'Top of the line' quality as an optimal solution for a multi-quality monopolist

In this paper we consider a finite variant of the Mussa-Rosen model of vertical differentiation. There is a multiquality monopolist and a finite number of buyers each purchasing at most one unit of the good. We are concerned here with the question when it is optimal for the monopolist to offer the 'top of the line' quality only. We use a simple Lagrange multipliers technique in order to derive sufficient conditions for optimality of this policy for a wide class of utility functions, including those of a Cobb-Douglas type. We also demonstrate that the offer of the 'top of the line' quality only, is always an optimal policy whenever buyers have linear utility functions.