The Role of Land Retirement Programs for Management of Water Resources

PRELIMINARY VERSION - PLEASE DO NOT CITE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 ∑ A_jjt ≤ A_i 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 d^t

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}

^{+ λ}2 i ^Ai ^- Σ Ajjt

(4)

Assuming interior solutions (i.e., g_ijt ≥ 0 and A_ijt ≥ 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^-ρ^t
jt

^{- λ}1αigijt

- λ2i≥ 0, λ2i≥ 0, j

2i            2i        _∂Aijt 2i

=0

(5.a)

(5.b)

(5.c)