MATCHINGS, COVERS, AND JACOBIAN MATRICES

It is well-known that the nonsingularity of the Jacobian matrix is a n ecessary and sufficient condition for the local uniqueness of the solu tion of a system of equations. A necessary condition for this nonsingu larity is the existence of a normalization of the equations. If the Ja cobian matrix is large and sparse, as it is, for instance, for macroec onometric models, the verification of this necessary condition is not immediate. The paper shows how this problem can be efficiently investi gated by means of a graph-theoretic approach. In particular, this is d one by seeking a maximum cardinality matching in a bipartite graph. Th e case where a normalization does not exist often constitutes a heavy challenge to the model builder and is a situation which, again, is ana lyzed using properties connecting covers to matchings in bipartite gra phs. The whole approach is qualitative, and this not obvious condition can be checked prior to the quantification of the model, i.e., at the stage of the theoretical formulation of the model where numerical met hod cannot yet been used.