# 6th Viennese Workshop

The Sixth Viennese Workshop on Optimal Control, Dynamic Games, Nonlinear Dynamics and Adaptive Systems was held in May 1997. As can be seen from its title four different main fields have been covered. A selection of the presented papers has been published in two volumes of the Annals of Operations Research.

The focus of the articles lies on optimal control and dynamic games. They represent the state of the art in the application of optimal control theory to dynamic economic decision problems. As will be described below in more detail, there are several focuses in different areas of application ranging from classical advertising models and growth models to applications in crime prevention. With respect to the decision situation we have arranged the articles in three groups:

Decision problems with one decision maker in a deterministic world: deterministic optimal control

Decision problems with one decision maker in a stochastic world: stochastic optimal control

Game situations with at least two decision makers: dynamic games

The first block of papers use advanced techniques to analyse extensions of classical advertising models: A review of second order sufficient optimality conditions is given by Maurer and their applicability is demonstrated in two models of marketing and production. Barucci and Gozzi develop a vintage model of advertising capital by extending the Nerlove-Arrow dynamics to a continuum of goods. Favaretto and Viscolani investigate a discrete advertising model for a seasonal product where each period consists of a production subinterval and a sale subinterval while the firm can carry out advertising during the whole period.

In economics a classical but still very active area of research is the investigation of balanced economic growth paths. Wöhrmann investigates the effect of government spending and taxation by extending the two sector endogenous growth model of Lucas, and allowing for government productive investment in physical and human capital. In a numerical investigation, Greiner and Semmler study the balanced growth path as well as the transitional dynamics in an endogenous growth model with government capital market borrowing.

A rather new field of application that has however received increased interest lately is the area of safety and security. Kort, Haunschmied and Feichtinger investigate the optimal investment of a firm in equipment for protection against criminals under consideration of reputation effects of the firm in the criminal world. On the other hand, Hartl, Kort and Novak consider a firm which can reduce the probability of causing an environmental disaster by investing in safety thus reducing the risk of bankruptcy.

The section on stochastic optimal control starts with two papers on piecewise-deterministic Markov processes. Farid and Davis deal with the problem of consumption and exploration of non-renewable resources by modelling via piecewise-deterministic Markov processes, where new resources are found at random times. Jean-Marie and Tidball use impulse controls of piecewise deterministic processes to approximate the solution of multi-item single machine stochastic scheduling problems.

In cases where only few experiments are made the optimization of the expected value is often criticized. Dai Pra, Di Masi and Trivellato propose pathwise optimality concepts for stochastic infinite horizon economic control problems where only a single try (trajectory) is performed and thus the mean value approach is questionable.

The final block of contributions on stochastic optimal control extends different well known deterministic models and their results to the stochastic case. Katayama and Otha investigate the effect of uncertainty in Hartwicks rule that a country extracting an exhaustible resource and investing in reproducible capital abroad will enjoy constant consumption forever. Montrucchio and Privileggi show that also in a stochastic setting for every dynamical system a concave model exists for which the dynamical system is an optimal solution. Using this result they construct stochastic OC problems with fractal attractors. Lee and Leitmann investigate the allocation of new scientists to teaching an research careers in a stochastic context and derive robust controls for this problem.

In the last decades differential games have proved to be a very useful modelling tool to analyze the dynamic interaction between rational decision makers. The various fields of economics where this technique has been applied with great success include among others capital accumulation, resource extraction or monetary policies. The section on dynamic games in this volume demonstrates the wide range of applications of this tool. Vallee, Deissenberg and Basar study optimal cheating strategies in reversed Stachelberg games. As an example of such games -- where the leader may implement a different strategy than he has announced -- they consider the problem of unemployment and inflation. Neck models the interaction of the government and the central bank in Austria where he calibrates the model using an econometric model of the Austrian economy. Engwerda, Aarle and Plasmans analyze stabilization in the Economic and Monetary Union by solving a differential game with three players: the governments of two EMU countries and the European Central Bank.

Jorgensen and Yeung introduce an overlapping generations structure into a dynamic resource extraction game and study the effect of inter- and intragenerational competition in this framework. Capital accumulation games are analyzed by Figuieres et al.~and Stimming. Figuieres, Garderes, Michel and Rychen compare open-loop solutions of a capital accumulation game with reversible investment to the behavior induced by a central planer. An open-loop and a feedback equilibrium if capital accumulation is subject to pollution control are derived and compared in the contribution of Stimming.

In a more theoretical contribution, Tarasyev proposes grid approximations for constructing value functions and optimal feedbacks in problems of guaranteed control and gives an application in a capital accumulation game. In the last paper of the volume Pasetta deals with a dynamic extension of the divide the money game, where bribing of authorities might change the rules of the game. She does not use the technique of differential games but rather the theory of C$^*$ algebras to characterize the equilibria of the game.

The papers of the second volume are a selection of contributions to the Sixth Viennese Workshop on Optimal Control, Dynamic Games, Nonlinear Dynamics and Adaptive Systems held in Vienna in May 1997. The focus of the articles in this volume lies on Nonlinear Dynamics and Adaptive Learning. The analysis of nonlinear dynamical system in economic contexts has been one of the major new developments in economics during the last two decades. It has been shown that nonlinear dynamics arise not only in descriptive models but also as optimal paths in many economic settings. Also the analysis of expectation formation and learning in dynamic economic models leads almost generically to nonlinear dynamical systems and the growing importance of this field has further underlined the importance of a sound understanding of nonlinear dynamic phenomena for economic research. The contributions in this volume -- which all deal with nonlinear dynamical systems -- are grouped in three sections depending on whether they present nonlinear descriptive models, optimization models with complex optimal paths or learning models.

Descriptive dynamic models with nonlinearities: nonlinear dynamics

Optimization problems and dynamic games with nonlinear optimal solutions: complex optimal dynamics

Models of adaptive learning in games and application of these algorithms for optimization: learning and adaptive systems

In the first section on nonlinear dynamics we have included several descriptive models with interesting and complex dynamic properties. The variety of topics addressed in these papers illustrates the large number of fields in economics where nonlinear phenomena are of importance. They reach from endogenous growth models to price dynamics and business cycles.

Considering Pohjola's 1-D version of Goodwin's growth cycle model Sordi illustrates that the discretization of continuous-time, highly aggregated models is not only theoretically unsatisfactory, but can also be useless. Moreover, a 2-D version of the model that more closely resembles the original of Goodwin's work exhibits chaos only for unrealistic parameter values. In the paper by Chiarella and Khomin they investigate Cagan's model of monetary dynamics. Assuming that inflationary expectations are formed as a weighted average of fundamentalist and chartists expectations the resulting dynamic behaviour is studied. Chiarella and Flaschel study a high dimensional open economy monetary growth model with various forms of sluggish price and quantity adjustments. Many classical 2-D macromodels are submodels of the full 8-D model, and it is claimed that this model constitutes a rare example of an integrated and consistent dynamic macroeconomic model of high dimension. An increase of the adjustment parameters give rise to Hopf bifurcations; adding two extrinsic nonlinearitics leads to further interesting complex behaviour. In Borisov, Hutschenreiter and Kryazhimskii the steady balanced growth solutions of an endogenous growth model with knowledge exchanging economies are analyzed.

Asada reconsiders the theory of investment and finance with imperfect capital market from an analytical point of view. Krawiec and Szydlowski study the Kaldor-Kalecki business cycle model with a time delay due to a lag between investment decision and expenditure. They show that those gestation lags may generate complex behaviour and in particular persistent limit cycles. Matsumoto presents an unstable cobweb model augmented by bounds for the growth rate of the output. It is demonstrated that the long-run average profit obtained in perpetual disequilibrium can be greater than the profit in a steady state. In Bacsi and Vizvari the authors analyze a discrete-time piecewise linear cobweb model putting a lower and an upper limit to the prices. Varying the price elasticity a wide range of complex behaviour is generated. In the contribution of Weddepohl a discrete-time tatonnement process shows convergence only at low rates of adjustment and cycles and chaotic behaviour at higher rates, where prices in turn go up and down. Leonard and Nishimura address the question how wrong perceptions of the demand function influence the quantity dynamics in a Cournot duopoly.

The papers in the section 'Complex Optimal Dynamics' present studies where nonlinear complex dynamics emerge as the optimal path of a dynamic optimization problem or a dynamic game. It is well known that standard dynamic economic problems may have complex optimal paths and the three contributions in this section extend this knowledge by presenting new techniques for analyzing nonlinear optimal dynamics and proposing new areas where complex optimal paths arise. We could have included these paper in volume ?? dedicated to optimal control problems and dynamical games but decided to include them here because the main focus of these contributions is on the nonlinear phenomena resulting from the optimization exercise.

Wirl considers 1-D optimal control problems in which a second state captures a dynamic externality or spillovers. It is shown that planning attempts to internalize this externality may lead to stable limit cycles. Brito presents a general methodology for the local dynamics of 3-D optimal control models. In particular, the advantage of Brito's method lies in the explicit derivation of the eigenvalues and on the low-dimensionality of its parametrization, that allows for a geometrical approach. In Dockner, Plank and Nishimura the authors analyze a class of capital accumulation games in which current profits of each firm depend not only on its own capital stock but also on those of its rival. The existence of multiple Markov Perfect Equilibria is proved. While the strict equilibrium is characterized by simple dynamics the indifferent equilibria may exhibit Li and York-Chaos.

The analysis of learning, evolution and adaptation in the context of economic problems has attracted growing attention during the last years. In particular, game theorists have studied various models with this spirit in order to gain criteria of equilibrium selection which are derived from the evolving behavior of boundedly rational agents. Whereas the focus of most studies in this field is on dynamic equilibrium selection it has been shown that coordination failures and complex dynamics may arise if boundedly rational agents repeatedly play a game. In the section on 'Learning and and Adaptive System' both topics are addressed.

An innovative approach to dynamic equilibrium selection based on a spatial model is presented by Hofbauer. In Bischi, Gallegati and Naimzada it is shown by a global analysis of the behavior of adaptive firms in a Cournot duopoly that the assumption of the homogeneity of firms is not as innocent as usually assumed. Dawid shows in the context of the battle of the sexes game how simple word of mouth learning may lead to complex dynamics and persistently false anticipations of the agents.

The fact that dynamical systems which model learning and evolution in nature may also be successfully used as an optimization algorithm is well known and the basis for many modern heuristics. Two examples of such applications conclude the section on 'Learning and Adaptive Systems'. Bomze and Stix use an algorithm based on the optimization properties of the replicator dynamics to solve maximum clique problems. Bullnheimer, Hartl and Strauss show that the so called 'Ant System' is a promising heuristic for generating good solutions of vehicle routing problems.