign a value to cR to reflect the importance of ecosystem inputs. The efficiency
parameter is found from the definition of the production function given the distributional shares,
dR 
X R ,0
R
R
LaR ,0 K Rb ,0 NAT0c
R
.
(3.2.4)
The calibration required we construct an aggregate measure of the relative importance of the Bering Sea
ecosystem to Alaskan recreation. In a two-step procedure we first assumed that the importance of Alaska's
natural resources is realized through factor productivity enhancements NAT in the industry. In attempt to
determine a value for these enhancements we assumed they accrue only to capital in the industry. We then
calculated an "un-enhanced" measure of capital services for the industry as a fixed proportion of XR given the
percentage of "Other Expenditures" to "Total Expenditures" of Federal, State, and nongovernmental agencies
directly associated with environmental stewardship in Alaska, (Table 3, page 10, Colt, 2001). Productivity
enhancements attributable to the natural resources were then found as the residual of XR net of labor and unenhanced capital inputs (an amount equal to 3.8% of industry output).
Second, we calculated the percentage of visitor recreation that takes place in the South West region (our
study area) using data from McDowell Group (1998). Within this percentage (5%) we assumed only killer
whales N8, stellar sea lions N5 and sea otters N7 are of importance to the industry (we make this assumption on
the substantial cruise industry in the region that markets whale watching, and the high profile of the marine
23
24
mammals, especially endangered stellar sea lions). Further, we assumed an equal weight to each species. Under
these assumptions the aggregate measure of natural resource inputs to the industry becomes,
ECOR  ECORNET  ECORBSAI ,
(3.2.5)
ECORBSAI  aSL N 5  aO N 7  aKW N8 ,
(3.2.6)
and
where ECORNET are all resources apart from those in the BSAI, ECORBSAI. Coefficients aSL, aO and aKW assign
relatively equal weights to sea lion, sea otter and killer whale populations. In an important assumption we
assume the stock of "other" natural resources in Alaska, ECORNET, remains constant at 1997 levels through out
our analysis.
Investment is treated commonly in all sectors. The fixed proportion coefficients for investment demands
are calculated as a fixed proportion of total output attributable to regional factor employment XiDF, ITi = aiINXiDF
aiIT 
ITi ,0
X iDF
,0
i  {CV, CP, P, R, C} ,
(3.2.7)
where ITi,0 are benchmark investment demands of industry i.
Following standard practice, estimates of elasticities of substitution are taken from the literature and
displayed in Table 3. The elasticity between labor and capital in composite goods  C is the average of all
reported industries other than agriculture, forestry and fishery industries as reported in Ballard et al.
For households, both the elasticity between current consumption goods  1R and the elasticity between
current and future consumption  2R are averages of values in Ballard et al. Both are based on the assumption of
a savings elasticity of 0.4. In the specification of the Stone-Geary utility function estimates of the income
elasticity of demand for food  HF , I and the marginal utility of income with respect to income MUIT are taken
from De Melo and Tarr (1992).
24
25
For trade related elasticities central estimates of De Melo and Tarr (1992) were employed.14 The
average of all industries other than agriculture was employed for the elasticity of substitution in demand
between regional goods and imports in composite goods  CC , while agriculture was used for fishery demand (and
factor)  FC (  FK ,  FL ) elasticities. Similarly, the elasticities of transformation of composite goods  CT , the
fishery  FT , and recreation  RT are from De Melo and Tarr’s estimates for agriculture (fishery and recreation)
and the average of all else for composite goods.
14
A limited sensitivity test using the low and high elasticity estimates of De Melo and Tarr were performed with only
minor changes in results. The more rigorous Monte Carlo approach following Abler, Rodrigues and Shortle (1999) is left
for future research.
25
Table 1. Initial Variables and Parameters for the Marine Ecosystem
Phytoplank.
Zooplank.
Pollock
Steller sea
lion
Killer whale
42.222b
1 unit =
1x1012 ind.
210.816c
1 unit =
1x109 ind.
3d
1 unit =
1000 ind.
0.1236154e
1 unit =
1 ind
0.007723
1 unit =
0.1 ind.
Sea otter
Urchin
Kelp
0.050631g
1 unit =
100 ind.
10.7692h
1076.92i
1 unit =
1 x 107 ind.
1 unit =
1x104 ind
o
255,500
kg unit-1 y-1
p
330,000
kg unit-1 y-1
21024q
kg unit-1
1810w
kcal kg-1
717x
kcal kg-1
821y
kcal kg-1
Variables
Populations
Ni a
(units km-2)
Biomass or
Biomass
Flow
xij
Parameters
Embodied
Energy
ei
435.6
kg unit-1
1782.7k
kg unit-1 y-1
8030
kg unit-1 y-1
2663
kgy-1
400r
kcal kg-1
559s
kcal kg-1
1107t
kcal kg-1
2000u
kcal kg-1
j
l
m
f
401.098n kg
unit-1 y-1
(Steller)
221.601 kg
unit-1 y-1 (otter)
NAv
650ab
kcal kg-1
yr-1
Light
Absorption
e0i
40868z
kcal kg-1 yr-1
NA
Resting
Metabolic
Rate bi
1068126 ac
kcal unit-1 yr-1
178270ad
kcal unit-1 yr-1
1346631ae
kcal unit-1 yr-1
731326af
kcal yr-1
195535ag
kcal unit-1 yr-1
32193000ah
kcal unit-1 yr-1
67732500ai
kcal unit-1 yr-1
2049840aj
kcal unit-1
yr-1
Weight
wi
Predation
dijas
Plant
Congestion
435.6ak
kg unit-1
3.757al
kg unit-1
1000am
kg unit-1
250an
kg
399.6ao
kg unit-1
2800ap
kg unit-1
87600aq
kg unit-1
21024ar
kg unit-1
426.4804
2.7064
1.3335
0.4459
NA
0.0669
2.0816
22.9661
iat
0.014148
NA
NA
NA
NA
NA
NA
870.814
Var. Resp.
riau
(kcal yr-1)
15.6236
0.0664
0.0250
0.1066
1.3085
0.0005
0.0006
0.0084
aa
NA
aa
NA
26
aa
NA
aa
NA
aa
NA
aa
Table 2. Notes
NA – not applicable or not needed.
a
Individuals are aggregated into population units and the units are divided by ocean surface area to yield
population units per square kilometer. Pelagic populations are divided by 1.3x106 km2, the approximate area of
the EBS, and nearshore populations are divided by 26,000 km2, the approximate area along the Aleutian Islands.
Killer whales are divided by both areas.
b
An aggregate of multiple phytoplankton producer and saprophage species (Petipa et al. 1970, Table 1). The
data are from the Black Sea but assumed to be transferable to the EBS. Populations in Petipa et al. are given in
individuals per square meter; thus, when extrapolating to the EBS, the number of individuals is in an
unmanageable sextillions. Consequently for phytoplankton and other species in Table 2 populations are
converted to population units, then placed on a square kilometer basis.
c
An aggregate of multiple zooplankton herbivore species (Petipa et al. 1970, Table 1) The data are from the
Black Sea but assumed to be transferable to the EBS.
d
Pollock biomass estimates for 1966 including juveniles are 3,900,000 metric tones (Ianelli et al. 2002).
Assuming pollock are 1 kg on average, this is 3.9x109 individuals which converts to 3.9x106 population units.
On a km2 basis: 3.9x106 units/1,300,000 km2 =3. Recall, 1,300,000 is the ecosystem size in km2.
e
1966 Stellar sea lion population was estimated to be 160,700 (Appendix D 2000), and on a km2 basis:
160,700/1,300,000 km2.
f
Population assumed to be roughly the same in 1966 as that in the early 1980’s. Based on 1024 individuals
(SAFE 2000, Appendix D). Because killer whale habitat includes both ocean and nearshore systems, the
population was divided by 1,300,000 + 26,000 to put on a square kilometer basis.
g
Based on 131,631 individuals extrapolated from Estes and Duggins (1995) estimates of populations in
Aleutians island groups.
h
Individuals from multiple sea urchin species at 153 randomly selected sites in the Aleutians (Estes and
Duggins 1995).
i
Kelp density of multiple species is about 10% of urchin at the same 153 sites in the Aleutians (Estes and
Duggins 1995).
j
A weighted average of phytoplankton species’ body weights (4.35615x10-10 kg., Petipa et al. 1970, Table 1), in
units of 1 x 1012 phytoplankton.
k
A weighted average of zooplankton species indicates an individual weighs 3.757x10-6 gm. and consumes 130%
of its weight in phytoplankton per day (Petipa et al. 1970, Table 1). This yields a consumption of 1782.7 kg unit1
yr-1.
l
Trites et al. (1997) p. 186. Pollock eat mostly zooplankton (Witherell 2000) although adults may eat smaller
fish including juvenile pollock.
m
From Appendix D, SAFE, in 1990s Steller diet was 76% fish, of which 69% was groundfish and we assume
60% was pollock. Therefore, of the 5840 kg/yr taken by an individual sea lion (based on Rosen and Trites,
2000), the pollock consumption was (.76) (.60) (5840) = 2663kg/yr.
n
Killer whale prey includes sperm and baleen whales, pinnepeds, seabirds, fish, turtles, otter, and based on the
stomach content of one whale, pigs; however, there is no consensus on the importance of any one prey
(Jefferson et al. 1991). We assume that around 1980 the proportion of Steller sea lions in the killer whale diet
was the same as the proportion of the Steller sea lion population in the sum of the populations of Steller sea
lions, harbor seals, Northern fur seals and walruses in the EBS region as reported in Trites et al. (1997). This
amount is about 10% of the total diet, or 8,021,970 Kcal per year (the total is based on the daily killer whale
energy requirement (Estes et al. 1998)). As the embodied energy, per kg, of sea lions is 2,000 Kcal, and the
average sea lion weights 250 kg, 16.04 sea lions are eaten by a killer whale each year. In terms of biomass this is
therefore 4,010.985 kg of sea lions per killer whale, in population units this becomes 401.0985 kg. Otter are
assumed to make up 5% of the total killer whale biomass demand, or 4,010,985 Kcal per year15. Given an
(assumed) embodied energy of 1,810 Kcal for otter, the biomass demand of a killer whale for otter is 2,216.0138
kg, or 221.6014 kg in population units. o Otter eat 20-30% of body weight per day and on average an adult
weighs 28 kg. (Costa 1978). Otter eat mostly sea urchins (Mason and Macdonald 1986), and here they are
assumed to eat only sea urchins.
15
Estes et al. indicate killer whales did not consume significant numbers of otter until recently.
27
p
Urchin weighing 0.00876 kg are assumed to grow by 38% in one year to 0.01201 (Estes and Duggins 1995,
Table 11). This implies production of 0.003329 and if they consume ten times their production implies 0.03329
of biomass flow per individual. This is rounded to 333000 per population unit.
q
Average biomass of an urchin is 0.00876 kg (Estes and Duggins 1995) and multiplied by the urchin population
(vii) yields 943382 kg for the population. Assuming prey biomass is 1.2 times predator biomass (Kerr 1974),
and assuming 5% of predation on kelp is by sea urchin, yields a biomass for kelp of 943382 (1.2)/.05. Per
population unit this is 21024 kg.
r
Weighted average of caloricity measures of three phytoplankton species groupings (Petipa et al. 1970, Table
7).
s
Weighted average of caloricity measures of three zooplankton species groupings (Petipa et al. 1970, Table 7).
t
In a captive situation, 7.2kg d-1 of pollock was fed to sea lions and its energy content was 4.72 kJ g-1 (Rosen
and Trites 2000); therefore, the kcal embodied energy in pollock is (4.72 kJ g-1) (1Mcal/4.184MJ) (1
MJ/1000kJ)(1000kcal Mcal-1)(1000g kg-1) = 1128 kcal kg-1.
u
Estimated based on blubber content in a sea lion versus otter which have no blubber. (Costa 1978) (See w).
v
Not needed because killer whales are at the top of the food web and are not prey.
w
Estes et al. (1998).
x
Costa (1978).
y
Lembi and Waalan (1988).
z
A rough rule of thumb is that 10% of the energy taken at one trophic level is passed on to the next trophic level
(See, e.g., Pauly and Christensen 1995). Petipa et al. suggest a 20% transfer rule for ocean communities.
Therefore, equate 20% of the energy taken by phytoplankton to the energy taken by zooplankton: (20%) N1 x10
e01 = N2 x21 e1 and solve to obtain e01 = 15150. kcal kg-1 yr-1. (Note N1 is from b, x10 from j, N2 from c, x21 from k
and e1 is from r.
aa
Not applicable because only plants photosynthesize.
ab
Using the 20% transfer rule (See z.), equate 20% of the energy taken by kelp to the energy taken by urchin:
(20%) N2 x20 e02 = N6 x62 e2 and solve to obtain e02 = 650 kcal kg-1 yr-1.
ac
An average of respiration as a % of body weight over multiple phytoplankton species yields 6%. (Petipa et al.
1970, Table 2). Incoming phytoplankton energy is e01 x10 = (40868)(435.6), and 6% of this is 1068126 kcal yr-1.
ad
An average of respiration as a % of body weight over multiple zooplankton species yields 25%. (Petipa et al.
1970, Table 2). Calculations are similar to ac.
ae
Pollock are assumed to have an average respiration of 30%. Their incoming energy from zooplankton is (8030
kg unit-1 y-1)(559 kcal kg-1) which is then multiplied by 30%.
af
For mammals, resting metabolic rate in kcal d-1 (M) is related to body weight (W) by the formula M =
67.61W0.756  5% (Kleiber 1975). Using 250 kg as sea lion weight and extrapolating to one year yields 1603786
kcal yr-1. The RMB used in the simulations is lowered by (76%)(60%) to reflect that sea lions are preying on
more than just Pollock.
ag
Use the formula from af and an average weight of 3996 kg, reduced by 85% to reflect the model only
accounting for 15% of killer whale energy intake
ah
Use the formula from af and an average weight of 28 kg and a +5% because otter have high metabolic rates
(Costa 1978).
ai
Similar to the estimate in ad except urchin are assumed to respirate at about 25%.
aj
Calculated as in ac except algae respiration (kelp) is assumed to be 15% of the value of photosynthesis (Petipa
et al. 1970, Table 2).
ak
Phytoplankton are plants; therefore, weight is given in (ix).
al
Average of multiple zooplankton herbivore species (Petipa et al. 1970, Table 1).
am
Average of adult and juvenile, both are taken by fisheries and Steller sea lions. (See d.)
an
Based on weights of immature sea lions in Rosen and Trites (2000) and adult weights in Whitaker (1980).
ao
Average of male and female adults is 3996 kg (Estes et al. 1998).
ap
Average of male and female adults is 28 kg (Costa 1978).
aq
Urchins at six locations in the Aleutians averaged 8.76 gm each with a wide variance (Estes and Duggins
1995, Table 2).
ar
Kelp are plants; therefore, weight is given in q.
28
as
In population units km-2. Calculated from the short-run equilibrium (i.e., biomass clearing) conditions using
benchmark values for populations, biomasses and biomass flows (i.e., demands) from the first two table rows.
at
Calculated using the plant congestion conditions and assuming that at the benchmark values for populations,
biomasses and biomass flows, the plants fully occupy the available water space.
au
In kcal yr-1. Derived from calibration. The benchmark biomasses and biomass flows were used as parameters
in the eight net energy objective functions set to zero and in the nine first-order conditions to derive values for
the variable respiration terms, ri, and the energy prices, eij. The derived energy prices are benchmark energy
prices in the simulation.
29
Table 2
Value of Benchmark Variables, in Million $
Variable
Value
KF
365.608
LF
293.567
QMF
0.244
QF
659.420
KR
894.368
LR
766.398
QR
1660.766
KC
10311.029
LC
8755.514
QMC
10938.005
QC
30004.549
XDF
32.080
XDR
752.798
XDC
20104.217
a
b
c
c
d
e
f
g
h
i
j
k
l
m
Definition
Variable
Value
Fishery Capital
KAK
11263.681
Fishery Labor
K
307.325
Fish Imports
LAK
9625.415
Aggregate Fish
Output
L
190.064
Recreation Capital
CTAF
24.443
Recreation Labor
CTR
737.244
Aggregate Recreation
Output
Composite Goods
Capital
Composite Goods
Labor
Composite Goods
Imports
Aggregate Composite
Goods Output
Regional Fishery
Demand
Regional Recreation
Demand
Regional Composite
Goods Demand
CTAC
19925.646
S
201.764
IF
7.638
IR
15.554
IC
178.572
XEF
627.340
XER
907.968
XEC
9900.331
n
o
p
q
r
s
t
u
v
w
x
y
z
aa
Definition
Regional Capital
Endowment
Foreign Capital
Regional Labor
Endowment
Foreign Labor
Household Fish
Demand
Household Recreation
Demand
Household Composite
Goods Demand
Household Savings
Fishery Investment
Recreation Investment
Composite Goods
Investment
Fish Exports
Recreation Exports
Composite Goods
Exports
Table 1 Notes
a
KF source: The fishery is modeled as a single, vertically integrated industry consisting of catcher vessels (CV),
catcher processors (CP) and, motherships and inshore processors (P). The sector and regional profiles follow the
Steller Sea Lion Supplemental Environmental Impact Statement (SEIS, U.S. Department of Commerce, 1991).
Given our assumption of zero profits in the industry, F payments for capital services are determined as a
residual of the value of total output of the fishery XF net of payments to labor LF ((see notes b and c) Regionally
and foreign supplied capital, KDF and KMF are the Alaskan and Non-Alaskan ownership proportions of KF,
assumed to be given by the ratios of Alaska Payments to Labor and Non-Alaska Payments to Labor to Total
Payments to Labor (Table 2.1.1-30, Northern Economics Inc. and EDAW, 2001; see note b). We find capital in
real terms in the same fashion as labor, discussed in Section 4.2.
b
LF source: CV labor payments are found by multiplying Groundfish Total Payments to Labor (Table 2.1.1-24,
Northern Economics Inc. and EDAW, 2001) by both the proportion of BSAI vessels and pollock's proportion of
total groundfish catches. CP payments to labor are found by scaling BSAI Total Payments to Labor (Table
2.2.1-16, Northern Economics Inc. and EDAW, 2001) by pollock's proportion of total groundfish catches. P
payments to labor are found by scaling Groundfish Processing Payments to Labor (Table 3.1-18, Northern
Economics Inc. and EDAW, 2001; and Table 2.3.6-17, Northern Economics 2001) by BSAI pollock's
proportion of total processed groundfish. Regional factor differentiation for each of the component sectors
30
followed a similar process. Regionally supplied CV labor is found from weighting Alaska Payments to Labor
(Table 2.1.1-30, Northern Economics Inc. and EDAW, 2001) by both the proportion of BSAI vessels and
pollock's proportion of total groundfish catches. Foreign supplied labor is also determined by the same
weighting of Non-Alaska Payments to Labor from the same table. Regionally and foreign supplied labor for CP
are found by weighting total CP labor employment by the relevant Alaskan and Non-Alaskan proportions of
total CP labor payments as given in Table 2.2.1-21, Northern Economics Inc. and EDAW (2001). We assume
processors employ only domestic labor.
c M
Q F and QF sources: QF as in b, is an aggregation of CV, CP, and P output. CV output is from the BSAI Pollock
Exvessel Value of landings, Table 2.1.1-21, Northern Economics Inc. and EDAW (2001). In order to construct a
single catcher vessel sector the five major vessel classes (trawlers: greater than or equal to 125 feet; 60 to 125
feet; AFA-eligible greater than or equal to 60 feet; non-AFA greater than or equal to 60; and those less than 60
feet in length) are aggregated into a single uniform class. Regretfully, this procedure neglects any economies of
scale that may be present in larger vessels and further assumes a constant efficiency of effort across fishery
participants. CP output is defined as BSAI Wholesale value of Pollock, Table 2.2.1-15, Northern Economics
Inc. and EDAW (2001). In a similar fashion as with CV, with identical restrictions, we aggregate the three major
catcher processor classes (Surimi, Fillet, and Head and Gut catcher processors) into a single sector. The final
component of the sector, P is classified as all relevant inshore processors (Bering Sea, Alaskan Peninsula and
Aleutian Islands, Kodiak and South Central Alaska) and motherships are aggregated into a single measure. The
value of mothership processing is given by BSAI Wholesale value of Pollock (Table 2.3.6-15, Northern
Economics Inc. 2001). All pollock processed at Bering Sea and Alaskan Peninsula and Aleutian Island inshore
processors are from the BSAI and corresponding values given by Pollock Wholesale Production Value (Table
1.6-4, Northern Economics Inc. 2001). The BSAI proportion of groundfish landings are used to scale output
values of Kodiak and South Central inshore processors. QMF is assumed to be 1% of CTAF (see note r) in the
absence of observed data, allowing some substitution possibilities for households.
d
KR source: Determined as a residual of the value of XR net of LR (see notes e and f).
e
LR source: Estimated Total Employment Income From Tourism For The State Of Alaska, Table 11, Colt
(2001).
f
XR source: Expenditures for Wildlife Related Recreation, By State Where Recreation Took Place (Alaska),
Table 48, Bureau of Census, 1996.
g
KC source: Found through a two step process. First, QDC is found as the residual of gross state product for all
private industries in Alaska (Bureau of Economic Analysis, 1997a), net of FXF and XR (note that their are no
imports for either of these industries, so Qi = QDi = Xi for each sector). Second, given the zero profit assumption,
KC is simply the residual of QDC net of LC below.
h
LC source: Determined as the residual of ωLAK net of LR and LDF (see notes peb, b, and ii.)
i M
Q C source: Determined from the market balance condition: QC = XC, which is identical to QMC + QDC = XEC +
XDC, allowing QMC = XEC + XDC - QDC. See notes g, m, and aa.
j
XC source: Found as the sum of XDC and XEC. See notes m and aa.
k D
X F source: Determined as the sum of CTAFCTAF and IF. See notes r and v.
l D
X R source: Determined as the sum of CTCTR and IR. See notes s, and u.
m D
X C source: Determined as the sum of CTACCTAC and IC. See notes t and w.
n
KAK source: Found as the sum of all regional payments to capital from the fishery KDF, recreation KR, and
from composite goods KC (see notes a, d, and g).
o
K source: Determined as the imports of capital in the fishery. See note a.
LAK source: Wage and salary disbursements by place of work, Alaska (Bureau of Economic Analysis, 1997b.)
q
L source: Determined as imported labor in the fishery. See note b.
p
CTAF source: All fish consumption in the region is assumed to be comprised of Pollock. For the amount
consumed we use 1997 per household expenditures on Fish and Seafood for the Western United States (Series
CXUFHM50804, Consumer Expenditure Survey, Bureau of Labor Statistics Data, http://data.bls.gov/). To
convert this into the relevant value for Alaska we multiplied it by Alaska's consumption units for 1997
(consumption units are households, derived by dividing Alaska's population (608846 in 1997, Bureau of
r
31
Economic Analysis, 1997b) by the average number of people per household (2.74 in 1997 Consumer
Expenditure Survey, Bureau of Labor Statistics Data, http://data.bls.gov/) to find consumption units).
s T
C R source: Expenditures for Wildlife Related Recreation, By Participants State of Residence (Alaska), Table
49, Bureau of Census, 1996.
t T
C AC source: Found as the residual of regional household income IAK (the sum of ωKAK and ωLAK from notes n
and p) net of S, CTR, and CTF (notes r and s).
u
S source: Savings for the region are derived from the 1997 Net Change in Consumer Total Assets and
Liabilities for the Western United States (Series CXUNC000804, Consumer Expenditure Survey, Bureau of
Labor Statistics Data, http://data.bls.gov/). To convert this into the relevant value for Alaska we multiplied it by
Alaska's consumption units for 1997 (consumption units are households, derived by dividing Alaska's
population (608846 in 1997, Bureau of Economic Analysis, 1997b) by the average number of people per
household (2.74 in 1997 Consumer Expenditure Survey, Bureau of Labor Statistics Data, http://data.bls.gov/) to
find consumption units).
v
IF source: See note x.
w
IR source: See note x.
x
IC source: Investment expenditures are limited to represent regional investment, and avoid the complexity
foreign investment would add to the analysis. Due to the structure of our model, wherein capital services are a
flow variable, net investment is defined as sum of fixed investment expenditures (net of expenditures to replace
depreciated capital) and changes in inventories. Under a focus on regional investment, aggregate regional
investment equates aggregate regional savings (see note u). Sector investment is found as a proportion of
aggregate investment/savings. Sector investment coefficients are found from the ratio of sector output to
aggregate regional output.
y E
X F source: Determined as a residual of the value of XF net of XDF. See notes c and k.
z E
X R source: Determined as a residual of the value of XR net of XDR. See notes f and l.
aa E
X C source: Determined as the sum of Alaska's major exporting industries (not including those of specific
interest): Farms, Mining, Lumber and Wood products, Petroleum products, and Pipelines (ex. natural gas),
Bureau of Economic Analysis (1997a).
32
Table 3
Parameter
a
Calibrated Economic Parameters
Definition
Quota intercept
b
Quota slope
dF
Harvest function
catchability coefficient
Harvest function
exponent
Season function
parameter
Season function
parameter
Recreation production
parameter
Recreation production
parameter
Recreation production
parameter
Recreation production
parameter
Composite production
efficiency parameter
Composite production
labor share parameter
Composite import
efficiency parameter
Composite import
share parameter
Composite export
efficiency parameter
Composite export
share parameter
aF
dmF
amF
aR
bR
cR
dR
C
C
CC
CC
CT
CT
Value
Parameter
6502115.654

475200.088
FC
2.24x10-7
FT
0.8997
FT
1.988
RT
0.445
RT
0.346
FK
0.404
FK
0.250
FL
3.801
FL
1.986
R
0.437
R
1.942
F
0.400
C
1.892
F
0.363
nAK
C
F
Definition
Fishery import
efficiency parameter
Fishery import
share parameter
Fishery export
efficiency parameter
Fishery export
share parameter
Recreation export
efficiency parameter
Recreation Export
share parameter
Fishery capital import
efficiency parameter
Fishery capital import
share parameter
Fishery labor import
efficiency parameter
Fishery labor import
share parameter
Household share
of recreation
Household share of current
consumption
Household subsistence
level of fish
Household subsistence
level of composite goods
Stone Geary
exponent
Benchmark rate of growth of
regional capital
Value
1.012
0.004
1.333
0.879
1.069
0.508
1.663
0.752
1.941
0.600
0.036
0.937
22.377
14307.968
3.676x10-4
7.165x10-4
Elasticity Definitions
C
 1R
 2R
 FH,I
MU IT
 CC
Composite goods: labor
and capital
Households: current
consumption goods
Households: current and
future consumption
Income elasticity of the
demand for food
Marginal utility of
income with respect to
income
Composite demand:
regional goods and
imports
0.8672
 FC
0.867
 CT
1.60042
 FT
0.3
 RT
Fish demand: regional goods and
imports
Composite supply: regional
goods and exports
Fish supply: regional goods and
imports
Recreation supply: regional
goods and imports
-1.09
 FK
Fishery capital demand: regional
factors and imports
1.7
2.12
 FL
Fishery labor demand: regional
factors and imports
1.7
33
1.42
2.79
3.9
2.79
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