Christian Bidard

The Spectrum of Random Matrices

The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Bródy (1997) noticed that, when the entries of the matrix are independently and identically distributed, the speed of convergence increases statistically with the dimension of the matrix. As the speed depends on the ratio between the subdominant and the dominant eigenvalues, Bródys conjecture amounts to stating that this ratio tends to zero when the dimension tends to infinity. The paper provides a simple proof of the result. Some mathematical and economic aspects of the problem are discussed.

Duality between prices and techniques

Abstract In this paper a number of properties of price vectors generated by a technique of the Leontief-Sraffa type at different profit rates are obtained. It is then proved that for any given set of vectors satisfying those properties the technique which generates them as price vectors can be determined. Hence all the economically relevant aspects of a technique can be obtained from a number of price vectors and wage rates.

'Our daily bread': Maurice Potron, from Catholicism to mathematical economics.

Abstract Maurice Potron (1872–1942) is a French Jesuit and mathematician whose main source of inspiration in economics is the encyclical Rerum Novarum. With virtually no knowledge in economic theory, he wrote down a linear model of production in which he formalized the notions of just prices and just wages. As early as 1911, he used the Perron–Frobenius theorem to prove the existence of a positive solution and established a duality result between the quantity side and the price side of the model. He returned to economics in the 1930s, but in both periods he failed to make a lasting impression upon economists. JEL Classification Code: B3

THE CORN-GUANO MODEL

The model presented here can be seen as an extension, specifically designed to deal with the issue of exhaustible resources, of the well-known ‘corn model’. We believe that useful insights are obtained by concentrating on a benchmark case, that of an economy with only one reproducible ‘good’, corn, and only one ‘resource’, guano. The corn‐guano model aims at delineating some major issues related to the introduction of exhaustible resources within a postSraffian model. In this paper we concentrate on the dynamics of the price variables. The ultimate goal concerns more generally the treatment of intertemporal economies outside a regular growth path, i.e. it concerns the degree of generality of the post-Sraffian theory itself.

Fixed capital and internal rate of return

Abstract For a neo-Austrian process of production, more generally multisectoral fixed capital models, it is shown that the maximum rate of growth and the minimum rate of interest are both equal to the internal rate of return. This von Neumann type equality is here obtained in absence of truncation, i.e., free disposal of fixed capital. If several internal rates of return coexist, the economically significant one is the maximal one. This result provides theoretical grounds for the internal rate of return criterion in infinite horizon.

The number and type of long-term equilibria

We consider a balanced-growth model with finitely many processes, a uniform rate of profits, and a given composition of final demand. Two types of equilibria are distinguished, “black” and “white”: the color of an equilibrium is defined by the relative sign of two determinants associated with the operated methods, and can be interpreted as a local property of a cross-dual dynamic process. Under standard economic assumptions, and flukes apart, the number of white equilibria exceeds that of black equilibria by one. In particular the total number of long-term equilibria is odd.

Potron and the Perron–Frobenius Theorem

Maurice Potron is an important precursor of the study of linear models of production and, in particular, of input–output analysis. We show that, contrary to Abraham-Frois and Lendjels interpretation which we consider as unfaithful to Potrons model, there is a clear connection between his theory and the Perron-Frobenius theorem.

Reproduction and Temporary Disequilibrium: A Classical Approach

We build a bisector reproduction model with classical features in which the capitalists aim at maximizing accumulation of their profits. At variance with gravitation models, it is assumed that they invest their profits in their own industry. Their plans are based on actual productions and expected prices. Effective prices and effective allocations of resources are determined by a market-clearing mechanism. A simple law on the formation of expectations allows us to define the dynamics of disequilibria, which let appear endogenous self-sustained fluctuations, around a long-run path. The long-run rate of growth and the amplitude of the fluctuations depend on the initial conditions.

Laws on Long‐term Prices

Long-term prices depend on distribution in a complex way, especially when choice of technique is allowed. It is shown, however, that the movement of prices in terms of wage obeys certain laws. More precisely, the movement is characterized in terms of linear programming problems. Necessary or sufficient conditions connected with the convexity of the wage-profit curves are also obtained. But, with regard to the relative prices of commodities, they vary arbitrarily, so that the Wicksell price effects are not under control. Copyright 1998 by Blackwell Publishers Ltd and The Victoria University of Manchester

Further Reflections on the Corn–Guano Model

The corn–guano model is the simplest model with exhaustible resources. On the constant rate of profit hypothesis, a change of the numeraire from corn to corn-and-labour affects the price trajectories. In particular, the corn price at the exhaustion date is no longer equal to its long-term level. The system of intertemporal prices admits one degree of freedom. Only one path, called the natural path, admits a positive price, wage and royalty for any number of periods before exhaustion. But, for a given exhaustion date, the non-natural paths close to it are also associated with positive prices, wages and royalties. In this paper we study the characteristics of the natural path and the properties of the non-natural paths. The results are partly extended to multisector models.