An objective function arising in the optimization problem of a quasilinear complex system with dependent priorities is considered. In the case of three priorities based on the results of one experiment, sufficient conditions are obtained for all stationary points of the objective function under consideration to be local maximum points.

The paper considers a Cobb-Douglas function with n random elasticities. A theorem on the concavity and strict concavity of such a function is proved.

The main result of this paper is the proof of the strict concavity of some function of integral form depending on three random variables, which we call priorities. This function is an objective function in the so-called model with priorities, in which the arbiter, following expert opinions, distributes funds among the enterprises and institutions under his jurisdiction. This result implies an important corollary about the existence and uniqueness of a local maximum point (which is also a global maximum point) of the objective function. This is a significant generalization of the corresponding result of N.V. Neumezhitskaia, S.I. Uglich and T.A. Volosatova, published in December 2020.