The asymptotic behavior of dynamic producer-consumer systems

Dynamic Arrow-type price dynamics are investigated in a continuous time framework. The existence of a unique equilibrium is first proved under realistic conditions. Then, the local asymptotic stability of the equilibrium in the presence of instantaneous price and output information is shown. Continuously distributed time lags are then assumed in obtaining and implementing price and output information, or equivalently, it may be assumed that the firms and/or the market wants to react to long term effects rather than to follow sudden changes in price or outputs. If a time lag is assumed only in estimating the demand in the market, then the local asymptotic stability of the equilibrium is preserved. If the producers also use delayed information, then instability may occur. Stability conditions are derived and in the case of bifurcation the possibility of the birth of limit cycles is explored.     

Heuristic algorithms for delivered price spatially competitive network facility location problems

We review previous formulations of models for locating a firm's production facilities while simultaneously determining production levels at those facilities so as to maximize the firm's profit. We enhance these formulations by adding explicit variables to represent the firm's shipping activities and discuss the implications of this revised approach. In these formulations, existing firms, as well as new entrants, are assumed to act in accordance with an appropriate model of spatial equilibrium. The firm locating new production facilities is assumed to be a large manufacturer entering an industry composed of a large number of small firms. Our previously reported proof of existence of a solution to the combined location-equilibrium problem is briefly reviewed. A heuristic algorithm based on sensitivity analysis methods which presume the existence of a solution and which locally approximate price changes as linear functions of production perturbations resulting from newly established facilities is presented. We provide several numerical tests to illustrate the contrasting locational solutions which this paper's revised delivered price formulation generates relative to those of previous formulations. An exact, although computationally burdensome, method is also presented and employed to check the reliability of the heuristic algorithm.     

Chaos in duopoly pricing

The dynamics of two competing firms in a market is studied in terms of Cournot's duopoly theory. Assuming iso-elastic demand and constant unit production costs the iterative mapping

for the outputs of the two firms ensues. The two constants are the unit production costs. The fixed point, the Cournot equilibrium, has earlier been assumed to be the only interesting feature of this model. It is, however, shown that the model can produce persistent motion, periodic or chaotic.     

On Cournot-Nash-Walras Equilibria and Their Computation

This paper concerns a model of Cournot-Nash-Walras (CNW) equilibrium where the Cournot-Nash concept is used to capture equilibrium of an oligopolistic market with non-cooperative players/firms who share a certain amount of a so-called rare resource needed for their production, and the Walras equilibrium determines the price of that rare resource. We prove the existence of CNW equilibria under reasonable conditions and examine their local stability with respect to small perturbations of problem data. In this way we show the uniqueness of CNW equilibria under mild additional requirements. Finally, we suggest some efficient numerical approaches and compute several instances of an illustrative test example.     

Centralised reallocation of emission permits using DEA

In this paper a data envelopment analysis (DEA) approach to the problem of emission permits reallocation is presented. It can be used with conventional command and control as well as with an allowance market. It uses a centralized point of view, which represents the common good. In the model it is assumed that firms produce two types of outputs: desirable outputs (i.e. good outputs with positive value for consumers) and undesirable outputs (i.e. bad outputs with negative value for consumers, such as emissions of pollutants). The proposed approach has three phases, which correspond to three objectives that are pursued lexicographically. The three objectives are maximizing aggregated desirable production, minimizing undesirable total emissions and minimizing the consumption of input resources. The relative priority of these objectives is defined by the regulator. The whole approach is units-invariant and does not require information on input and output prices. The approach is applied on a dataset from the Swedish pulp and paper industry.     

Signaling advertising by multiproduct firms

We consider the use of advertising expenses as quality signals in multiproduct firms, extending previous results on single product firms. In our model, a firm introduces sequentially two products whose qualities are positively correlated. We investigate whether there exist information spillovers from the first to the second market. We show that, when correlation is high, the equilibrium in market 2 depends on the quality reputation the firm has gained in market 1. Moreover, if a firm with a high-quality product 1 wants to separate from its low-quality counterpart, it needs to advertise more in this market than if the qualities of the two products are unrelated. This advertising level signals not only high quality in the first market, but also the likely quality of the second product. Thus, advertising in the first market has information spillovers in the second market.     

R&;D Equilibrium Strategies with Surfers

A typical assumption in the game-theoretic literature on research and development (R&D) is that all firms belonging to the industry under investigation pursue R&D activities. In this paper, we assume that the industry is composed of two groups; the first (the investors) is made of firms that have R&D facilities and are involved in this type of activity. The second group corresponds to firms that are inactive in R&D (the surfers). The latter group benefits from its competitors’ R&D efforts, thanks to involuntary spillovers. This division of the industry is in line with actual practice, where indeed not all firms are engaged in costly and risky R&D. We adopt a two-stage game formalism where, in the first stage investors decide on their levels of investment in R&D, and in the second stage all firms compete à la Cournot in the product market. We characterize and analyze the unique subgame perfect Nash equilibrium. Research supported by NSERC, Canada. F. Ben Abdelaziz is on leave at The College of Engineering, American University of Sharjah, UAE.     

On securitization,market completion and equilibrium risk transfer

We propose an equilibrium framework within which to price financial securities written on non-tradable underlyings such as temperature indices. We analyze a financial market with a finite set of agents whose preferences are described by a convex dynamic risk measure generated by the solution of a backward stochastic differential equation. The agents are exposed to financial and non-financial risk factors. They can hedge their financial risk in the stock market and trade a structured derivative whose payoff depends on both financial and external risk factors. We prove an existence and uniqueness of equilibrium result for derivative prices and characterize the equilibrium market price of risk in terms of a solution to a non-linear BSDE.     

Duopoly competition in networks

This paper presents a new approach to modeling competition between firms in network-based industries, i.e. industries where the firms' technology decisions correspond to choices of networks. Industries having this structure include transportation, telecommunications, and some service industries. Competition is studied between two firms who make both network design decisions and price decisions for services. This situation is modeled as a game, an equilibrium solution corresponding to a Nash equilibrium is defined, and properties of the solution are characterized. Necessary and sufficient conditions are shown for equilibrium solutions and existence of equilibrium solutions is demonstrated. Among the results is that each firm will maximize its own profit by minimizing total industry cost of providing services. An example demonstrating results is presented.     

The law of one price in data envelopment analysis: Restricting weight flexibility across firms

The Law of One Price (LoOP) states that all firms face the same prices for their inputs and outputs under market equilibrium. Taken here as a normative condition for ‘efficiency prices’, this law has powerful implications for productive efficiency analysis, which have remained unexploited thus far. This paper shows how LoOP-based weight restrictions can be incorporated in Data Envelopment Analysis (DEA). Utilizing the relation between industry-level and firm-level cost efficiency measures, we propose to apply a set of input prices that is common for all firms and that maximizes the cost efficiency of the industry. Our framework allows for firm-specific output weights and for variable returns-to-scale, and preserves the linear programming structure of the standard DEA. We apply the proposed methodology to the evaluation of the research efficiency of economics departments of Dutch Universities. This application shows that the methodology is computationally tractable for practical efficiency analysis, and that it helps in deepening the DEA analysis.     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.     

Limit pricing under third-degree price discrimination

This paper shows how a multimarket incumbent can use low pre-entry prices for entry deterrence. We consider an incumbent who operates in two independent markets and has private information about his production cost. In one of the markets, there is a potential entrant offering a differentiated product. The most reasonable perfect Bayesian equilibrium is either the least-cost separating equilibrium or the pooling equilibrium where both types of incumbents set the low-cost monopoly prices. This equilibrium may involve a downward distortion in the pre-entry prices of both markets. Our model has interesting implications for antitrust regulation as well as for international trade policy. First, predatory tests based on a single market are inadequate for a multimarket incumbent. Second, a lower price in a foreign market is neither a necessary nor a sufficient condition for the existence of entry deterrence in a foreign market.     

Modified base-stock policies for semiconductor production system with dependent yield rates

We consider a two-stage production system faced by semiconductor manufacturing which produces a hierarchy of multiple grades of outputs. In the first stage, a single type of input (wafer) is used to produce multiple types of semi-finished parts with dependent yield rates, and in the second stage, each type of semi-finished parts can be transformed into a corresponding type of final products, or downgraded to a type of lower grade final products. Random customer demands are faced on the final products, and demands of different types of final products are not allowed to be substituted. The advantage of this production system is that it can prevent unhealthy ordering from customers who intentionally send out false demand signals for high grade products and revise the orders to lower grade products when the delivery time is close, which was observed in semiconductor manufacturing. The objective of the study is to plan the quantity of the input at the first stage and the respective downgrade quantities at the second stage so as to meet the required service level at the minimum cost. With some common assumptions, we propose a modified base-stock policy for this two-stage production system and show that the occurrence of nil excess inventory above the base-stock level follows a renewal process. We further extend the modified base-stock policy to a better policy that invokes risk pooling over multiple grade products. The performance of these two polices are evaluated via simulation to provide managerial insights.     

Two-tier equilibria for supply chains with a single manufacturer and multiple asymmetric retailers

We derive sufficient conditions for the existence and uniqueness of the Stackelberg–Nash–Cournot equilibria for a supply chain problem with a single manufacturer and multiple asymmetric retailers and characterize the first and second order derivatives of the total equilibrium quantities. The Stackelberg manufacturer is assumed to supply a homogeneous product to all retailers with the retail price determined by a general nonlinear inverse demand function. We provide several extensions of our previous results [G.J. Kyparisis, C. Koulamas, A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers, Operations Research Letters 39 (2011) 471–474] obtained for a similar supply chain with symmetric retailers.     

Supply chain coordination for newsvendor-type products with two ordering opportunities and demand information update

This paper studies a coordination issue with two ordering opportunities in a two-echelon supply chain, where one manufacturer sells a single newsvendor-type product through one buyer. The manufacturer does not hold inventory and activates production or order with an infinite capacity and a fixed setup cost in response to the buyer’s order. The buyer places two orderings during the selling period of the product: one happens at the beginning of the period and the other at some specified time within the selling period. The whole selling period is divided into two stages or sub-periods by the buyer’s second order. The stochastic demands in the two sub-periods are assumed to be auto-correlated. The excess demand before the second order is partially backordered, whereas the excess demand at the end of the selling season is utterly lost. Under both the centralized and decentralized settings, we develop the models of how the buyer determines his two-ordering policies. We analyse the existence and uniqueness of the optimal solutions to the models and present the corresponding analytical solutions. Furthermore, we propose an improved revenue-sharing contract that can realize the perfect coordination of the supply chain and study how the revenue-sharing policies affect the supply chain members’ decisions. Finally, we show the superiority of the presented two-ordering strategy through numerical examples.     

A mathematical model for continuous crystallization

J. Banasiak We discuss a mixed‐suspension, mixed‐product removal crystallizer operated at thermodynamic equilibrium. We derive and discuss the mathematical model based on population and mass balance equations and prove local existence and uniqueness of solutions using the method of characteristics. We also discuss the global existence of solutions for continuous and batch mode. Finally, a numerical simulation of a continuous crystallizer in steady state is presented. Copyright © 2015 John Wiley & Sons, Ltd.     

Integrating operations and marketing decisions using delayed differentiation of products and guaranteed delivery time under stochastic demand

In this research, we integrate the issues related to operations and marketing strategy of firms characterized by large product variety, short lead times, and demand variability in an assemble-to-order environment. The operations decisions are the inventory level of components and semi-finished goods, and configuration of semi-finished goods. The marketing decisions are the products price and a lead time guarantee which is uniform for all products. We develop an integrated mathematical model that captures trade-offs related to inventory of semi-finished goods, inventory of components, outsourcing costs, and customer demand based on guaranteed lead time and price.The mathematical model is a two-stage, stochastic, integer, and non-linear programming problem. In the first stage, prior to demand realization, the operation and marketing decisions are determined. In the second stage, inventory is allocated to meet the demand. The objective is to maximize the expected profit per-unit time. The computational results on the test problems provide managerial insights for firms faced with the conflicting needs of offering: (i) low prices, (ii) guaranteed and short lead time, and (iii) a large product variety by leveraging operations decisions.     

Competitive capacity expansion under demand uncertainty

We analyze the capacity expansion behavior of firms in a duopoly faced with an uncertain new market. The market demand may be high or low with a given probability mass function. Firms obtain private information about the market size and build their capacity before the market demand is known. Once the demand is revealed, firms enter a capacity constrained price competition phase which determines their revenues. Two scenarios are considered: first, when firms choose their capacities simultaneously in the investment phase, and second, when they do so sequentially. For each case, we determine the unique symmetric Nash equilibrium. Excess capacity can occur in equilibrium in the industry. It is seen that preempting the competitor in the capacity expansion phase offers first mover benefits. We argue that the sequential moves game is more prone to equilibrium excess capacity compared to the simultaneous case. We show that preemption is a good strategy if the investing environment is either highly optimistic or highly pessimistic. If the industry outlook is only moderately optimistic, a capacity planner is still better off preempting his competitor, however, the industry may encounter overcapacity as a consequence.     

Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses

In this paper, we study a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. We show the existence of a bounded positive invariant and attracting set. The possibility of existence and uniqueness of positive equilibrium are considered. The asymptotic behavior of the positive equilibrium and the existence of Hopf-bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. The existence and non-existence of periodic solutions are established under suitable conditions. The permanence conditions are also established. We obtained sufficient conditions to ensure the global stability of the unique positive equilibrium, by using suitable Lyapunov functions, LaSalle invariance principle and Dulac’s criterion. We obtained also sufficient conditions for the global stability of the prey-extinction equilibrium when the unique positive equilibrium is not feasible. Finally, numerical simulations are presented to illustrate the analytical results.     

Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems

We are concerned with the problem of existence, uniqueness and qualitative properties of solutions to the radially symmetric variational problem where is the ball of centered at the origin and with radius , the map is a normal integrand, and is a convex function of the second variable. This kind of problems, with non-convex lagrangians with respect to , arise in various fields of applied sciences, such as optimal design and nonlinear elasticity. Received June 18, 1998; in final form August 26, 1999 / Published online September 14, 2000