The Fundamental Welfare Theorems The so-called Fundamental Welfare Theorems of Economics tell us about the relation between market equilibrium and Pareto efficiency. The First Welfare Theorem: Every Walrasian equilibrium allocation is Pareto efficient. The Second Welfare Theorem: Every Pareto efficient allocation can be supported as a Walrasian equilibrium. The theorems are certainly not true in the unconditional form in which we ve stated them here. A better way to think of them is this: Under certain conditions, a market equilibrium is efficient and Under certain conditions, an efficient allocation can be supported as a market equilibrium. Nevertheless, these two theorems really are fundamental benchmarks in microeconomics. We re going to give conditions under which they re true. One (very stringent) set of conditions will enable us to prove the theorems with the calculus (Kuhn- Tucker, Lagrangian, gradient) methods we ve used very fruitfully in our analysis of Pareto efficiency. Additionally, for each theorem we ll provide a much weaker set of conditions under which the theorem remains true. Assume to begin with, then, that the consumers preferences are very nice viz., that they re representable by utility functions u i that satisfy the following condition: u i is continuously differentiable, strictly quasiconcave, and x i R l + : u i k(x i ) > 0 ( ) The First Welfare Theorem: If ( p, ( x i ) n 1) is a Walrasian equilibrium for an economy E = ((u i, x i )) n 1 that satisfies ( ), then ( xi ) n 1 is a Pareto allocation for E. Because ( p, ( x i ) n 1) is a Walrasian equilibrium for E, each x i maximizes u i subject to x i 0 and to the budget constraint p x i p x i. Therefore, for each i N there is a λ i 0 that satisfies the first-order marginal conditions for i s maximization problem: k : u i k λ i p k, with equality if x i k > 0, (1) where of course the partial derivatives u i k are evaluated at xi k. In fact, because each ui k > 0, we have λ i > 0 and p k > 0 for each i and each k. The market-clearing condition in the definition of Walrasian equilibrium therefore yields
x i k = x i k for each k (because each p k > 0). (2) For each i we can define λ i = 1, because λ i > 0, and then we can rewrite (1) as λ i k : λ iu i k p k, with equality if x i k > 0, (3) for each i N. But (2) and (3) are exactly the first-order conditions that characterize the solutions of the Pareto maximization problem (P-max). And since each u i k > 0, a solution of (P-max) is a Pareto allocation. In the proof, we could have instead expressed the first-order marginal conditions for the individual consumers maximization problems in terms of marginal rates of substitution: k, k : MRS i kk = p k p k (4) (written for the interior case, to avoid a lot of inequalities), which yields the (Equal MRS) condition i, j, k, k : MRSkk i = MRSj kk. (5) The conditions (5) and (2) characterize the Pareto allocations. Now let s see if we can prove the First Welfare Theorem without any of the assumptions in ( ). It turns out that we can t quite do that; however, the only thing we need to assume is that each u i is locally nonsatiated. The First Welfare Theorem: If ( p, ( x i ) n 1) is a Walrasian equilibrium for an economy E = ((u i, x i )) n 1 in which each ui is locally nonsatiated, then ( x i ) n 1 is a Pareto allocation for E. Suppose ( x i ) n 1 is not a Pareto allocation i.e., some allocation ( x i ) n 1 is a Pareto improvement on ( x i ) n 1: (a) n 1 xi n 1 xi (b1) i N : u i ( x i ) u i ( x i ) (b2) i N : u i ( x i ) > u i ( x i ). Because ( p, ( x i ) n 1) is a Walrasian equilibrium for E, each x i maximizes u i on the budget set B( p, x i ) := { x i R l + p x i p x i }. Therefore, (b2) implies that and since each u i is locally nonsatiated, (b1) implies that p x i p x i, (6) 2
p x i p x i for each i. (7) ((7) follows from a duality theorem that we ll state and prove momentarily.) Summing the inequalities in (6) and (7) yields i.e., p x i p p x i, (8) x i p Since p R l +, it follows from (9) that there is at least one k for which p k > 0 and x i. (9) x i k x i k. (10) Since p k > 0, the market-clearing equilibrium condition yields n xi k = n xi k, and (10) therefore yields n xi k n xi k. Therefore there are no Pareto improvements on ( xi ) n 1, and it s therefore a Pareto allocation. 3
The Second Welfare Theorem: Let ( x i ) n 1 be a Pareto allocation for an economy E in which the utility functions u 1,..., u n all satisfy ( ) and in which the total endowment of goods is x R l ++. Then there is a price-list p R l + such that for every ( x i ) n 1 that satisfies x i = x and i : p x i = p x i, (11) ( p, ( x i ) n ) 1 is a Walrasian equilibrium of the economy Ẽ = ((u i, x i )) n 1. i.e., ( p, ( x i ) n 1) is a Walrasian equilibrium of the economy in which each consumer has the utility function u i and the initial bundle x i. Let ( x i ) n 1 be a Pareto allocation for E. We first show that the conclusion holds for ( x i ) n 1 = ( x i ) n 1 i.e., if each consumer s initial bundle is x i. Since ( x i ) n 1 is a Pareto allocation, it is a solution of the Pareto maximization problem (P-max), and it therefore satisfies the first-order marginal conditions i, k : λ i, σ k 0 : λ i u i k σ k, with equality if x i k > 0. (12) Each λ i is the Lagrange multiplier for one of the utility-level constraints in (P-max), and since u i k > 0 for each i and each k, it s clear that each constraint s Lagrange multiplier must be positive: relaxing any one of the constraints will allow u 1 to be increased. For each k, let p k = σ k. Now (11) yields, for each i, λ i : k : u i k 1 λ i p k, with equality if x i k > 0. (13) These are exactly the first-order marginal conditions for consumer i s utility-maximization problem at the price-list p. Therefore, since each consumer s initial bundle is x i, each consumer is maximizing u i at x i ; moreover, the first equation in (11) ensures that all markets clear at ( x i ) n 1. Thus, we ve shown that all the conditions in the definition of Walrasian equilibrium are satisfied for ( p, ( x i ) n 1). If each consumer s initial bundle is x i instead of x i, then the first equation in (11) again ensures that markets clear, and the second equation in (11) implies that each consumer s budget constraint is the same for initial bundle x i as for initial bundle x i viz., n 1 p xi n 1 p xi = n 1 p xi and therefore x i maximizes u i subject to the budget constraint n 1 p xi n 1 p xi. Just as with the First Welfare Theorem, the Second Theorem is true under weaker assumptions than those in ( ) but in the case of the Second Theorem, not that much weaker: we can dispense with the differentiability assumption, and we can weaken the convexity assumption and the assumption that utility functions are strictly increasing. 4
The Second Welfare Theorem: Let ( x i ) n 1 be a Pareto allocation for an economy E in which each u i is continuous, quasiconcave, and locally nonsatiated. For each i, let x i = x i. Then there is a price-list p R l + such that for every ( x i ) n 1 that satisfies x i = x and i : p x i = p x i, (14) ( p, ( x i ) n ) 1 is a Walrasian equilibrium of the economy Ẽ = ((u i, x i )) n 1. The proof of the Second Theorem at this level of generality is not nearly as straightforward as the proof of the First Theorem. The proof requires a significant investment in the theory of convex sets, as well as some additional mathematical concepts. This is a case in which (for this course) the cost of developing the proof outweighs its value. 5