CMB signal in WMAP 3yr data with FastICA

We present an application of the fast Independent Component Analysis (FastICA) to the WMAP 3yr data with the goal of extracting the CMB signal. We evaluate the confidence of our results by means of Monte Carlo simulations including CMB, foreground contaminations and instrumental noise specific of each WMAP frequency band. We perform a complete analysis involving all or a subset of the WMAP channels in order to select the optimal combination for CMB extraction, using the frequency scaling of the reconstructed component as a figure of merit. We found that the combination KQVW provides the best CMB frequency scaling, indicating that the low frequency foreground contamination in Q, V and W bands is better traced by the emission in the K band. The CMB angular power spectrum is recovered up to the degree scale, it is consistent within errors for all WMAP channel combination considered, and in close agreement with the WMAP 3yr results. We perform a statistical analysis of the recovered CMB pattern, and confirm the sky asymmetry reported in several previous works with independent techniques.Comment: 10 pages, 7 figures, submitted to MNRA

Hicks on Walrasian equilibrium in the 1930s and beyond

After many explorations in different directions during the early 1930s, in 1934 Hicks ends up by advocating an interpretation of Walrasian equilibrium and capital theory along stationary lines, but the suggested interpretation is at variance with the view endorsed by the last Walras and by Pareto at the turn of the century. In the second half of the 1930s, during the long gestation of Value and Capital (VC), Hicks\u2019s ideas on equilibrium and capital progressively change and mature, to eventually culminate, with the publication of VC in 1939, in the rediscovery of a method of analysis and an equilibrium concept, Hicks\u2019s temporary equilibrium, that are substantially similar to the method of analysis and equilibrium concept put forward by the last Walras and by Pareto about forty years before. Yet this direct link with the Walrasian tradition is not overtly recognised by Hicks in VC: in particular, the essentially Walrasian character of the equilibration process supporting Hicks\u2019s temporary equilibrium concept is carefully disguised under Marshallian garments. This fact will not only delay Hicks\u2019s own recognition of the limits of the VC approach, which will start to be questioned by him only in the mid-1950s, but will also concur to spreading unsubstantiated ideas about the origins and theoretical foundations of the neo-Walrasian research programme. The aim of this paper is to clarify the theoretical reasons behind the winding path followed by Hicks over the 1930s, especially as far as the Walrasian conception of equilibrium and equilibration is concerned, and to identify the roots of Hicks\u2019s ambiguity about the theoretical ancestry of the VC model in his previous intellectual histor

Equilibrio, disequilibrio e tempo in Walras

Walras\u2019s theory of general economic equilibrium goes through many changes over the twenty-six years elapsing from the publication of Walras\u2019s first significant theoretical contribution in 1874 and the appearance, in 1900, of the fourth edition of his masterpiece, the El\ue9ments d\u2019\ue9conomie politique pure. This paper reconstructs the main steps characterizing the twisted evolution of Walras\u2019s ideas concerning equilibrium, disequilibrium, and time in economics. Early in his scientific career, Walras perceives, albeit confusedly, the existence of theoretical, analytical, and epistemological difficulties undermining his analysis of the t\ue2tonnement processes associated with his equilibrium models of exchange, production, and capital formation; yet he refrains from facing such issues squarely for fear that this would compel him to give up all realistic justification in support of his theory. Concerning the exchange model, under the pressure of external criticism, Walras soon enough yields to the logical requirements of the analysis, by explicitly endorsing, since 1885, the so-called \u201cno-trade-out-of-equilibrium\u201d assumption, an assumption which forces him to interpret the t\ue2tonnement process in exchange as a purely virtual equilibration process in \u201clogical\u201d time, supporting an \u201cinstantaneous\u201d equilibrium notion. Concerning the production and capital formation models, however, Walras strenuously strives to preserve the pseudo-realistic assumptions permeating his original view of the t\ue2tonnement construct: up to the third edition of the El\ue9ments (1896), in fact, Walras keeps to the idea that the t\ue2tonnement process in production should be viewed as an actual equilibration process in \u201creal\u201d time, supporting a \u201cstationary\u201d equilibrium notion. Yet, due to his eventual realization of the logical contradictions engendered by such interpretation of both the t\ue2tonnement process and the associated equilibrium notion, hardly \u2013 if at all \u2013 compensated by a few dubious gains in terms of descriptive realism, in the fourth edition of the El\ue9ments Walras eventually resolves to give up all pseudo-realistic pretence concerning the equilibration process: in 1900, in fact, he adopts a novel, openly unrealistic assumption, the so-called \u201chypoth\ue8se des bons\u201d, to the effect of turning all t\ue2tonnement processes, including those concerning production, into virtual processes in \u201clogical\u201d time, supporting a generalized notion of \u201cinstantaneous\u201d equilibrium of the \u201ctemporary\u201d type JEL classification codes: B13, B21, B31, B41, C62, D5

Jevons, Jenkin, and Walras on demand-and-supply analysis in the theory of exchange

In his economic writings Jevons insists on the allegedly fundamental role played by the so-called "laws of supply and demand" in his theory of exchange; yet no demand-and-supply analysis is actually employed in deriving such theory, as developed in Chapter 4 of The Theory of Political Economy (TPE). This is all the more puzzling in the light of the following two facts: 1) in his 1868 correspondence with Jevons, Fleeming Jenkin provides a complete geometrical solution of the exchange equilibrium problem based on the use of demand and supply curves, but his suggestion is wholly neglected by Jevons in the first edition of TPE (1871); 2) in his 1874 open letter to Jevons, Walras explicitly criticizes his correspondent for his defective treatment of the "laws of supply and demand", suggesting an alternative analytical solution of the exchange problem based on the use of demand and supply functions; yet Jevons entirely disregards Walras's remarks in preparing the second edition of TPE (1879). This paper compares Jevons's, Jenkin's and Walras's approaches to the exchange equilibrium problem, explaining the analytical and epistemological reasons that underlie Jevons's neglect of his correspondents' criticism and advic

Marshall vs. Walras on Equilibrium and Disequilibrium

In all introductory and intermediate textbooks in microeconomics, price theory is at first developed in a partial equilibrium framework which, though invariably Marshallian from the point of view of the graphical apparatus, is almost always Walrasian from the point of view of the analysis. In discussing the well-known Marshallian graphical treatment of the equilibrium problem of a single, isolated, assumedly "competitive" market for a producible consumers' good, a few diligent textbook writers 2 occasionally point out that placing the price and quantity variables on the ordinate and abscissa axes, respectively, is somewhat incongruous with the role played by the same two variables in the analytical development of the theory: for, from a Walrasian perspective, price is the independent variable, while quantity is the dependent one (whatever this may mean). Such incongruousness is sometimes historically justified by recalling that price had indeed been regarded as the independent variable by Marshall, to whom the demand-and-supply graphical apparatus can be traced back; but this suggestion, never elaborated upon, hovers about as a mysterious reference to a by now forgotten past. As a matter of fact, in the partial equilibrium framework of an isolated market, to which microeconomic primers confine most of their discussions of price theory, almost all methodological, epistemological, as well as analytical distinctions between Marshall's and Walras's approaches are skipped over. Moreover, when intermediate and advanced microeconomic textbooks eventually deal with the issue of price formation in a multi-market framework, general equilibrium theory is invariably presented as the natural extension of partial equilibrium analysis, as if Walrasian and Marshallian approaches to price theory only differed in scope and intended applications, being otherwise essentially similar in their foundations and results 3 . In this paper we want to oppose the received view on the basic equivalence of the two traditional approaches to price theory. Specifically, we want to show that Marshall's analysis of the equilibration process and his related interpretation of the equilibrium concept are essentially different from, and irreducible to, Walras's analysis and interpretation. Further, we want to show that the patent difference in scope of their respective theories (that is, partial vs. general analysis), far from being an accidental outcome of history or the innocuous consequence of the idiosyncratic preferences of the two economists, is in effect the unavoidable and irremediable by-product of their different analytical assumptions and explanatory aims. To this end, we shall first identify a common ground for our discussion, that is, a model economy that, being explicitly examined by both authors in their respective writings, will allow us to contrast their analyses and to precisely single out what distinguishes them from one another. Such model economy is represented by the two-commodity, pure-exchange economy which is dealt with by both Walras and Marshall right at the beginning of their respective expositions of price theory, under the assumption that there exists a fixed finite number of traders in the economy. Yet, while Walras's analysis is entirely developed under the assumption of an arbitrary finite number of traders (greater than or equal to two), in Marshall's case one finds a distinction between two sorts of economies: the former, called by Marshall a "barter" economy and more recently referred to as an "Edgeworth Box" economy, is an economy where the number 1 I would like to thank Michel De Vroey, Antonio Guccione, and Enrico Minelli for their comments and suggestions. The usual disclaimer applies