PRELIMINARY VERSION - PLEASE DO NOT CITE
The water planner’s problem is to maximize the net present value of the total profits in
the region subject to equation (1) and ∑ Ajjt ≤ Ai in order to determine the optimal water use and
j
land allocations among crops as:
N
max ∫ Σ e - A- [Pyt - C ( Gt, di ) gjjt dt
gijt, Aijt 0 i,j
(3)
where ρ is the discount rate. By augmenting the Hamiltonian, we can write the present-value
Lagrangian with the information in the inequality constraint as (Chiang, 1992, p. 278):
Λ = Σ e^ρtAj∏ijte^ρt + λ - ∑ jig, + R
j, j L j, j
+ λ2 i Ai - Σ Ajjt
(4)
Assuming interior solutions (i.e., gijt ≥ 0 and Aijt ≥ 0 ), we have the following conditions for the
maximum principle along with the equation of motion for G in (1):
дЛ ∂∏ e ^ρ- λ>α= 0
∂gijt ∂gijt
∂Λ
dAijt
= πijte-ρt
jt
- λ1αigijt
- λ2i≥ 0, λ2i≥ 0, j
2i 2i ∂Aijt 2i
=0
(5.a)
(5.b)
— = ∑ A..te -ρt π = - dλ .
(5.c)
∂G i j ijt ∂G dt
,j
Equation (5.a) states that under the social optimality the marginal benefit of groundwater
use is equal to the marginal cost of groundwater extraction plus the shadow price of effectively
used groundwater. This shadow price reflects the cost imposed on the future generation by using
water now. Equation (5.b) with λ1 obtained from (5.a) implies that
ρt ∂πit ρt
πijte ρ - gijt—l-e ρ - λ2i ≥ 0, where λ2i is the shadow price of land availability constraint.
∂git
This indicates that the farmers allocate the land to the crop with the highest ratio of the benefits
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