Abstract
This chapter shows that there is a natural extension of the conventional price tâtonnement procedure of pure exchange economies which significantly increases the stability properties of such adjustment processes from a local as well as a global point of view. This extension is motivated by the observation that market price adjustments should not only depend on the levels of excess demand, but also on their direction (and magnitude) of change. Taking these additional forces appropriately into account implies an adjustment process which is formally similar to the Generalized Newton Methods which have been construed in the search for price mechanisms that are ‘universally stable’. Furthermore, this adjustment process also generalizes the stability proof for the ordinary tâtonnement procedure in the case of gross substitutes (by means of a suitable Liapunov function) in a straightforward way.
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Notes
- 1.
See also Ingrao and Israel (1990) for a recent survey on the dynamic properties of the ‘invisible hand’.
- 2.
With finitely many commodities and a continuum of individuals.
- 3.
- 4.
cf. Hildenbrand (1989) for a detailed description of this ‘Law’.
- 5.
- 6.
See Kaldor (1940) for an interesting dynamic exploitation of the non-uniqueness of equilibria.
- 7.
See Flaschel (1991) with respect to possible modifications for production economies and an alternative analysis of the local stability properties of such an extension.
- 8.
See the next section for a brief explanation of the employed notation.
- 9.
See Arrow and Hahn (1971) and Hahn (1982) for details and various special stability results that exist for such ‘numéraire processes’ and note here that variable adjustment speeds – as they are assumed by these authors – make the use of the conventional numéraire p ⋅p = 1 much less plausible.
- 10.
See Jordan (1983) in particular.
- 11.
We shall neglect here their adjustment rule for boundary values of \({\mathcal{R}}_{++}^{n}\) for simplicity.
- 12.
Note here, that they, too, employ a numéraire tâtonnement approach – due to the fact that they use adjustment functions G i instead of fixed parameters for the various components of the excess demand function.
- 13.
or by a suitable redefinition of the function X.
- 14.
See Flaschel (1991) for details in this matter.
- 15.
Note that this regularity condition must hold true for all sufficiently large parameter values g i . This can be shown by means of continuity arguments applied to the expression \(\det (< g >)\det (< g {> }^{-1}I - X'({p}^{{_\ast}}))\) – due to the assumed regularity of the matrix X′ at each of the equilibrium points of X.
- 16.
cf. for example Dieudonné (1960, pp. 268/9).
- 17.
cf. Hirsch and Smale (1974, p. 193).
- 18.
cf. Dieudonné (1960, p. 155).
- 19.
for suitable choices of the parameters g i for each excess demand function X, see Flaschel (1991) for details.
- 20.
An obvious example for a too demanding definition is Jordan’s (1983, p. 253) concept of a market mechanism which keeps markets in equilibrium once they have reached it – independent of what happens in the other markets. If this definition were economically sensible it would exclude our proposal for a price mechanism – and indeed any derivative feedback mechanism – from the set of economically meaningful adjustment processes.
- 21.
We shall see in the proof of this proposition that (13.6) is well-defined on \({\mathcal{R}}_{++}^{n}\). The invariance of \({\mathcal{R}}_{++}^{n}\) can then be obtained from the condition that \(\|X({p}^{n})\| \rightarrow \infty \) if the sequence p n approaches the boundary of \({\mathcal{R}}_{++}^{n}\).
- 22.
Note that these expression are in addition equal to \(\ll X(p) - X({p}^{{_\ast}}),p - {p}^{{_\ast}}\gg =\ll X(P) - X({P}^{{_\ast}}),P - {P}^{{_\ast}}\gg \) and are thus related to the notion of monotonicity.
- 23.
See Fig. 13.1 for an example where such an upper bound will exist for the whole positive orthant \({\mathcal{R}}_{++}^{n}\) and not only for sets C of the above kind and note that we had k= 0 in the preceding proposition.
- 24.
ε > 0, since C is a compact set which contains no other equilibrium of X.
- 25.
cf. Hirsch and Smale (1974, p. 190) for a description of this intuitive concept.
- 26.
not necessarily by increasing it further!
- 27.
When it is believed that it can, however remotely, mimic what goes on in actual markets, see Arrow and Hahn (1971, p. 265)!
- 28.
See, however, Fig. 13.4 for a simple, more elaborate choice of adjustment speeds.
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Flaschel, P. (2010). Dressing the Emperor in a New Dynamic Outfit. In: Topics in Classical Micro- and Macroeconomics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00324-0_13
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