After taking into account positive transaction costs via tackling static nonlinear programming with dynamic programming, this paper reaches three main conclusions: 1) The aggregate excess demand function of each market will be discontinuous at many price points, but the transfiguration of the graph of aggregate excess demand of each commodity will be continuous with respect to prices of other commodities, and at those price points incurring discontinuity, the corresponding traders' utility or profit will be indifferent between different choices. 2) Via a new approach able to tackle the discontinuous aggregate excess demand, it is proved that, there must exist a market equilibrium price vector in a perfectly competitive economy with positive transaction costs even if each consumer's preference isn't convex. 3) Transaction cost probably spoils the satisfaction of those two fundamental theorems of welfare economics, as well as it ruins the social aggregate welfare through three approaches.
