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This book presents the state of the art in the relatively new field of dynamic economic modelling with regime switches. The contributions, written by prominent scholars in the field, focus on dynamic decision problems with regime changes in underlying dynamics or objectives. Such changes can be externally driven or internally induced by decisions. Utilising the most advanced mathematical methods in optimal control and dynamic game theory, the authors address a broad range of topics, including capital accumulation, innovations, financial decisions, population economics, environmental and resource economics, institutional change and the dynamics of addiction. Given its scope, the book will appeal to all scholars interested in mathematical and quantitative economics.



Chapter 1. Managing, Inducing, and Preventing Regime Shifts: A Review of the Literature

How do economic agents manage anticipated shifts in regimes? How do they try to influence or prevent the arrival of such shifts? This paper provides a selective survey of the analysis of regime shifts from an economic viewpoint, with particular emphasis on the use of the techniques of optimal control theory and differential games. We examine the concepts of regime shifts, thresholds, and tipping points in both deterministic and stochastic settings, with or without ambiguity aversion. Applications to the analysis of political regime shifts are reviewed, with a focus on the role of policy instruments such as repression, redistribution, and gradual democratization. Other dynamic games involving regime shifts that we survey include games of resource exploitation and games in industrial organization theory (R&D races, sabotage against rivals to prevent entry). We compare regime shifts under a Big Push and gradual regime changes.
Ngo Van Long

Chapter 2. Institutional Change, Education, and Population Growth: Lessons from Dynamic Modelling

This is one of the first papers that links population growth, education, and institutional change within a dynamic optimization. The basic premise is, following interesting papers of Boucekkine with different co-authors dealing with the Arab Spring, that elites manage first the ruling and then the transition to a ‘democratic’ government. We are less optimistic concerning the economic efficiency of domestic resource use and, more importantly, extend this framework by accounting for (endogenous) population growth. These extensions render the survival of any elite much less feasible than without population growth. Only a (cynical) elite worrying about the size of the population allows for a long-run and interior outcome. Therefore, the rulers and the elite have to account for a second phase in which they lose control over a country’s (financial) resources. If the elite lacks sufficient stakes in the second phase, it will ‘take the money and run’, i.e., no investment, at least close to the (endogenous) terminal time.
Gustav Feichtinger, Andreas Novak, Franz Wirl

Chapter 3. Poverty Traps and Disaster Insurance in a Bi-level Decision Framework

In this paper, we study mechanisms of poverty traps that can occur after large disaster shocks. Our starting point is a stylized deterministic dynamic model with locally increasing returns to scale possibly generating multiple equilibria paths with finite upper equilibrium. The deterministic dynamics is then overlayed by random dynamics where catastrophic events happen at random points of time. The number of events follows a Poisson process, whereas the proportional capital losses (given a catastrophic event) are beta distributed. In a setup with fixed insurance premium per unit of insured capital, a fraction of the capital might be insured, and this fraction may change after each event. We seek for an optimal strategy with respect to the insured fraction. Falling below the instable equilibrium of the deterministic dynamics introduces the possibility of ending up in a poverty trap after the disaster shocks. We show that the trapping probability (over an infinite time horizon) is equal to one when the stable upper equilibrium of the deterministic dynamics is finite. This is true regardless of the chosen amount of insured capital. Optimization then is done with the expected discounted capital after the next catastrophic event as the objective. Our model may also be useful to assess risk premia and creditworthiness of borrowers when a sequence of shocks at uncertain times and of uncertain size occurs.
Raimund M. Kovacevic, Willi Semmler

Chapter 4. Rationally Risking Addiction: A Two-Stage Approach

We extend the Becker-Murphy rational addiction model to account for a period before the onset of addiction. While during the first stage of recreational consumption of the addictive good does not imply negative effects, the second stage is analogous to the classical Becker-Murphy model. In line with neurological research, the onset of addiction is a random event positively related to the past consumption of the addictive good. The resulting multistage optimal control model with random switching time is analyzed by way of a transformation into an age-structured deterministic optimal control model. This enables us to analyze in detail the anticipation of the second stage, including the possible emergence of a Skiba point. A numerical example demonstrates that it is optimal to stop consuming the addictive good in case of an early onset (i.e. at a low level of cumulative consumption) of addiction. A late onset tends to lead into long-run addiction.
Michael Kuhn, Stefan Wrzaczek

Chapter 5. Modeling Social Status and Fertility Decisions Under Differential Mortality

Unlike most non-human social animals, the social status of humans does not consistently correlate with higher fertility and in many cases appears to suppress fertility. This discrepancy has been employed as an argument against the use of evolutionary biology to understand human behavior. However, some literature suggests that social status and its implications for survival during high-mortality events may imply that status-seeking at a cost to fertility may be an optimal strategy over the long term. Here, we propose a theoretical model, in which each generation trades-off between social status and fertility under different economic and environmental constraints. To our knowledge, the model we present here is the first to connect individual decisions of generations, strategies to maximize long-term biological fitness, and key environmental and economic conditions in a coherent stylized modeling framework. We use it, in particular, to explicate the conditions, under which the strategy of having a lower number of offspring with higher social status may result in higher biological fitness over the long term. Furthermore, we delineate sets of economic and environmental conditions, for which the dynasty shrinks and grows. As adaptation of individual preferences is costly, limited and may take generations, we argue that a sudden change in environmental or economic conditions may shift a dynasty from a growing to a declining trajectory, which may be irreversible. Also, we show that in some cases, a slight change in environmental conditions can lead to a regime switch of an optimal strategy maximizing biological fitness.
Sergey Orlov, Elena Rovenskaya, Matthew Cantele, Marcin Stonawski, Vegard Skirbekk

Chapter 6. Financing Climate Change Policies: A Multi-phase Integrated Assessment Model for Mitigation and Adaptation

We present an extended integrated assessment model (IAM) that optimizes climate financing policies over multiple phases of discrete policy action. We build on Semmler et al. (Control systems and mathematical methods in economics. Springer, 2018) which develops a single-phase model of the optimal allocation of infrastructure expenditure to carbon-neutral physical capital, climate change adaptation, and emissions mitigation. That model is solved by discretizing the optimal control problem and applying large-scale optimization techniques. A new algorithm, the arc parameterization method (APM), allows us to extend the model to multiple regimes, operationalized through a 3-phase policy environment. The first policy regime is defined by a limited set of policy tools. In the second regime, green bonds (i.e. climate-focused financing) are introduced, and in the final regime, green bonds are repaid with a new tax. We demonstrate that this multi-phase model incorporating climate, fiscal and financial policies is superior to single-phase models.
Willi Semmler, Helmut Maurer, Tony Bonen

Chapter 7. On Scientific Innovations and Constraints: A Dynamic Analysis

This paper models scientific productivity and reward, as well as the emergence and dynamics of new scientific fields. The crucial features are the individual reward that depends in addition on own abilities and the collective knowledge of a field. The overall knowledge about a field leads to increasing, at low levels (i.e., at the beginning) and at high level to decreasing. This set up renders individual as well as collective decisions, planned or as a market outcome, non-trivial. We suggest, formulate (and partly solve) problems about individual and collective behavior of scientists and study the conditions that a field emerges.
Yuri Yegorov, Franz Wirl

Chapter 8. On the Structure and Regularity of Optimal Solutions in a Differential Game with Regime Switching and Spillovers

Dynamic games with multiple regimes frequently appear in economic literature. It is well known that, in regime-switching systems, novel effects emerge which do not appear in smooth problems. In this contribution, we explore in detail a differential game with regime switching and spillovers and study the behavior of optimal trajectories both in cooperative and non-cooperative differential game settings. We explicitly derive analytic solutions and point out cases where these solutions cannot be obtained via the application of maximum principle. We demonstrate that our approach that is based on an extension of the classical maximum principle agrees with the intuition given by the hybrid maximum principle while being more convenient for solving problems at hand.
Anton Bondarev, Dmitry Gromov

Chapter 9. Optimal Switching from Competition to Cooperation: A Preliminary Exploration

In this paper, we tackle a generic optimal regime switching problem where the decision-making process is not the same from one regime to another. Precisely, we consider a simple model of optimal switching from competition to cooperation. To this end, we solve a two-stage optimal control problem. In the first stage, two players engage in a dynamic game with a common state variable and one control for each player. We solve for open-loop strategies with a linear state equation and linear-quadratic payoffs. More importantly, the players may also consider the possibility to switch at finite time to a cooperative regime with the associated joint optimization of the sum of the individual payoffs. Using theoretical analysis and numerical exercises, we study the optimal switching strategy from competition to cooperation. We also discuss reverse switching.
Raouf Boucekkine, Carmen Camacho, Benteng Zou

Chapter 10. Delaying Product Introduction in a Duopoly: A Strategic Dynamic Analysis

We analyze the optimal timing for the introduction of a new product in a duopoly. Two incumbent firms are active on a homogeneous product market and one of these firms has an option to additionally introduce a new product, thereby incurring costs of product adoption. We assume that the innovator can commit on the time of product introduction and numerically derive the optimal introduction time as well as the associated Markov-perfect equilibria for investment in production capacities. We find that depending on the initial capacities for the established product and the size of the adoption costs, three scenarios are possible for the innovator: innovating immediately, delaying introduction, and abstaining from product introduction. In case of delayed introduction, the innovator strategically reduces capacities on the established market prior to product introduction, whereas the dynamics of the non-innovator’s capacity is ambiguous. Furthermore, in this case, the firm commits to a market introduction time such that at the time of market introduction it has incentives to further delay the product adoption.
Herbert Dawid, Serhat Gezer

Chapter 11. On the Cournot-Ramsey Model with Non-linear Demand Functions

We investigate the Cournot–Ramsey differential game using a homogeneous good oligopoly with three different specifications of market demand: convex, linear and concave, allowing us to obtain fully analytical solutions in the presence of linear cost functions. The analysis of the static game predicts that the ranking of equilibrium quantities across the three market configurations is changing as the choke price changes, and this exerts an impact on the steady-state solution of the differential game, in particular on the attainment of the Ramsey golden rule. We also show that, the game being state-redundant, the degenerate feedback strategies emerging under open-loop information do not include the Ramsey equilibrium.
Luca Lambertini, George Leitmann

Chapter 12. Optimal Taxation with Endogenous Population Growth and the Risk of Environmental Disaster

This study considers a market economy where firms produce goods from labor and capital and households supply labor, rear children, save in capital, promote their members’ health and longevity by health care, and derive utility from their consumption and children, without caring of their adult offspring. There is a risk that population growth and capital accumulation trigger a lethal environmental disaster. Optimal policy is solved by a game where the government is the leader and the representative household the follower. The solution yields precautionary taxes on both capital income and health care.
Tapio Palokangas

Chapter 13. A Regime-Switching Model with Applications to Finance: Markovian and Non-Markovian Cases

A Markov regime-switching model may capture abrupt changes in the financial market efficiently, which are generated by inner or outer effects in an economy. These systems are governed by both continuous and discrete time dynamics, for which they are also called hybrid systems and have many applications in science and technology. In this work, we present a survey of financial applications under a specific semimartingale result of Markov chains. First, we present a robust portfolio strategy, which is also a zero-sum stochastic differential game. Second, optimal portfolio formulas of two investors’ collaboration are established by a nonzero-sum game. Both of these applications are solved by the Dynamic Programming Principle (DPP) approach in stochastic games. Our third result is an optimal consumption problem of a cash flow with regimes and time delay. This last financial problem is represented by the results of the necessary and sufficient stochastic Maximum Principle (MP).
E. Savku, G.-W. Weber
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