WATER RESOURCES RESEARCH, VOL. 16, NO. 4, PAGES 638-642, AUGUST 1980 Competition Versus Optimal Control in GroundwaterPumping MICHA GISSER Department of Economics, University of New Mexico, Albuquerque, New Mexico 87131 DAVID A. SANCHEZ Department of Mathematics, University of New Mexico, Albuquerque, New Mexico 87131 This article considers one of the most important issues in water resources research, namely, the man- agementof groundwater. Economists have long taken it for granted that the temporal allocation of groundwater would lead to welfarelosses if left to the free marketbecause all farmers pump from a com- mon aquifer. Hence water economists studied extensively optimal controlof temporalgroundwater allo- cation.They neverpaused to compare the temporalallocation yieldedby optimal controlwith the free market. In thisarticle we proveby comparing the two strategies analytically that if the storage capacity of the aquifer is relativelylarge,the difference between them is so small that it can be ignoredfor practical consideration. INTRODUCTION This paper discusses the economic aspects of a model for theproblems of farmers pumping groundwater outof an aqui- fer. This hasalways beenregarded as a typicalexample of an externality (when all producers increase their output as a whole, the cost function of each individual producer is af- fected), but thisviewpoint needs some elaboration. One could consider the case of pumping an aquiferaslying between fish- ery harvesting at one extremeand the cuttingof privately owned timber at the other. In the caseof fisheries the fishing grounds area nonexclusive resource which consequently leads to dissipation of rent aswell asa possibly inefficient distribu- tion of effort over time. In the case of privately ownedtimber, nonowners are excluded from cutting; hencethere is no dis- sipation of rent,and owners may spread theirproduction over time in an optimal manner. Exclusiveness is present at a very large extent in the aquifer, since only farmers who own land overlying the aquifercan pumpwater and otherfarmers are excluded from the resource. The effect of this exclusiveness was discussed by Cheung [1970]. But exclusiveness in the case of farmers using groundwater for irrigation is not ascomplete asin the case of privatelyownedtimber. The contrast of the pumpingof an aquiferwith the cutting of timber revolves aboutthe temporal allocation of groundwa- ter. If each farmer owned his own little aquifer, there would be no problem of externality, and his effortovertime would maximize the present value of all future income streams de- rived from irrigation. This behavior would be identical with that of the timber owner. However, given the usual situation wheremany farmers pump water from the common aquifer, then the individual farmer cannot expect to have more water in storage for him next year if he pumps less thisyear. Con- sequently, instead of maximizing present value,farmers sim- ply pumpwater eachyear, satisfying the condition that the marginal cost of pumping equals the valueof the marginal physical product(VMP) of water. The type of externality described above hasbeendiscussed in the literature [Burt, 1964, 1967, 1970; Burt and Stauber, 1971; Cummings and McFarland, 1974; Brown and Deacon, 1972]. Economists have applied various optimal control (or, Copyright¸ 1980 by the American Geophysical Union. Paper number 80W0464. 0043-1397/80/080W-0464501.00 equivalently, dynamic programing) methods in order to de- rive an optimal water useover time. Brown and Deacon [1972] derived a formula for a tax that should be imposed on groundwater (pumped) in order to yield the optimal control solution. On the other hand, Gisser and Mercado [1973, 1972] studied the temporal allocation of groundwaterunder com- petitive conditions (no control), and so the important policy question occurs: Is the difference between the optimal control solutionand the competitivesolutionlarge enoughto justify the use of control?This questionis even more relevant when we take into consideration that optimal control is not costless. Before we embarkedon this project we shared the opinion of other economists that an optimal control strategywould yield higher present value of future income streams as com- pared to no-control strategy. We have selectedthe Pecos River Basin for our empirical comparison. We simulatedthe control and no-control(competition) strategies for given esti- mates of economicand hydrologic parameters. We obtained resultsthat are practically identical for the two strategies. Theseresults, which are of more interest to the operations re- search profession than to water economists, were published in anotherjournal [Gisser and Sanchez,1980]. In this paper we provide the analytical results which are of more interest to water economists. In what follows we shall proceedas follows. First, we shall describe briefly a model which integratesthe demand function for irrigation water with the hydrologictheory, assuming a single-cell aquifer. Second, we shallprove that the difference in temporal alloca- tion yielded by competition, on one hand, and optimal con- trol, on the other, reduces to expressions which incorporate economic and hydrologic parameters that are negligible for all practical purposes whenthe areatimes storativity of the aqui- fer is reasonably large. THE BA$1C MODEL AND COMPETITION (No CONTROL) For the benefit of the reader we reproduce the basic ele- ments of the model developedby Gisser and Mercado [1973]. Figure 1 illustrates the relationships betweenthe variouswa- ter flows and the aquifer. We assume that the demand for irrigation water is a nega- tively sloped linear function as follows: W= g + top (•) 638