The optimal momentum of population growth and decline.

About 50 years ago, Keyfitz (1971) asked how much further a growing human population would increase if its fertility rate were immediately to be reduced to replacement level and remain there forever. The reason for demographic momentum is an age-structure inertia due to relatively many potential parents because of past high fertility. Although nobody expects such a miraculous reduction in reproductive behavior, a gradual decline in fertility in rapidly growing populations seems inevitable. As any delay in fertility decline to a stationary level leads to an increase in the momentum, it makes sense to think about the timing and the quantum of the reduction in reproduction. More specifically, we consider an intertemporal trade-off between costly pro- and anti-natalistic measures and the demographic momentum at the end of the planning period. This paper uses the McKendrick-von Foerster partial differential equation of age-structured population dynamics to study a sketched problem in a distributed parameter control framework. Among the results obtained by applying an appropriate extension of Pontryagin's Maximum Principle are the following: (i) monotony of adaptation efforts to net reproduction rate and convex decrease/concave increase (if initial net reproduction rate exceeds 1/is below 1); and (ii) oscillating efforts and reproduction rate if, additionally, the size of the total population does not deviate from a fixed level.

The optimal lockdown intensity for COVID-19.

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.

How long should the COVID-19 lockdown continue?

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.

Leading bureaucracies to the tipping point: An alternative model of multiple stable equilibrium levels of corruption.

We present a novel model of corruption dynamics in the form of a nonlinear optimal dynamic control problem. It has a tipping point, but one whose origins and character are distinct from that in the classic Schelling (1978) model. The decision maker choosing a level of corruption is the chief or some other kind of authority figure who presides over a bureaucracy whose state of corruption is influenced by the authority figure's actions, and whose state in turn influences the pay-off for the authority figure. The policy interpretation is somewhat more optimistic than in other tipping models, and there are some surprising implications, notably that reforming the bureaucracy may be of limited value if the bureaucracy takes its cues from a corrupt leader.

The Impact of Policies Influencing the Demography of Age-Structured Populations: Lessons from Academies of Sciences.

In this paper, we assess the role of policies aimed at regulating the number and age structure of elections on the size and age structure of five European Academies of Sciences. We show the recent pace of ageing and the degree of variation in policies across them and discuss the implications of different policies on the size and age structure of academies. We also illustrate the potential effect of different election regimes (fixed vs. linked) and age structures of election (younger vs. older) by contrasting the steady-state dynamics of different projections of Full Members in each academy into 2070 and measuring the size and age-compositional effect of changing a given policy relative to a status quo policy scenario. Our findings suggest that academies with linked intake (i.e., where the size of the academy below a certain age is fixed and the number of elections is set to the number of members becoming that age) may be a more efficient approach to curb growth without suffering any ageing trade-offs relative to the faster growth of academies electing a fixed number of members per year. We further discuss the implications of our results in the context of stable populations open to migration.

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.

Numerical solution of a conspicuous consumption model with constant control delay.

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account.This non-standard optimal control problem is difficult to solve analytically, and solutions depend on the variable model parameters. Therefore, we use a numerical result-driven approach. We propose a structure-exploiting direct method for optimal control to solve this challenging optimization problem. In particular, we discretize the uncertainties in the model formulation by using scenario trees and target the control delays by introduction of slack control functions.Numerical results illustrate the validity of our approach and show the impact of uncertainties and delay effects on optimal economic strategies. During the recession, delayed optimal prices are higher than the non-delayed ones. In the normal economic period, however, this effect is reversed and optimal prices with a delayed impact are smaller compared to the non-delayed case.

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.

On the age dynamics of learned societies-taking the example of the Austrian Academy of Sciences.

In a hierarchical organisation of stable size the annual intake is strictly determined by the number of deaths and a statutory retirement age (if there is one). In this paper we reconstruct the population of the Austrian Academy of Sciences from 1847 to 2005. For the Austrian Academy of Sciences we observe a shift of its age distribution towards older ages, which on the one hand is due to rising life expectancy, i.e., a rising age at death, as well as to an increased age at entry on the other hand. Therefore the number of new entrants has been fluctuating considerably-especially reflecting several statutory changes-and the length of tenure before reaching the age limit has declined during the second half of the last century.Based on alternative scenarios of the age distribution of incoming members-including a young, an old, the 'current' and a mixed-age model-we then project the population of the Austrian Academy and its ageing forward in time. Our results indicate that the 'optimum policy' would be to elect either young or old aged new members.

A model of chaotic drug markets and their control.

This paper explores the idea that drug markets may be chaotic in a mathematical sense by considering a discrete-time model of populations of drug users and drug sellers for which initiation into either population is a function of relative numbers of both populations. The structure of the system follows that considered in an arms control context by Behrens, Feichtinger and Prskawetz (Behrens D.A., Feichtinger G., & Prskawetz A. (1997). Complex Dynamics and Control of Arms Race. European Journal of Operations Research, 100, 192-215). The model presented in this paper summarizes prerequisites for possible chaotic behavior of the number of addicts and drug dealers frequenting a local drug market. Interestingly, even if the uncontrolled market dynamics do not exhibit chaotic patterns, a static intervention like removing a constant fraction of addicts each time period can easily create chaos--but even if static control would create chaos, dynamic controls can be crafted that avoid it. Especially OGY controls seem to work well for this example.

Age-structured optimal control in population economics.

This paper brings both intertemporal and age-dependent features to a theory of population policy at the macro-level. A Lotka-type renewal model of population dynamics is combined with a Solow/Ramsey economy. We consider a social planner who maximizes an aggregate intertemporal utility function which depends on per capita consumption. As control policies we consider migration and saving rate (both age-dependent). By using a new maximum principle for age-structured control systems we derive meaningful results for the optimal migration and saving rate in an aging population. The model used in the numerical calculations is calibrated for Austria.