Authors: Dockner E., Feichtinger G. and Novak A.J.
Abstract: A non-linear version of Parlar's production and marketing model is analysed. The optimal production and advertising policies are derived using the Maximum Principle. A stability analysis of the canonical system is carried out to determine the behaviour of production and marketing decisions over time. As the main result we show that the model might exhibit stable limit cycles as optimal policies. Using the Hopf Bifurcation Theorem the existence of a stable limit cycle is shown in a numerical example.
Authors: Feichtinger G. and Novak A.J.
Abstract: As discussed in Rauscher (1990) the optimal policies of consumption and exploitation of a renewable resource by an indebted country may be complex; cyclical and unstable paths cannot be excluded. Nevertheless, until now the existence of stable cycles has still been in question. In this note a numerical example leading to such stable cycles is presented.
Authors: Feichtinger G. and Novak A.J.
Abstract: In an article by Ngo Van Long it is shown that the optimal policy in exploiting the fish stocks of a predator-prey system by a country may be rather complex: closed orbits, oscillating convergence or monotonic convergence are possible. Nevertheless, the question about the existence of stable cycles is still unresolved. In this note, a numerical example leading to stable cycles is presented.
Authors: Feichtinger G., Kaitala V. and Novak A.J.
Abstract: We consider a resource management problem for which the management objective is to sustain both employment and stock level in an open-access common property fishery. The socially optimal policy includes utilities derived from the employment level, the resource level, and from the regulation actions. The optimal management policy includes stable limit cycles in the resource level, employment, and the regulation policy.
Authors: Novak A.J. and Feichtinger G.
Abstract: In this paper we analyze the joint determination of optimal consumption and allocation of time between learning (accumulation of human capital), working for wages (used for consumption and accumulation of financial capital) and leisure. Using Hopf bifurcation theory we are able to show that cyclical training, working, leisuring and consumption is optimal under certain constellations of parameters.
Authors: Hartl R.F., Mehlmann A. and Novak A.J.
Abstract: In this paper, we present a new approach for modelling the dynamic intertemporal confrontation between vampires and humans. It is assumed that the change of the vampiristic consumption rate induces costs and that the vampire community also derives some utility from possessing humans and not only from consuming them. Using the Hopf bifurcation theorem, it can be shown that cyclical bloodsucking strategies are optimal. These results are in accordance with empirical evidence.
Keywords: Maximum principle; limit cycles; economics of human resources; vampire myths."To the feather-fool and lobcock, the pseudo-scientist and materialist, these deeper and obscurer things must, of course, appear a grandma 's tale. " From The Vampire in Europe, by Montague Summers.
Authors: Novak A.J. and Feichtinger G.
Abstract: A new optimal control model of chemotherapeutic cancer treatment is presented including both the number of cancer cells and the level of toxicity as state variables. It is shown that under certain conditions periodic drug administration is optimal. Although the model is highly stylized and too simple to describe realistic chemotherapeutic treatment patterns it may be seen as a first step to prove periodic treatment strategies to be superior to therapies with monotonic decreasing or increasing dosage rates.
Authors: Feichtinger G., Novak A.J. and Wirl F.
Abstract: This paper starts from well-behaved (i.e. concave) one-state-variable optimal control models. The crucial feature is that indigenous growth is present. The replacement of the control by an adjustment process and presumably penalizing these adjustments may convert stable (and only stable) fixed point equilibria into limit cycles. Indeed, given positive growth and proper externalities (such that discounting exceeds growth), one can generate stable limit cycles with ease, e.g., for a separable framework. Examples from such diverse fields as renewable resources, optimal saving and public choice highlight the economic applicability.
Keywords: Stable limit cycles; Hopf bifurcation; Renewable resources; Optimal saving; Subsidies.JEL: C61, C62
Authors: Feichtinger G. and Novak A.J.
Abstract: Empirical evidence indicates that a given moderate number of ads per year may achieve higher effect when concentrated in flights than when spread equally. In the control -theoretic literature a few approaches have been developed for which the optimal policy is a pulsing schedule. The present communication analyses a two-state diffusion advertising model for repeat purchasing. Recognizing the interaction between customers and potential buyers, the optimal advertising policy turns out to be a persistent periodic oscillation. This provides a further interesting example in marketing for the existence of a stable limit cycle.
Keywords: Optimal control; Advertising by word-of-mouth; Stable limit cycle; Hopf bifurcation; Diffusion model;
Authors: Feichtinger G. and Novak A.J.
Abstract: Ancient Chinese history reveals many examples of a cyclical pattern of social development connected with the rise and the decline of dynasties. In this paper, a possible explanation of the periodic alternation between despotism and anarchy by a dynamic game between the rulers and the bandits is offered. The third part of the society, the farmers, are dealt with as a renewable resource which is exploited by both players in a different manner. It is shown that the Nash solution of this one-state differential game may be a persistent cycle. Although we restrict the analysis to open-loop solutions, this result is of interest for at least two reasons. First, it provides one of the few existing dynamic economic games with periodic solutions. Second, and more important, the model is an example of a three-dimensional canonical system (one state, two costates) with a stable limit cycle as solution. As far as we see, our model provides up to now the simplest (i.e., lowest dimensional) case of a persistent periodic solution of an intertemporal decision problem.
Keywords: Differential games,open-loop Nash solutions, Hopfbifurcations, stable limit cycles.
Authors: Feichtinger G. and Novak A.J.
Abstract: The aim of the present paper is to illustrate how extremely complex patterns may be generated in a simple model of educational planning. In particular, we will show that certain dependencies of the flow rates on the teacher/student ratio imply nonlinearities which are substantial enough to generate erratic behaviour of the time paths. The main message is that chaos in educational planning may result from assumptions which are indeed qualitatively realistic but which are quantitatively exaggerated.
Authors: Novak A.J., Kaitala V. and Feichtinger G.
Abstract: In this paper we analyse a fishery resource exploitation model in which a single firm or a cartel has leased the rights to manage the resources independently. Two variables, resource level and the capital level, determine the dynamics of the resource system. The leasing contract includes an incentive for the agent to maintain the resource level high. The main result is that sole-agent resource management and efficiency of the resource use do not necessarily imply that the fishery is stabilized at a unique steady state level. Instead, the optimal resource exploitation may lead to periodic capital investments in fishing vessels and gear which in turn causes cycles in the resource economy. We show analytically that nonzero discount rate and low capital depreciation rate both favor the conditions under which periodic optimal solutions may occur. Simulation results related to a Baltic herring fishery are used to illustrate the results.
Authors: Feichtinger G. and Novak A.J.
Abstract: In this paper we present a simple time-continuous behavioural model of habit formation. Addictive behaviour is damped by a threshold which adapts itself to the habit. This adaptive behaviour of the threshold may lead to periodic fluctuations of the consumption rate, the habit and the threshold. It turns out that both a low adjustment rate of the threshold as well as a steep consumption function favour oscillatory patterns.
Keywords: nonlinear dynamical systems, limit cycles, threshold models, habit formation, addiction.J.E.L. classification: C62, D11
Authors: Chand S., Moskowitz H., Novak A.J., Rekhi I., Sorger G.
Abstract: Management of process improvement activities is an essential part of the manufacturing strategy of a firm to remain globally competitive in the long run. This paper considers a manufacturing environment where process improvement activities require use of the productive capacity of the firm in addition to other investments. Thus the firm must allocate its productive capacity between production activities and improvement activities. The output of production activities is used to meet customer demand. Process improvement activities improve the quality of the output, which in turn leads to lower quality related costs (both internal and external) and possibly lower per-unit production cost. It is assumed that the demand function is downward sloping and that revenue is a concave function of output. A continuous-time, finite-horizon, profit maximization, resource allocation model is developed to find an optimal time path for process improvement activities and production activities. Computational results are provided to study the effect of various problem parameters on the optimal decisions.
Authors: Wirl F., Novak A.J., Feichtinger G. and Dawid H.
Abstract: This paper introduces a differential game that captures the strategic interactions between the ruling class, their hidden misbehavior (corruption, adultery, drinking, drugs, etc.), and the uncovering pursued by a tabloid press with market power. Aside from investigating these topical issues, the purpose of this paper is to draw attention to two formal aspects that are so far by and large neglected in the literature on differential games: First, the open loop Nash equilibrium need not be unique, i.e. an entire family of solutions istead of a single pair of strategies may exist, similar to nonlinear Markov strategies in linear quadratic differential games; yet contrary to nonlinear Markov strategies, all different Nash equilibrium strategies converge, if they converge at all, to the same and unique long run strategies. Second, the long run strategies need not be constants but may follow a (stable) limit cycle. These two results are established for a simple (but nonlinear), one state variable differential game so that this potential complexity holds already in the lowest conceivable dimension of one state and two costates. Therefore, a limit cycle, which is derived using the Hopf bifurcation theorem and then numerically calculated, is only possible, if the open loop Nash equilibrium is not unique. ln fact, indeterminacy is a generic property of the investigated game.
Authors: Pötscher B.M. and A.J. Novak
Abstract: Frequently, parameter estimation is preceded by a data-driven model selection procedure. In such a case traditional distribution theory based on an a priori given model is no longer valid. In Pötscher (1991) the asymptotic distribution of estimators that are preceded by model selection has been studied. In this paper we study the small sample distribution and, in particular, evaluate the accuracy of the approximation provided by the asymptotic distribution in small samples. We also show, how one of the assumptions in Pötscher (1991) can be weakened substantially.
Keywords: Model selection, post-model-selection estimator, AIC.
Authors: Asada T., Semmler W. and Novak A.J.
Abstract: The paper shows that the Romer (1990) model of endogenous growth, its version for the social planning problem, has a unique steady state and is saddle path stable. The steady state can be monotonically reached by an appropriate choice of the initial values of the control variables if the initial values of the state variables lie in the vicinity of the steady state. For the original version as well as for generalizations of it we demonstrate that Hopf bifurcation can be excluded. The technique employed here may be useful in studying other variants of endogenous growth models. We also demonstrate in the paper that the market variant of the Romer model as, for example, presented in Benhabib, Perli and Xie (1994), does admit Hopf-bifurcation and stable periodic solutions.
Authors: Feichtinger G., Jorgensen S. and Novak A.J.
Abstract: The paper is concerned with a celebrated collection of love poems, the 14th century Italian poet Francis Petrarch's Canzoniere. A striking feature of these poems is the emotional ups and downs experienced by Petrarch and his platonic mistress Laura. Recently, attempts have been made to model these emotional swings by catastrophe theory or nonlinear differential equations. This paper takes a different approach. Starting with a pair of differential equations that model the dynamics of the emotions of the two individuals, we formulate an optimal control problem. A key hypothesis of this problem is that Petrarch was a rational addict of his desire for Laura. With specific functional forms and parameter values we identify a stable limit cycle that gives a representation of the oscillating emotions of Laura and Petrarch.
Keywords: Differential equations; Optimal control theory; Addictive behavior; Limit cycle; Love affairs.
Authors: Hartl R. F., Kort P.M. and Novak A.J.
Abstract: This paper studies the optimal behavior of a firm over time that faces the probability of causing an environmental disaster by its activities. Here, we can think of explosions in the chemical industry, oil tankers that lose oil, etc. In the static literature, a distinction is made between small firms and large firms. Small firms are calles undercapitalized in the sense that the firm cannot pay for the accident because the accidental damage exceeds the value of the firm. As soon as this happens, the firm is declared bankrupt. This gives an incentive to spend part of the safety budget on distributing dividends to the shareholders under the motto: pay dividends as long as it is still possible.
In this paper, we extend the static framework to a dynamic one. The major reason is to find out under what circumstances it can be optimal for a small firm to expand in order to become a large firm that has the means to fully pay for the damage caused by an environmental accident. Admittedly, this paper is only a first step in reaching this goal. Here, we determine the optimal safety expenditures and the convergence behavior of long-run investment policies.
Author: Novak A.J.
Abstract: 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.
Keywords: Hopf bifurcation theory; Stable limit cycles; Investment.
Author: Novak A.J.
Abstract: In many manufacturing applications the regulation of a quality process to attain a given target is of central interest. In a recent paper Liu and Nam (1999) present a model of optimal quality regulation. The underlying quality process evolves due to regulation actions and superimposed random disturbances. Optimal regulation is sought as to minimize the regulation costs and the mean squared deviation from the desired target over a finite time horizon. Unfortunately the model is incorrectly analyzed in Liu and Nam (1999) and we therefore present the correct results in the following paper.
Keywords: Quality improvement; Optimal control; Diffusion models
Authors: G. Feichtinger, Hartl R.F., Kort P.M. and Novak A.J.
Abstract: In some countries, for instance Egypt, terrorists try to hurt the country's income from the tourism industry by violent actions against tourists. Another example are actions of the Kurds to bring tourism down in the east of Turkey. This paper is a first attempt to model some relevant aspects in that prey-predator relation. The country tries to maximize profits from the tourism industry, where profit is defined by the difference between revenue from the tourism industry and the sum of expenditures on tourism industry investments and expenditures on enforcement associated with reducing terrorism. It turns out that for reasonable parameter values the optimal trajectory exhibits a cyclical strategy. The interpretation is that, after starting out with a low number of tourists and terrorists, tourism investments are undertaken to increase tourism. This attracts terrorists which reduces the effect of tourism investments. Therefore investment declines and so does the number of tourists. This makes it less attractive for terrorists to act so we are back in the original situation, where the whole thing starts again.
Keywords: Hopf bifurcation, Limit cycles, Tourism industry, Law enforcement.
Authors: Liski M., Kort P., Novak A.J.
Abstract: We consider optimal fishery management under the assumption of increasing returns that is supported by previous empirical evidence. We improve the tractability and realism of the previous approaches by introducing flow adjustment costs on changes in harvest rate. Our framework is the first to provide a link between stable limit cycle policies and increasing returns in harvesting. The type of the harvest policy depends on flow adjustment costs: for relatively costly adjustments the usual steady state harvest policy is conceivable, whereas for relatively cheap adjustments the harvest policy is cyclical. We also show a connection between chattering control policies and limit cycles, which helps us to develop a clear economic meaning for cyclical harvesting.
Keywords: Nonconvexities; Adjustment costs; Hopf bifurcation; Renewable resources.
Authors: Dawid H., Feichtinger G., Novak A.J.
Abstract: In this paper we use a differential game analysis to study the dynamic strategic interaction between a criminal gang extorting money from local shop-owners and the local police force. In particular, we are interested in characterizing which factors are important in determining whether the capital stock of local shop-owners keeps growing in spite of extortion or the criminal activity leads to a phase of stagnation of the local economy. A Markov perfect equilibrium of the game is characterized in order to address this question and several policy implications are derived to facilitate growth in regions affected by extortion.
JEL Classification: C73, D92, K42.
Keywords: Extortion, Differential Games, Capital Accumulation.
Authors: Kort P.M., Greiner A., Feichtinger G., Haunschmied J.L., Novak A.J. and Hartl R.F.
Abstract: Efficient investment programs in touristic infrastructure have to take into consideration that any kind of tourism reduces the environmental quality. Since pollution shows negative repercussions as concerns the attractiveness of a touristic region, tourism planners have to determine a trade-off between adequate services for tourists and a clean environment. To deal with this problem in a dynamic context, a three-state optimal control model is formulated. It turns out that, even if pollution reduction is not a goal in itself, the profit-maximizing tourism industry should care for ecological conservation. The paper further shows that persistent periodic investment policies are optimal for realistic parameter sets, and provides an economic intuition for such behavior. From an economic point of view, this result implies that expansionary periods with high investment are followed by periods of stagnation with low investment.
Keywords: Tourism Industry, Environment, Optimal Control, Pontryagin's Maximum Principle, Limit Cycles.
Authors: Hartl Richard F., Kort Peter M. and Novak A.J.
Abstract: This paper considers a dynamic model of the firm in which the firm has debt financing as an additional means to provide funds for investments. It is assumed that it is more difficult for the firm to get a loan when the debt-equity ratio is large. Consequently it is imposed that the interest rate increases with this ratio. The model under consideration has been studied before by Steigum (1983, Econometrica). He obtained that the problem reduces to a strictly concave optimal control problem with a unique saddle point. In this paper we prove this result in a different way and provide a generalization to his model by extending the objective functional such that positive utility is assigned to dividends as well as equity. Despite the fully concave setting, we show that the optimal solution of a mainly equity oriented firm has history dependent equilibria.
Authors: Hartl R.F., Novak A.J., Rao A.G. and Sethi S.P.
Abstract: We consider a market consisting of two populations, termed rich and poor for convenience. If a product is priced such that it is very expensive for the poor, but affordable to the rich, then it becomes a status symbol for the poor and this makes it more desirable for the poor. At a lower price the product is affordable by both populations. However, as more of the poor buy the product, it ceases to be a status symbol, and becomes less appealing to the rich. We present a two-state non-linear optimal control problem that aims to obtain profit-maximizing prices over time in this environment. We find that there are three categories of optimal price paths. One is status symbol pricing with high initial price, declining over time. The other two are mass market pricing, with price declining in one, and increasing and then decreasing in the other.
Keywords: Optimal control, marketing, pricing, market diffusion, aspirational group.
Authors: Faria Joao R and Novak A.J.
Abstract: A naive model of cannibalism assumes that it decreases intra-species competition, increasing per capita supply of resources. As a result, there is a cycle between prey population and carrying capacity of the environment, which oscillates around cannibals' population. The model is applied to examine the behaviour of firms in an oligopolistic industry. From this exercise emerges a general model of cannibalism which shows the economic rationale for the existence of cannibalistic strategies and has stronger properties than the naive model.
JEL Classification: L11, L19, L29, J51.
Keywords: Cannibalism, Entry and exit, Predation, Monopolization strategies.
Authors: Feichtinger G., Novak A.J.
Abstract: The purpose of this paper is to study efficient measures to combat terror. To choose efficient actions against terror organizations, the response of terrorists as well as the public has to be taken into consideration. Thus, an appropriate framework to study the intertemporal strategic interactions of Western governments, and terror organizations is dynamic game theory.
Using the open-loop Nash solution concept it turns out that the system may exhibit long-run persistent oscillations. However, transitory behavior is nonunique.
Keywords: Differential games, Counterterror measures, Limit cycles, Indeterminacy.
JEL Classification: C73, H56.
Authors: Wirl F., Novak A.J., Hof F.X.
Abstract: This paper departs from the standard open-economy Ramsey model and introduces additional concerns for wealth, status and Easterlin's (2001) hypothesis that consumption changes, in particular increases, are important and not only the level. These extensions induce first of all interior steady states, which are lacking in the standard model, multiple steady states (separated by thresholds leading to history dependence) and limit cycles. The existence of cyclical consumption patterns could provide a so far ignored source for real business cycles. Surprisingly, introducing status conferred by private wealth or conspicuous consumption has no effect despite the involved externalities (the outcomes are observationally equivalent) as long as the social influence associated with the status externalities remains moderate.
Keywords: wealth, valuing consumption increases, social influence, stability, limit cycles.
JEL Classification: E21, C61, D91.
Authors: Novak A.J., Feichtinger G., Leitmann G.
Abstract: The question of how best to prosecute the 'war on terror' leads to strategic interaction in an intertemporal setting. We consider a non-zero sum differential game between a government and a terrorist organisation. Due to the state-separability of the game we are able to determine Nash and Stackelberg solutions in analytic form. Their comparison as well as the sensitivity analysis deliver interesting insight into the design of efficient measures to combat terror.
Keywords: Differential games, Counterterror measures, Nash solution, Stackelberg solution, State-separable games.
JEL Classification: C73, H56.
Authors: Feichtinger G., Novak A.J., Veliov V.M.
Abstract: The main result in this short note is that the integral form of the Leitmann-Stalford sufficiency conditions can be verified for a class of optimal control problems whose Hamiltonian is not concave with respect to the state variable. The main requirement for this class of problems is that the dynamics is sufficiently dissipative. An application to a Stackelberg differential game between a producer and a developer is exemplified. Using our result we show that the necessary conditions implied by Pontryagin's maximum principle are also sufficient. This allows a complete characterization of the solution.
Keywords: Optimal control, sufficient optimality conditions, differential games, Stackelberg equilibrium.
Authors: Jo„o Ricardo Faria, Damien Besancenot, Andreas J. Novak
Abstract: This paper deals with two elements of Thomas Kuhn (1962) ideas regarding paradigm: Depletion and resiliency. The possibility of paradigm depletion taking resilience into account, given the hierarchy among scientists, is modeled as a Stackelberg differential game between editors [leaders] and authors [followers]. A number of results emerge from the model: i) Paradigm depletion can be optimal; ii) The optimal editorís shadow price of potential knowledge must be non-positive, if it is positive, the editor is just a keeper of the orthodoxy rather than a scientist; iii) Editorís and/or researcherís impatience is always bad for science; iv) In equilibrium editorís behavior does not matter for optimal research effort, while only editorís behavior matter for the paradigm.
Keywords: Paradigm; Economics of science; Research effort.
JEL Classification: C79; J39; Q39.
Authors: Jonathan P. Caulkins, Gustav Feichtinger, Richard F. Hartl, Peter M. Kort,. Andreas J. Novak, Andrea Seidl.
Abstract: Becker and Murphy (J Polit Econ 96(4):675-700,1988) have established the existence of unstable steady states leading to threshold behavior for optimal consumption rates in intertemporal rational addiction models. In the present paper a simple linear-quadratic optimal control model is used to illustrate how their approach fits into the framework of multiple equilibria and indifference-threshold points. By changing the degree of addiction and the level of harmfulness we obtain a variety of behavioral patterns. In particular we show that when the good is harmful as well as very addictive, an indifference-threshold point, also known in the literature as a Skiba point, seperates patterns converging to either zero or maximal consumption, where the latter occurs in the case of a high level of past consumption. This implicitly shows that an individual needs to be aware in time of these characteristics of the good. Otherwise, he/she may start consuming so much that in the end he/she is totally addicted.
Keywords: Optimal control, indifference points, history dependence, rational addiction.
Authors: Jonathan P. Caulkins, Gustav Feichtinger, Dieter Grass, Richard F. Hartl, Peter M. Kort, Andreas J. Novak, Andrea Seidl.
Abstract: We present a novel model of corruption dynamics in the form of a nonlinear optimal dynamic control problem. It has a tipping point, but one whose origins and character are dinstinct from that in the classic Schelling (1978) model. The decision maker choosing a level of corruption is the chief or leader who presides over a bureaucracy whose state of corruption is influenced by the leader's actions, and whose state in turn influences the pay-off for the leader. The policy interpretation is somewhat more optimistic than in other tipping models, and there are some surprising implications, notable that reforming the bureaucracy may be of limited value if it takes its cues from a corrupt leader.
Keywords: Corruption, tipping point, Skiba point, optimal control.
JEL Classification: C79; J39; Q39.
Authors: Wirl F., Novak A.J.
Abstract: This paper provides a new and surprising reason for growth, namely costs. More precisely, adding adjustment costs of the control to a one-dimensional, strictly concave optimal control problem does not affect the steady state(s). Then, sufficiently high adjustment costs turn an interior and saddlepoint stable steady state of the original, one-state variable model into a source that can lead to unbounded growth. Given a version of the open economy Ramsey model, the initial conditions determine whether unbounded growth or impoverishment results. Related to this threshold property, the strict concave two-state variable control model allows for thresholds even if it has a unique and stable steady state.
Keywords: growth, adjustment costs, thresholds, strict concave dynamic optimization, thresholds.
JEL Classification: E21, C61.
Authors: Jonathan P. Caulkins, Gustav Feichtinger, Richard F. Hartl, Peter M. Kort, Andreas J. Novak, Andrea Seidl.
Abstract: We consider a semi-rational addiction model in which the user has perfect foresight over all things within the user's control, but not necessarily with respect to exogenous parameter shocks, e.g., those stemming from changes in national policy. We show that addictive substances are more likely to have state-dependent solution trajectories, and that in turn can create path dependence at the macro-policy level; in particular, legalization may be an irreversible experiment. Also, in this model, shifting from a nuanced policy that differentiates between high and low intensity users, to a tougher one where the government makes life hard for every user reduces initiation considerably. However, it also may have perverse effects. In particular, we show that making the policy tougher in this way could drive some people from a "happy" stable saddle point equilibrium with moderate consumption into increasing rather than reducing their consumption and addiction stock. So implementing zero tolerance policies may increase rather than reduce aggregate drug use, depending on the population's distribution of parameter values and initial consumption stocks. Further, we consider the impact of announcing a policy change.