L ≤ H(X,q(zq,N)), NumTrips ≤ DMAX , NumTrips =
D *
NvN
(7)
The first constraint simply limits landings to an amount weakly less than individual harvest.16
The second constraint states that the number of trips taken by a vessel must not exceed the total
number of available opportunities in a season, DMAX . Note, however, that NumTrips is not
included as a separate control variable in our statement of the problem. The rationale is found in
the third constraint of (7). We assume, given the homogeneity of vessels in our model, that both
passengers and trips are spread evenly across the fleet. Combining this assumption with the
endogenously determined number of vessels and anglers per trip (and the constant variable cost
of trips by a vessel) yields the per-vessel trip count indicated in (7).
The task of the social planner is to choose the time paths of fishing days, landings, angler
density, the number of vessels and vessel inputs so as to maximize the discounted present value
of the flow of net benefits. The constrained current value Hamiltonian, where we have
substituted for the third constraint from (7), is:
D*
D D ʌ
-------, w, r )
NvN
HC = ∫MB(D,H(X,q(zt,N)),L,ʌ'(z,,N))dD-N ∙c(zq,z,,N,
0
+λ[g(X)-D*(φ(H(X,q(zq,N))-L)+L)]+μ1[H(X,q(zq,N))-L]
(8)
+ μ2
D--
max NvN
A glance at (4) and (8) reveals that the Hamiltonian (excepting the constraints) is linear
inNv . This linearity is particularly simple in that it plays no role in the equation of motion and
only affects net benefits through a positive effect on fixed costs. The implication of this linearity
16 In reality individual landings may exceed individual harvest if fishermen are able to trade their catch with other
passengers. In this case, the constraint could be modified to apply to the sum of individual landings and harvest.
However, given our current assumption of identical preferences and skill across anglers (and non-stochastic catch),
the individual and aggregate constraints are equivalent.
10