Life-cycle behaviour in the face of large shocks to health

Typically, life-cycle models of optimal health behaviour are based on an ex-ante view, i.e. a representative individual faces a depreciation of a health stock, a mortality risk, or is subject to the accumulation of health deficits. The optimal behaviour over the life-cycle is described in light of these average processes. However, in the real world health does not develop smoothly over the life-course, but is subject to smaller or greater shocks. Large shocks, such as accidents and/or the onset of (life-threatening and/or chronic) diseases, have the propensity to affect survival probability, income and well-being as well as their dynamics and, thus, the entire life-course. In consequence, individuals are prone to alter life-cycle behaviour in the face of health shocks. This includes both the (ex-post) adjustment of behaviour in response to a shock and anticipatory/preventive behaviour (ex-ante). In this project, we study the implications of large health shocks for the individual and its life- course. We compare different types of anticipation (full anticipation, myopic), and consider the effects of health/disability insurance and a longevity insurance through the annuity market. In order to analyze these questions we use optimal control theory with a random stopping time, where the stopping rate is determined by the probability of the health shock. In order to analyze and solve the model we have to extend and adapt existing mathematical methods. In addition to a theoretical investigation of the model, we are planning to calibrate and solve numerically the model for different types of shock (diseases) and different specifications of health insurance and longevity insurance. This enables us to understand the quantitative implications of the various shocks to health for health and saving behaviours. Both is important and necessary to understand the impact of policy-making and the relative efficiency of policies toward the design of the various types of insurance.

Individual life-cycle models in economics (health economics, in our particular case) usually include risks to health (diseases, accidents) on a representative level (i.e., by a continuous mortality or morbidity rate) or completely stochastically (adding a random variable to the dynamic model). However, with this type of models it is not possible to observe the real effect of large shocks to health, which typically does not happen continuously. As a result, we developed a new modeling approach for these life-cycle models allowing for stochastic switches of the dynamics (i.e., evolution of state variables describing life conditions) and the objective function (i.e., intertemporal utility function). Since the classical methods are difficult to work with in our model set up, we first developed a new transformation method to an age-structured optimal control model, which allows additional economic insights in the optimal solution and to solve the two-stage optimal control model not sequentially, but simultaneously. For this new transformation we developed and implemented a suitable numerical method. We applied our model to the life-cycle model on cancer. We distinguish between general health care and shock (i.e., onset of the disease) specific prevention, acute and chronic care. This allows to analyse how the health risk shapes individual behaviour with respect to the different types of health care and how health shocks change the optimal behaviour over the life-cycle. Applying the framework to the established rational addition model by the famous economist Gary Becker, distinguishing periods without and with addiction. In this model we are able to show different long-run optimal solution (carrying over from the 'history dependence' phenomenon of deterministic optimal control models) depending on the stochastic escalation time of the addiction. Third we mention the application to the COVID-19 pandemic, for which we modeled the unknown arrival / approval time of the vaccination. In this model we were able to show that it is optimal to even intensify the lockdown at the approval time of the vaccine. This is due to the effect that (almost) everybody will be infected without a vaccine, which means that people should get infected such that the capacity of the health care system is not exceeded. If, on the other hand, a vaccination is available, people can receive protection against the virus without infection.