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In the digital economy it is frequently observed that products become more valuable the larger is the number of people that use it. To account for such network effects, we introduce a new diffusion equation in a dynamic model of the firm with the aim to obtain the advertising policy that maximizes firm profits. Also an advertising budget is introduced. First, we find that introduction of the network effect leads to a critical mass of the sales level. Only if the sales are greater or equal to this level, is it optimal for the firm to have a substantial sales level in the long run. Second, introduction of an advertising budget could result in the existence of a Stalling equilibrium. This new type of equilibrium, at which the advertising amount is at its upper bound, serves as the critical mass threshold separating growth and decline. Third, the advertising policy is continuous in the sales level either when the discount rate is large and the rate of forgetfulness is low, when the Stalling equilibrium exists, or when the rate of forgetfulness is that high that the firm is always in decline. Furthermore, existence of the Stalling equilibrium involves an important message to the management of the firm in that a slight increase of the advertising budget would open up an avenue of growth in sales.

We present and analyze a life cycle model of a scientific career to investigate how the budget available for research related activities affects a scientist’s long run success in academia. Recognizing that reputation has a positive impact on obtaining research funds as well as on producing scientific output, we determine the optimal efforts of a scientist with respect to applying for research grants and costly science related activities to improve reputation, such as networking and inviting guest researchers, under the assumption that the total extent of these activities is limited by a time budget. We find that the optimal solution is history-dependent and study four different career paths which differ with respect to the initial reputation and the initial financial means to identify the determinants of success and failure in academia. It is shown that the extent to which a scientist finds the results of his or her work rewarding, has a substantial impact on whether the scientist will build up a successful career.

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.

In this article, we intend to explore the role of using an”infinite time horizon” framework to address the issues of sustainability and long-term strategies in the control of biological processes. We use two case study models to explain why considering a fixed or moving endpoint does not lead to the desired long-term effects. The first biological model considered concerns the spread of an infectious disease and its treatment as an infinite horizon optimal control problem. The second one deals with the metronomic chemotherapy cancer treatment over the remaining lifetime horizon of the patient. The latter is consistent with the conception of cancer as a chronic disease. Both models show structural differences in the choice of the objective functional, the first one uses a stabilization functional containing a weight function, the second one contains a damage functional which involves a density function.

This paper describes the process of capital accumulation subject to the following characteristics: (i) convex returns to (human) capital and (ii) the need to self-finance investments. Our setup is applicable to some peculiarities in the arts, sports and science, inter alia, coined the Matthew effect in Merton (1968) and explains, e.g., why prominent researchers get disproportional credit for their work. The potential young artist’s (athlete’s or scientist’s) optimal strategies include quitting, or continuing and even expanding one’s human capital in the respective profession. Both outcomes are separated by a threshold level in human capital. In addition, we find that it can be optimal to stay in business although consumption falls and stays at the subsistence level forever (we call this outcome a Sisyphus point). This possibility is also interesting from a theoretical point-of-view, as the optimal control problem may turn abnormal, i.e., the objective does not enter the Hamiltonian.

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.

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 , 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 evidence that there are fundamental disagreements.

A substantial increase of green investments is still required to reach the Paris Agreement’s emission targets. Yet, capital markets to expedite innovative green investments are generically constrained. Literature has shown that governments could de-risk such investments. Empirical beta pricing and yield estimates reveal some public involvement in the green bond market, especially for long maturity bonds. We provide empirical evidence that Governments and Multilateral organizations can de-risk green investments by supporting the issuance of green bonds in contrast to private green bonds - that show higher yields, volatility and beta prices - and conventional energy bonds, that are more volatile due to oil price variations. Since lower betas also mean lower capital costs, we use these empirical results and run a dynamic model with two types of firms, modeling the economic behavior of innovators (renewable energy firms) and incumbents (fossil fuel firms). Our model reveals that de-risked interest rates, from the cost side, and mark-ups and profit flows, from the market side, improve new entrants’ performance. As innovators normally have low mark-ups, inducing higher profit flows and lower capital costs allows a better default risk control and debt repayment which keep the firms in the market for a longer period of time and maintain low debt levels. Subsidies and carbon taxation can complement the role of the de-risked interest rate and expedite the energy transition.

In abnormal optimal control problems it is necessary to basically ignore the objective for certain state values in order to be able to determine the optimal control. In the past, abnormal problems were considered to be degenerated problems that did not fit to any real application. In the present paper we discuss reasons for the occurrence of abnormality. We show that abnormality can be an integral part of a meaningful problem rather than to be a sign for degeneracy.

Nations are struggling to decide when 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. We examine this choice within a simple optimal dynamic 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 di_erent 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 to essentially eradicate the virus or merely to atten the curve so fewer people urgently need healthcare when hospitals are already _lled to capacity.

One of the principal ways nations are responding to the COVID-19 pandemic is by lockingdown portions of their economies to reduce infectious spread. This is expensive in terms oflost jobs, lost economic productivity, and lost freedoms. So it is of interest to ask: What isthe optimal intensity with which to lockdown, and how should that intensity vary dynamicallyover the course of an epidemic? This paper explores such questions with an optimal controlmodel that recognizes the particular risks when infection rates surge beyond the healthcaresystem's capacity to deliver appropriate care. The analysis shows that four broad strategies canbe optimal, ranging from brief lockdowns that only \smooth the curve" to sustained lockdownsthat 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 afterinitial restrictions have been lifted is not necessarily a sign of failure. Relatively small changesin judgments about how to balance health and economic harms can alter dramatically whichstrategy is optimal. Indeed, there are constellations of parameters for which two or even three ofthese distinct strategies can all be optimal for the same set of initial conditions; these correspondto so-called triple Skiba points. The performance of trajectories can be highly nonlinear in thestate 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 epidemiologycan prefer dramatically di_erent policies. Conversely, favoring very di_erent policies is notevidence that there are fundamental disagreements.

A substantial increase of green investments is still required to reach the Paris Agreement’s emission targets. Yet, capital markets to expedite green invest-ments are generically constrained. Literature has shown that governments could de-risk such investments. Empirical beta pricing and yield estimates reveal some public involvement in the green bonds market, especially for long ma-turity bonds. We provide empirical evidence that Governments and Multilateral organizations can de-risk green investments by supporting the issuance of green bonds in contrast to private green bonds - that show higher yields, volatility and beta prices - and conventional energy bonds, that are more volatile due to oil price variations. Since lower betas also mean lower capital costs, we use those empirical results and run a dynamic model with two types of firms, modeling the economic behavior of innovators (renewable energy firms) and incumbents (fos-sil fuel firms). The simulations of our model show that de-risked interest rates help to phase in renewable energy firms in the market and avoid a sharp debt increase. However, when the new entrants carry negative pay-offs for a longer time, it might not be sufficient to keep the debt low and to avoid a shake-out in the market. Subsidies and carbon taxation can complement the role of the de-risked interest rates and expedite the energy transition. Beside deterministic model variants, we also explore a stochastic version of the model.

This paper addresses the spatial evolution of countries accounting for economics, geography and (military) force. Economic activity is spatially distributed following the AK model with the output being split into consumption, investment, transport costs and military (for defense and expansion). The emperor controls the military force subject to the constraints imposed by the economy but also the geography (transport costs, border length) and the necessity to satisfy the needs of the population. The border changes depending on how much pressure the emperor can muster to counter the pressure of neighboring countries. The resulting dynamic process determines a country’s size over time. The model leads to multiple steady states, large empires and small countries being separated by a threshold and collapse. The resulting patterns can be linked to historical observations.