By Marco Lehmann-Waffenschmidt

ISBN-10: 3540686622

ISBN-13: 9783540686620

ISBN-10: 3540686649

ISBN-13: 9783540686644

This paintings makes use of numerous version frameworks to check the evolution of equilibria in an open loop evolving financial system within which the version features evolve with none directional regulations with the exception of continuity. employing mathematical equipment, it really is proven that equilibria can regularly be tailored in a piecewise slow, non bang-bang manner.

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**Extra info for Economic evolution and equilibrium**

**Example text**

This means that pk ∈ V ζ for all k ≥ max(k(h), (h)). In other words, there is a tail (pk ) which is in V ζ , and this proves the compactness of K ζ = (V ζ )co . In the next step we extend the mapping fζ to the boundary ∂Δn−1 in a continuous way so that it remains unchanged on K ζ and thus particularly ˚n−1 the number also on ζ −1 (0n ). Since K ζ is a compact subset of Δ dist (K ζ , ∂Δn−1 ) =: γζ is well–deﬁned and positive. Consider the inscribed closed simplex Δn−1 with ζ ζ ζ = (5/6)γζ which still contains K .

In complete analogy to Subsection (ii) one sees that the one-parametrization (gs )s∈[0,1] leads to the continuous one-parametrization (r ◦ gs )s∈[0,1] : n n n T × [0, 1] −→ T of equilibrium equivalent self–mappings of T . From the considerations in (ii) follows directly that n,1−γ ∪s∈[0,1] F ix (gs ) = ∪s∈[0,1] F ix (gs ) = ∪s∈[0,1] F ix (r ◦ gs ) ⊂ T γ × [0, 1]. Consequently, ∪s∈[0,1] ζs−1 (0) = ∪s∈[0,1] ϕ(F ix (gs )) = ∪s∈[0,1] ϕ(F ix (r ◦ gs )) n,1−γ is contained in the compact subspace ϕ(T γ have achieved our aim.

8, pp. 235-241). While Mas-Colell, however, uses individual preferences and endowments as primitives, here we basically will follow the lines of Dierker (1974, Chapter 12) who directly builds on demand functions (see also Hildenbrand 1974). Below we will discuss these diﬀerences in greater detail. The space of characteristics of the agents is given by D0 × (Rn+ \{0n }). D0 denotes the set of continuous individual demand functions ˚n−1 × R++ → Rn+ f :Δ ˚n−1 × R++ , and a which satisfy Walras’ law: p · f (p, w) = w for all (p, w) ∈ Δ slightly strengthened desirability condition: ˚n−1 × R++ which converges to a pair for any sequence (pk , wk ) in Δ n−1 (p, w) ∈ ∂Δ × R++ the norm f (pk , wk ) of the demand vectors grows beyond all ﬁnite bounds.

### Economic evolution and equilibrium by Marco Lehmann-Waffenschmidt

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