A new theoretical approach to the calculation of the utilizability function is developed and applied to the case of the annual performance of solar collectors of high concentration ratio. The utilizability is calculated in mathematical closed form as an integral over the contributions of the daylight hours of one representative day, equinox. The daily variations in the instantaneous solar radiation (due, for example, to varying degrees of cloudiness) are represented by random fluctuations superimposed on the clear day radiation pattern. The results are in good agreement with the corresponding utilizability curves that are based on detailed meteorological data. A simple and practical analytic expression for the utilizability is derived for the limiting case of clear climates, and turns out to provide rather accurate predictions for the utilizability for all climates as well. The presentation is divided into two separate papers: the current paper, in which the details of the theory and derivation are presented, and the following paper, for the reader who may not not be interested in all the details of the derivation, in which the solved numerical examples and the comparisons with results based on detailed meteorological data are presented.