Fundamental theorems of welfare economics

Implications of the First Theorem

The First Theorem is often taken to be an analytical confirmation of Adam Smith's "invisible hand" hypothesis, namely that competitive markets tend toward an efficient allocation of resources. The theorem supports a case for non-intervention in ideal conditions: let the markets do the work and the outcome will be Pareto efficient. However, Pareto efficiency is not necessarily the same thing as desirability; it merely indicates that no one can be made better off without someone being made worse off. There can be many possible Pareto efficient allocations of resources and not all of them may be equally desirable by society.

This appears to make the case that intervention has a legitimate place in policy – redistributions can allow us to select from all efficient outcomes for one that has other desired features, such as distributional equity. The shortcoming is that for the theorem to hold, the transfers have to be lump-sum and the government needs to have perfect information on individual consumers' tastes as well as the production possibilities of firms. An additional mathematical condition is that preferences and production technologies have to be convex.

Proof of the First Theorem

The first fundamental theorem was first demonstrated graphically by economist Abba Lerner and mathematically by economists Harold Hotelling, Oskar Lange, Maurice Allais, Kenneth Arrow and Gérard Debreu. The theorem holds under general conditions.

The formal 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. An equilibrium in this sense either relates to an exchange economy only or presupposes that firms are allocatively and productively efficient, which can be shown to follow from perfectly competitive factor and production markets.

Suppose that consumer i has wealth w i such that Σ i w i = p ⋅ ω + Σ j p ⋅ y j ∗ where ω 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:

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:

To see why, imagine that x i ≥ i x i ∗ but p ⋅ 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 ≥ i x i ∗ for all i and x i > i x i ∗ for some i. By the above, we know p ⋅ x i ≥ w i for all i and p ⋅ x i > w i for some i. Summing, we find:

Because y ∗ is profit maximizing, we know Σ j p ⋅ y j ∗ ≥ Σ j p ⋅ y j , so Σ i p ⋅ x i > p ⋅ ω + Σ j p ⋅ 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 Theorem formally states that, under the assumptions that every production set Y j is convex and every preference relation ≥ 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 equilibria with transfers.

The proof proceeds in two steps: first, we prove that any Pareto-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 Σ i w i = p ⋅ ω + Σ j p ⋅ y j ∗ (where ω is the aggregate endowment of goods and y j ∗ is the production of firm j) such that:

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 ⋅ x i ≥ 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 sum of all V i . V i is convex due to the convexity of the preference relation ≥ i . V is convex because every V i is convex. Similarly Y + { ω } , 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 + { ω } 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 ≠ 0 and a number r such that p ⋅ z ≥ r for every z ∈ V and p ⋅ z ≤ r for every z ∈ Y + { ω } . 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 ≥ i x i ∗ for all i then p ⋅ ( Σ i x i ) ≥ 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 ⋅ ( Σ i x i ′ ) ≥ r . Taking the limit as x i ′ → x i does not change the weak inequality, so p ⋅ ( Σ i x i ) ≥ 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 ⋅ ( Σ i x i ∗ ) ≥ r . We also know that Σ i x i ∗ ∈ Y + { ω } , so p ⋅ ( Σ i x i ∗ ) ≤ r as well. Combining these we find that p ⋅ ( Σ 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 ⋅ ( Σ i x i ∗ ) = r and Σ i x i ∗ = ω + Σ j y j ∗ we know that for any firm j:

which implies p ⋅ y j ≤ p ⋅ y j ∗ . Similarly we know:

which implies p ⋅ x i ≥ p ⋅ 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 ⋅ 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 ⋅ x i ≥ w i " imples "if x i > i x i ∗ then p ⋅ 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 ≥ i is continuous. Then, if there exists a consumption vector x i ′ such that x i ′ ∈ X i and p ⋅ 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 ⋅ x i = w i , and x i exists. Then by the convexity of X i we have a bundle x i ″ = α x i + ( 1 − α ) x i ′ ∈ X i with p ⋅ x i ″ < w i . By the continuity of ≥ i for α close to 1 we have α x i + ( 1 − α ) 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.

Related theorems

Because of welfare economics' close ties to social choice theory, Arrow's impossibility theorem is sometimes listed as a third fundamental theorem.

The ideal conditions of the theorems, however are an abstraction. The Greenwald-Stiglitz theorem, for example, states that in the presence of either imperfect information, or incomplete markets, markets are not Pareto efficient. Thus, in real world economies, the degree of these variations from ideal conditions must factor into policy choices. Further, even if these ideal conditions hold, the First Welfare Theorem fails in an overlapping generations model.