Scalarization techniques or relationship between a social welfare function and a Pareto optimality concept

This paper discusses some scalarization techniques and one application of the multi-objective optimization problem into a mathematical economics. A function of social welfare and a concept of Pareto optimality in a finite pure exchange economy are considered. Here it is proved that a set of Pareto optimality allocations is path-connected and uncountable, if the utility functions of economical agents are monotone, concave and strictly quasi-concave. In the end, it is proved that there exist non-negative weights and Pareto optimality allocations such that the social welfare function has maximum not proving continuity of this function.