Parametric regularity of exchange equilibria ($D_{(p,t)}Z\ne 0\text{ on }{Z}^{-1}(0)$)

If zero is a regular value of the joint smooth aggregate excess demand $Z(p,t)$, almost every parameter gives a regular economy. The regular level set theorem makes the zero set a manifold. Its projection to parameter space fails to have full rank exactly where the derivative with respect to prices is singular; Sard theorem makes these exceptional parameter values a null set.