Alphonse P. Magnus

Creative destruction, investment volatility, and the average age of capital

In this article, a new numerical procedure is used to compute the equilibrium of a vintage capital growth model with nonlinear utility, where the scrapping time is nonconstant. We show that equilibrium investment and output converge nonmonotonically to the balanced growth path due to replacement echoes. We find that the average age of capital is inversely related to output, which is consistent with recent micro evidence reinforcing the importance of the embodied question. We also find that an unanticipated permanent increase in the rate of embodied technological progress causes labor productivity to slowdown in the short run.

Generalized Jacobi weights, Christoffel functions, and Jacobi polynomials

The authors obtain upper bounds for Jacobi polynominals which are uniform in all the parameters involved and which contain explicit constants. This is done by a combination of some results on generalized Christoffel functions and some estimates of Jacobi polynomials in terms of Christoffel functions.

On Freud's equations for exponential weights

Let {pn}n = 0∞ be the sequence of orthonormal polynomials associated with the weight exp(−f(x)), x ϵ(−∞, ∞), where f is a polynomial of even degree with positive leading coefficient. The coefficients of the three-term recurrence relation an + 1 Pn + 1(x) = (x − bn) pn(x) − anpn − 1(x), are shown to be unique “admissible” solution of the equations Fn(a, b) = 0, n = 1, 2,…,Gn(a, b) = 0, n = 0, 1 2,…, already considered by Freud for f(x) = x2m. Using these equations, we prove an important special case of Freuds Conjecture. More precisely, we establish the asymptotic behaviour of {an} and {bn} for the weight exp(−f(x)). Further, we suggest extensions of the method used here, which should lead to a proof in the general case f(x) = ¦x¦α, α > 1.

Special nonuniform lattice ( snul ) orthogonal polynomials on discrete dense sets of points

Difference calculus compatible with polynomials (i.e., such that the divided difference operator of first order applied to any polynomial must yield a polynomial of lower degree) can only be made on special lattices well known in contemporary q-calculus. Orthogonal polynomials satisfying difference relations on such lattices are presented. In particular, lattices which are dense on intervals (q = 1) are considered.

Optimal selection of orthogonal polynomials applied to the integration of chemical reactor equations by collocation methods

In this paper, we analyse some properties of the orthogonal collocation in the context of its use for reducing PDE (partial differential equations) chemical reactor models for numerical simulation and/or control design. The approximation of the first order derivatives is first considered and analysed with respect to the transfer of the stability properties of the transport component from the PDE model to its approximated ODE (ordinary differential equations) model. Then the choice of the collocation points as zero of Jacobi polynomial is analysed and interpreted as an optimal choice with respect to a weighted norm. Finally, some guidelines for the use of orthogonal collocation are proposed and the results are illustrated on a simulation example

Numerical solution by iterative methods of a class of vintage capital models

We build up an iterative numerical procedure in order to solve vintage capital growth models with nonlinear utility functions and Leontieff technologies, a class of models intensively used in the literature since the early 1990s. The numerical procedure is of the relaxation type and uses a step-by-step maximization scheme for updating, The procedure is close to the cyclic coordinate descent algorithm as described in the computational mathematics literature. We explain why and how our numerical scheme is suitable to handle the considered class of models

Sieved Orthogonal Polynomials and Discrete Measures With Jumps Dense in An Interval

We investigate particular classes of sieved Jacobi polynomials for which the weight function vanishes at the zeros of a Chebyshev polynomial of the first kind. These polynomials are then used to give a proof, using only orthogonal polynomials on (-1,1), that the discrete orthogonal polynomials introduced by Lubinsky have converging recurrence coefficients. We construct similar discrete measures with jumps dense in (—1,1) and use sieved ultra- spherical polynomials to show that their recurrence coefficients converge.

Dynamic Core-Theoretic Cooperation in a Two-Dimensional International Environmental Model

This article deals with cooperation issues in international pollution problems in a two dimensional dynamic framework implied by the accumulation of the pollutant and of the capital goods. Assuming that countries do reevaluate at each period the advantages to cooperate or not given the current stocks of pollutant and capital, and under the assumption that damage cost functions are linear, we define at each period of time a transfer scheme between countries, which makes cooperation better for each of them than non-cooperation. This transfer scheme is also strategically stable in the sense that it discourages partial coalitions.

On the Use of the Carathéodory-Fejér Method for Investigating ‘1/9’ and Similar Constants

Error norms of best rational approximations of exp(-t) on [0, ∞) are known to decrease like ln, where n is the degree of the approximant and l is the famous number ‘1/9’ = 1/9.28902549192081891875544943595174506... Trefethen and Gutknecht have demonstrated this effect on the sequence of singular values of a Hankel matrix, as an example of their use of the Caratheodory-Fejer method. It is shown here how the rate of decrease of these singular values can be estimated from their symmetric functions. The examples of rational approximation of exp(-tm) on [0, ∞), m=2,3 are also explored. The relation with the extremal polynomials method is briefly discussed.

Calculation of the eigenvalues of Schrödinger equations by an extension of Hill's method

The eigenfunctions of the one dimensional Schrodinger equation Ψ″ + [E − V(x)]Ψ=0, where V(x) is a polynomial, are represented by expansions of the form ∑k=0∞ckϕk(ω, x). The functions ϕk (ω, x) are chosen in such a way that recurrence relations hold for the coefficients ck: examples treated are Dk(ωx) (Weber-Hermite functions), exp (−ωx2)xk, exp (−cxq)Dk(ωx). From these recurrence relations, one considers an infinite bandmatrix whose finite square sections permit to solve approximately the original eigenproblem. It is then shown how a good choice of the parameter ω may reduce dramatically the complexity of the computations, by a theoretical study of the relation holding between the error on an eigenvalue, the order of the matrix, and the value of ω. The paper contains tables with 10 significant figures of the 30 first eigenvalues corresponding to V(x) = x2m, m = 2(1)7, and the 6 first eigenvalues corresponding to V(x) = x2 + λx10 and x2 + λx12, λ = .01(.01).1(.1)1(1)10(10)100.