PRELIMINARY VERSION - PLEASE DO NOT CITE
with the crop production to the costs imposed on future water users. Equation (5.c) gives the
dynamics of shadow price of the groundwater stock. The steady-state total extraction and
∂gijt ∂G ∂C
groundwater stock are derived with the assumption of —jjt- = — = 0 as: λ =--> Ajjgi /ρ
∂t ∂t ∂G i,j
and ∑ Ajαigj = R.
j,j
Under the private optimal solution, the farmers maximize the expected profits and
determine the groundwater use such that the marginal benefit of the water use equals the
marginal private cost (i.e., p df- - C(G , d ) = 0). The farmers do not take into account the
j ∂gjjt t j
additional cost of groundwater use, namely the cost imposed on the future water use. Thus, the
optimal water use under the private optimality is higher than the social optimality. Under the
private optimality, the land allocation decision is made such that πjjte-ρt -λ2j ≥ 0. This implies
that the farmers allocate the land to the crop with the highest profit.
Land Retjrement Programs for Managjng Groundwater
We now develop a least-cost land retirement policy to achieve a given level of
groundwater stock. We consider a policy that targets achieving a given level of stock ( G ) in N
years. We define Ljt as the acreage to be retired from farmer j at year t such that
∑ ( Aijt + Lit ) ≤ Ai. Since the retired land may not be brought back to the production, we define a
j
new state variable Γit as the size of the parcel that is available for the land retirement at time t.
_ . „ . „ _ . d Γ . .. _ - _....... „
The equation of motion for Γ., is —- = -L,, with Γ∩ = A. This indicates that the amount of
it it i0 i
dt
land available over time decreases by Lit .