Fundamental Theorems of Welfare Economics - Proof of The Second Fundamental Theorem

Proof of The Second Fundamental Theorem

The second fundamental theorem of welfare economics states that, under the assumptions that every production set is convex and every preference relation 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.

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, a price vector p, and a vector of wealth levels w (achieved by lump-sum transfers) with (where is the aggregate endowment of goods and is the production of firm j) such that:

i. for all (firms maximize profit by producing )

ii. For all i, if then (if is strictly preferred to then it cannot cost less than )

iii. (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 making it a price quasi-equilibrium. Later we will strengthen this to make a price equilibrium.

Define to be the set of all consumption bundles strictly preferred to by consumer i, and let V be the sum of all . is convex due to the convexity of the preference relation . V is convex because every is convex. Similarly, the union of all production sets plus the aggregate endowment, is convex because every is convex. We also know that the intersection of V and must be empty, because if it were not it would imply there existed a bundle that is strictly preferred to by everyone and is also affordable. This is ruled out by the Pareto-optimality of .

These two convex, non-intersecting sets allow us to apply the separating hyperplane theorem. This theorem states that there exists a price vector and a number r such that for every and for every . In other words, there exists a price vector that defines a hyperplane that perfectly separates the two convex sets.

Next we argue that if for all i then . This is due to local nonsatiation: there must be a bundle arbitrarily close to that is strictly preferred to and hence part of, so . Taking the limit as does not change the weak inequality, so as well. In other words, is in the closure of V.

Using this relation we see that for itself . We also know that, so as well. Combining these we find that . We can use this equation to show that fits the definition of a price quasi-equilibrium with transfers.

Because and we know that for any firm j:

for

which implies . Similarly we know:

for

which implies . 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 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 then " imples "if then ". For this to be true we need now to assume that the consumption set is convex and the preference relation is continuous. Then, if there exists a consumption vector such that and, a price quasi-equilibrium is a price equilibrium.

To see why, assume to the contrary and, and exists. Then by the convexity of we have a bundle with . By the continuity of for close to 1 we have . This is a contradiction, because this bundle is preferred to and costs less than .

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 . One way to ensure the existence of such a bundle is to require wealth levels to be strictly positive for all consumers i.