Introduction
This question involves an optimal control problem in the context of renewable resource management—specifically, the fishing industry. The goal is to determine the optimal harvesting strategy (xt) that maximizes utility over time while accounting for population dynamics of the fish stock. We are also asked to derive the transversality condition and solve for the optimal catch (xt), assuming utility is u(xt) = ln(xt).
Model Setup
State Variable: Pt = fish population at time t
Control Variable: xt = fishing intensity or harvest rate
Equation of Motion: Pt’ = a + bPt − xt
Utility Function: u(xt) = ln(xt)
Objective: Maximize discounted utility:
V = ∫0∞ e−rtln(xt) dt
Part a) Transversality Condition
In infinite horizon optimal control problems, the transversality condition ensures the solution is optimal and the value of the co-state variable becomes negligible at infinity.
Transversality Condition:
limt→∞ λ(t) * P(t) * e−rt = 0
This means that at the infinite time horizon, the present value of the shadow price (λ) of the remaining resource should vanish.
Part b) Finding the Optimal xt (Harvesting Strategy)
We use the Hamiltonian approach to solve this.
1. Define the Hamiltonian:
H = e−rtln(xt) + λ(t)(a + bPt − xt)
2. First-Order Condition:
∂H/∂xt = e−rt(1/xt) − λ(t) = 0
Solving for λ(t):
λ(t) = e−rt / xt → (1)
3. Costate Equation:
λ̇(t) = rλ(t) − ∂H/∂Pt = rλ(t) − λ(t)*b = λ(t)(r − b)
Therefore,
dλ/λ = (r − b) dt
Integrate:
ln(λ) = (r − b)t + C → λ = C * e(r−b)t → (2)
4. Combine (1) and (2):
e−rt / xt = C * e(r−b)t → xt = e−rt / (C * e(r−b)t)
Simplify:
xt = (1/C) * e−(2r−b)t
Final Form of Optimal xt:
xt = A * e−(2r−b)t, where A = 1/C is a positive constant to be determined from initial or boundary conditions.
Economic Interpretation
- If the discount rate r is high (i.e., future is valued less), then the exponent −(2r − b) becomes large in magnitude, making xt decline rapidly. This implies that the planner will front-load the fishing effort — fish more now, less later.
- If b (fish population growth response) is high, then the decline in xt will be slower — implying sustainable harvesting is more feasible.
- The function xt declines exponentially if 2r − b > 0. If 2r − b < 0, then harvesting may grow over time — which can be dangerous if not regulated properly.
Summary
- The transversality condition ensures that the planner doesn’t overvalue the remaining fish stock at infinity.
- The optimal harvesting path follows an exponential function driven by discount rate and fish regeneration speed.
- This model reflects classic natural resource optimization logic, balancing utility from consumption today against preservation for future use.
Conclusion
Using optimal control theory, we derived the optimal catch (xt) strategy for a fishery system. The model incorporates economic preferences (discounting), biological growth of fish (parameter b), and consumption utility. The solution shows that when future is heavily discounted, immediate harvesting is prioritized, while when fish regenerate quickly, a sustainable path becomes more desirable. Such frameworks are vital for designing policies on natural resource use, balancing current economic needs with long-term sustainability.