Finance:Fundamental theorems of welfare economics

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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]

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.[2]

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.[3]

Proof of the first theorem

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.[3]

The formal 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.[3]

Given a set G of types of goods we work in the real vector space over G, G and use boldface for vector valued variables. For instance, if G={butter,cookies,milk} then G would be a three dimensional vector space and the vector 1,2,3 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 wi such that Σiwi=𝐩𝐞+Σj𝐩𝐲𝐣* where 𝐞 is the aggregate endowment of goods (i.e. the sum of all consumer and producer endowments) and 𝐲𝐣* is the production of firm j.

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

if x𝐢>i𝐱𝐢* then 𝐩x𝐢>w𝐢

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

if x𝐢i𝐱𝐢* then 𝐩x𝐢w𝐢

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

An allocation is a pair (𝐗,𝐘) where 𝐗ΠiIG and 𝐘ΠjJG, i.e. 𝐗 is the 'matrix' (allowing potentially infinite rows/columns) whose ith column is the bundle of goods allocated to consumer i and 𝐘 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 (𝐗,𝐘) that Pareto dominates (𝐗*,Y*). This means that x𝐢i𝐱𝐢* for all i and x𝐢>i𝐱𝐢* for some i. By the above, we know 𝐩x𝐢wi for all i and 𝐩x𝐢>wi for some i. Summing, we find:

Σi𝐩x𝐢>Σiwi=Σj𝐩𝐲𝐣*.

Because 𝐘* is profit maximizing, we know Σj𝐩yj*Σjpyj, so Σi𝐩x𝐢>Σj𝐩y𝐣. But goods must be conserved so Σix𝐢>Σjy𝐣. Hence, (𝐗,𝐘) is not feasible. Since all Pareto-dominating allocations are not feasible, (𝐗*,𝐘*) must itself be Pareto optimal.[3]

Note that while the fact that 𝐘* 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 𝐘* 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 Yj is convex and every preference relation i is convex and locally nonsatiated, any desired Pareto-efficient allocation can be supported as a price quasi-equilibrium with transfers.[3] 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 (x*,y*), a price vector p, and a vector of wealth levels w (achieved by lump-sum transfers) with Σiwi=pω+Σjpyj* (where ω is the aggregate endowment of goods and yj* is the production of firm j) such that:

i. pyjpyj* for all yjYj (firms maximize profit by producing yj*)
ii. For all i, if xi>ixi* then pxiwi (if xi is strictly preferred to xi* then it cannot cost less than xi*)
iii. Σixi*=ω+Σjyj* (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 (pxiwi) making it a price quasi-equilibrium. Later we will strengthen this to make a price equilibrium.[3] Define Vi to be the set of all consumption bundles strictly preferred to xi* by consumer i, and let V be the sum of all Vi. Vi is convex due to the convexity of the preference relation i. V is convex because every Vi is convex. Similarly Y+{ω}, the union of all production sets Yi plus the aggregate endowment, is convex because every Yi is convex. We also know that the intersection of V and Y+{ω} 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 p0 and a number r such that pzr for every zV and pzr for every zY+{ω}. In other words, there exists a price vector that defines a hyperplane that perfectly separates the two convex sets.

Next we argue that if xiixi* for all i then p(Σixi)r. This is due to local nonsatiation: there must be a bundle x'i arbitrarily close to xi that is strictly preferred to xi* and hence part of Vi, so p(Σix'i)r. Taking the limit as x'ixi does not change the weak inequality, so p(Σixi)r as well. In other words, xi is in the closure of V.

Using this relation we see that for xi* itself p(Σixi*)r. We also know that Σixi*Y+{ω}, so p(Σixi*)r as well. Combining these we find that p(Σixi*)=r. We can use this equation to show that (x*,y*,p) fits the definition of a price quasi-equilibrium with transfers.

Because p(Σixi*)=r and Σixi*=ω+Σjyj* we know that for any firm j:

p(ω+yj+Σhyh*)r=p(ω+yj*+Σhyh*) for hj

which implies pyjpyj*. Similarly we know:

p(xi+Σkxk*)r=p(xi*+Σkxk*) for ki

which implies pxipxi*. 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 wi=pxi* 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 xi>ixi* then pxiwi" imples "if xi>ixi* then pxi>wi". For this to be true we need now to assume that the consumption set Xi is convex and the preference relation i is continuous. Then, if there exists a consumption vector x'i such that x'iXi and px'i<wi, a price quasi-equilibrium is a price equilibrium.

To see why, assume to the contrary xi>ixi* and pxi=wi, and xi exists. Then by the convexity of Xi we have a bundle x'i=αxi+(1α)x'iXi with px'i<wi. By the continuity of i for α close to 1 we have αxi+(1α)x'i>ixi*. This is a contradiction, because this bundle is preferred to xi* and costs less than wi.

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 wi to be strictly positive for all consumers i.[3]

Because of welfare economics' close ties to social choice theory, Arrow's impossibility theorem is sometimes listed as a third fundamental theorem.[4] [dubious ]

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

See also

References

  1. http://web.stanford.edu/~hammond/effMktFail.pdf
  2. Stiglitz, Joseph E. (1994), Whither Socialism?, MIT Press, ISBN 978-0-262-69182-6 
  3. 3.0 3.1 3.2 3.3 3.4 3.5 3.6 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, https://archive.org/details/isbn_9780198089537 
  4. * Feldman, Allan M. (2008), "Welfare Economics", The New Palgrave: A Dictionary of Economics 4, http://www.dictionaryofeconomics.com/article?id=pde2008_W000050, retrieved 9 June 2014 
  5. Stiglitz, Joseph E. (March 1991), "The Invisible Hand and Modern Welfare Economics", NBER Working Paper No. W3641, doi:10.3386/w3641