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 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 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. The theorem supports a case for non-intervention in ideal conditions: let the markets do the work and the outcome will be desirable.

These ideal conditions, however, collectively known as Perfect Competition, do not exist in the real world. 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. [ [http://ssrn.com/abstract=227430 Joseph E.,The Invisible Hand and Modern Economics(March 1991). NBER Working Paper No. W3641.] ]

The second theorem states that out of the infinity 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 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 on the economy since taxes change the relative remunerations of the factors of production, distorting the structure of production. Additionally, the government would need to have perfect knowledge of consumers' preferences and firms' production functions (which are in fact unknowable [ [http://www.econlib.org/library/Essays/hykKnw1.html The Use of Knowledge in Society] ] ) in order to choose the transfers correctly. In addition, this remedy cannot be expected to work if large numbers of people do not understand the economy, and how to make effective use of any transfers they receive.

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 Gerard Debreu, although the restrictive assumptions necessary for the proof mean that the result may not necessarily reflect the workings of real economies. 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." 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 $Sigma\; \_i\; w\_i\; =\; p\; cdot\; omega\; +\; Sigma\; \_j\; p\; cdot\; y^*\_j$ 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 . Then by local nonsatiation we could find $x\text{'}\_i$ 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 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 $p\; eq\; 0$ 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 $x\text{'}\_i$ arbitrarily close to $x\_i$ that is strictly preferred to $x\_i^*$ and hence part of $V\_i$, so $p\; cdot\; (Sigma\; \_i\; x\text{'}\_i)\; geq\; r$. Taking the limit as $x\text{'}\_i\; ightarrow\; x\_i$ 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 $h\; eq\; j$

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 $k\; eq\; i$

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 $x\text{'}\_i$ such that $x\text{'}\_i\; in\; X\_i$ and , 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 $x"\_i\; =\; alpha\; x\_i\; +\; (1\; -\; alpha)x\text{'}\_i\; in\; X\_i$ with . By the continuity of $geq\; \_i$ for $alpha$ close to 1 we have $alpha\; x\_i\; +\; (1\; -\; alpha)x\text{'}\_i\; >\_i\; x\_i^*$. 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 $x\text{'}\_i$. 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".

ee also

*Convex preferences

References

* Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green (1995), "Microeconomic Theory", Chapter 16. Oxford University Press, ISBN 0195102681.