The Role of Land Retirement Programs for Management of Water Resources

The water planner’s problem is:

N

max ∫Σ (Ajt ^- Lit )∏j_te^-ρtd^t
gijt, Aijt, Lit 0 i,j

(6)

subject to

=— = ^-∑ (Ajt ^- Lit )^αi^gijt ⁺ R
dt i,j

(7)

∑ ( Ajt + Lit ) ≤ A_i
j

d Гу. _τ
~dΓ - L^it

^Lit ^{≤ Γ}it

(8)

(9)

(10)

GN=G.

(11)

Equation (8) is the total land availability constraint for each farmer. The dynamics of the land
available for retirement and the constraint on the maximum amount of land that can be retired
over time are given by equations (9) and (10), respectively. The target groundwater stock to be
achieved in N years is given by equation (11). Augmenting the Hamiltonian leads to the
following Lagrangian with the inequality constraint (Chiang, 1992, p. 278):

Λ = ∑ (A_ij — Li )π_ij_te ^ρt + λ — ∑ (A_ij — Li )α_ig_iit + R + λ₂i A_i — ∑ (A_ii + L_it )

ijt it ijt                  1                      ijt it i ijt                       2i i                    ijt it

i^{, j}                   L  ^{i, j}                  JL ^j

-λiLit + λ.,.l∣ i — Lit)

(12)

We obtain the following conditions for the maximum principle along with the equation of motion

for G in (7):

∂Λ             ^∂πijt

Ξ— = (  ^— L^it ) Ξ— ^e

^∂gijt                   ^∂gijt

—ρt

^{— λ}ι( Ajt ^— Lit )^αι = 0

(13.a)