Fundamental theorems of welfare economics

There are two fundamental theorems of welfare economics. The first states that any competitive equilibrium or Walrasian equilibrium leads to a Pareto 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 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.

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 most real world economies, the degree of these variations from ideal conditions must factor into policy choices.

The second theorem states that out of all possible Pareto efficient outcomes one can achieve any particular one 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 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 fundamental theorem

The "First fundamental theorem of welfare economics" states that any Walrasian equilibrium is Pareto-efficient. This 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 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.

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 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 $\Sigma _{i}w_{i}=p\cdot \omega +\Sigma _{j}p\cdot y_{j}^{*}$ (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 sum 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 $Y+\{\omega \}$, 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 $Y+\{\omega \}$ 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 $z\in Y+\{\omega \}$. 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 $\Sigma _{i}x_{i}^{*}\in Y+\{\omega \}$, 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.

See also

• Convex preferences