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Lecture 7: General Equilibrium - Welfare HS 12 Overview 1 Setting the Stage 2 First Welfare Theorem 3 Second Welfare Theorem Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 2/11 Setting the Stage Again, we consider an exchange economy. In this case P P feasibility requires xi ∈ Rn+ and i∈I xi = i∈I ei . an allocation x∗ is a Walrasian equilibrium allocation (WEA) if there exists a Walrasian equilibrium price vector p∗ such that x∗ := (x1 (p∗ , p∗ · e1 ), x2 (p∗ , p∗ · e2 ), . . . , xI (p∗ , p∗ · eI )). As a there might be several equilibria p∗ for given endowments e, it is useful to define the following set: For an economy with initial endowments e, we denote the set of Walrasian equilibrium allocations by W (e). Similarly, let F (e) denote the set of feasible allocations. Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 3/11 Setting the Stage: Feasibility of WEAs Every Walrasian equilibrium allocation is feasible: W (e) ⊂ F (e). The proof is trivial (Try writing it down!) The same result holds in a economy with production. Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 4/11 First Welfare Theorem: Statement Every Walrasian equilibrium allocation is Pareto-efficient. The only assumption needed to prove this result is that all utility functions are strictly increasing. The textbook proves a stronger result (Theorem 5.6) – we will not discuss the core in this lecture. Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 5/11 First Welfare Theorem: Proof Consider x∗ ∈ W (e) and let p∗ be the associated vector of equilibrium prices. Consider any allocation x̃ (feasible or not) such that u i (x̃) ≥ u i (x∗ ) holds with at least one strict inequality. Then p∗ · x̃ ≥ p∗ · ei holds with at least one strict inequality. (Why?) Adding up the inequalities from the previous step yields: " # X X ∗ i i p · x − e > 0. i∈I i∈I This implies that x̃ is not feasible. (Why?) Hence, x∗ is Pareto-efficient. (Why?) The same logic works in an economy with production. Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 6/11 First Welfare Theorem: Discussion The First Welfare Theorem is a remarkable result, highlighting the role of the market in coordinating individual actions. All individuals strive only to achieve the best for themselves . . . nevertheless, the social outcome resulting from a competitive market system is Pareto-efficient. However, we may care about more than efficiency, such as “justness” or “fairness.” Indeed, many efficient allocations might not be desirable, because they violate appealing conditions of distributional justice. So we may wonder: Is there a fundamental conflict between what can be achieved through competitive markets and some notion of distributional justice? Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 7/11 Second Welfare Theorem: Statement Assume that all utility functions u i are continuous, strongly increasing, and quasiconcave on Rn+ . Assume in P strictly addition that i∈I ei 0. For every Pareto-efficient allocation x̄ there exists a redistribution of the initial endowments ē ∈ F (e) such that starting from these endowment x̄ is the unique Walrasian equilibrium allocation of the exchange economy: {x̄} = W (ē). Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 8/11 Second Welfare Theorem: Proof 1 Set ē = x̄. 2 The assumptions ensure the existence of a WEA x̂ for the exchange economy with endowments x̄. It remains to show that x̂ = x̄ must hold. 3 x̂ is feasible in the original economy and satisfies u i (x̂i ) ≥ u i (x̄i ) for all i ∈ I. (Why?) 4 5 Because x̄ is Pareto-efficient, it follows that u i (x̂i ) = u i (x̄i ) holds for all i ∈ I. Strict quasiconcavity of u i (and the fact that the consumer can afford both x̂i and x̄i under the Walrasian equilibrium prices) implies that x̂ = x̄ holds. Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 9/11 Second Welfare Theorem with Production A counterpart to the Second Welfare Theorem holds in economies with production. The formulation and proof (See Theorem 5.15) is somewhat different because appropriate assumptions on the production sets Y j have to be imposed. agents are not only endowed with goods but also with profit shares. the profit maximizing behavior of firm has to be taken into account. Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 10/11 Second Welfare Theorem: Discussion The Second Welfare Theorem tells us that we can reach any Pareto-efficient allocation in a decentralized way. Thus if we have a distributional objective (some notion of “fairness”, “justice”, or “sustainability”), we can attain this objective in a competitive market system as long as it also insists on efficiency. So a market economy is not only a tool for achieving some efficient allocation, it grants us the choice between all possible efficient allocations. Note, however, that this is not a piece of practical advice: if we had enough information to “use” the Second Welfare Theorem, we might as well forget about markets and impose that allocation directly . . .. Advanced Economic Theory Lecture 7: General Equilibrium - Welfare 11/11