Finance:Fundamental theorems of welfare economics

Short description: Fully competitive markets tend toward a Pareto efficient allocation of resources

There are two fundamental theorems of welfare economics. The first theorem states that a market will tend toward a competitive equilibrium that is weakly Pareto optimal when the market maintains the following two attributes:

1. Complete markets with no transaction costs, and therefore each actor also having perfect information.

2. Price-taking behavior with no monopolists and easy entry and exit from a market.

Furthermore, the first theorem states that the equilibrium will be fully Pareto optimal with the additional condition of:

3. Local nonsatiation of preferences such that for any original bundle of goods, there is another bundle of goods arbitrarily close to the original bundle, but which is preferred.

The second theorem states that out of all possible Pareto optimal outcomes one can achieve any particular one by enacting a lump-sum wealth redistribution and then letting the market take over.

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 and that total wealth is maximized. 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, Lionel McKenzie, 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 [math]\displaystyle{ (\mathbf{X^*},\mathbf{Y^*}, \mathbf{p}) }[/math] is a price equilibrium with transfers, then the allocation [math]\displaystyle{ (\mathbf{X^*},\mathbf{Y^*}) }[/math]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.

Given a set [math]\displaystyle{ G }[/math] of types of goods we work in the real vector space over [math]\displaystyle{ G }[/math], [math]\displaystyle{ \mathbb{R}^{G} }[/math] and use boldface for vector valued variables. For instance, if [math]\displaystyle{ G=\lbrace \text{butter}, \text{cookies}, \text{milk} \rbrace }[/math] then [math]\displaystyle{ \mathbb{R}^{G} }[/math] would be a three dimensional vector space and the vector [math]\displaystyle{ \langle 1, 2, 3 \rangle }[/math] would represent the bundle of goods containing one unit of butter, 2 units of cookies and 3 units of milk.

Suppose that consumer i has wealth [math]\displaystyle{ w_i }[/math] such that [math]\displaystyle{ \Sigma_i w_i = \mathbf{p} \cdot \mathbf{e} + \Sigma _j \mathbf{p} \cdot \mathbf{y^*_j} }[/math] where [math]\displaystyle{ \mathbf{e} }[/math] is the aggregate endowment of goods (i.e. the sum of all consumer and producer endowments) and [math]\displaystyle{ \mathbf{y^*_j} }[/math] is the production of firm j.

Preference maximization (from the definition of price equilibrium with transfers) implies (using [math]\displaystyle{ \gt _i }[/math] to denote the preference relation for consumer i):

if [math]\displaystyle{ \mathbf{x_i} \gt _i \mathbf{x^*_i} }[/math] then [math]\displaystyle{ \mathbf{p} \cdot \mathbf{x_i} \gt \mathbf{w_i} }[/math]

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

if [math]\displaystyle{ \mathbf{x_i} \geq _i \mathbf{x^*_i} }[/math] then [math]\displaystyle{ \mathbf{p} \cdot \mathbf{x_i} \geq \mathbf{w_i} }[/math]

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

An allocation is a pair [math]\displaystyle{ (\mathbf{X},\mathbf{Y}) }[/math] where [math]\displaystyle{ \mathbf{X} \in \Pi_{i \in I} \mathbb{R}^{G} }[/math] and [math]\displaystyle{ \mathbf{Y} \in \Pi_{j \in J} \mathbb{R}^{G} }[/math], i.e. [math]\displaystyle{ \mathbf{X} }[/math] is the 'matrix' (allowing potentially infinite rows/columns) whose ith column is the bundle of goods allocated to consumer i and [math]\displaystyle{ \mathbf{Y} }[/math] is the 'matrix' whose jth column is the production of firm j. We restrict our attention to feasible allocations which are those allocations in which no consumer sells or producer consumes goods which they lack, i.e.,for every good and every consumer that consumers initial endowment plus their net demand must be positive similarly for producers.

Now consider an allocation [math]\displaystyle{ (\mathbf{X},\mathbf{Y}) }[/math] that Pareto dominates [math]\displaystyle{ (\mathbf{X^*}, Y^*) }[/math]. This means that [math]\displaystyle{ \mathbf{x_i} \geq _i \mathbf{x^*_i} }[/math] for all i and [math]\displaystyle{ \mathbf{x_i} \gt _i \mathbf{x^*_i} }[/math] for some i. By the above, we know [math]\displaystyle{ \mathbf{p} \cdot \mathbf{x_i} \geq w_i }[/math] for all i and [math]\displaystyle{ \mathbf{p} \cdot \mathbf{x_i} \gt w_i }[/math] for some i. Summing, we find:

[math]\displaystyle{ \Sigma _i \mathbf{p} \cdot \mathbf{x_i} \gt \Sigma _i w_i = \Sigma _j \mathbf{p} \cdot \mathbf{y^*_j} }[/math].

Because [math]\displaystyle{ \mathbf{Y^*} }[/math] is profit maximizing, we know [math]\displaystyle{ \Sigma _j \mathbf{p} \cdot y^*_j \geq \Sigma _j p \cdot y_j }[/math], so [math]\displaystyle{ \Sigma _i \mathbf{p} \cdot \mathbf{x_i} \gt \Sigma _j \mathbf{p} \cdot \mathbf{y_j} }[/math]. But goods must be conserved so [math]\displaystyle{ \Sigma _i \mathbf{x_i} \gt \Sigma _j \mathbf{y_j} }[/math]. Hence, [math]\displaystyle{ (\mathbf{X},\mathbf{Y}) }[/math] is not feasible. Since all Pareto-dominating allocations are not feasible, [math]\displaystyle{ (\mathbf{X^*},\mathbf{Y^*}) }[/math] must itself be Pareto optimal.

Note that while the fact that [math]\displaystyle{ \mathbf{Y^*} }[/math] is profit maximizing is simply assumed in the statement of the theorem the result is only useful/interesting to the extent such a profit maximizing allocation of production is possible. Fortunately, for any restriction of the production allocation [math]\displaystyle{ \mathbf{Y^*} }[/math] and price to a closed subset on which the marginal price is bounded away from 0, e.g., any reasonable choice of continuous functions to parameterize possible productions, such a maximum exists. This follows from the fact that the minimal marginal price and finite wealth limits the maximum feasible production (0 limits the minimum) and Tychonoff's theorem ensures the product of these compacts spaces is compact ensuring us a maximum of whatever continuous function we desire exists.

Proof of the second fundamental theorem

The second theorem formally states that, under the assumptions that every production set [math]\displaystyle{ Y_j }[/math] is convex and every preference relation [math]\displaystyle{ \geq _i }[/math] 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 [math]\displaystyle{ (x^*,y^*) }[/math], a price vector p, and a vector of wealth levels w (achieved by lump-sum transfers) with [math]\displaystyle{ \Sigma _i w_i = p \cdot \omega + \Sigma _j p \cdot y^*_j }[/math] (where [math]\displaystyle{ \omega }[/math] is the aggregate endowment of goods and [math]\displaystyle{ y^*_j }[/math] is the production of firm j) such that:

i. [math]\displaystyle{ p \cdot y_j \leq p \cdot y_j^* }[/math] for all [math]\displaystyle{ y_j \in Y_j }[/math] (firms maximize profit by producing [math]\displaystyle{ y_j^* }[/math])

ii. For all i, if [math]\displaystyle{ x_i \gt _i x_i^* }[/math] then [math]\displaystyle{ p \cdot x_i \geq w_i }[/math] (if [math]\displaystyle{ x_i }[/math] is strictly preferred to [math]\displaystyle{ x_i^* }[/math] then it cannot cost less than [math]\displaystyle{ x_i^* }[/math])

iii. [math]\displaystyle{ \Sigma_i x_i^* = \omega + \Sigma _j y_j^* }[/math] (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 ([math]\displaystyle{ p \cdot x_i \geq w_i }[/math]) making it a price quasi-equilibrium. Later we will strengthen this to make a price equilibrium. Define [math]\displaystyle{ V_i }[/math] to be the set of all consumption bundles strictly preferred to [math]\displaystyle{ x_i^* }[/math] by consumer i, and let V be the sum of all [math]\displaystyle{ V_i }[/math]. [math]\displaystyle{ V_i }[/math] is convex due to the convexity of the preference relation [math]\displaystyle{ \geq _i }[/math]. V is convex because every [math]\displaystyle{ V_i }[/math] is convex. Similarly [math]\displaystyle{ Y + \{\omega\} }[/math], the union of all production sets [math]\displaystyle{ Y_i }[/math] plus the aggregate endowment, is convex because every [math]\displaystyle{ Y_i }[/math] is convex. We also know that the intersection of V and [math]\displaystyle{ Y + \{\omega\} }[/math] must be empty, because if it were not it would imply there existed a bundle that is strictly preferred to [math]\displaystyle{ (x^*,y^*) }[/math] by everyone and is also affordable. This is ruled out by the Pareto-optimality of [math]\displaystyle{ (x^*,y^*) }[/math].

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

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

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

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

[math]\displaystyle{ p \cdot (\omega + y_j + \Sigma_h y_h^*) \leq r = p \cdot (\omega + y_j^* + \Sigma_h y_h^*) }[/math] for [math]\displaystyle{ h \neq j }[/math]

which implies [math]\displaystyle{ p \cdot y_j \leq p \cdot y_j^* }[/math]. Similarly we know:

[math]\displaystyle{ p \cdot (x_i + \Sigma_k x_k^*) \geq r = p \cdot (x_i^* + \Sigma_k x_k^*) }[/math] for [math]\displaystyle{ k \neq i }[/math]

which implies [math]\displaystyle{ p \cdot x_i \geq p \cdot x_i^* }[/math]. 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 [math]\displaystyle{ w_i = p \cdot x_i^* }[/math] 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 [math]\displaystyle{ x_i \gt _i x_i^* }[/math] then [math]\displaystyle{ p \cdot x_i \geq w_i }[/math]" imples "if [math]\displaystyle{ x_i \gt _i x_i^* }[/math] then [math]\displaystyle{ p \cdot x_i \gt w_i }[/math]". For this to be true we need now to assume that the consumption set [math]\displaystyle{ X_i }[/math] is convex and the preference relation [math]\displaystyle{ \geq _i }[/math] is continuous. Then, if there exists a consumption vector [math]\displaystyle{ x'_i }[/math] such that [math]\displaystyle{ x'_i \in X_i }[/math] and [math]\displaystyle{ p \cdot x'_i \lt w_i }[/math], a price quasi-equilibrium is a price equilibrium.

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

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 [math]\displaystyle{ x'_i }[/math]. One way to ensure the existence of such a bundle is to require wealth levels [math]\displaystyle{ w_i }[/math] 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.

See also

• Convex preferences
• Varian's theorems – a competitive equilibrium is both Pareto-efficient and envy-free.
• General equilibrium theory