We start with a study of the piece-wise linear function determining how the maximum of the objective varies with the right-hand members of the constraints. Generation of this function needs dual prices. The latter being used only in their validity sets, the two proposed procedures avoid the main difficulty to which the Kornai-Liptak procedure is subject. Basically, the subsystems propose at each step new production processes out of various projects which are often not well specified. The central agency sends prices or resource allocations.