The Role of Land Retirement Programs for Management of Water Resources



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2. Theoretical Model

We consider an aquifer serving as a source of groundwater for irrigation to M farmers
with
Ai acreage producing j crops. The per-acre groundwater extracted by farmer i at time t is
denoted as
gijt . The pumping lift, di , is defined as the initial distance from the surface to the
water level. It varies across heterogeneous farmers for a given level of groundwater stock
Gt .

The cost of groundwater extraction, C(Gt,di) , depends on the groundwater stock and pumping
lift. It is assumed to be convex in
Gt (CG0 and CGG0). Let zi represent the soil
characteristics. The production function depends on the applied water (
gijt ), water-use efficiency

f            ∂ 2 f

(αi ), and soil characteristics (zi ) as: yit = f (αigijt,zi) with 0 and —ʃ 0. The water-
g          g2

use efficiency defines the fraction of the water that is actually utilized by a crop. The product

αigijt represents the amount of applied water that is effectively used (Caswell et al., 1990).

The differential equation describing the groundwater dynamics is the net gain to the
aquifer, provided that the natural recharge (
R) is higher than the total extraction:

dG = -Ai^g,, + R.                                              (1)

dt        i, j

The per-acre profit of farmer i at time t for crop j is given by

πijt =Pjyijt -C(Gt,di)gijt                                                                       (2)

where Pj is the output price. Let Aijt be the acreage to be allocated to crop j such that

a∣∣,Aj.

j

Optimal Land and Groundwater Allocations



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