Subsistence induced and complementarity induced irrelevance in preferences

In a two-good setting we axiomatize (a) preferences with subsistence consumption and (b) a generalized version of Leontief preferences. Our axiomatization allows for different levels of subsistence and captures the presence of poverty and prosperity. Our axioms are based on the irrelevance of one of the goods at certain consumption bundles. For subsistence, the irrelevance is induced by the subsistence requirement and for generalized Leontief, it is induced by complementarity. We capture this difference using the notion of unhappy sets.