Economic Evolution and Equilibrium: Bridging the Gap by Marco Lehmann-Waffenschmidt PDF

By Marco Lehmann-Waffenschmidt

This paintings makes use of a variety of version frameworks to review the evolution of equilibria in an open loop evolving financial system within which the version features evolve with none directional regulations aside from continuity. using mathematical tools, it's proven that equilibria can constantly be tailored in a piecewise sluggish, non bang-bang manner.

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Additional info for Economic Evolution and Equilibrium: Bridging the Gap (Lecture Notes in Economics and Mathematical Systems)

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E. Thus, agent i’s individual demand set for any given price vector p ∈ S++ l gi (p) = {x ∈ R+ |px ≤ pωi and ∀y∈Rl+ py ≤ pωi implies y i x}, is a singleton. Given a continuous and monotone preference relation on Rl the closed upper contour set, or indifference-or-preference set, of any point x ∈ Rl is denoted by ψ(x) := {y ∈ Rl |x y}. Given an l-tuple of pairs ( i , ωi )li=1 with i ∈ P 0mo and ωi ∈ Rl++ for all sco i, . . , l the function l−1 fi : S++ −→ Rl p → {x ∈ Rl+ |px ≤ pωi , and py ≤ pωi for any y ∈ Rl+ implies y i x} − ωi = gi (p) − wi is the individual excess demand function derived from ( i , ωi ), and l fi (p) f (p) := i=1 is the market excess demand function derived from the l-consumer exchange economy ( i , ωi )li=1 (cf.

The equilibrium set is given by the zero set ζ −1 (0n ) = of ζ. Consider now the derived mapping n −1 i=1 ζi (0) ˚n−1 → Rn fζ : Δ ⎛ ⎞ p1 + p1 ζ1 (p) ⎜ ⎟ .. p→⎝ ⎠ . Pn + pn ζ(p) Clearly, ζ −1 (0n ) = Fix (fζ ). Note that Walras’ law and boundedness from below imply that pi ζi (p) is bounded from above for i = 1, . . , n. Thus fζ is bounded from below and from above as well. This will become important later. Due to Walras’ law the image of fζ lies in the affine hyperplane H in Rn through Δn−1 . For h ∈ {1, .

Iii) Likewise it is entirely straightforward to check that an evolution of large exchange economies induces a continuous one-parametrization of equilibrium equivalent self–mappings. The composition ×(ζs )s∈[0,1] ζ tr ˚n−1 × [0, 1] idΔ˚n−1 −→ ˚n−1 × MT → ˚n−1 → Δ Δ MT × Δ Rn (p, s) → (p, μs )} → (μs , p) → ζμs (p) clearly is continuous and satifies the uniformized desirability condition as exchange-II–evolutions do it. The latter is due to the additional boundary assumption on the individual demand functions of D0 .

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