5. Seguiment dels resultats
4.2.6 Infeccions susceptibles de tractament al domicil
Optimal allocation of groundwater is a multistage decision process. At each stage, e.g. each year, a decision must be made regarding the level of groundwater use which will maximize the present value of economic returns to the basin. The intial conditions for each stage may be di¤erent due to changes in either the economic or hydrologic parameters of the basin under consideration. However, in most of the dynamic mathematical models employed in the literature reviewed in chapter 2 [see for example, Burt, 1964; Bredehoeft and Young, 1970; Gisser and Mercado, 1973; Gisser and Sanchez, 1980 (a, b); Noel, 1980; Feinerman and Knapp, Allen and Gisser 1984; Nieswiadomy, 1985; Worthington et al., 1985; Zimmerman, 1990, Knapp
and Olson; 1995] groundwater is modelled as a stock to be depleted in a mining era before
this literature are the assumptions of …xed economic relations and/or exogenous rates of change through time4. However, as indicated in the second part of table 2.3.1 of chapter 2, converging lines of evidence from sensitivity analyses show that bene…ts from groundwater management are highly sensitive to the functional form and price elasticity of the relevant demand function used in the optimal control model. Hence, paying attention to temporal changes in the bene…t functions that govern depletion modelling, is potentially important.
More complex and realistic representations of increasing resource scarcity incorporate op- portunities for adaptation to rising resource prices. That is, in the long-run perspective, shift away from water intensive production activities, adoption of new techniques or backstop tech- nologies, substitution of alternative inputs, and production of a di¤erent mix of products o¤er rational responses to increasing scarcity. Kamien and Schwartz (1981), Rossana (1985) and
Tomiyama (1985), developed theoretical two-stage optimal control models that accounted for
sequential changes through time. Kim et al. (1989) generalized these models to (n) stages. In particular, Kim et al. developed the technique of multistage optimal control in the context of groundwater mining for agricultural production. Rather than directly stating aggregate de- mand, their model states disaggregate categories of demand curves for di¤erent crops, with the intertemporal relative allocation of the resource among categories of products being de…ned as groundwater prices increase through time. Their approach provides the possibility of a more detailed description of natural resource depletion, where allocation across products of resource use can be portrayed in an intertemporal context. In the present chapter, we employ this tech- nique to describe the chronological pattern of groundwater use by di¤erent economic sectors in order to de…ne optimally the quantity of the resource that should be produced when the avail- able backstop technology is adopted at some endogenously de…ned time. Including in a control model the opportunity for this type of adaptation strengthens its ability to describe economic processes associated with natural resource depletion. The additional detail, further can inform public policy decisions concerning natural resource allocation among economic sectors, optimal timing of adoption of an available backstop technology and de…nition of optimal quantity of
4As already mentioned in footnote 38 of chapter 2, two notable exceptions exist in the literature. Burness and
Brill (1992)considered endogenous irrigation technology choice, andShah et al. (1995)integrated conservation technology di¤usion within the exhaustible groundwater model. Their results indicated that without intervention the gradual adoption process of the conservation technology will be slower than is socially optimal.
the resource to be produced by this technology for each of the di¤erent users.
The essence of the above arguments that call for an endogenous formulation of the bene…t function in an optimal control model, is the following. Projected results from optimal control models become more tenuous the further one moves into the future. In particular, one of the di¢culties of using long-run optimal control is the need to provide the mathematical model with estimates of willingness to pay (WTP) not only relevant for the present but also relevant for some time in the future. However as time passes by, WTP for the resource in question may change as the scarcity and hence the relevant price for the resource changes. (Price equals marginal extraction cost in a competitive-commonality situation where rights are not exclusively assigned, and marginal extraction costs plus scarcity rents if extraction is optimally controlled). Two approaches exist towards estimation of the relevant demand functions. The …rst is the traditional approach, where a functional form is chosen and used in the estimation of the demand function based on past and/or current observations. This demand function is assumed to be relevant for the entire time span over which the problem is solved. However, discontinuities may occur because the WTP of various demanders/users of the resource may be bounded from above at di¤erent shadow prices. With groundwater shadow prices increasing through time as the resource’s scarcity increases, some users of the resource may exit the market and their WTP for the resource be driven down to zero. That is, there is a choke price for the resource (which varies among di¤erent resource users) above which demand for groundwater becomes zero. Thus, the estimated bene…t functions based on current observations, may bear little relation to the actual bene…t function when future economic, hydraulic and agronomic conditions are much di¤erent. The piecewise approach adopted in our analysis, takes account of some foreseeable discontinuities in the derivatives of the demand functions. Both of these approaches are extrapolations into the future, but the piecewise approach o¤ers some advantages over the traditional approach because it uses more of the available information. If the incorporated foreseeable changes do take place, results from the piecewise approach will be more relevant in a long-run perspective.
More speci…cally, the use of the piecewise demand approach enables us to establish optimal intertemporal mining paths for di¤erent economic sectors, model the possibility of demand
adaptation to resource depletion and derive accurate bene…ts from groundwater management at steady-state conditions. Below we brie‡y discuss these strengths of the piecewise approach if compared with the traditional approach based on aggregate water demand. Equations (3.1) and (3.2) represent the demand for water by the agricultural and the domestic economic sector, respectively,
WA=a1 b1P (3.1)
WD =a2 b2P (3.2)
where (WA); (WD) are groundwater quantities demanded by the agricultural and domestic sectors respectively,(P)is groundwater price. Optimality dictates that the price of groundwater should equal the marginal bene…t derived from groundwater use. However, in the absence of clear property rights price equals marginal extraction costs. Parameters (a1); (a2); (b1) and
(b2) are coe¢cients of the ordinary (uncompensated) sector speci…c demand curves for water.
Figure 3.2.1 presents these demand curves as AA0 and DD0, respectively. Horizontal sum-
mation of the two demand curves gives the kinked demand curve DBC, which represents the multiple-sector piecewise aggregate water demand. Several strengths of the approach based on multiple-sector aggregate demand become evident from this …gure. In the descriptive mode, two opportunities for a more detailed analysis exist. First, di¤erentiating by economic sector permits estimation of the instantaneous relative allocation of groundwater between the agricul- tural and the domestic sectors during the groundwater depletion era. In particular, the model generates an endogenous switch time, the year at which groundwater is used only by the do- mestic sector rather than the agricultural and domestic sectors, simultaneously. For example, when the price of groundwater at some time (or the marginal cost of water extraction in the absence of a groundwater market) increases to(P1) in …gure 3.2.1, as a consequence of greater
pumping lifts, agricultural use of groundwater becomes zero and remains zero thereafter. The shift exclusively to the domestic sector provides an example of a sectoral pattern that occurs through time. The pattern is one of a shrinking number of existing economic sectors in a region’s economy as the price of groundwater pumping increases secularly with mining.
Second, prices that are higher than(P1) result in distinct extraction volumes depending on
which notion of aggregate demand is used; this proves to be crucial in choosing the exact volume of water to be produced by the backstop technology at steady-state conditions. For example, if the unit price of desalination is(P2), the multiple sector notion results in an accurate prediction
of(W2)rather than an underestimate of(W20). Thus, water prices above(P1)generate di¤erent
estimates of social bene…ts. These a¤ect both descriptive and normative analyses. In particular, the kinked representation of aggregate demand provides a larger, and more accurate estimate of the surplus from the economy-wide groundwater use relative to the traditional approach. This additional information may prove crucial as groundwater mining becomes pervasive and policy makers attempt to reform groundwater institutions or decide when it is optimal to adopt an expensive backstop source of water. For example, if the unit price of desalination is(P3) then
the adoption of the available backstop technology is feasible only if the multi-sector notion of the aggregate demand curve is used. Hence, accurate estimates of the value of social bene…ts derived from groundwater use for the economy as a whole, may help to balance equitably and e¢ciently any future reforms. When data is available, an approach based on a multiple-
sector piecewise aggregate demand for groundwater o¤ers a superior alternative to the approach commonly adopted in the literature based on undi¤erentiated aggregate demand.