Sunshine-Mantel-Debreu theorem

In microeconomics and especially in the theory of general equilibrium, the Sonnenschein-Mantel-Debreu theorem is a proposition that goes back to Hugo Sonnenschein, Rolf Mantel and Gérard Debreu . It simply means that the aggregated excess demand functions , which belong to a model of general equilibrium constructed with common assumptions, only have a few specific, general properties, but otherwise no concrete statements about their shape are possible.

classification

After a large number of existential sentences for Walrasian (i.e. competitive) equilibria had been formulated, especially from the mid-1950s, starting with Arrow and Debreu (1954), and it had also been shown on various occasions that these models regularly at most finitely many equilibria the question arose whether further conclusions about the nature of the resulting equilibrium can be derived from the parameters underlying the economy. In particular, this problem takes on the following form in an equilibrium analysis using excess demand functions: Which properties can be derived from the pure exchange economy obeying common assumptions

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

to characterize the aggregated excess demand

${\displaystyle \mathbf {z} (\mathbf {p} )\equiv \sum _{i=1}^{I}\mathbf {z} ^{i}(\mathbf {p} )=\sum _{i=1}^{I}\left[\mathbf {x} ^{i}(\mathbf {p} ,\mathbf {p} \cdot \mathbf {e} ^{i})-\mathbf {e} ^{i}\right]}$

derive? Here, the vector-valued Marshallian demand of a consumer describes his excess demand, his initial equipment and his utility function. ${\displaystyle \mathbf {x} ^{i}}$ ${\displaystyle \mathbf {z} ^{i}}$ ${\displaystyle \mathbf {e} ^{i}}$ ${\displaystyle u^{i}}$

Various general properties of the aggregated excess demand can be derived from common assumptions, as the following lemma shows.

Let the I -element set of all consumers give their utility functions continuous, strictly quasi-concave and non-falling . Let an initial configuration be given for each consumer (see footnote for an explanation of the spelling). Then: ${\displaystyle {\mathcal {I}}=\{1,\ldots ,I\}}$ ${\displaystyle u^{i}(\mathbf {x} ^{i})\in \mathbb {R} }$ ${\displaystyle \mathbf {e} ^{i}=(e_{1}^{i},\ldots ,e_{n}^{i})\in \mathbb {R} ^{n}}$

1. The individual excess demands are in each case steady in , homogeneous from grade zero in and satisfy the Walras law, that is, it applies to all . ${\displaystyle \mathbf {z} ^{i}(\mathbf {p} )}$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {p} \cdot \mathbf {z} ^{i}(\mathbf {p} )=0}$ ${\displaystyle \mathbf {p} }$
2. The aggregated excess demand is steady in , homogeneous from grade zero in and satisfies Walras' law, that is, it applies to all . ${\displaystyle \mathbf {z} (\mathbf {p} )}$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {p} \cdot \mathbf {z} (\mathbf {p} )=0}$ ${\displaystyle \mathbf {p} }$

The Sonnenschein-Mantel-Debreu theorem simply states that it is not possible to derive further properties of the aggregated excess demand without making more restrictive assumptions.

theorem

Theorem: be continuous in , homogeneous of degree zero in and satisfy Walras' law. Then for every k there are consumers with a continuous, strictly quasi-concave and non-falling utility function and an equipment vector , whose excess demand function is, for all prices for which it is true that for all . ${\displaystyle \mathbf {z} (\mathbf {p} )}$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \mathbf {z} (\mathbf {p} )}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \mathbf {e} ^{i}=(e_{1}^{i},\ldots ,e_{n}^{i})\in \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {z} (\mathbf {p} )}$ ${\displaystyle \mathbf {p} }$ ${\displaystyle p_{i}/\left\Vert \mathbf {p} \right\Vert \geq 0}$ ${\displaystyle i\in {\mathcal {I}}}$

This is both a generalization and an adaptation to the Arrow-Debreu framework , which is based on an earlier proof by Mantel (1976), which in turn goes back to the preliminary work of Sonnenschein (1973). Mantel's version of the theorem was as follows (with H being the set of all for which that holds , and with the set of all that generate non-negative costs for any price vectors, i.e. ): ${\displaystyle \mathbf {p} \in \mathbb {R} ^{n}}$ ${\displaystyle \sum _{i=1}^{n}p_{i}=1}$ ${\displaystyle P^{*}}$ ${\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {p} \in P}$ ${\displaystyle \mathbf {p} \cdot \mathbf {x} \geq 0}$

Theorem (Mantel 1976): Be compact and convex . Let further be a twice continuously partially differentiable function and let the Walras law apply to all , that is . Also , be,, independent vectors. Then there is a real and a convex cone and thus n unsaturated consumers with a strictly concave, homogeneous utility function and initial endowments , whose individual surplus demands add up to P over Z. ${\displaystyle P\subset H}$ ${\displaystyle z:P\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {p} \in P}$ ${\displaystyle \mathbf {p} \cdot \mathbf {z} (\mathbf {p} )=0}$ ${\displaystyle \mathbf {w} _{j}\in P^{*}}$ ${\displaystyle j=1,\ldots ,n}$ ${\displaystyle k>0}$ ${\displaystyle X\subset P^{*}}$ ${\displaystyle u^{i}:X\rightarrow \mathbb {R} }$ ${\displaystyle k\cdot \mathbf {w} _{i}}$

literature

• William DA Bryant: General equilibrium. Theory and evidence. World Scientific, Hackensack 2010, ISBN 978-981-281-834-8 (e-book: ISBN 978-981-281-835-5 ).
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .
• David M. Kreps: Microeconomic Foundations I. Choice and Competitive Markets Princeton University Press, Princeton 2012, ISBN 978-0-691-15583-8 .
• James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin a. a. 2007, ISBN 978-3-540-31407-3 (also online: doi : 10.1007 / 978-3-540-32223-8 ).
• Wayne Shafer and Hugo Sonnenschein: Market demand and excess demand functions. In: Kenneth J. Arrow and Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Vol. 2. North Holland, Amsterdam 1982, ISBN 978-0-444-86127-6 , pp. 671-693 (also online: doi : 10.1016 / S1573-4382 (82) 02009-8 ).

Individual evidence

1. ↑ Kenneth J. Arrow and Gerard Debreu: Existence of an Equilibrium for a Competitive Economy. In: Econometrica. 22, No. 3, 1954, pp. 265-290 ( JSTOR 1907353 ).
2. ↑ On this, for example, Gerard Debreu: Existence of general equilibrium. In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, doi : 10.1057 / 9780230226203.0523 (online edition) for an overview-like historical classification, and Ders .: Existence of competitive equilibrium. In: Kenneth J. Arrow and Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Vol. 2. North Holland, Amsterdam 1982, ISBN 978-0-444-86127-6 , pp. 697-743 (also online: doi : 10.1016 / S1573-4382 (82) 02010-4 ) for a formal presentation of several different Models.
3. ↑ Gerard Debreu: Existence of competitive equilibrium. In: Kenneth J. Arrow and Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Vol. 2. North Holland, Amsterdam 1982, ISBN 978-0-444-86127-6 , pp. 697-743 (also online: doi : 10.1016 / S1573-4382 (82) 02010-4 ).
4. ↑ ^{a b} To explain this definition: Consider an economy made up of n markets. The prices on these markets are summarized in a price vector , with . In economics there are still consumers, for whom the index quantities (the quantity of all consumers) are defined. A person's consumption profile is - it provides information about the amount of each of the n goods that person i consumes. The preference structure of each individual is in turn expressed in its utility function . The initial stocks of the respective goods are given by an equipment vector. is the equipment of a person (with regard to all goods). ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$ ${\displaystyle \mathbf {p} \neq \mathbf {0} }$ ${\displaystyle I}$ ${\displaystyle {\mathcal {I}}=\{1,\ldots ,I\}}$
   ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle \mathbf {x} ^{i}=(x_{1}^{i},\ldots ,x_{n}^{i})\in \mathbb {R} ^{n}}$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle u^{i}=u^{i}(\mathbf {x} ^{i})\in \mathbb {R} }$
   ${\displaystyle \mathbf {e} =(e_{1},\ldots ,e_{n})\in \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {e} ^{i}=(e_{1}^{i},\ldots ,e_{n}^{i})\in \mathbb {R} ^{n}}$ ${\displaystyle i\in {\mathcal {I}}}$
5. ↑ See Kreps 2012, p. 316; Mas-Colell / Whinston / Green 1995, p. 581 f. Here modified because of the economy not defined on the basis of the preference-indifference relation R ; note for understanding that is continuous R is continuous (another theorem by Debreu, cf. e.g. Kreps 2012, p. 35); further is strictly quasi-concave R is strictly convex; furthermore is strictly monotonically increasing R is strictly monotonous. Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd edition Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 , p. 17. The continuity of the utility function assumed here could be replaced by the weaker assumption that R is unsaturation . ${\displaystyle u(\mathbf {x} )}$ ${\displaystyle \Rightarrow }$ ${\displaystyle u(\mathbf {x} )}$ ${\displaystyle \Leftrightarrow }$ ${\displaystyle u(\mathbf {x} )}$ ${\displaystyle \Leftrightarrow }$
6. ↑ See Kreps 2012, p. 317; for the deviations see above. Similar to Mas-Colell / Whinston / Green 1995, p. 602. The theorem itself follows Gerard Debreu: Excess demand functions. In: Journal of Mathematical Economics. 1, No. 1, 1974, pp. 15-21, doi : 10.1016 / 0304-4068 (74) 90032-9 .
7. ^ Rolf R. Mantel: Homothetic preferences and community excess demand functions. In: Journal of Economic Theory. 12, No. 2, 1976, pp. 197-201, doi : 10.1016 / 0022-0531 (76) 90073-9 .
8. ^ Hugo F. Sonnenschein: Do Walras' identity and continuity characterize the class of community excess demand functions? In: Journal of Economic Theory. 6, No. 4, 1973, pp. 345-354, doi : 10.1016 / 0022-0531 (73) 90066-5 .
9. ↑ So a function whose all partial derivatives of the first and second order are continuous.