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Fundamental theorems of welfare economics

  (Redirected from First Welfare Theorem)

There are three 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 three attributes:[1]

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.

The third theorem [also referred to as Arrow's theorem] focuses on defining the social welfare of a society. It determines if there is a way to acquire the genuine interests of society regarding the distributions of e.g. (wealth/income) from the specified preferences of individuals. The third theorem states that there is no Arrow Social Welfare equilibrium that will satisfy Pareto consistency.

Four conditions of Arrow's theorem: [2]

  1. Universality: A function should always be satisfied no matter the individuals' preferences.
  2. Pareto Consistency: Social preferences must be consistent with Pareto criterion.
  3. Independence: Individual's preferences must be independent of each other.
  4. Non-dictatorship: There should be no dictatorship in the individuals in society's preferences.


Implications of the first theoremEdit

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.[3]

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.[4]

Proof of the first theoremEdit

The first fundamental theorem was first demonstrated graphically by economist Abba Lerner[citation needed] and mathematically by economists Harold Hotelling, Oskar Lange, Maurice Allais, Lionel McKenzie, Kenneth Arrow and Gérard Debreu. The theorem holds under general conditions.[4]

The formal statement of the theorem is as follows: If preferences are locally nonsatiated, and if   is a price equilibrium with transfers, then the allocation  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.[4]

Given a set   of types of goods we work in the real vector space over  ,   and use boldface for vector valued variables. For instance, if   then   would be a three dimensional vector space and the vector   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   such that   where   is the aggregate endowment of goods (i.e. the sum of all consumer and producer endowments) and   is the production of firm j.

Preference maximization (from the definition of price equilibrium with transfers) implies (using   to denote the preference relation for consumer i):

if   then  

In other words, if a bundle of goods is strictly preferred to   it must be unaffordable at price  . Local nonsatiation additionally implies:

if   then  

To see why, imagine that   but  . Then by local nonsatiation we could find   arbitrarily close to   (and so still affordable) but which is strictly preferred to  . But   is the result of preference maximization, so this is a contradiction.

An allocation is a pair   where   and  , i.e.   is the 'matrix' (allowing potentially infinite rows/columns) whose ith column is the bundle of goods allocated to consumer i and   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   that Pareto dominates  . This means that   for all i and   for some i. By the above, we know   for all i and   for some i. Summing, we find:


Because   is profit maximizing, we know  , so  . But goods must be conserved so  . Hence,   is not feasible. Since all Pareto-dominating allocations are not feasible,   must itself be Pareto optimal.[4]

Note that while the fact that   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   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 theoremEdit

The second theorem formally 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.[4] 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  , 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.[4] 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:


which implies  . Similarly we know:


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.[4]

Related theoremsEdit

Because of welfare economics' close ties to social choice theory, Arrow's impossibility theorem is sometimes listed as a third fundamental theorem.[5]

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.[6] Further, even if these ideal conditions hold, the First Welfare Theorem fails in an overlapping generations model.

See alsoEdit


  1. ^
  2. ^ Feldman, Allan M. (2008), Palgrave Macmillan (ed.), "Welfare Economics", The New Palgrave Dictionary of Economics, Palgrave Macmillan UK, pp. 9–10, doi:10.1057/978-1-349-95121-5_1417-2, ISBN 9781349951215, retrieved 2019-04-24
  3. ^ Stiglitz, Joseph E. (1994), Whither Socialism?, MIT Press, ISBN 978-0-262-69182-6
  4. ^ a b c d e f g Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995), "Chapter 16: Equilibrium and its Basic Welfare Properties", Microeconomic Theory, Oxford University Press, ISBN 978-0-19-510268-0
  5. ^ * Feldman, Allan M. (2008), "Welfare Economics", The New Palgrave: A Dictionary of Economics (online ed.), 4, pp. 889–95, retrieved 9 June 2014
  6. ^ Stiglitz, Joseph E. (March 1991), "The Invisible Hand and Modern Welfare Economics", NBER Working Paper No. W3641, doi:10.3386/w3641