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 A_i acreage producing j crops. The per-acre groundwater extracted by farmer i at time t is
denoted as g_ijt . The pumping lift, d_i , 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 G_t .

The cost of groundwater extraction, C(G_t,d_i) , depends on the groundwater stock and pumping
lift. It is assumed to be convex in G_t (C_G < 0 and C_GG > 0). Let z_i represent the soil
characteristics. The production function depends on the applied water (g_ijt ), water-use efficiency

∂f            ∂ ² f

(α_i ), and soil characteristics (z_i ) as: y_it = f (α_ig_ijt,z_i) 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

α_ig_ijt 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 P_j is the output price. Let A_ijt be the acreage to be allocated to crop j such that

∑ ^a∣∣, ≤ Aj.

j

Optimal Land and Groundwater Allocations

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