Independence of downstream and upstream benefits in river water allocation problems

Abstract

We consider the problem of sharing water among agents located along a river, who have quasi-linear preferences over water and money. The benefit of consuming an amount of water is given by a continuous, concave benefit function. In this setting, a solution efficiently distributes water over the agents and wastes no money. Since we deal with concave benefit functions, it is not always possible to follow the usual approach and define a cooperative river game. Instead, we directly introduce axioms for solutions on the water allocation problem. Besides three basic axioms, we introduce two independence axioms to characterize the downstream incremental solution, introduced by Ambec and Sprumont (2002), and a new solution, called the UTI incremental solution. Both solutions can be implemented by allocating the water optimally among the agents and monetary transfers between the agents.We also consider the particular case in which every agent has a satiation point, constant marginal benefit, equal to one, up to its satiation point and marginal benefit of zero thereafter. This boils down to a water claim problem, where each agent only has a nonnegative claim on water, but no benefit function is specified. In this case, both solutions can be implemented without monetary transfers.

The Economics of Transboundary River Management

“…In a different paper, Van den Brink et al (2014) weaken some assumptions on the benefit function used by Ambec and Sprumont (2002). Specifically, they replace differentiability by continuity and strict concavity by concavity (allowing for satiation).…”

“…Specifically, they replace differentiability by continuity and strict concavity by concavity (allowing for satiation). In addition, Van den Brink et al (2014) weaken the two axioms used by Ambec and Sprumont (2002 Both solutions will also be characterized in the next section in the setting of the river claims problem.…”

“…A special case of the river sharing problem was observed by Van den Brink et al (2014). This case features linear benefit functions up to a satiation point, as shown in Figure 2.…”

“…Ambec and Sprumont, 2002;Ambec and Ehlers, 2008b;Khmelnitskaya, 2010;Van den Brink et al, 2012;Béal et al, 2012) model river sharing as a cooperative game, where the axioms are imposed on the distribution of welfare to the agents. Recently, Van den Brink et al (2014) argued that, instead, the axioms should be imposed directly on the allocation of welfare derived from water use, which allows a closer link between the axioms and actual water allocation. In response, Ansink and Weikard (2013) took this argument one step further and imposed axioms directly on the allocation of water.…”

“…Both solutions are characterized by Ansink and Weikard (2013) using the three characterizing axioms of sequential sharing rules plus one axiom that reflects the nature of the Priority rule and its inverse. Both solutions are also characterized by Van den Brink et al (2014) using an appropriate conversion of the axioms used in Results 10 and 11 to the domain of river claims problems (e.g.…”