Welfare theorems

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

Introductory presentation

Both theorems apply under the condition of perfect competition , in which all market participants behave as price takers and there are no externalities . Under these conditions, a condition in which supply and demand match in all markets is called competitive equilibrium. The reality is more complicated, however the following results are considered important starting points for further research.

First welfare theorem

With perfect competition, every (general) competitive equilibrium is a Pareto optimum .

In other words, in a competitive equilibrium, nobody can be better off without another being worse off. This sentence goes back in particular to the work of Kenneth Arrow and Gérard Debreu , after decisive preparatory work by Léon Walras . He formalizes Adam Smith's idea that markets function like an invisible hand .

Second welfare theorem

Under certain restrictive conditions, every Pareto-optimal allocation can be realized as a competitive equilibrium , that is, there are initial equipment and prices that guarantee that given a Pareto-optimal allocation all households maximize their utility and all companies maximize their profit and all plans are compatible.

According to this, the two cardinal questions of economics, namely efficiency and fairness of distribution , can be separated from one another: In order to achieve the Pareto optimum that appears fair, one does not need to abolish the market economy, it is sufficient to adapt the initial equipment of the market participants.

Formal representation

Agreements and Definitions

Basics; notation

Consider an economy made up of n markets. The prices on these markets are summarized in a price vector , with . In economics there are also consumers and companies, whereby the index quantities (the quantity of all consumers) and (the quantity of all producers) are defined for these two groups . Consumers and producers are now considered one after the other, then the initial equipment of the economy: ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$ ${\displaystyle \mathbf {p} \neq \mathbf {0} }$ ${\displaystyle I}$ ${\displaystyle J}$ ${\displaystyle {\mathcal {I}}=\{1,\ldots ,I\}}$ ${\displaystyle {\mathcal {J}}=\{1,\ldots ,J\}}$

• A person's consumption profile is - it provides information about the amount of each of the n goods that person i consumes. The quantity covers all possible consumption profiles of i (consumption possibility quantity). The preference structure of each individual is in turn expressed in its utility function . ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle \mathbf {x} ^{i}=(x_{1}^{i},\ldots ,x_{n}^{i})\in \mathbb {R} ^{n}}$ ${\displaystyle X_{i}\subset \mathbb {R} ^{I}}$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle u^{i}=u^{i}(\mathbf {x} ^{i})\in \mathbb {R} }$
• A company's production is given by technology . The quantity includes all possible technologies of company j (production possibilities quantity). ${\displaystyle j\in {\mathcal {J}}}$ ${\displaystyle \mathbf {y} ^{j}=(y_{1}^{j},\ldots ,y_{n}^{j})\in \mathbb {R} ^{n}}$ ${\displaystyle Y_{j}\subset \mathbb {R} ^{J}}$
• The initial stocks of the respective goods are given by an equipment vector. We agree further than the equipment of one person (regarding all goods). ${\displaystyle \mathbf {e} =(e_{1},\ldots ,e_{n})\in \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {e} ^{i}=(e_{1}^{i},\ldots ,e_{n}^{i})\in \mathbb {R} ^{n}}$ ${\displaystyle i\in {\mathcal {I}}}$

Definition - economy

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

${\displaystyle \mathbf {\mathcal {E}} =\left[\left(X_{i},u^{i}\right)_{i\in {\mathcal {I}}},\left(Y_{j}\right)_{j\in {\mathcal {J}}},\mathbf {e} \right]}$

characterize.

Definition - Allocations and Permitted Allocations

An allocation vector in turn gives a concrete “state” of (with specific consumption and production vectors for each consumer or producer). Such an allocation is said to be permissible if it applies to every resource that the total amount consumed corresponds to the initial equipment plus the total amount produced, i.e. if ${\displaystyle n(I+J)}$ ${\displaystyle \left[\left(\mathbf {x} ^{i}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} }$

${\displaystyle \sum _{i=1}^{I}\mathbf {x} ^{i}=\mathbf {e} +\sum _{j=1}^{J}\mathbf {y} ^{j}}$

Definition - Pareto efficiency of allocations

Moreover, an allocation is Pareto-efficient if there is no way to redistribute resources between consumers in such a way that everyone has at least the same benefit, but at least one person even experiences an increase in benefit. Formally, the allocation is Pareto-efficient if and only if it is permissible (see above) and no other permissible allocation exists, so that for all and for certain . ${\displaystyle \left[\left(\mathbf {x} ^{i}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \left[\left(\mathbf {\tilde {x}} ^{i}\right)_{i\in {\mathcal {I}}},\left(\mathbf {\tilde {y}} ^{j}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle u^{i}\left(\mathbf {\tilde {x}} ^{i}\right)\geq u^{i}\left(\mathbf {x} ^{i}\right)}$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle u^{i}\left(\mathbf {\tilde {x}} ^{i}\right)>u^{i}\left(\mathbf {x} ^{i}\right)}$ ${\displaystyle i\in {\mathcal {I}}}$

Definition - competitive economy with private property

A special economy is now considered, namely a competitive system in which all companies (and their profits) represent private property, i.e. the profits are part of the aggregate consumption budget. Since it is a competitive economy, goods are also traded on a decentralized basis on competitive markets, with the market players acting as price takers: Consumers maximize their benefits, producers their profits. The assumption of private property results formally that the consumer's budget is made up of two components: on the one hand, a share of the initial equipment, and on the other hand, a share of the producers' profits. This share amounts to ( would be, for example, the share that person i can claim in the profits of producer 4). According to the requirements is and . Such an economy can then be called a tuple ${\displaystyle \mathbf {e} ^{i}\in \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {\theta } ^{i}\in \mathbb {R} ^{J}}$ ${\displaystyle \theta ^{i}=(\theta ^{i1},\ldots ,\theta ^{iJ})}$ ${\displaystyle \theta ^{i4}}$ ${\displaystyle \mathbf {e} =\sum _{i=1}^{I}\mathbf {e} ^{i}}$ ${\displaystyle \sum _{i=1}^{I}\mathbf {\theta } ^{i}=\mathbf {1} }$

${\displaystyle \mathbf {\mathcal {E}} ^{p}=\left[\left(X_{i},u^{i}\right)_{i\in {\mathcal {I}}},\left(Y_{j}\right)_{j\in {\mathcal {J}}},\left(\mathbf {e} ^{i},\mathbf {\theta } ^{i}\right)_{i\in {\mathcal {I}}}\right]}$

describe.

Definition - (price taker) competitive equilibrium for ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

For the competitive economy with private property , a competitive equilibrium is defined as a tuple ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} ^{*}\right]}$

to which the following properties apply:

1. The allocation is permitted in. ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$
2. Given the equilibrium market prices, every company maximizes its profit, that is, for all : for all . ${\displaystyle j\in {\mathcal {J}}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {y} ^{j}\leq \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$ ${\displaystyle \mathbf {y} ^{j}}$
3. Each person maximizes their utility, given the equilibrium market prices and their consumption budget. More precisely: Let be the set of all consumption vectors that meet the budget condition: ${\displaystyle {\mathcal {X}}}$ ${\displaystyle \mathbf {x} ^{i}}$

   ${\displaystyle {\mathcal {X}}=\left\{\mathbf {x} ^{i}\left|\mathbf {p} ^{*}\cdot \mathbf {x} ^{i}\leq \mathbf {p} ^{*}\cdot \mathbf {e} ^{i}+\sum _{j=1}^{J}\theta ^{ij}\cdot \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}\right.\right\}}$

Then it is and it applies: for everyone . ${\displaystyle \mathbf {x} ^{i*}\in {\mathcal {X}}}$ ${\displaystyle u\left(\mathbf {x} ^{i*}\right)\geq u\left(\mathbf {x} ^{i}\right)}$ ${\displaystyle \mathbf {x} ^{i}\in {\mathcal {X}}}$

Such a balance is known as Walrasian balance.

First law

First law of welfare economics: Let the considered economy. Let the order of preference underlying each individual utility function ( ) not be saturated locally (or, in the special case: strictly monotonic). Keep going ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$ ${\displaystyle u^{i}:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ ${\displaystyle i\in {\mathcal {I}}}$

${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} ^{*}\right]}$

a Walrasian balance.

Then the derived Walrasian equilibrium allocation is

${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$

for efficient Pareto. ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

A somewhat more general definition makes use of the concept of a quasi-equilibrium explained below; it then does not require a specific economy , as it does here , but applies in general. Please refer to a footnote for this. ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

Additional definitions

Definition - quasi (price taker) competitive equilibrium with transfers for ${\displaystyle \mathbf {\mathcal {E}} }$

We are expanding the narrow stipulations of Walrasian equilibrium again by transferring the equilibrium concept to the “more abstract” economy . The prerequisite for the (price-taker) competitive equilibrium is that each individual has only as much available as is made up of (the value) of his original goods and the proportionate corporate profits to which he is entitled. The equilibrium concept dealt with in this section knows another component of the determination of wealth: the transfer payment. This can practically be imagined, for example, as a one-off (positive or negative) tax through which a social planner “shifts” funds between consumers before competitive activity in the economy. ${\displaystyle \mathbf {\mathcal {E}} }$ ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

One now first defines a measure for the individual prosperity for all consumers. This is done using the vector . ${\displaystyle w^{i}\in \mathbb {R} }$ ${\displaystyle \mathbf {w} =(w^{i},\ldots ,w^{I})\in \mathbb {R} ^{I}}$

For the competitive economy then ${\displaystyle \mathbf {\mathcal {E}} }$

${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} ^{*}\right]}$

a quasi (price-taker) competitive equilibrium with transfers if and only if there is a tuple with such that: ${\displaystyle \mathbf {w} }$ ${\displaystyle \sum _{i=1}^{I}w^{i}=\mathbf {p} ^{*}\cdot \mathbf {e} +\sum _{j=1}^{J}\mathbf {p} ^{*}\cdot \mathbf {y} _{j}^{*}}$

1. The allocation is permitted in. ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$
2. Given the equilibrium market prices, every company maximizes its profit, that is, for all : for all . ${\displaystyle j\in {\mathcal {J}}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {y} ^{j}\leq \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$ ${\displaystyle \mathbf {y} ^{j}}$
3. Each person maximizes their utility, given the equilibrium market prices and their consumption budget. More precisely: Let be the set of all consumption vectors that meet the budget condition: ${\displaystyle {\mathcal {X}}}$ ${\displaystyle \mathbf {x} ^{i}}$

   ${\displaystyle {\mathcal {X}}=\left\{\mathbf {x} ^{i}\left|\mathbf {p} ^{*}\cdot \mathbf {x} ^{i}\leq w^{i}\right.\right\}}$

Then it is and it applies: for everyone . ${\displaystyle \mathbf {x} ^{i*}\in {\mathcal {X}}}$ ${\displaystyle u\left(\mathbf {x} ^{i*}\right)\geq u\left(\mathbf {x} ^{i}\right)}$ ${\displaystyle \mathbf {x} ^{i}\in {\mathcal {X}}}$

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

Second law

Second Law of Welfare Economics: Let the considered economy. Let the order of preference underlying each individual utility function ( ) be locally unsaturated (or, in the special case: strictly monotonic) and also be convex. Keep on being convex for everyone . ${\displaystyle \mathbf {\mathcal {E}} }$ ${\displaystyle u^{i}:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle Y_{j}}$ ${\displaystyle j\in {\mathcal {J}}}$

Then there is a price vector for every Pareto-optimal allocation , so that a quasi (price-taker) competitive equilibrium with transfers forms. ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {p} ^{*}\neq \mathbf {0} }$ ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} ^{*}\right]}$

proofs

Proof of the First Law

Proof by contradiction: Assume that the resulting from the price-takers-competitive equilibrium allocation for not is Pareto optimal. Then by definition there is a permissible allocation for with ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$ ${\displaystyle \left[\left({\tilde {\mathbf {x} }}^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {\tilde {y}} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

1. ${\displaystyle u^{i}(\mathbf {\tilde {x}} ^{i*})\geq u^{i}(\mathbf {x} ^{i*})}$for all i and
2. ${\displaystyle u^{i'}(\mathbf {\tilde {x}} ^{i'*})>u^{i'}(\mathbf {x} ^{i'*})}$for at least one individual . ${\displaystyle i'\in {\mathcal {I}}}$

It must be shown that such a permissible allocation does not exist. To do this, proceed step by step.

• a) Since the (price-taker) competitive equilibrium is for (with the budget), it must also apply that . (For if instead , there would be in an environment to one that strictly opposite is preferred [local non-saturation] and so also sufficient budget condition - but then would not the optimal consumption bundle, see item 3 in the definition of (Preisnehmer-). To put it bluntly, a Pareto-superior allocation must be too expensive , otherwise the Pareto-inferior allocation could not be balanced.) ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} ^{*}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {e} ^{i}+\sum _{j=1}^{J}\theta ^{ij}\cdot \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$ ${\displaystyle \mathbf {p} ^{*}\cdot {\tilde {\mathbf {x} }}^{i*}\geq \mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}}$
  ${\displaystyle \mathbf {p} ^{*}\cdot {\tilde {\mathbf {x} }}^{i*}<\mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}}$ ${\displaystyle \mathbf {\tilde {x}} ^{i*}}$ ${\displaystyle \mathbf {\tilde {\tilde {x}}} ^{i*}}$ ${\displaystyle \mathbf {\tilde {x}} ^{i*}}$ ${\displaystyle \mathbf {\tilde {x}} ^{i*}}$

• b) From 2. it follows that , because strictly prefers to . If it were the same size or even smaller than , it would certainly be chosen in equilibrium - in contradiction to property 1 in the definition of the (price-taker) competitive equilibrium . ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {\tilde {x}} ^{i'*}>\mathbf {p} ^{*}\cdot \mathbf {x} ^{i'*}}$ ${\displaystyle i'}$ ${\displaystyle \mathbf {\tilde {x}} ^{i'*}}$ ${\displaystyle \mathbf {x} ^{i'*}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {\tilde {x}} ^{i'*}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {x} ^{i'*}}$ ${\displaystyle \mathbf {\tilde {x}} ^{i'*}}$

• c) According to the prerequisite, for every producer j is the profit-maximizing production volume at the price , which is why it is necessary , because instead , property 2 in the definition of the (price-taker) competitive equilibrium would not be fulfilled. ${\displaystyle \mathbf {y} ^{j*}}$ ${\displaystyle \mathbf {p} ^{*}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}\geq \mathbf {p} ^{*}\cdot \mathbf {\tilde {y}} ^{j*}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}<\mathbf {p} ^{*}\cdot \mathbf {\tilde {y}} ^{j*}}$ ${\displaystyle \left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}}$

• d) Each person is in their given budget amount. ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}=\mathbf {p} ^{*}\cdot \mathbf {e} ^{i}+\sum _{j=1}^{J}\theta ^{ij}\cdot \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$

• e) Since according to the prerequisite and for all (see the definition of the competitive exchange economy), summing over the equation in d) yields that too ${\displaystyle \sum _{i=1}^{n}\mathbf {e} ^{i}=\mathbf {e} }$ ${\displaystyle \sum _{i=1}^{I}\mathbf {\theta } ^{ij}=1}$ ${\displaystyle j\in {\mathcal {J}}}$ ${\displaystyle \sum _{i=1}^{I}\mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}=\mathbf {p} ^{*}\cdot \mathbf {e} +\sum _{j=1}^{J}\mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$

• f) From c) and e) it follows that . ${\displaystyle \sum _{i=1}^{I}\mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}\geq \mathbf {p} ^{*}\cdot \mathbf {e} +\sum _{j=1}^{J}\mathbf {p} ^{*}\cdot \mathbf {\tilde {y}} ^{j*}}$

• g) a) and b) inserted in f) result ${\displaystyle \sum _{i=1}^{I}\mathbf {p} ^{*}\cdot \mathbf {\tilde {x}} ^{i*}>\mathbf {p} ^{*}\cdot \mathbf {e} +\sum _{j=1}^{J}\mathbf {p} ^{*}\cdot \mathbf {\tilde {y}} ^{j*}}$

However, according to the definition of the admissibility of allocations (see above), it follows from this that , contrary to the assumption, q is not admissible. e. d. ${\displaystyle \left[\left({\tilde {\mathbf {x} }}^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {\tilde {y}} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$

Proof of the Second Law

The following evidence largely follows the widespread evidence method from Mas-Colell / Whinston / Green 1995, p. 552 ff.

Be a crowd for everyone ${\displaystyle i\in {\mathcal {I}}}$

${\displaystyle V^{i}\equiv \left\{\mathbf {x} ^{i}\in \mathbb {R} ^{n}\left|u^{i}\left(\mathbf {x} ^{i}\right)>u^{i}\left(\mathbf {x} ^{i*}\right)\right.\right\}}$

defined (the upper contour set of or the set of all consumption vectors that create a higher utility than ). If one adds this amount over all i, one obtains ${\displaystyle \mathbf {x} ^{i*}}$ ${\displaystyle \mathbf {x} ^{i*}}$

${\displaystyle V=\sum _{i=1}^{I}V^{i}=\left\{\mathbf {x} \in \mathbb {R} ^{n}\left|\mathbf {x} ^{1}\in V^{1},\ldots ,\mathbf {x} ^{n}\in V^{n}\;{\textrm {mit}}\;\mathbf {x} =\sum _{i=1}^{I}\mathbf {x} ^{i}\right.\right\}}$,

that is, the set of all individual consumption plans (combined into a vector ) through which all individuals are strictly better off than with . Analog is a lot for everyone ${\displaystyle \mathbf {x} }$ ${\displaystyle \mathbf {x} ^{i*}}$ ${\displaystyle j\in {\mathcal {J}}}$

${\displaystyle Y=\sum _{j=1}^{J}\mathbf {y} ^{j}=\left\{\mathbf {y} \in \mathbb {R} ^{n}\left|\mathbf {y} ^{1}\in Y^{1},\ldots ,\mathbf {y} ^{n}\in Y^{n}\;{\textrm {mit}}\;\mathbf {y} =\sum _{j=1}^{J}\mathbf {y} ^{j}\right.\right\}}$

defines the set of all production plans at the macroeconomic level. This aggregated amount of production can be shifted by the equipment vector, whereby the aggregated amount of consumption possibilities, ${\displaystyle \{\mathbf {e} \}}$

${\displaystyle A=Y+\{\mathbf {e} \}}$,

receives.

• a) Each is a convex set . ${\displaystyle V^{i}}$

Be and both elements of . Then by definition and . Now be without loss of generality . The convexity property of the order of preference implies that for anything too . Since preference orders are also transitive , it is also true that . So is a convex set. ${\displaystyle \mathbf {\tilde {x}} ^{i}}$ ${\displaystyle \mathbf {\hat {x}} ^{i}}$ ${\displaystyle V^{i}}$ ${\displaystyle u^{i}\left(\mathbf {\tilde {x}} ^{i}\right)>u^{i}\left(\mathbf {x} ^{i*}\right)}$ ${\displaystyle u^{i}\left(\mathbf {\hat {x}} ^{i}\right)>u^{i}\left(\mathbf {x} ^{i*}\right)}$ ${\displaystyle u^{i}\left(\mathbf {\tilde {x}} ^{i}\right)\geq u^{i}\left(\mathbf {\hat {x}} ^{i}\right)}$ ${\displaystyle \lambda \in [0,1]}$ ${\displaystyle u^{i}\left[\lambda \mathbf {\tilde {x}} ^{i}+(1-\lambda )\mathbf {\hat {x}} ^{i}\right]\geq u^{i}\left(\mathbf {\hat {x}} ^{i}\right)}$ ${\displaystyle u^{i}\left[\lambda \mathbf {\tilde {x}} ^{i}+(1-\lambda )\mathbf {\hat {x}} ^{i}\right]>u^{i}\left(\mathbf {x} ^{i*}\right)}$ ${\displaystyle V^{i}}$

• b) V is convex as the sum of convex sets, just like Y, even after shifting to A. ${\displaystyle \mathbf {e} }$
• c) It is . ${\displaystyle V\cap A=\emptyset }$

Because of the Pareto optimality of the allocation (according to the prerequisite), (there must be no possible "offer" that is also contained in V , otherwise there would be an allocation that can be produced with the given technology and equipment and would be preferred to the other; but the initial allocation might not even be Pareto-optimal). ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle V\cap A=\emptyset }$

• d) There is a price vector and a such that 1. for all and 2. for all . ${\displaystyle \mathbf {p} ^{*}\in \mathbb {R} ^{n}\setminus \{\mathbf {0} \}}$ ${\displaystyle r\in \mathbb {R} }$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {z} \geq r}$ ${\displaystyle \mathbf {z} \in V}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {z} \leq r}$ ${\displaystyle \mathbf {z} \in A}$

This follows from a going back to Minkowski version of separation set of disjoint convex sets in the normalized space by real affine hyperplanes , after disjoint non-empty and convex for two subsets of , A and B, is that a hyperplane exists, the A of B separates, that is, there exists one such that , or, to put it another way, there exists a non-empty one and one such that it applies to all and to all that . (For the proof of the theorem cf. abbreviated Mas-Colell 1995, p. 948 and completely Moore 1999, p. 297 ff.) ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {p} ^{*}\in \mathbb {R} ^{n}\setminus \{\mathbf {0} \}}$ ${\displaystyle \sup _{\mathbf {a} \in A}\mathbf {p} ^{*}\cdot \mathbf {a} \leq \inf _{\mathbf {b} \in B}\mathbf {p} ^{*}\cdot \mathbf {b} }$ ${\displaystyle \mathbf {p} ^{*}\in \mathbb {R} ^{n}}$ ${\displaystyle \alpha \in \mathbb {R} }$ ${\displaystyle \mathbf {a} \in A}$ ${\displaystyle \mathbf {b} \in B}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {a} \leq \alpha \leq \mathbf {p} ^{*}\cdot \mathbf {b} }$

• e) If for all i, then also . ${\displaystyle u^{i}\left(\mathbf {x} ^{i}\right)\geq u^{i}\left(\mathbf {x} ^{i*}\right)}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{i=1}^{I}\mathbf {x} ^{i}\right)\geq r}$

For non-saturation because there are for each consumer a consumption bundle in an arbitrarily small neighborhood around , with the and so . Consequently, now also and therefore according to Minkowski's theorem , so that with for all i in the limit value too . ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle {\tilde {\mathbf {x} }}^{i}}$ ${\displaystyle \mathbf {x} ^{i}}$ ${\displaystyle u^{i}\left({\tilde {\mathbf {x} }}^{i}\right)>u^{i}\left(\mathbf {x} ^{i}\right)}$ ${\displaystyle {\tilde {\mathbf {x} }}^{i}\in V^{i}}$ ${\displaystyle \mathbf {\tilde {x}} \in V}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{i=1}^{I}\mathbf {\tilde {x}} ^{i}\right)\geq r}$ ${\displaystyle {\tilde {\mathbf {x} }}^{i}\rightarrow \mathbf {x} ^{i}}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{i=1}^{I}\mathbf {x} ^{i}\right)\geq r}$

• f) It applies . ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{j=1}^{J}\mathbf {y} ^{j*}+\mathbf {e} \right)=r}$

According to the previous point is . At the same time, however, is ( ) and so . It follows that and because of too . ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{i=1}^{I}\mathbf {x} ^{i*}\right)\geq r}$ ${\displaystyle \sum _{i=1}^{I}\mathbf {x} ^{i*}=\sum _{j=1}^{J}\mathbf {y} ^{j*}+\mathbf {e} }$ ${\displaystyle \in A}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{i=1}^{I}\mathbf {x} ^{i*}\right)\leq r}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{i=1}^{I}\mathbf {x} ^{i*}\right)=r}$ ${\displaystyle \sum _{i=1}^{I}\mathbf {x} ^{i*}=\sum _{j=1}^{J}\mathbf {y} ^{j*}+\mathbf {e} }$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{j=1}^{J}\mathbf {y} ^{j*}+\mathbf {e} \right)=r}$

• g) It applies . ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {y} ^{j}\leq \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$

It is for everyone . For each and for all , that is true . Together this implies , so too . From this it follows in turn for all and for all that . ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {y} ^{j}\leq \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$ ${\displaystyle \mathbf {y} ^{j}\in Y^{j}}$ ${\displaystyle j\in J}$ ${\displaystyle \mathbf {y} ^{j}\in Y^{j}}$ ${\displaystyle \left(\mathbf {y} ^{j}+\sum _{k\neq j}\mathbf {y} ^{k*}\right)\in Y}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\mathbf {y} ^{j}+\sum _{k\neq j}\mathbf {y} ^{k*}+\mathbf {e} \right)\leq r=\mathbf {p} ^{*}\cdot \left(\sum _{j=1}^{J}\mathbf {y} ^{j*}\right)}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\mathbf {y} ^{j}+\sum _{k\neq j}\mathbf {y} ^{k*}\right)\leq r=\mathbf {p} ^{*}\cdot \left(\mathbf {y} ^{j*}+\sum _{j=1}^{J}\mathbf {y} ^{j*}\right)}$ ${\displaystyle \mathbf {y} ^{j}\in Y^{j}}$ ${\displaystyle j\in J}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {y} ^{j}\leq \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$

• h) If so , then too . ${\displaystyle u^{i}\left(\mathbf {x} ^{i}\right)>u^{i}\left(\mathbf {x} ^{i*}\right)}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {x} ^{i}\geq \mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}}$

If so , then according to e) and f) also and therefore . ${\displaystyle u^{i}\left(\mathbf {x} ^{i}\right)>u^{i}\left(\mathbf {x} ^{i*}\right)}$ ${\displaystyle \mathbf {p} ^{*}\cdot \left(\sum _{k\neq i}\mathbf {x} ^{k*}+\mathbf {x} ^{i}\right)\geq r=\mathbf {p} ^{*}\cdot \left(\sum _{k\neq i}\mathbf {x} ^{k*}+\mathbf {x} ^{i*}\right)}$ ${\displaystyle \mathbf {p} ^{*}\cdot \mathbf {x} ^{i}\geq \mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}}$

Since it is also admissible according to the prerequisite, the choice of (for all ) guarantees the existence of the quasi (price-taker) competitive equilibrium with transfers , q. e. d. ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle w^{i}=\mathbf {p} ^{*}\cdot \mathbf {x} ^{i*}}$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} ^{*}\right]}$

literature

• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-19-507340-1 , chapter 16.
• Allan M. Feldman: Welfare Economics. In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, Internet http://www.dictionaryofeconomics.com/article?id=pde2008_W000050&edition=current (online edition).
• James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin a. a. 2007, ISBN 978-3-540-31407-3 (also as an e-book: doi : 10.1007 / 978-3-540-32223-8 ).
• Hal Varian : Intermediate Microeconomics. A modern approach. 8th edition WW Norton, New York and London 2010, ISBN 978-0-393-93424-3 , chapter 31. (For a simple "Edgeworth" economy, see Edgeworth box . )

Web links

• Michael Powell: Econ 201C: General Equilibrium and Welfare Economics ( Memento from January 6, 2009 in the Internet Archive ) (PDF file). Lecture notes. ( Archived version ( Memento from January 21, 2014 in the Internet Archive ) from January 21, 2014)

Remarks

1. ↑ See Feldmann 2008.
2. ^ Similar to Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 , p. 212.
3. ↑ Simplified to Mas-Colell / Whinston 1995, pp. 546-549.
4. ↑ Let the considered economy. Let the order of preference underlying each individual utility function ( ) not be saturated locally (or, in the special case: strictly monotonic). Go on: a quasi (price taker) competitive equilibrium with transfers. Then the equilibrium allocation derived from this is: for Pareto-efficient. See Mas-Colell / Whinston / Green 1995, pp. 549 f. ${\displaystyle \mathbf {\mathcal {E}} }$ ${\displaystyle u^{i}:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} ^{*}\right]}$ ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} }$
5. ↑ According to the definition , there is what obviously fulfills the above requirement . ${\displaystyle w^{i}=\mathbf {p} ^{*}\cdot \mathbf {e} ^{i}+\sum _{j=1}^{J}\theta ^{ij}\cdot \mathbf {p} ^{*}\cdot \mathbf {y} ^{j*}}$ ${\displaystyle \sum _{i=1}^{I}w^{i}}$
6. ↑ See Mas-Colell / Whinston / Green 1995, p. 552; Moore 2007, p. 213.
7. ↑ The evidence follows the step-by-step evidence procedure in Sam Bucovetsky: General Equilibrium and Welfare (Chapter 16). Lecture notes. Internet archive link ( Memento of the original dated February 2, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF file). The concrete procedure itself is based on Mas-Colell / Whinston 1995, p. 549 f., Where, however, a slightly generalized version is proven and partial steps are left out. The idea of ​​proving contradiction goes back to Kenneth Arrow : An extension of the basic theorems of classical Welfare Economics. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California, Berkeley in 1951. See Alexandre B. Cunha. ' Working paper. P. 1 and Feldmann 2008. 
8. ↑ A set V is convex if and . ${\displaystyle \mathbf {a} ,\mathbf {b} \in V}$ ${\displaystyle \lambda \in [0,1]}$ ${\displaystyle \Rightarrow }$ ${\displaystyle \lambda \mathbf {a} +(1-\lambda )\mathbf {b} \in V}$
9. ↑ See Knut Sydsaeter, Arne Strøm and Peter Berck: Economists' mathematical manual. 4th edition. Springer, Berlin a. a. 2005, ISBN 978-3-540-26088-2 (also as e-book: doi : 10.1007 / 3-540-28518-0 ), p. 90.
10. ↑ See Mas-Colell 1995, p. 948 and James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin a. a. 2007, ISBN 978-3-540-31407-3 (also as e-book: doi : 10.1007 / 978-3-540-32223-8 ), p. 172.
11. ↑ James C. Moore: Mathematical methods for economic theory. Vol. 1. Springer, Berlin a. a. 1995, ISBN 3-540-66235-9 .