In this paper we show that if all agents are equipped with well-behaved discrete concave production functions, then a feasible price allocation pair is a market equilibrium if and only if it solves a linear programming problem. Using this result we are able to obtain a necessary and sufficient condition for existence that requires an equilibrium price vector to satisfy finitely many inequalities. A necessary and sufficient condition for the existence of market equilibrium when the maximum value function is Weakly Monotonic at the initial endowment that follows from our results is that the maximum value function is partially concave at the initial endowment. We also provide a discussion of the results and an alternative solution concept. The alternative solution concept is however, informationally and computationally inefficient.
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