ON THE SUSTAINABLE PROGRAM IN SOLOW'S MODEL

ABSTRACT. We show that our general result (Withagen and Asheim [1998]) on the converse of Hartwick's rule also applies for the special case of Solow's model with one capital good and one exhaustible resource. Hence, the criticism by Cairns and Yang [2000] of our paper is unfounded.     

THE CONVERSE OF HARTWICK'S RULE AND UNIQUENESS OF THE SUSTAINABLE PATH

ABSTRACT. We provide a constructive proof of the converse of Hartwick's rule for a generalization of Solow's model with one capital good and one exhaustible resource. We also show that the sustainable path is unique.     

Constant Sustainable Consumption Rate in Optimal Growth with Exhaustible Resources

The problem of optimal growth with an exhaustible resource deposit under R. M. Solow's criterion of maximum sustainable consumption rate, previously formulated as a minimum-resource-extraction problem, is shown to be a Mayer-type optimal-control problem. The exact solution of the relevant firstorder necessary conditions for optimality is derived for a Cobb-Douglas production function, whether or not the constant unit resource extraction cost vanishes. The related problem of maximizing the terminal capital stock over an unspecified finite planning period is investigated for the development of more efficient numerical schemes for the solution of multigrade-resource deposit problems. The results for this finite-horizon planning problem are also important from a theoretical viewpoint, since they elucidate the economic content of the optimal growth paths for infinite-horizon problems.     

An incentive model for closed-loop supply chain under the EPR law

Motivated by the collection outsourcing phenomena under Extended Producer Responsibility (EPR), this paper studies a contract design problem for a manufacturer who consigns the used product collection to a collector, while the manufacturer only has incomplete information on the collector's cost. On the basis of the incentive theory, optimal contracts are developed to minimize the cost and satisfy the collection constraints prescribed by EPR. Properties of the contract parameters are derived, and issues such as information rent and information value are also explored. The impacts of EPR are analysed by comparing whether or not EPR law is implemented, and more managerial insights are further obtained through numerical examples.     

Investment decisions in the rent-to-own industry in the absence of inventory

This study addresses the product investment decision faced by firms in the rent-to-own industry. In this setting, a customer arrives according to a random process and requests one unit of a product to rent (and eventually own should he/she choose to make all the required payments). At the time of request, if the product is available in inventory, the firm enters into a contractual agreement (by accepting the customer's offer) and rents the merchandise. More interesting and the case considered here, if the requested item is not in inventory, the firm must decide whether to purchase the item in order to rent it out or to simply reject the request. The customer's offer specifies the desired maximum contract length and the payment frequency—from which the firm determines the fixed periodic payment charged. The firm makes its investment decision based on the characteristics of the offer as well as those of the product (eg, initial and resale values, useful life and carrying costs) in essence performing a complicated cost benefit analysis. An extension is also considered whereby instead of simply rejecting the request the firm can adjust the required payment amount. Dynamic programming techniques are used to address the problem and to solve for the firm's optimal decision.     

A two-echelon inventory model for a deteriorating item with stock-dependent demand,partial backlogging and capacity constraints

This study investigates a two-echelon supply chain model for deteriorating inventory in which the retailer’s warehouse has a limited capacity. The system includes one wholesaler and one retailer and aims to minimise the total cost. The demand rate in retailer is stock-dependent and in case of any shortages, the demand is partially backlogged. The warehouse capacity in the retailer (OW) is limited; therefore the retailer can rent a warehouse (RW) if needed with a higher cost compared to OW. The optimisation is done from both the wholesaler’s and retailer’s perspectives simultaneously. In order to solve the problem a genetic algorithm is devised. After developing a heuristic a numerical example together with sensitivity analysis are presented. Finally, some recommendations for future research are presented.     

PRODUCTION COST UNCERTAINTY AND INDUSTRY EQUILIBRIUM FOR NON-RENEWABLE RESOURCE EXTRACTION

In this paper we examine production cost uncertainty in a non-renewable resource industry model. A rational expectations, m-firms industry equilibrium is characterized, and the effects of production cost uncertainty on industry rent and firms' profits are examined.     

FORECASTING MINERAL SUPPLY FROM MINERAL RENT DATA

An optimal control model of exhaustible resources is used to clarify the long run relationship between mineral rent and depletion cost at the industry level. A standard first order condition of the time rate of change of rents is reformulated to reveal that rent data may be used to help forecast the rise in extraction costs resulting from resource depletion. This application of the theory of exhaustible resources is illustrated using historical mineral industry rent and extraction cost data. A forecast of U.S. coal extraction costs, following the method proposed in this paper, suggests that future rates of extraction cost increases will be similar to rates experienced in the past.     

Suboptimality Estimates for the Semi-Discretized LQR Problem for Parabolic PDEs

The linear quadratic regulator problem (LQR) for parabolic partial differential equations (PDEs) has been understood to be an infinite-dimensional Hilbert space equivalent of the finite-dimensional LQR problem known from mathematical systems theory. The matrix equations from the finite-dimensional case become operator equations in the infinite-dimensional Hilbert space setting. A rigorous convergence theory for the approximation of the infinite-dimensional problem by Galerkin schemes in the space variable has been developed over the past decades. Numerical methods based on this approximation have been proven capable of solving the case of linear parabolic PDEs. Embedding these solvers in a model predictive control (MPC) scheme, also nonlinear systems can be handled. Convergence rates for the approximation in the linear case are well understood in terms of the PDE's solution trajectories, as well as the solution operators of the underlying matrix/operator equations. However, in practice engineers are often interested in suboptimality results in terms of the optimal cost, i.e., evaluation of the quadratic cost functional. In this contribution, we are closing this gap in the theory. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Distribution Free Newsboy Problem: Review and Extensions

We present here a new, very compact, proof of the optimality of Scarf's ordering rule for the newsboy problem where only the mean and the variance of the demand are known. We then extend the analysis to the recourse case, where there is a second purchasing opportunity; to the fixed ordering cost case, where a fixed cost is charged for placing an order; to the case of random yields; and to the multi-item case, where multiple items compete for a scarce resource.     

A consistent second-order plate theory for monotropic material

Mathematical homogenization (or averaging) of composite materials, such as fibre laminates, often leads to non-isotropic homogenized (averaged) materials. Especially the upcoming importance of these materials increases the need for accurate mechanical models of non-isotropic shell-like structures. We present a second-order (or: Reissner-type) theory for the elastic deformation of a plate with constant thickness for a homogeneous monotropic material. It is equivalent to Kirchhoff's plate theory as a first-order theory for the special case of isotropy and, furthermore, shear-deformable and equivalent to R. Kienzler's theory as a second-order theory for isotropy, which implies further equivalences to established shear-deformable theories, especially the Reissner-Mindlin theory and Zhilin's plate theory. Details of the derivation of the theory will be published in a forthcoming paper [3]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dempster's rule of combination

The theory of belief functions is a generalization of probability theory; a belief function is a set function more general than a probability measure but whose values can still be interpreted as degrees of belief. Dempster's rule of combination is a rule for combining two or more belief functions; when the belief functions combined are based on distinct or “independent” sources of evidence, the rule corresponds intuitively to the pooling of evidence. As a special case, the rule yields a rule of conditioning which generalizes the usual rule for conditioning probability measures. The rule of combination was studied extensively, but only in the case of finite sets of possibilities, in the author's monograph A Mathematical Theory of Evidence. The present paper describes the rule for general, possibly infinite, sets of possibilities. We show that the rule preserves the regularity conditions of continuity and condensability, and we investigate the two distinct generalizations of probabilistic independence which the rule suggests.     

Improving the efficiency of the purchasing process using total cost of ownership information: The case of heating electrodes at Cockerill Sambre S.A.

Improving the efficiency of the purchasing process provides important opportunities to increase a firm's profitability. In this paper we introduce a mathematical programming model that uses total cost of ownership information to simultaneously select suppliers and determine order quantities over a multi-period time horizon. The total cost of ownership quantifies all costs associated with the purchasing process and is based on the activities and cost drivers determined by an activity based costing system. Our approach is motivated by the purchasing problem of heating electrodes at Cockerill Sambre, a Belgian multinational steel producer. In this case quality issues account for more than 70% of the total cost of ownership making the quality of a supplier a critical success factor in the supplier selection process.     

On a generalization of W.A.J. Luxemburg's asymptotic problem concerning the Laplace transform

In this paper we prove a more general case of Luxemburg's asymptotic problem concerning the Laplace transform: The problem deals with the conservation of a certain asymptotic behavior of a function at infinity, under analytic transformation of its Laplace transform. The theory of commutative Banach algebras tells us that the problem is equivalent to a family of special cases of the original problem, viz. a set of convolution integral equations, parametrized by a complex variable λ. For ∥ λ ∥ large enough, we may use Luxemburg's original result, and for other λ we modify the integral equations, and apply a modification of Luxemburg's result.     

Denumerable controlled Markov chains with strong average optimality criterion: Bounded & unbounded costs

This paper studies discrete-time nonlinear controlled stochastic systems, modeled by controlled Markov chains (CMC) with denumerable state space and compact action space, and with an infinite planning horizon. Recently, there has been a renewed interest in CMC with a long-run, expected average cost (AC) optimality criterion. A classical approach to study average optimality consists in formulating the AC case as a limit of the discounted cost (DC) case, as the discount factor increases to 1, i.e., as the discounting effectvanishes. This approach has been rekindled in recent years, with the introduction by Sennott and others of conditions under which AC optimal stationary policies are shown to exist. However, AC optimality is a rather underselective criterion, which completely neglects the finite-time evolution of the controlled process. Our main interest in this paper is to study the relation between the notions of AC optimality andstrong average cost (SAC) optimality. The latter criterion is introduced to asses the performance of a policy over long but finite horizons, as well as in the long-run average sense. We show that for bounded one-stage cost functions, Sennott's conditions are sufficient to guarantee thatevery AC optimal policy is also SAC optimal. On the other hand, a detailed counterexample is given that shows that the latter result does not extend to the case of unbounded cost functions. In this counterexample, Sennott's conditions are verified and a policy is exhibited that is both average and Blackwell optimal and satisfies the average cost inequality.     

Gröbner Bases for Ideals of σ-PBW Extensions

In this article, we introduce the σ-PWB extensions and construct the theory of Gröbner bases for the left ideals of them. We prove the Hilbert's basis theorem and the division algorithm for this more general class of Poincaré–Birkhoff–Witt extensions. For the particular case of bijective and quasi-commutative σ-PWB extensions, we implement the Buchberger's algorithm for computing Gröbner bases of left ideals.     

On the Hamilton's isoperimetric ratio in complete Riemannian manifolds of finite volume

We study a family of geometric variational functionals introduced by Hamilton, and considered later by Daskalopulos, Sesum, Del Pino and Hsu, in order to understand the behavior of maximal solutions of the Ricci flow both in compact and noncompact complete Riemannian manifolds of finite volume. The case of dimension two has some peculiarities, which force us to use different ideas from the corresponding higher-dimensional case. Under some natural restrictions, we investigate sufficient and necessary conditions which allow us to show the existence of connected regions with a connected complementary set (the so-called “separating regions”). In dimension higher than two, the associated problem of minimization is reduced to an auxiliary problem for the isoperimetric profile (with the corresponding investigation of the minimizers). This is possible via an argument of compactness in geometric measure theory valid for the case of complete finite volume manifolds. Moreover, we show that the minimum of the separating variational problem is achieved by an isoperimetric region. The dimension two requires different techniques of proof. The present results develop a definitive theory, which allows us to circumvent the shortening curve flow approach of the above mentioned authors at the cost of some applications of the geometric measure theory and of the Ascoli-Arzela's Theorem.     

A Quantum Dynamical Approach to Matrix Khrushchev's Formulas

Khrushchev's formula is the cornerstone of the so‐called Khrushchev theory, a body of results which has revolutionized the theory of orthogonal polynomials on the unit circle. This formula can be understood as a factorization of the Schur function for an orthogonal polynomial modification of a measure on the unit circle. No such formula is known in the case of matrix‐valued measures. This constitutes the main obstacle to generalize Khrushchev theory to the matrix‐valued setting, which we overcome in this paper. It was recently discovered that orthogonal polynomials on the unit circle and their matrix‐valued versions play a significant role in the study of quantum walks, the quantum mechanical analogue of random walks. In particular, Schur functions turn out to be the mathematical tool which best codify the return properties of a discrete time quantum system, a topic in which Khrushchev's formula has profound and surprising implications. We will show that this connection between Schur functions and quantum walks is behind a simple proof of Khrushchev's formula via “quantum” diagrammatic techniques for CMV matrices. This does not merely give a quantum meaning to a known mathematical result, since the diagrammatic proof also works for matrix‐valued measures. Actually, this path‐counting approach is so fruitful that it provides different matrix generalizations of Khrushchev's formula, some of them new even in the case of scalar measures. Furthermore, the path‐counting approach allows us to identify the properties of CMV matrices which are responsible for Khrushchev's formula. On the one hand, this helps to formalize and unify the diagrammatic proofs using simple operator theory tools. On the other hand, this is the origin of our main result which extends Khrushchev's formula beyond the CMV case, as a factorization rule for Schur functions related to general unitary operators.© 2016 Wiley Periodicals, Inc.     

Differential calculus over general base fields and rings

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller's C^k_c-theory.     

Asymptotic behavior of dynamic equations ontime scales

As a way to unify a discussion of many kinds of problems for equations in the contionous and discrete case(but also in order to reveal discrepancies between both cases), a theory of "time scales" was proposed and developed by Sulbach and Hilger. In our paper we investigate the asymptoic behaviour of so-called dynamic equations on time scales, and sych dynamic equations are differentialequations in the continous case and difference equations in the discrete case. We offer a perturbation result that leads to a time scales version of Levinson's Fundamental Lemma. Crucial are a dichotomy condition and a growth condition on the perturbation. Also, in the case that Levinson's result cannot be applied immediately, we suggest several preliminary transformations that might lead to a situation where Levinson's lemma is applicable. Such tranformations have been suggested by Harris and Lutz in the continuous case and by Benzaid and Lutz in the discrete case. Both those cases are covered by our theory, plus cases "in between". Examples for such cases will also be discussed in this paper.