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## Appendix 1

The current value Hamiltonian corresponding to the capture fish problem can be specified as:

where λ is the scarcity value of the capture fish stock or the marginal value assigned by the planner to the marginal reductions in the fish stock. The maximum principle provides the following first-order condition with respect to the flow variable, catch (*h*):

From equation (25), to maximize intertemporal benefits from the fishery, the net marginal benefit, (*p*_{h}
(*h,z*)*h + p*(*h + z*) – *c*(*x*)), must be equal to the scarcity value of the stock (λ). Clearly the net marginal surplus depends on catch, capture fish stock, and biomass production from the aquaculture. The co-state equation corresponding to equation (24) is:

Equation (26) stipulates that in dynamic equilibrium, returns on investing the proceeds from harvesting a kilogram of fish (δ λ) must be equal to the opportunity cost of catching the fish, which includes capital gain (
$\dot{\lambda}$
) and some stock effect (λ *g*_{x}
– *c*_{x}
(*x*)*h*). In the steady state, there is no gain in fish stock and the shadow price (i.e.,
$\dot{x}=\dot{\lambda}=0)$
, hence the dynamic equation
$\dot{x}=g(x,k(m))-h$
and equation (26) give us *g*(*x, k*) = *h* and λ = – *c*_{x}
(*x*)*h*/δ – *g*_{x}
(ċ), respectively. Assume that the aquaculture production depends on the cage size, so that the production function is *z* = *z*(*m*). Using this expression and substituting the above values for *h* and λ in equation (25), we have:

## Appendix 2

The current value Hamiltonian corresponding to the farmer's problem is:

where μ is the shadow value of the cage area. The maximum principle generates the following first-order condition with respect to the flow variable *v*:

In intertemporal equilibrium, the farmer will expand the cage marginally if the marginal benefit of the expansion (i.e., measured by the shadow value of the total cage area, μ) is at least equal to the marginal cost of the expansion (i.e., τ_{
v
}). The co-state equation defining the dynamic equilibrium is:

Similarly, from equation (30), in a dynamic equilibrium the capital gain from investing in an extra unit of the cage area (
$\dot{\mu}$
) plus the marginal benefit from fish harvest attributable to the marginal increment in the cage area (i.e., (*p*_{z}
(*h,z*(*m*))*z*(*m*) + *p*(*h,z*(*m*)))*z*_{m}
(*m*) – ς_{
m
} (*m*)) should balance the marginal opportunity of interest earnable on μ (i.e., δ μ). In the steady state, we have
$\dot{\mu} =\dot{m}=v=0$
. Combining equations (29) and (30), and using *h* = *g*(*x, k*(*x*)), we obtain the following:

## Appendix 3

The current value Hamiltonian corresponding to the social planner's problem is

The maximum principle generates the following first-order conditions:

Equation (33) differs from (25) by *p*_{h}
(*h,z*(*m*))*z*(*m*) < 0, indicating that, in intertemporal equilibrium, the socially-optimum catch level is lower than the ‘private’ equilibrium catch level of the capture fisheries’ manager. Consequently, decentralizing the two subsectors could result in over-harvesting of the capture fish stock due to imperfect competition in the market for fish. The corresponding co-state equations (for the co-state variables *x* and *m*, respectively) are:

While equations (26) and (35) are the same, (30) and (36) are obviously different. Since *p*_{z}
(*h,z*(*m*))*z*_{m}
(*m*)*h* < 0 and λ_{
s
}
*g*_{k}
(ċ)*k*_{m}
(ċ) = – ω λ_{
s
}
*g*_{k}
(ċ) – γ λ_{
s
}
*g*_{k}
(ċ) < 0, in dynamic equilibrium the net benefit from marginally expanding the cage is overstated, which favors increment in the cage size in equation (30). Thus, properly accounting for the resource use externality (pollution) discourages expansion of aquaculture.

In the steady state,
$\dot{m}=\dot{x} =\dot{\lambda}_{s}=\dot{\mu}_{s}=v=0$
, and *h = g*(*x,k*
_{0} – *m*(ω + γ )), hence the tax expression becomes:

Also we have the following expressions for the shadow values:

Substituting these values in equations (35) and (36), the equations that relate the optimum stock and cage size become:

## Appendix 4

The Lagrangian function corresponding to the FDI problem is:

where ψ is a Lagrange multiplier, and

The first-order conditions with respect to the flow variables, following the maximum principle, are:

The co-state equations are

Again, in the steady state,
$\dot{m}=\dot{x}=\dot{\lambda}_{s} = \dot{\mu}_{s}=v=0$
, implies:

In addition to the first-order conditions, we have the following transversality condition:

Since ψ = 1, it follows that ∂ L/∂ ψ = 0, and the tax expression is:

## Appendix 5

Table A1. Optimum wild fish catch, aquaculture harvest, and tax if local capital finances aquaculture and externalities are internalized

Table A2. Optimum wild fish catch, aquaculture harvest, and tax if foreign capital finances aquaculture and externalities are internalized