Optimal Choice of Characteristics for a non-excludable Good

I consider a model where a principal decides whether to produce one unit of an indivisible good (e.g. a private school) and which characteristics it will contain (emphasis on language or science). Agents (parents) are differentiated along two substitutable dimensions: a vertical parameter that captures their privately known valuation for the good (demand for private education), and an horizontal parameter that captures their observable differences in preferences for the characteristics. I analyze the optimal mechanism offered by the principal to allocate the good and show that the principal will produce a good with characteristics more on the lines of the preferences of the agent with the lowest valuation. Furthermore, if the principal has also a private valuation for the good, he will bias the choice of the characteristics against his own preferences.