Abstract
Since the early investigations of Arrow et al. in the late fifties and early sixties with their optimistic views on the stability properties of general equilibrium systems, there have been numerous contributions which have favored the opposite view – due to the counterexamples found shortly afterwards and due to the theorems of ‘Debreu-Sonnenschein’ type which have been formulated in the sequel. A typical statement in this regard is that of Frank Hahn (1970, p. 2): ‘What has been achieved is a collection of sufficient conditions, one might almost say, anecdotes, and a demonstration by Scarf and later by Gale, that not much more could be hoped for.1, 2 Similar statements can be found in a variety of publications on the stability issue, cf. for example Dierker (1974, p. 115) and in particular the recent survey article of Kirman (1989)3. This latter article also discusses one important possible route of escape from (non-uniqueness and) instability in the context of general equilibrium models, i.e., the approach by Hildenbrand and others who introduce further restrictions on the distribution of the characteristics of economic agents in order to avoid the above problems.
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Notes
- 1.
See, however, Keenan (1990) for a quite new attempt to obtain global stability for pure price mechanisms on the basis of a Morishima-type sign pattern of the Jacobian of the excess demand function (for all prices).
- 2.
I thank Dierker and Saari for helpful comments during the period when this chapter took shape. Of course, usual caveats apply.
- 3.
cf. also Kirman (1988).
- 4.
It has also been noticed recently, that there exists a close relationship between Walras’ price-quantity tâtonnement process for production economies and the stability analysis for so-called classical long-term positions, cf. in particular Duménil and Lévy (1989), Goodwin (1989), e.g., Essay 1, and Flaschel and Semmler (1987) for such observations. This (formal) similarity in the type of price-quantity adjustment considered by Walras’ and the Classics allows that results which have been obtained with respect to one approach may be applicable to the other approach if the differences in their concepts of ‘equilibrium’ are taken into account in an appropriate way. In the present article we shall study this cross-dual price-quantity adjustment process within the framework of Walrasian equilibrium analysis. [For the alternative approach the reader is referred to Flaschel and Semmler (1987, 1988)].
- 5.
cf. e.g., Flaschel and Semmler (1987).
- 6.
i.e., fixed capital and the like is here excluded from consideration.
- 7.
Profits π are neglected in Mas-Colell’s disequilibrium investigation of this basic situation.
- 8.
Note here, that our stability results can also be applied to such points of rest (which are no equilibria, since profits are at minimum here). These points of rest are of importance in the literature on public utilities and marginal cost pricing.
- 9.
The second of the above two matrices will in general be of the form:
$$\left (\begin{array}{cc} A & - B\\ B' & C \end{array} \right ),$$cf. Mas-Colell (1986, p. 55) for further details. Note also, that we will make use of subscripts in order to denote partial derivatives in the following.
- 10.
- 11.
which in this case is independent of adjustment speeds (D-stability) because of the negative quasi-definiteness of the matrix J. In the case d p > 0 we will have a unique bifurcation point with regard to the parameter β for any α – instead of D-stability – which separates stable from unstable spirals.
- 12.
In the case of d p > 0 the second adjustment coefficient β must be chosen sufficiently large in order to obtain local asymptotic stability (if l′(y) > 0 holds true).
- 13.
Note that we have neglected – as in Mas-Colell (1986, pp. 64/4) – the influence of π on d in our Fig. 14.2. Due to this fact we could make use of ‘Debreu-Sonnenschein’ theorems on the arbitrary nature of demand functions as they are formulated in the context of pure exchange economies, cf. Dierker (1974, pp. 56 ff.) for example and also Kirman (1989). Note further, that there can be no equilibrium in the presence of increasing returns due to the neoclassical assumption of a pure price-taker behavior.
- 14.
- 15.
Such an observation quite naturally gives rise to the question of global stability (treated in Flaschel (1990) for stable as well as unstable types of equilibria), since it appears as plausible that adjustment speeds γ should not be chosen to large in order to make sense economically.
- 16.
The proposed dynamics is therefore also of interest for studies of marginal cost pricing in the presence of increasing returns to scale.
- 17.
- 18.
Note here, that the Jacobian at points of rest in the lower part of Fig. 14.3 is given by \(\frac{X'({p}^{{_\ast}})} {1-\gamma X'({p}^{{_\ast}})}\) which must be negative for each γ larger than \(\frac{1} {X'({p}^{{_\ast}})}\).
- 19.
See Kamiya (1989) for a recent example of this kind.
- 20.
The matrix of adjustment coefficients [see (14.12)] is now suppressed in the above presentation by an appropriate choice of the function F.
- 21.
Ω denotes the set of regular n ×n matrices l : detl≠0.
- 22.
See, e.g., Zurmühl (1964) for a list of such norms.
- 23.
See also Woods (1978, Sect. 8.5) for remarks on the limitations of this approach.
- 24.
A formulation of the general kind: γ(F, F′) – together with appropriate assumptions on such a function γ – may also be of use in developing this matter further.
- 25.
Note here, that part of the results of this section will depend on the particularly simple economy chosen to discuss these matters (cf. Flaschel and Semmler (1987) for more general investigations).
- 26.
In the present case, only one of the two eigenvalues of J will tend to zero as γ2 approaches infinity.
- 27.
det(l)≠0 and also \(\det (I -\left [\begin{array}{cc} 0 & 0\\ 0 & {\gamma }_{2} \end{array} \right ]l)\neq 0\).
- 28.
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Flaschel, P. (2010). Stability: Independent of Economic Structure? A Prototype Analysis. In: Topics in Classical Micro- and Macroeconomics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00324-0_14
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