Optimization, Dynamics, and Economic Analysis

The aim of this paper is to make transparent the close relationship between copositivity (a property of symmetric matrices which is an extension of positive-semidefiniteness), and standard quadratic problems which form a central class in quadratic optimization. Apart from theoretical interest, the results presented may have immediate consequences in implementing efficient global optimization procedures, e.g., along the lines of primal-dual interior-point methods which have become increasingly popular recently. Accounts on algorithms and important real-life applications are also provided.

Stability analysis of parametric control problems has recently been extended to control problems subject to pure state constraints. This paper illustrates the numerical aspects of stability analysis via a specific numerical example: the optimal control of the state constrained Van-der-Pol oscillator. The multiple shooting method is used to determine a nominal solution of high accuracy that satisfies first order necessary conditions. Second order sufficient conditions can be checked by solving an associated Riccati equation. Finally, sensitivity differentials of optimal solutions with respect to parameters are computed.

The optimal solution set M(t) to some parametric optimization problem $$\begin{array}{*{20}{c}} {minimize f(t,x)} & {subject to x \in D(t)} \\ \end{array}$$ is said to be roughly stable w.r.t. the roughness degree r > 0 at $$ \bar t \in T$$ if for all ∈ > 0 there is a neighborhood $$ V\left( {\bar t} \right) \subset T$$ of $$ \bar t$$ such that $$ \left( {{ \cup _{t \in V\left( {\bar t} \right)}}M\left( t \right)} \right) < r + \in $$ diam This paper states some sufficient conditions for this kind of generalized stability. One of the most important assumptions is that f is strictly roughly convexlike w.r.t. the roughness degree r. The result is applied to some optimal control problems, in particular, to a shipping problem.

The new concept in the current state of the public discussion on environmental policy is the concept of eco-efficiency, in which not only efficiency with respect to input and output productivity but efficiency from an environmental point of view is taken into account simultaneously. We propose data envelopment analysis as an approach for measurement of eco-efficiency. First, we measure the eco-efficiency in two steps: We estimate the technical efficiency (as the relation of the desirable outputs to the inputs) and the so-called ecological efficiency (as the relation of desirable to the undesirable outputs) separately. Then taking the results of both models as the output variables for the new data envelopment analysis model, the indicator for eco-efficiency can be obtained. Second, we formulate a model by taking into account simultaneously the inputs, the desirable outputs (goods) and the undesirable outputs (bads). The bads are treated as the inputs in the sense that we wish to expand desirable outputs and reduce the undesirable outputs and inputs. For illustration and comparison of both approaches a numerical example is provided.

This paper analyses the effects of the new activity-based hospital financing system on hospital performance and hospital costs in Austria. The research concentrates on differences in response among publicly-owned and privately-owned, not for profit hospitals by exploring hospital data from 1994 to 1997 with regard to credit point optimisation, shifts in performance from the inpatient to the outpatient care unit and from the outpatient care unit to physicians with their own practice, and changes in average length of stay. Techniques applied range from index figures to weighting figures and, at least, Data Envelopment efficiency scores.

We briefly summarize the algorithm OPTGAME 1.0 to solve two-person discrete-time LQ dynamic games exactly and discrete-time non-linear quadratic games approximately by means of an appropriate linearization procedure. Furthermore, we consider the relations between different solutions of these dynamic games, in particular between open-loop and feedback Nash and Stackelberg equilibria and Pareto-optimal solutions, and derive conditions for differences between the results of these solution concepts for a given game to become very small.

This paper is concerned with n-person games which typically occur in mathematical conflict models [cf. [4], [7],[8]. These games are so called cost-games, in which every actor tries to minimize his own costs and the costs are interlinked by a system of linear inequalities. It is shown that, if the players cooperate, i.e., minimize the sum of all the costs, they achieve a Nash Equilibrium. In order to determine Nash Equilibria, the simplex method can be applied with respect to the dual problem. An important special case is discussed and numerical examples are presented.

In this paper we investigate the dynamics of the traditional cobweb model where producers are risk averse and seek to learn the distribution of asset prices. We consider the subjective estimates of the statistical distribution of the market prices based on L.-step backward time series of market clearing prices. With constant absolute risk aversion, the cobweb model becomes nonlinear. Sufficient conditions on the local stability of the unique positive equilibrium of the nonlinear model are derived and, consequently, we show that the local stability region is proportional to the lag length L. When the equilibrium loses its local stability, we show that, for L = 2, the model has a strong 1: 3 resonance bifurcation and a family of fixed points of order 3 becomes unstable on both sides of criticality. For general lag lengths, numerical simulations suggest that the model displays a variety of complex dynamics.

The paper reconsiders models of AS-AD growth from a disequilibrium perspective which in fact is easier to treat than AS-AD which admits disequilibrium in the labor market but not in the product market. We assume sluggish wage and price adjustments and a Metzlerian inventory dynamics. We obtain fluctuating rates of capacity utilization for both labor and capital. For exogenous productivity growth, the model implies six laws of motion. Stability, proved for sluggish adjustment speeds, gets lost through Hopf-bifurcations, implying the existence of attracting or repelling periodic motions close to such bifurcation values. We show that these results remain valid for endogenous growth, with new features concerning the production of technological change (which is subject to hysteresis).

In this paper we focus on the emergence of trading patterns in a market where two groups of farmers compete. The members of each group share the transportation costs and we study the effect of differences in these costs on the market evolution. The agents do not have sufficient information to be able to always correctly judge whether it is advantageous to trade or not. Rather they use a simple decision rule where they observe the utility of some other farmer in the population and change their action with some probability if their own utility is smaller than the observed one. For the resulting market dynamics the basins of attraction of the coexisting equilibria are analyzed using the concept of critical curves.

This paper discusses recent work on expectation formation in the familiar cobweb ‘hog cycle’ model. A convenient feature of the model is the fact that it has a unique rational expectations equilibrium (REE). Attention will be focused on whether this unique REE can be learned under bounded rationality, with agents using simple habitual rule of thumb forecasting rules, or whether other ‘approximate rational expectations equilibria’ can arise, where forecasts are, although not perfect, approximately correct. In particular, the possibility of ‘chaotic equilibria’ under bounded rationality will be discussed. Both the homogeneous agent case and the heterogeneous case, with evolutionary competition between different forecasting rules, will be discussed. Recent experimental work on expectations formation in the cobweb economy suggests excess volatility and endogenous, possibly chaotic price fluctuations driven by the interaction of competing boundedly rational agents.

Kamien & Schwartz’ famous maintenance model is taken as a basis to analyze an agency problem in continuous time. The owner of the system to be maintained delegates the maintenance job to an agent. The discount rate of the agent is assumed to be higher than that of the owner. The so-called first-best problem assumes the agent be able to commit himself to an agreed upon effort level for the whole contract period. The analysis shows that both parties can capitalize on the different discount rates. The optimal effort level turns out to be increasing during every given finite contract period. Both parties tend to choose the contract period as long as possible.The “second-best” problem assumes the agent’s effort level is unobservable, such that the owner has to rely on an incentive payment scheme to govern the agent’s activity. The only piece of information that can serve as a basis for an incentive contract is the death of the system, triggering the end of compensation payments to the agent. Assuming the principal decides on and commits himself to the compensation scheme, the second-best problem seems to be a Stackelberg differential game. It degenerates, however, into a parameter optimization problem, due to the participation and incentive compatibility constraints. The higher discount rate of the agent requires a higher cost for the owner. Some analysis of the sensitivity of the solution with respect to the agent’s characteristics concludes the article.

In a recent paper Faria and Andrade (1998) present a model of two different representative agents, borrowers and lenders, and investigate conditions such that the borrower’s problem results in a cyclical relationship between capital and loans by applying Hopf bifurcation theory. Nevertheless, the question about the existence of stable cycles is still unresolved. In this note a numerical example leading to stable cycles is presented.

We analyze in this paper the dynamics of the production rates in creative professions from a purely theoretical point of view. The analysis is an extension of a previous work in which self-esteem was assumed to be the unique source of motivation. By contrast, in this paper we assume that the individual is stimulated by a mix of intrinsic and extrinsic motivation and we show that extrinsic motivation is a stabilizing factor. Our theory explains the fluctuating patterns of production frequently observed in artists and scientists and the absence of such fluctuations in other professions.

This paper studies a deterministic one-sector capital accumulation model with endogenous labor supply. Because of a production externality (learningby-investing) the economy may experience endogenous growth. It is shown that the distribution of capital among the agents has an effect on the growth rate of per-capita output. There exists a continuum of balanced growth paths with different growth rates. A higher growth rate can be achieved when income inequality is greater, that is, when the income distribution is more strongly dispersed. The paper shows that countries with identical production technologies and identical preferences may have different GDP growth rates because wealth is distributed differently among their inhabitants. JEL Classification Numbers:041, D31.

Most of the literature on global stability of optimal public capital often treats private and public capital as physically identical goods. As a result, one important characterlistic of public capital,”non-rivalness,” will be neglected. This is clearly a serious analytical defect. To tackle the problem, a three-sector optimal growth model, where one of the sectors is a public sector and produces public capital, is set up, and the global stability of the optimal path of public capital will be demonstrated.

This paper investigates organised wage bargaining coupled with asymmetric private information. This attempts to rationalise the persistence of unemployment. The present strategy of (European) unions of continuing to demand high wages despite substantial unemployment may be rational, even if the unions account for the unemployed. In fact, a union accounting for the gains of all - those employed and those unemployed (who must receive unemployment benefits) - and not only for the employed workers raises its demands and thus unemployment! Other interesting features are that increasing the efficiency of workers at the ‘top’ increases unemployment, unless the unions reduce their demands. In contrast increasing efficiency at the ‘bottom’ mitigates unemployment, ceteris paribus.

Traditionally investment decisions of firms have been studied under the assumption of perfectly competitive markets. If firms, however, compete in oligopolistic product markets capacity choices and hence investment decisions of one firm influnence the strategies chosen by the rival firms. Based on this observation Spence in his seminal paper introduced a class of dynamic games which are frequently referred to as capital accumulation or investment games. In this paper we present a survey of discrete time investment games. We concentrate on two different classes of games. In one class firms invest in a single capital stock that constitues a pure public good while in the second class each firm accumulates a private capital stock. For these classes of games we summarize existence and stability results for Open-Loop (OLE) and Markov Perfect Equilibria (MPE).

This paper considers a capital accumulation model that was previously analyzed by Barucci (1998). The specific feature of the model is that revenue is a convex function of the capital stock. We extend Barucci’s work by giving a full analytical characterization of the case where a saddle point with a positive capital stock level exists. Furthermore we also analyze the other cases.

This note revisits the area of differential bargaining games, a field that attracted some interest in the late 1970s and early 1980s. This was also the period of time during which Gustav Feichtinger started his work on differential games and their applications. The bargaining model under consideration, as its predecessors, leads to “conservation of outcome”. This means that negotiators are indifferent between settling at the outset of the game or waiting till offer equals demand. The phenomenon of instantaneous agreement has later been noted in discrete-time, complete information bargaining games with alternating offers.

Reverse logistics is the term for logistic environments with reuse of product and materials. In these systems products can be manufactured and the returned units be remanufactured. These can be considered as new. The operation of such a system is described by the return process of used products and the disposal activity. The effectivity of a reverse logistics system can be measured by its costs. The costs are inventory holding costs, manufacturing and remanufacturing costs. We investigate two stores. The demand is satisfied from the first store, where the manufactured and remanufactured items are stored. The returned products are collected in the second store and then remanufactured. The costs of this system consist of the linear holding costs for these two stores and the convex non-decreasing manufacturing and remanufacturing costs. There is no delay between the using and return process.

We consider a production planning problem for a general jobshop producing a number of products and subject to breakdown and repair of machines. The machine capacities are modeled as Markov chains. The objective is to choose the rates of production of the final products and intermediate parts on the various machines over time in order to meet the demand for the system’s production at the minimum long-run average cost of production and surplus. The problem is formulated as a stochastic dynamic program. We prove a verification theorem and derive the optimal feedback control policy in terms of the directional derivatives of the so-called potential function. Finally, we construct a potential function in the special case of a jobshop producing only one final product.

The paper shows the insufficiency of traditional production and cost theory to capture production and cost phenomena. It is argued that already for the one-person situation production theory is not able to incorporate the organizational structure and, in addition, it is unable to deal with non-cost criteria, the cost value problem, and the aggregation problem. For the multi-person situation, transactional and transformational interactions cannot be described. As a consequence, the paper extends traditional production theory and shows that the theory of hierarchical planning provides a framework that is broad enough to amend the deficiencies mentioned. In particular, it is able to capture principal agent relationships and provides a general framework for modern cost and managerial accounting.

We start with a brief review of a descriptive multi population (multi market) advertising diffusion model, where word of mouth communication takes place both within single markets and between individuals from different markets. Assuming that a monopolist for one market who seeks to maximize profits has naive expectations about the states of the neighbouring markets, we derive an optimal advertising effort for this particular monopolist. Even if the according advertising strategy is not truly optimal in the sense that it does not take into account the variations in the neighbouring markets, in numerical analyses it has so far always proven to be better than constant or random advertising efforts. The technique proposed here may hence be seen as a fair trade-off between accuracy and practicability for multi state optimal control problems as the one presented in this paper.

This paper describes the development of life expectancy at birth and at age 60 in Austria, Germany, the Netherlands, Japan and Switzerland from 1955 to 1995. With the help of the programme LIFETIME it analyses contributions of mortality differences by age groups and causes of death to the change in life expectancy at birth in the four decades for Austria and by age groups in all the countries chosen.The most important finding is that Austrian men aged 60 to 79 years have the most promising prospects of improvements in mortality reduction.

This paper develops a quantitative simulation model linking population parameters and education to land degradation, food production and distribution, and resulting in the proportion of the population which is food insecure. This model is inspired by the Vicious Circle Model of Dasgupta and others, but can be applied more generally to interactions between these variables. The model chooses a population-based approach which groups individuals into eight categories as defined by rural/urban place of residence, literacy status and food security status. Using the tools of multi-state population projections, each group is simulated by age and sex. The model links this population module to an agricultural production function and a food distribution function which considers the fact that not all people have equal access to the food produced. This model has been applied to several African countries. Here it is illustrated with an application to Burkina Faso.

One of the most challenging subjects in economic growth theory is the study of economic development ranging from Malthusian stagnation to the modern growth regime within a unified model. Recent work in this area includes Becker et al. (1999), Galor and Weil (1998), Goodfriend and McDermott (1995), Jones (1999) and Kremer (1993). While the Malthusian regime is characterised by low and non-growing per capita output and high mortality and fertility, constant growth rates of per capita output and low levels of fertility and mortality will prevail in the modern growth regime. Moreover, the relation between population growth and per capita income will be positive in the Malthusian regime and turn negative in the modern growth regime. “Demographic transition” is the theoretical concept that explains the change in the relation between per capita income and population growth.

In the very long run both the development of population size and the rise of income per capita have been characterised by stagnation or, at most, by very modest growth with cyclical ups and downs. It was not until two centuries ago that the age of industrialisation changed this trend and initiated an unprecedented increase in the standard of living (income per capita) as well as the population size. Industrialisation occurred first in Great Britain, and later in other western European countries, North America, Australia, and Japan - and eventually, most recently, in many Asian and Latin American countries. The coincidence of rapid economic and demographic growth has been a common feature in the initial stage of the modern process of development in these various countries.

Pseudo-stable and quasi-stable population models were developed by Feichtinger (1979), Coale (1972), Dinkel (1989) and others. Building on an article by Bennett and Horiuchi (1981), Preston and Coale (1982), Arthur and Vaupel (1984), and Kim (1986) further extended these models to develop general relationships for arbitrary population surfaces over age and time. In the spirit of Feichtinger and these other demographers, we present and prove in this chapter some formulas that capture the impact of mortality improvement on population growth. Goldstein and Schlag (forthcoming) also investigate this question but from a different perspective.

This paper seeks to integrate two competing notions of what drives initiation into illicit drug use, the so-called “snowball model” of sellers recruiting new customers and the “social contagion” model of current users recruiting friends into drug use. The model hypothesizes that both occur, but in distinct phases of the drug epidemic. The data are insufficient to validate the model, but inasmuch as it is valid the principal policy conclusion is that interventions that reduce the “snowball effect” in Phase I may substantially reduce consumption in Phase II and overall.

In this paper we deal with structural properties of cointegrated systems in ARMA form, where the integration and cointegration orders are arbitrary. In particular we analyze the relation between the structural properties and the Smith-McMillan form.

Albert Einstein’s request to those who create models in physics or even more general in the empirical sciences was: “ Make things as simple as possible, but not simpler! ” The opinion of further famous physicists including Werner Heisenberg was that connections in physics are based on simple principles. These ideas lead us to the following questions, which we want to ask in connection with the natural sciences as well as with other empirical sciences, including economics:

This paper provides a Bayesian analysis of a semiparametric autoregressive conditional heteroscedasticity (ARCH) model. We propose a semiparametric ARCH model using a Dirichlet process prior and show a Markov chain Monte Carlo method for the posterior inference. The model is estimated with a data set of monthly exchange rate for the Deutsche Mark to the U. S. Dollar.

Utilizing Lyapunov stability theory constructively we deduce constant screening policies for use by a public health authority in the control of an infectious disease such as gonorrhea.