Thus, the use of Brouwer's fixed-point theorem is essential for showing that the equilibrium exists in general. Fundamental theorems of welfare economics In welfare economics, one possible concern is finding a Pareto-optimal plan for the economy. Intuitively, one can consider the problem of welfare economics to be the problem faced by a master planner for the whole economy: given starting endowment for the entire society, the planner must pick a feasible master plan of production and consumption plans . The master planner has a wide freedom in choosing the master plan, but any reasonable planner should agree that, if someone's utility can be increased, while everyone else's is not decreased, then it is a better plan. That is, the Pareto ordering should be followed. Define the Pareto ordering on the set of all plans by iff for all . Then, we say that a plan is Pareto-efficient with respect to a starting endowment , iff it is feasible, and there does not exist another feasible plan that is strictly better in Pareto ordering. In general, there are a whole continuum of Pareto-efficient plans for each starting endowment . With the set up, we have two fundamental theorems of welfare economics: Proof idea: any Pareto-optimal consumption plan is separated by a hyperplane from the set of attainable consumption plans. The slope of the hyperplane would be the equilibrium prices. Verify that under such prices, each producer and household would find the given state optimal. Verify that Walras's law holds, and so the expenditures match income plus profit, and so it is possible to provide each household with exactly the necessary budget. Convexity vs strict convexity The assumptions of strict convexity can be relaxed to convexity. This modification changes supply and demand functions from point-valued functions into set-valued functions (or "correspondences"), and the application of Brouwer's fixed-point theorem into Kakutani's fixed-point theorem. This modification is similar to the generalization of the minimax theorem to the existence of Nash equilibria. The two fundamental theorems of welfare economics holds without modification. Equilibrium vs "quasi-equilibrium" The definition of market equilibrium assumes that every household performs utility maximization, subject to budget constraints. That is, The dual problem would be cost minimization subject to utility constraints. That is,for some real number . The duality gap between the two problems is nonnegative, and may be positive. Consequently, some authors study the dual problem and the properties of its