We consider a special class of the fixed point problems for piecewise constant mappings of a simplex into itself. These are polyhedral complementarity problems arising in studying the classical exchange model and its variations. We study the problems that stem from the consideration of models with fixed budgets and possessing a certain property of monotonicity (logarithmic monotonicity). Our considerations are purely mathematical and not associated with the economic models that gave rise to these mathematical objects. The class of regular mappings is investigated, and their potentiality is proved.