Fundamental theorems of welfare economics

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There exist two fundamental theorems of welfare economics. The first states that any competitive equilibrium leads to an efficient allocation of resources. The second states the converse, that any efficient allocation can be sustainable by a competitive equilibrium. Despite the apparent symmetry of the two theorems, in fact the first theorem is much more general than the second, requiring far weaker assumptions.

The first theorem appears to make a case for non-intervention: let the markets do the work and the outcome will be desirable. The theorem is often taken to be an analytical confirmation of Adam Smith's "invisible hand" hypothesis, namely that competitive markets tend toward the efficient allocation of resources. However, the economic concept of efficiency is not the only thing that a society might care about. In particular, the theorem says nothing about the distributional equity of the outcome.

The second theorem states that out of all possible efficient outcomes (of which there may be many) one can achieve any particular efficient outcome by enacting a lump-sum wealth redistribution and then letting the market take over. This appears to make the case that intervention has a legitimate place in policy -- redistributions can allow us to select from among all efficient outcomes for one that has other desired features, such as distributional equity. However, it is unclear how any real-world government might enact such redistributions. Lump-sum transfers are difficult to enforce and virtually never used, and proportional taxes may have large distortionary effects. Additionally, the government would need to have perfect knowledge of consumers' preferences and firms' production functions in order to choose the transfers correctly.

Proof of the first fundamental theorem

The first fundamental theorem of welfare economics states that any Walrasian equilibrium is Pareto-efficient. The only assumption needed (in addition to complete markets and price-taking behavior) is the relatively weak assumption of local nonsatiation of preferences. In particular, no convexity assumptions are needed. More formally, the statement of the theorem is as follows: If preferences are locally nonsatiated, and if (x*, y*, p) is a price equilibrium with transfers, then the allocation (x*, y*) is Pareto optimal.

Suppose that consumer i has wealth [w_i] such that [\Sigma _i w_i = p \cdot \omega + \Sigma _j p \cdot y^*_j] where [ \omega ] is the aggregate endowment of goods and [y^*_j] is the production of firm j.

Preference maximization (from the definition of price equilibrium with transfers) implies:

:if [x_i >_i x^*_i] then [p \cdot x_i > w_i]

In other words, if a bundle of goods is strictly preferred to [x^*_i] it must be unaffordable at price p. Local nonsatiation additionally implies:

:if [x_i \geq _i x^*_i] then [p \cdot x_i \geq w_i]

To see why, imagine that [x_i \geq _i x^*_i] but [p \cdot x_i < w_i]. Then by local nonsatiation we could find [x'_i] arbitrarily close to [x_i] (and so still affordable) but which is strictly preferred to [x^*_i]. But [x^*_i] is the result of preference maximization, so this is a contradiction.

Now consider an allocation [ (x, y) ] that Pareto dominates [(x^*, y^*)]. This means that [x_i \geq _i x^*_i] for all i and [x_i >_i x^*_i] for some i. By the above, we know [p \cdot x_i \geq w_i] for all i and [p \cdot x_i > w_i] for some i. Summing, we find:

:[\Sigma _i p \cdot x_i > \Sigma _i w_i = p \cdot \omega + \Sigma _j p \cdot y^*_j]

Because [ y_j ] is profit maximizing we know [\Sigma _j p \cdot y^*_j \geq \Sigma _j p \cdot y_j ], so [\Sigma _i p \cdot x_i > p \cdot \omega + \Sigma _j p \cdot y_j]. Hence, [(x,y)] is not feasible. Since all Pareto-dominating allocations are not feasible, [(x^*,y^*)] must itself be Pareto optimal.

Proof of the second fundamental theorem

The second fundamental theorem of welfare economics states that, under the assumptions that every production set [Y_j] is convex and every preference relation [\geq _i] is convex and locally nonsatiated, any desired Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers. Further assumptions are needed to prove this statement for price equilibriums with transfers. We will proceed in two steps: first we prove that any Parento-efficient allocation can be supported as a price quasi-equilibrium with transfers, then we give conditions under which a price quasi-equilibrium is also a price equilibrium.

Let us define a price quasi-equilibrium with transfers as an allocation [(x^*,y^*)], a price vector p, and a vector of wealth levels w (achieved by lump-sum transfers) with [\Sigma _i w_i = p \cdot \omega + \Sigma _j p \cdot y^*_j] (where [ \omega ] is the aggregate endowment of goods and [y^*_j] is the production of firm j) such that:

:i. [p \cdot y_j \leq p \cdot y_j^*] for all [y_j \in Y_j] (firms maximize profit by producing [y_j^*])

:ii. For all i, if [x_i >_i x_i^*] then [p \cdot x_i \geq w_i] (if [x_i] is strictly preferred to [x_i^*] then it cannot cost less than [x_i^*])

:iii. [\Sigma_i x_i^* = \omega + \Sigma _j y_j^*] (budget constraint satisfied)

The only difference between this definition and the standard definition of a price equilibrium with transfers is in statement (ii). The inequality is weak here ([p \cdot x_i \geq w_i]) making it a price quasi-equilibrium. Later we will strengthen this to make a price equilibrium.

Define [V_i] to be the set of all consumption bundles strictly preferred to [x_i^*] by consumer i, and let V be the union of all [V_i]. [V_i] is convex due to the convexity of the preference relation [\geq _i]. V is convex because every [V_i] is convex. Similarly [Y + \], the union of all production sets [Y_i] plus the aggregate endowment, is convex because every [Y_i] is convex. We also know that the intersection of V and [Y + \] must be empty, because if it were not it would imply there existed a bundle that is strictly preferred to [(x^*,y^*)] by everyone and is also affordable. This is ruled out by the Pareto-optimality of [(x^*,y^*)].

These two convex, non-intersecting sets allow us to apply the separating hyperplane theorem. This theorem states that there exists a price vector [p \neq 0] and a number r such that [p \cdot z \geq r] for every [z \in V] and [p \cdot z \leq r] for every [z \in Y + \]. In other words, there exists a price vector that defines a hyperplane that perfectly separates the two convex sets.

Next we argue that if [x_i \geq _i x_i^*] for all i then [p \cdot (\Sigma _i x_i) \geq r]. This is due to local nonsatiation: there must be a bundle [x'_i] arbitrarily close to [x_i] that is strictly preferred to [x_i^*] and hence part of [V_i], so [p \cdot (\Sigma _i x'_i) \geq r]. Taking the limit as [x'_i \rightarrow x_i] does not change the weak inequality, so [p \cdot (\Sigma _i x_i) \geq r] as well. In other words, [x_i] is in the closure of V.

Using this relation we see that for [x_i^*] itself [p \cdot (\Sigma _i x_i^*) \geq r]. We also know that [\Sigma _i x_i^* \in Y + \], so [p \cdot (\Sigma _i x_i^*) \leq r] as well. Combining these we find that [p \cdot (\Sigma _i x_i^*) = r]. We can use this equation to show that [(x^*,y^*,p)] fits the definition of a price quasi-equilibrium with transfers.

Because [p \cdot (\Sigma _i x_i^*) = r] and [\Sigma _i x_i^* = \omega + \Sigma _j y_j^*] we know that for any firm j:

:[p \cdot (\omega + y_j + \Sigma_h y_h^*) \leq r = p \cdot (\omega + y_j^* + \Sigma_h y_h^*)] for [h \neq j]

which implies [p \cdot y_j \leq p \cdot y_j^*]. Similarly we know:

:[p \cdot (x_i + \Sigma_k x_k^*) \geq r = p \cdot (x_i^* + \Sigma_k x_k^*)] for [k \neq i]

which implies [p \cdot x_i \geq p \cdot x_i^*]. These two statements, along with the feasibility of the allocation at the Pareto optimum, satisfy the three conditions for a price quasi-equilibrium with transfers supported by wealth levels [w_i = p \cdot x_i^*] for all i.

We now turn to conditions under which a price quasi-equilibrium is also a price equilibrium, in other words, conditions under which the statement "if [x_i >_i x_i^*] then [p \cdot x_i \geq w_i]" imples "if [x_i >_i x_i^*] then [p \cdot x_i > w_i]". For this to be true we need now to assume that the consumption set [X_i] is convex and the preference relation [\geq _i] is continuous. Then, if there exists a consumption vector [x'_i] such that [x'_i \in X_i] and [p \cdot x'_i < w_i], a price quasi-equilibrium is a price equilibrium.

To see why, assume to the contrary [x_i >_i x_i^*] and [p \cdot x_i = w_i], and [x_i] exists. Then by the convexity of [X_i] we have a bundle [x_i = \alpha x_i + (1 - \alpha)x'_i \in X_i] with [p \cdot x_i < w_i]. By the continuity of [\geq _i] for [\alpha] close to 1 we have [\alpha x_i + (1 - \alpha)x'_i >_i x_i^*]. This is a contradiction, because this bundle is preferred to [x_i^*] and costs less than [w_i].

Hence, for price quasi-equilibria to be price equilibria it is sufficient that the consumption set be convex, the preference relation to be continuous, and for there always to exist a "cheaper" consumption bundle [x'_i]. One way to ensure the existence of such a bundle is to require wealth levels [w_i] to be strictly positive for all consumers i.

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