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We study medical progress within a two-sector economy of overlapping generations subject to endogenous mortality. Individuals demand health care with a view to lowering mortality over their life-cycle. We characterise the individual optimum and the general equilibrium, and study the impact of a major medical innovation leading to an improvement in the effectiveness of health care. We find that general equilibrium effects dampen strongly the increase in health care usage following medical innovation. Moreover, an increase in savings offsets the negative impact on GDP per capita of a decline in the support ratio. Finally, we show that the reallocation of resources between the final goods and health care sector, following the innovation, plays a crucial role in shaping the general equilibrium impact.

In determining the durability of its product a firm faces a trade off. Performing a policy of planned obsolescence by making their products less durable implies that the consumer needs to replace them earlier, which thus enhances demand. However, a lower quality of the product will result in a lower reputation, which in turn will affect demand negatively. In many cases, the government protects the consumer by implementing a warranty period. Our paper studies how a firm should optimally deal with this trade off and react to government policy. We obtain the following results. First, we find that the length of the warranty period has an inverted U-shaped effect on the product life time. Second, if more consumers are aware of the existence of a warranty period and ask for a free product replacement, this will increase the product life time. Third, increasing uncertainty about the breakdown of the product also has an inverted U-shaped effect on the product lifetime.

Most nations have responded to the COVID-19 pandemic by locking down parts of their economies starting in early 2020 to reduce the infectious spread. The optimal timing of the beginning and end of the lockdown, together with its intensity, is determined by the tradeoff between economic losses and improved health outcomes. These choices can be modelled within the framework of an optimal control model that recognises the nonlinear dynamics of epidemic spread and the increased risks when infection rates surge beyond the healthcare system’s capacity. Past work has shown that within such a framework very different strategies may be optimal ranging from short to long and even multiple lockdowns, and small changes in the valuation on preventing a premature death may lead to quite different strategies becoming optimal. There even exist parameter constellations for which two or more very different strategies can be optimal. Here we revisit those crucial questions with revised parameters reflecting the greater infectivity of variants such as the “UK variant” of the SARS-CoV-2 virus and describe how the new variant may affect levels of mortality and other outcomes.

During the COVID-19 pandemic countries invested significant amounts of resources into its containment. In early stages of the pandemic most of the (nonpharmaceutical) interventions can be classified into two groups: (i) testing and identification of infected individuals, (ii) social distancing measures to reduce the transmission probabilities. Furthermore, both groups of measures may, in principle, be targeted at certain subgroups of a networked population. To study such a problem, we propose an extension of the SIR model with additional compartments for quarantine and different courses of the disease across several network nodes. We develop the structure of the optimal allocation and study a numerical example of three symmetric regions that are subject to an asymmetric progression of the disease (starting from an initial hotspot). Key findings include that (i) for our calibrations policies are chosen in a “flattening-the-curve,” avoiding hospital congestion; (ii) policies shift from containing spillovers from the hotspot initially to establishing a symmetric pattern of the disease; and (iii) testing that can be effectively targeted allows to reduce substantially the duration of the disease, hospital congestion and the total cost, both in terms of lives lost and economic costs.

Nations struggled to decide when and how to end COVID-19 inspired lockdowns, with sharply divergent views between those arguing for a resumption of economic activity and those arguing for continuing the lockdown in some form. We examine the choice between continuing or ending a full lockdown within a simple optimal control model that encompasses both health and economic outcomes, and pays particular attention to when need for care exceeds hospital capacity. The model shows that very different strategies can perform similarly well and even both be optimal for the same relative valuation on work and life because of the presence of a so-called Skiba threshold. Qualitatively the alternate strategies correspond to trying essentially to eradicate the virus or merely to flatten the curve so fewer people urgently need healthcare when hospitals are already filled to capacity.

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.

We analyze a market in which advertising is the dominant marketing tool to create market share. We assume that an incumbent firm dominates the market during an initial stage, and that a new competitor is going to enter the market. In particular, we analyze the different advertising policies that the incumbent firm can adopt, before and after the entry of the rival. We explore three possible behaviours. In the first scenario the firm knows that the competitor will arrive at a given instant. In the second one we assume the original firm to be surprised, in the sense that it does not anticipate the entry of the opponent either because it does not expect the competitor to arrive, or it is not prepared to react before the entry takes place. Finally, in the third scenario, the original firm knows that the competitor will enter at a constant rate. We characterize a differential game model and compare the firms’ behaviours in a strategic perspective.

In general, the spreading of gender egalitarianism has often been associated with a decline in fertility. However, recently a rebound in fertility has been observed in several industrialized countries. A possible explanation of this trend may be the spread of egalitarian values that induced institutional changes - such as expansion of child care facilities and father leave - and also changes in norms and values - such as gender equity in the distribution of domestic work - that foster the combination of parenthood and the egalitarian lifestyle. To study the diffusion from traditional to egalitarian gender-behavior and its impact on fertility a two-dimensional system of nonlinear ordinary differential equations is used. It is shown that the long-run development of the total fertility within a population not only depends on key parameters such as the pace of diffusion of egalitarianism and the extent to which social interactions affect the egalitarians’ birth rates, but also on the initial number of traditionalists and egalitarians. One of the main purposes of the present paper is to illustrate how bifurcation theory can be used to study the process of increasing gender equality and its implications on fertility.

One of the principal ways nations are responding to the COVID-19 pandemic is by locking down portions of their economies to reduce infectious spread. This is expensive in terms of lost jobs, lost economic productivity, and lost freedoms. So it is of interest to ask: What is the optimal intensity with which to lockdown, and how should that intensity vary dynamically over the course of an epidemic? This paper explores such questions with an optimal control model that recognizes the particular risks when infection rates surge beyond the healthcare system's capacity to deliver appropriate care. The analysis shows that four broad strategies emerge, ranging from brief lockdowns that only ``smooth the curve'' to sustained lockdowns that prevent infections from spiking beyond the healthcare system's capacity. Within this model, it can be optimal to have two separate periods of locking down, so returning to a lockdown after initial restrictions have been lifted is not necessarily a sign of failure. Relatively small changes in judgments about how to balance health and economic harms can alter dramatically which strategy prevails. Indeed, there are constellations of parameters for which two or even three of these distinct strategies can all perform equally well for the same set of initial conditions; these correspond to so-called triple Skiba points. The performance of trajectories can be highly nonlinear in the state variables, such that for various times t, the optimal unemployment rate could be low, medium, or high, but not anywhere in between. These complex dynamics emerge naturally from modeling the COVID-19 epidemic and suggest a degree of humility in policy debates. Even people who share a common understanding of the problem's economics and epidemiology can prefer dramatically different policies. Conversely, favoring very different policies is not evident that there are fundamental disagreements. (c) 2021 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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.

We present models of differential terror queue games, wherein terrorists seek to determine optimal attack rates over time, while simultaneously the government develops optimal counterterror staffing levels. The number of successful and interdicted terror attacks is determined via an underlying dynamic terror queue model. Different information structures and commitment abilities derive from different assumptions regarding what the players in the game can and cannot deduce about the underlying model. We compare and explain the impact of different information structures, i.e., open loop, closed loop, and asymmetric. We characterize the optimal controls for both the terrorists and the government in terms of the associated state and costate variables and deduce the costate equations that must be solved numerically to yield solutions to the game for the different cases. Using recently assembled data describing both terror attack and staffing levels, we compare the differential game models to each other as well as to the optimal control model of Seidl et al. (Eur J Oper Res 248:246–256, 2016). The paper concludes with a discussion of the lessons learned from the entire modeling exercise.

The paper presents a transformation of a multi-stage optimal control model with random switching time to an age-structured optimal control model. Following the mathematical transformation, the advantages of the present approach, as compared to a standard backward approach, are discussed. They relate in particular to a compact and unified representation of the two stages of the model: the applicability of well-known numerical solution methods and the illustration of state and control dynamics. The paper closes with a simple example on a macroeconomic shock, illustrating the workings and advantages of the approach.

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.

Some areas of science face the problem that many academics prefer the private sector over academia. This negatively affects the quality and the quantity of the research output as well as the availability of competent lecturers in these areas. The present paper investigates by means of an optimal control model how the reward of competencies in research and teaching in the private sector affects investments into these skills as well as the decision on whether and when to optimally leave academia. As the decision between academia and industry is obvious if a scientist has strong preference for either, we focus on scenarios where this is not the case. We notably show that the dynamic trade-off between academia and industry results in various forms of brain drain. In this regard, we first confirm that if academic competencies are well rewarded in the private sector, the most competent academics will leave academia. Further, we find scenarios where a scholar with intermediate competencies will try to improve his or her skills as much as possible before leaving academia and scenarios in which it is optimal to not put much effort into work and let competencies slowly depreciate before leaving. Even if scientists are highly skilled and motivated to stay, if poor working conditions do not support knowledge acquisition, competencies will inevitably fall and academia will consist solely of mediocre scholars. The results suggest that brain drain can be destructive for academia in the long run.

In this article, we investigate the effect of demography on wealth inequality. We propose an economic growth model with overlapping generations in which individuals are altruistic towards their children and differ with respect to the age of their parent. We denote the age gap between the parent and their child as generational gap. The introduction of the generational gap allows us to analyze wealth inequality not only across cohorts but also within cohorts. Our model predicts that a decline in fertility raises wealth inequality within cohorts and, simultaneously, it reduces inequality at the population level (across cohorts). In contrast, increases in life expectancy result in a non-monotonic effect on wealth inequality by age and across cohorts.

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.

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.

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 Pontryagin's 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.

This note generalizes the value of life to incorporate the preferences for descendants of a dynastically altruistic decision-maker. It derives the value of progeny and shows how it depends on the Benthamite vs. Millian nature of dynastic altruism.