Rent Dissipation in Chartered Recreational Fishing: Inside the Black Box

generated by a cost minimization problem subject to dual quality constraints with non-rivalrous
inputs - only here the optimal prices of quality replace the usual Lagrange multipliers. This
finding reflects how the family of conditions embodied in (16) jointly determines both the
optimal quality levels and the cost-minimizing input combinations. Equation (18) shows that the
relative combination of inputs is product of a mixture of two single-quality tangency conditions
where the relative influence of catch or non-catch quality in influencing the mix of inputs is a
product of their optimal marginal valuations.²⁰

The costate equation for the dynamic optimization problem is:

D^*

λ - δλ = -∫MBH (∙)HXdD - λ(g'(X) -D*φH_x).                 (19)

0

Considering this equation at the steady state ( λ = 0 ) yields the following solution for the costate
variable:

HX^D∫MBH(D,H(X,q(zq,N)),L,S(zs,N))dD

^λSS =           δ - g′( X ) + D∙φH₁

where for the sake of economy of notation it is understood that all control and state variables are
evaluated at their steady state levels. Several observations are warranted here. First, the capital
value of the fish stock is, predictably, inversely related to the discount rate of the social planner.
Second, if harvest rates have no impact on marginal benefits (MB_H = 0) or if increases in fish
stock stock density have negligible effects on catch rates (H_X = 0 ) then an extra unit of stock
has no long run value and the user cost is zero. Third, the effect of a higher mortality rate of
discards is to decrease the steady-state valuation of the fish stock due to the anticipated leakage

²⁰ We should note that analogous expressions to (17) and (18) can be derived for fixed inputs (and ratios of fixed and
variable inputs) as well.

16

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