Stefan Wrzaczek

Optimal choice of health and retirement in a life-cycle model

We examine within a life-cycle model the simultaneous choice of health care and retirement (together with consumption). Health tends to have an impact on retirement through morbidity, determining earnings and the disutility of work, and through longevity, determining the need to accumulate retirement wealth. Conversely, the age of retirement drives the demand for health care through the value of survival and the value of morbidity reductions. We characterise the optimal relationship between health expenditure and retirement and apply our model to analyse the effects of moral hazard in the annuity market. While moral hazard always induces excessive health investments and an excessive duration of working life, it also triggers an excessive level of consumption if the impact of health on the disutility of work is sufficiently large. We examine a transfer scheme and mandatory retirement as policies towards curtailing moral hazard. Numerical analysis illustrates the role of moral hazard in shaping the life-cycle allocation.

Externalities in a life-cycle model with endogenous survival

We study socially vs individually optimal life cycle allocations of consumption and health, when individual health care curbs own mortality but also has a spillover effect on other persons’ survival. Such spillovers arise, for instance, when health care activity at aggregate level triggers improvements in treatment through learning-by-doing (positive externality) or a deterioration in the quality of care through congestion (negative externality). We combine an age-structured optimal control model at population level with a conventional life cycle model to derive the social and private value of life. We then examine how individual incentives deviate from social incentives and how they can be aligned by way of a transfer scheme. The age-patterns of socially and individually optimal health expenditures and the transfer rate are derived. Numerical analysis illustrates the working of our model.

Quality effects in different advertising models - An impulse control approach

In this paper, we integrate quality as a control variable in three classical dynamic optimal control models of advertising, namely, Nerlove–Arrow, Vidale–Wolfe and Ozga models. Quality refers to design quality, which may deteriorate over time. We assume that decisions in quality improvement can only be made at some exogenously given instants of time, and consequently we use the formalism of impulse optimal control to determine optimal advertising and quality investments. We report numerical results for the three models and discuss the impact of adding quality on the results.

The reproductive value in distributed optimal control models.

We show that in a large class of distributed optimal control models (DOCM), where population is described by a McKendrick type equation with an endogenous number of newborns, the reproductive value of Fisher shows up as part of the shadow price of the population. Depending on the objective function, the reproductive value may be negative. Moreover, we show results of the reproductive value for changing vital rates. To motivate and demonstrate the general framework, we provide examples in health economics, epidemiology, and population biology.

R&D for green technologies in a dynamic oligopoly: Schumpeter, Arrow and inverted-U’s

We extend a well-known differential oligopoly game to encompass the possibility for production to generate a negative environmental externality, regulated through Pigouvian taxation and price caps. We show that, if the price cap is set so as to fix the tolerable maximum amount of emissions, the resulting equilibrium investment in green R&D is indeed concave in the structure of the industry. Our analysis appears to indicate that inverted-U-shaped investment curves are generated by regulatory measures instead of being a ‘natural’ feature of firms’ decisions.

Optimal control of a terror queue

The task of covert intelligence agents is to detect and interdict terror plots. Kaplan (2010) treats terror plots as customers and intelligence agents as servers in a queuing model. We extend Kaplan’s insight to a dynamic model that analyzes the inter-temporal trade-off between damage caused by terror attacks and prevention costs to address the question of how many agents to optimally assign to such counter-terror measures. We compare scenarios which differ with respect to the extent of the initial terror threat and study the qualitative robustness of the optimal solution. We show that in general, the optimal number of agents is not simply proportional to the number of undetected plots. We also show that while it is sensible to deploy many agents when terrorists are moderately efficient in their ability to mount attacks, relatively few agents should be deployed if terrorists are inefficient (giving agents many opportunities for detection), or if terrorists are highly efficient (in which case agents become relatively ineffective). Furthermore, we analyze the implications of a policy that constraints the number of successful terror attacks to never increase. We find that the inclusion of a constraint preventing one of the state variables to grow leads to a continuum of steady states, some which are much more costly to society than the more forward-looking optimal policy that temporarily allows the number of terror attacks to increase.

Optimal firm growth under the threat of entry.

Highlights • Anticipation is different in the open-loop commitment scenario compared to the case of Markovian strategies.• Entry costs level where entry accommodation passes into entry deterrence is lower in the Markov perfect case.• Incumbent’s capital stock level needed to deter entry is a hump shaped function of the entry time.

Coping with inefficiencies in a New Economic Geography model: The unintended consequences of policy interventions

This article introduces a social planner version of a central microfounded New Economic Geography model for explicitly answering whether the symmetric equilibrium of the decentralized market economy is socially desirable. We find that savings incentives are too weak, resulting in an inefficiently low capital stock and therefore an inadequate number of product varieties. We derive the appropriate subsidy and taxation scheme to remedy these distortions. Interestingly, implementing the associated policies crucially impacts on the stability of the symmetric equilibrium and has the potential to result in unintended agglomeration processes.

Social optimality in the constructed-capital model

In the constructed-capital model, the steady state is derived under the assumption that each individual behaves optimally. Contrasting to this decentralized approach, in this paper we derive the first-best outcome a central planner would choose. The results show that agglomeration is socially not optimal, irrespective of the level of trade barriers. Furthermore, the differences in the explicit solutions of both approaches are highlighted.

Optimizing Counter-terroristic Operations in an Asymmetric Lanchester Model

Abstract Effective counter-terroristic operations require efficient intelligence. Recently, Kress and Szechtmann (2009) analyzed the impact of the level of intelligence to the outcome of a fight between the government and a small insurgent group in a descriptive, asymmetric Lanchester-style model. The present paper uses optimal control theory, particularly Pontryagins maximum principle, to find the optimal intelligence policy and recruitment rate of the government forces. In particular, we intend to study multiple long-run steady states and complex behavior of the optimal solution paths. Moreover, a differential game between the government and the terrorist is proposed.