Local disaggregation of negative demand and excess demand functions

The literature on the characterization of aggregate excess and market demand has generated three types of results: global, local, or 'at a point'. In this note, we study the relationship between the last two approaches. We prove that within the class of functions satisfying standard conditions and whose Jacobian matrix is negative semi-definite, only n/2+1 agents are needed for the 'at' decomposition. We ask whether, within the same class, the 'around' decomposition also requires only n/2+1 agents.