# Open Collections

## UBC Theses and Dissertations ## UBC Theses and Dissertations

### The optimal allocation of the groundwater resource over time by the use of the user cost Alexander, Esther

#### Abstract

A mathematical model of maximizing the present-value of the groundwater reservoir is developed. According to it the optimal condition of the water use is: to ration the water in a way which makes the marginal net rent to the water equal to the marginal user cost each year. The model is valid for every amount of recharge. The rent function is derived from the agricultural production function and it is based on the main assumption that diminishing return to the water exists. Using the rent function the present-value expression for the reservoir has been established. It is differentiated in respect to the water used - the chosen independent variable of the rent function. In this way the result mentioned above has been achieved. The optimal sequence of the water use - which maximizes the present-value of the reservoir - has been calculated by salving the set of equations of the optimal conditions of each year. Both the optimal sequence of the water use and the user cost have been calculated explicitly and expressed in formulas by measurable parameters. It is argued that only a profit maximizing sole owner for whom the user cost is meaningful would use the water in the optimal sequence. If the reservoir is a common property resource a non-optimal water use would be practized. This non-optimal sequence of the water use is calculated by solving the set of equations of the non-optimal condition of the water use of each year. Considering the user cost zero the condition of the water use became: to use the quantity of water every year which makes the marginal net rent equal zero. The optimal condition of the water use developed in this work, which says marginal rent should be equal marginal user cost made possible to establish the non-optimal condition of the water use by turning the user cost to zero. The non-optimal water sequence is expressed too in an explicit formula by measurable parameters only. It is compared to the optimal water sequence. Also life-time of the reservoir and its final depth are compared for the cases of optimal and non-optimal water use. In the case of optimal water use (sole ownership) the water use is more gradual, the economical life of the reservoir is longer and its final depth is less deep than in the case of the non-optimal water use. A numerical example at the end of this work shows that - in these circumstance - the user cost has a significant magnitude.