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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.

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.

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This paper deals dynamically with the question of how recruitment to terror organizations is influenced by counter-terror operations. This is done within a optimal control model, where the key state is the (relative) number of terrorists and the key controls are two types of counter-terror tactics, one ("water") that does not one ("fire") that does provoke recruitment of new terrorists. The model is nonlinear and does not admit analytical solutions, but an efficient numerical implementation of Pontryagin's Minimum Principle allows for solution with base case parameters and considerable sensitivity analysis. Generally this model yields two different steady states, one where the terror-organization is nearly eradicated and one with a high number of terrorists. Whereas water strategies are used at almost any time, it can be optimal not to use fire strategies if the number of terrorists is below a certain threshold.

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This paper considers the problem of how to price a conspicuous product when the economy is in a recession that disrupts capital markets. A conspicuous product in this context is a luxury good for which demand is increasing in brand image. Brand image here means the ability of a consumer to impress observers by conspicuously displaying consumption of the good. Brand image is built up when the good is priced high enough to make it exclusive, and eroded if the good is discounted.Recession is modeled as having two effects: it reduces demand and it freezes capital markets so borrowing is not possible. In pricing the conspicuous product the firm faces the following trade-off. Reducing price helps maintain sales volume and cash flow in the face of reduced demand, but it also damages brand image and thus long term demand. Email address: kort@uvt.nl (P.M. Kort) September 13, 2010 interventions in capital markets and for timing of mergers and acquisitions. Preprint submitted to JEDC