Fundamental theorems of welfare economics

The Wohlfahrtstheoreme (also main theorems of welfare economics ) are two fundamental theorems of welfare economics from the microeconomic field of economics.

  • 2.1 arrangements and definitions
  • 2.2 First Law of Thermodynamics
  • 2.3 Additional definitions
  • 2.4 Second Law
  • 3.1 Proof of the First Fundamental Theorem
  • 3.2 Proof of the Second Law

Introductory presentation

Both theorems are valid under the assumption of an idealized market economy in which all market participants behave as price takers and there are no externalities. Under these conditions, it refers to a state in which supply and demand in all markets the same, as a competitive equilibrium. The reality is of course more complicated, but the following results are considered important starting points for further research.

First Wohlfahrtstheorem

"Every competitive equilibrium is a Pareto optimum. "

In other words, no one can be made better off in a competitive equilibrium, without another one worse. This set is due in particular on the work of Kenneth Arrow and Gerard Debreu. He formalized Adam Smith's notion that markets like an invisible hand to work.

Second Wohlfahrtstheorem

" Each Pareto optimum is a competitive equilibrium, as far as the initial equipment of market participants may be adjusted accordingly. "

Thereafter, these are the two cardinal questions of economics, namely efficiency and equity, separate from each other: in order to reach that of Pareto optimality, which seems fair, does not abolish the market economy, but it is sufficient to adapt the initial equipment of market participants.

Formal representation

Agreements and definitions

Consider whether an economy of n markets. The prices on these markets are summarized in a price vector, where. In economics, there are more consumers and companies, and for these two groups corresponding to the index sets ( the set of all consumers ) or ( the set of all producers) are defined. Will now consider successively consumers and producers, after the initial funding of the economy:

  • The consumption profile of a person - there is information on which person i amount consumed by each of the n goods. The preference structure of each individual will turn in his utility function expression.
  • Production of an entity is determined by the technology.
  • The initial stocks of the respective goods are given by a vector features. We agree further than the features of a person (as to all goods).

With the agreed definitions regarding the preference structure of individuals, the technological capacities of producers and resource stocks can be an economy by the tuple

Characterize.

Through an allocation vector, a concrete " state" of again being given ( with specific consumption and production vectors for each consumer or producer ). Such an allocation is called admissible if for each resource that the total amount consumed just corresponds to the initial endowment plus the total quantity produced, so therefore if

An allocation is also Pareto efficient if there is no way to reallocate the resources between consumers that everyone has at least the same benefits, at least one person but actually experiences an increase in utility. Formally, the allocation is Pareto- efficient if and only if it is admissible (see above) and there is no other admissible allocation such that for all and for certain.

Considering am now a special economy, namely a competitive system in which all firms (and their profits) constitute private property, that is, the profits are part of the aggregated consumption budget. Since this is a competitive economy, goods are also traded in a decentralized competitive markets, with market actors act as price takers: consumers maximize their utility, producers their profits. From the private ownership assumption results in formal, that the budget of consumers from of two components: first, from a share of the initial endowment, the other from a share of the profits of the producers. This proportion amounts to just with ( would be, for example, the proportion, the person can take in the profits of producer 4 lay claim i). In accordance with the requirements and is. Such an economy can then be given as tuples

Describe.

For the competitive economy with private property a competitive equilibrium is defined as a tuple

For the following properties apply:

Such a balance is called walrasianisches balance.

First Law of Thermodynamics

First law of welfare economics: Be the considered economy. Be the each individual utility functions () underlying preference ordering locally not saturated (or, in the special case: a strictly monotonic ). Let further

A walrasianisches balance.

Then the derived Walrasian equilibrium allocation

For Pareto- efficient.

A slightly more general definition draws on the process described hereinbelow concept of a quasi- equilibrium; it then requires not, as the specific economy, but is generally. For this purpose, reference is made to a footnote.

Additional Definitions

We are expanding the narrow requirements of the Walrasian equilibrium something, while we transfer the concept of equilibrium to the " abstract " economy. Is a prerequisite for the ( price taker ) competitive equilibrium for that every individual has available only as much as is composed of ( the value ) of its original goods equipment and proportionate corporate profits, which it is entitled. The equilibrium concept discussed in this section knows yet another component of welfare provision: the transfer payment. One can almost imagine this as a unique example (positive or negative ) control, a social planner " shifts " means between the consumer through the front of the competitive activity in the economy.

We now first define a measure of individual welfare for all consumers. This is done by means of the vector.

For the competitive economy then

A quasi- ( price-taker ) competitive equilibrium with transfers if and only if there is a tuple, so that:

In particular, the Walrasian equilibrium is a quasi- ( price-taker ) competitive equilibrium with transfers.

Second Law

Second Law of welfare economics: Be the considered economy. Be the each individual utility functions () underlying preference ordering locally not saturated (or, in the special case: strictly ) convex and beyond. Be more convex for all.

Then there exists at any Pareto- optimal allocation, a price vector, so that a quasi- ( price-taker ) competitive equilibrium forms with transfers.

Evidence

Proof of the First Fundamental Theorem

Proof by contradiction: Suppose that the resulting from the price-taker - competitive equilibrium allocation for is not pareto optimal. Then there is, by definition, an admissible allocation for with

Is to show that such permissible allocation does not exist. To this end, we proceed step by step.

  • A) Since the ( price taker ) competitive equilibrium is for ( with the budget ) must also apply that. ( In fact, if instead, there would be in an environment to one that is strictly against preferred [ local non- saturation ] and so also satisfies the budget condition - but then would not be the optimal consumption bundle, see point 3 in the definition of ( price-taker ) competitive equilibrium. crudely told so a pareto- superior allocation must be too expensive, otherwise the pareto inferior allocation may not even be the same weight. )
  • B ) It follows from 2 that, for strictly prefers opposite. Would be the same size or even smaller than, would certainly chosen in equilibrium - in contradiction to Property 1 in the definition of ( price-taker ) competition equilibrium.
  • C ) By hypothesis, for each producer j is the profit-maximizing production quantity for the price, why necessarily, because if instead, the property 2 in the definition of ( price-taker ) would not meet competitive equilibrium.
  • D) Each person is in their given budget by amount.
  • E) Since, by assumption, and for all (see the definition of competitive exchange economy ), provides summing of the equation in d) that also
  • F) from c) and e ) it follows that.
  • G) a) and b ) result used in f)

But it follows by the definition of the admissibility of allocations ( see above), that is not allowed, contrary to the assumption, qed

Proof of the second law

Primarily from the following proof follows the widely used method of proof of Mas-Colell/Whinston/Green 1995, pp. 552 ff

Be for all a lot

Defined ( or the upper contour set of the set of all consumption vectors, which have a higher benefit than donate ). Summing this quantity over all i, one obtains

That is, the set of all individual consumption plans ( combined into a vector), made ​​strictly better by all the individuals are as with. Analogously, for all a lot

Defines the set of all production plans at the aggregate level. This aggregate production quantity can be moved around the equipment vector, thereby, the aggregate amount of consumption possibilities

Receives.

  • A) A is a convex set.
  • B ) V is convex as the sum of convex sets, as well as Y, even after shift to A.
  • C ) It is.
  • D) There exists a price vector and a, so that for all 1 and 2 for all.
  • E) If for all i, and then.
  • F) applies.
  • G ) It is.
  • H) If, then.

Moreover, as allowed by assumption, thus ensuring the election of ( all ) the existence of quasi- ( price-taker ) competitive equilibrium with transfers, qed

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