Pricing a raindrop in a process-based model: general methodology and a case study of the Upper-Zambezi

A general approach is presented to value the stocks and flows of water as well as the physical structure of the basin on the basis of an arbitrary process-based hydrological model. This approach adapts concepts from the economic theory of capital accumulation, which are based on Lagrange multipliers that reflect market prices in the absence of markets. This permits to derive a financial account complementing the water balance in which the value of deliveries by the hydrological system fully balances with the value of resources, including physical characteristics reflected in the shape of the functions in the model. The approach naturally suggests the use of numerical optimization software to compute the multipliers, without the need to impose an immensely large number of small perturbations on the simulation model, or to calculate all derivatives analytically. A novel procedure is proposed to circumvent numerical problems in computation and it is implemented in a numerical application using AQUA, an existing model of the Upper-Zambezi River. It appears, not unexpectedly, that most end value accrues to agriculture. Irrigated agriculture receives a remarkably large share, and is by far the most rewarding activity. Furthermore, according to the model, the economic value would be higher if temperature was lower, pointing to the detrimental effect of climate change. We also find that a significant economic value is stored in the groundwater stock because of its critical role in the dry season. As groundwater comes out as the main capital of the basin, its mining could be harmful. (C) 2003 Elsevier Science Ltd. All rights reserved.